mm-4899 === Subject: Re: inverse of power series with prime power coefficients > The power series with prime coefficients > P(x) = 1 + 2x + 3x^2 + 5x^3 + 7x^4 + 11x^5 + 13x^6 + .... > and itsinverse > Q(x) = 1 / P(x) = 1 - 2x + x^2 - x^3 + 2x^4 - 3x^5 + 7x^6 - 10x^7 + > 13x^8 - 21x^9 + 26x^10 - 33x^11 + 53x^12 - .... > were studied in the mid-1990s, when Backhouse conjectured, and > Flajolet proved, that the limit of the absolute value of the ratio of > consecutive coefficients of Q(x) is equal to B=1.4560749...., > Backhouse's constant. (By comparison, the limit of the ratio of > consecutive primes is equal to 1.) > Consider instead the power series with prime power coefficients: > PP(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + 7x^5 + 8x^6 + 9x^7 + 11x^8 + > 13x^9 + 16x^10 + 17x^11 + 19x^12 + 23x^13 + 25x^14 + 27x^15 + 29x^16 + > 31x^17 + 32x^18 + 37x^19+ .... > I would expect itsinverseto be quite similar to the series Q(x) > above. But it appears to be quite different: > QQ(x) = 1 - 2x + x^2 - x^5 + 3x^6 - 3x^7 + 2x^9 - x^10 + x^15 - 4x^16 > + 8x^17 - 5x^18 - 13x^19+ 26x^20 - 9x^21 - 20x^22 + 25x^23 + 9x^24 - > 60x^25 + 65x^26 + .... > Note how many low terms are zero: x^3, x^4, x^8, and the four > consecutive terms x^11, x^12, x^13, x^14. I wonder if this is pure > coincidence or if it has some significance. There seem to be no other zero coefficients up to the x^300 term. > Also, whereas the terms of Q(x) alternate regularly between positive > and negative, there is no apparent pattern to the signs of the terms > of QQ(x). > Finally, the terms of QQ(x) appear to grow much more slowly than the > terms of Q(x). Perhaps as the series continues and the prime powers > become sparser compared to the primes, QQ(x) will start to look and > behave more like Q(x). But I am curious: does the limit of the > absolute value of the ratio of consecutive coefficients of QQ(x) > exist? If so, what is it? Looking at the first 300 or so terms, I see no pattern. > -- > Robert Israel isr...@math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada- Hide quoted text - - Show quoted text - Remember me? Or are my results too trivial Mr. Moderator? Welcome to the real mathematics and not the fake Matrix movie sorry excuse for chess you've been playing for the last 20 years. Musatov (Founder, Inverse 19 Mathematics) Not the only, but one without it would not exist. QED Efforts by mathematicians to determine the value of pi since have led to the conclusion that it ... Unlike rational numbers, pi cannot be represented by a common fraction, .... But what about an uncommon fraction? Will you look at my drawing and in particular how the twelve points are the outside points of the sphere are all rationally reached by decimal numbers? 6.2035884*2 and doubled an infinite number of times should yield a constant decimal binary expansion (what I mean by this is the numbers to the right of the decimal point should always end in 4, then 8, then 6, then 2, then 4, then 8, then 6, then 2...repeating for infinity). So in binary, by starting with this number we predict for n>2 the infinite binary tree will end with decimal progression: 2, 4, 8, 6 For n<2 the infinite binary tree will begin and end (depends on if you are increasing or decreasing the value) 1, 5, 5, 5 (repeating 5 infinitely) as we decrease to infinitely small. Musatov Inverse Counting Chain of Infinite Proportions (sample) 0.0000473372528076171875 0.000094674505615234375 0.00018934901123046875 0.0003786980224609375 0.000757396044921875 0.00151479208984375 0.0030295841796875 0.006059168359375 0.01211833671875 0.0242366734375 0.048473346875 0.09694669375 0.1938933875 0.387786775 0.77557355 1.5511471 3.1022942 6.2035884 12.4071768 24.8143536 49.6287072 99.2574144 198.5148288 397.0296576 794.0593152 1588.1186304 3176.2372608 6352.4745216 12704.9490432 25409.8980864 50819.7961728 101639.5923456 203279.1846912 406558.3693824 813116.7387648 1626233.4775296 3252466.9550592 6504933.9101184 13009867.8202368 26019735.6404736 52039471.2809472 104078942.5618944 208157885.1237888 416315770.2475776 832631540.4951552 1665263080.9903104 3330526161.9806208 6661052323.9612416 13322104647.9224832 26644209295.8449664 53288418591.6899328 106576837183.3798656 213153674366.7597312 426307348733.5194624 852614697467.0389248 1705229394934.0778496 3410458789868.1556992 6820917579736.3113984 13641835159472.6227968 27283670318945.2455936 54567340637890.4911872 109134681275780.9823744 218269362551561.9647488 436538725103123.9294976 873077450206247.8589952 1746154900412495.7179904 3492309800824991.4359808 6984619601649982.8719616 13969239203299965.7439232 27938478406599931.4878464 55876956813199862.9756928 111753913626399725.9513856 223507827252799451.9027712 447015654505598903.8055424 894031309011197807.6110848 1788062618022395615.2221696 3576125236044791230.4443392 7152250472089582460.8886784 14304500944179164921.7773568 28609001888358329843.5547136 57218003776716659687.1094272 114436007553433319374.2188544 228872015106866638748.4377088 457744030213733277496.8754176 915488060427466554993.7508352 1830976120854933109987.5016704 3661952241709866219975.0033408 7323904483419732439950.0066816 14647808966839464879900.0133632 29295617933678929759800.0267264 58591235867357859519600.0534528 117182471734715719039200.1069056 234364943469431438078400.2138112 468729886938862876156800.4276224 937459773877725752313600.8552448 1874919547755451504627201.7104896 3749839095510903009254403.4209792 7499678191021806018508806.8419584 (sample) If you would like to do the honors Mr. Plouffe, I endear you to calculate the angles and lengths of the number of ways, it is quite obviously possible to square a circle based on the geometry and numerical progression. As is self-evident to anyone with even a modest understanding of mathematics and geometry after visual inspection of the drawn proof (http://www.flickr.com/photos/24869933@N07/3619373444/sizes/l/) it is quite easily accomplished to connect and triangulate the outer points of the rectangles to the outer points of the square and circle and even by dimension sphere. Yes, the drawing clearly by definition of inverse symmetry (take the drawing cut it in half at the equator and turn the paper over and rotate it and compare the top and bottom mirrored symmetry and you will see it is an exact inverse of symmetry itself) the drawing explains how to properly and mathematically soundly represent a fourth dimension on a two dimensional plane. In plane language if you take the picture and separate the top half of the sphere by the bottom half of the sphere they are identical but only if the sphere existed on a complete three dimensional plane through expansion. Since the paper is flat, this is an impossibility, nonetheless it is completely expressed and retractable and countable infinitely. By the definition of our standard we have established any k-digit approximation to pi is rational. Which could have many practical scientific applications. For instance, search: ON THE THEOREM OF IVA$EV-MUSATOV IIIx e F(l) + F(2) + ... + F(p). This remark is the basis of our proof of Theorem .... -y(M + 1), x' + y(M + 1)] = 0 it is easy to check that Conditions (i), ...... Now set i = pi x. *jj, v. Since p, x is positive,. (i) fi is positive, ... plms.oxfordjournals.org/cgi/reprint/s3-53/1/143.pdf REPRESENTATION OF FUNCTIONSBY SERIES AND CLASSES £r(1)with measure E> 2£k -- £` and such that the trigonometric series for g(x) converges uniformly on [0, 2£k]. .... trigonometric series are due to Ivashev-Musatov [40] who has shown that there are null series (in the sense of ..... function £r(£q) e C(0, 1) with £r(0) = £r(1) = 0 such that ...... <*i *plonk* > muh brane i needs one- Hide quoted text - > - Show quoted text - NOTE THE PROGRESSION OF NUMBERS PELLS PROPORTION X^2=3 or 4 = 3 as the numbers progression, WOW if this does not proof this what will 1. Every odd number , number as Y value till infinity is exactly solvable with a proportionate result using the base Pells equation i.e 3,5,7,9,etc etc till infinty and the answer is as in 2(2^2)+1=3^2 and so on till infinity with a whole number integer. 2.Every even number , without exception, using Pells equation, the integer always is a .75 till infinity consistant as 63.75(2^4)=1=4^4 till infinity every integer will end with .75. is that the same proportion as 3 over 4, Y=3 for the least with numbers proportion There is a definite pattern by pells to numbers, and I note that in that pattern the number 19 proportion is strikingly proprtionate, and I clearly surmise that that circluar progression is the same as numbers for vector 19 progression. === Subject: Re: inverse of power series with prime power coefficients > On May 11, 5:37 pm, Robert Israel > The power series with prime coefficients > P(x) = 1 + 2x + 3x^2 + 5x^3 + 7x^4 + 11x^5 + > 13x^6 + .... > and itsinverse > Q(x) = 1 / P(x) = 1 - 2x + x^2 - x^3 + 2x^4 - > 3x^5 + 7x^6 - 10x^7 + > 13x^8 - 21x^9 + 26x^10 - 33x^11 + 53x^12 - .... > were studied in the mid-1990s, when Backhouse > conjectured, and > Flajolet proved, that the limit of the absolute > value of the ratio of > consecutive coefficients of Q(x) is equal to > B=1.4560749...., > Backhouse's constant. (By comparison, the limit > of the ratio of > consecutive primes is equal to 1.) > Consider instead the power series with prime > power coefficients: > PP(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + 7x^5 + 8x^6 > + 9x^7 + 11x^8 + > 13x^9 + 16x^10 + 17x^11 + 19x^12 + 23x^13 + > 25x^14 + 27x^15 + 29x^16 + > 31x^17 + 32x^18 + 37x^19+ .... > I would expect itsinverseto be quite similar to > the series Q(x) > above. But it appears to be quite different: > QQ(x) = 1 - 2x + x^2 - x^5 + 3x^6 - 3x^7 + 2x^9 - > x^10 + x^15 - 4x^16 > + 8x^17 - 5x^18 - 13x^19+ 26x^20 - 9x^21 - 20x^22 > + 25x^23 + 9x^24 - > 60x^25 + 65x^26 + .... > Note how many low terms are zero: x^3, x^4, x^8, > and the four > consecutive terms x^11, x^12, x^13, x^14. I > wonder if this is pure > coincidence or if it has some significance. > There seem to be no other zero coefficients up to > the x^300 term. > Also, whereas the terms of Q(x) alternate > regularly between positive > and negative, there is no apparent pattern to the > signs of the terms > of QQ(x). > Finally, the terms of QQ(x) appear to grow much > more slowly than the > terms of Q(x). Perhaps as the series continues > and the prime powers > become sparser compared to the primes, QQ(x) will > start to look and > behave more like Q(x). But I am curious: does the > limit of the > absolute value of the ratio of consecutive > coefficients of QQ(x) > exist? If so, what is it? > Looking at the first 300 or so terms, I see no > pattern. > -- > Robert Israel > isr...@math.MyUniversitysInitials.ca > Department of Mathematics > http://www.math.ubc.ca/~israel > University of British Columbia > Vancouver, BC, Canada- Hide quoted text - > - Show quoted text - > > Remember me? Or are my results too trivial Mr. > Moderator? > > Welcome to the real mathematics and not the fake > Matrix movie > sorry excuse for chess you've been playing for the > last 20 years. > > Musatov > > (Founder, Inverse 19 Mathematics) Not the only, but > one without it > would not exist. > > QED > > > Efforts by mathematicians to determine the value of > pi since have led > to the conclusion that it ... Unlike rational > numbers, pi cannot be > represented by a common fraction, .... > > But what about an uncommon fraction? > > Will you look at my drawing and in particular how the > twelve points > are the outside points of the sphere are all > rationally reached by > decimal numbers? > > 6.2035884*2 and doubled an infinite number of times > should yield a > constant decimal binary expansion (what I mean by > this is the numbers > to the right of the decimal point should always end > in 4, then 8, > then 6, then 2, then 4, then 8, then 6, then > 2...repeating for > infinity). > > So in binary, by starting with this number we predict > for n>2 the > infinite binary tree will end with decimal > progression: 2, 4, 8, 6 > > For n<2 the infinite binary tree will begin and end > (depends on if you > are increasing or decreasing the value) 1, 5, 5, 5 > (repeating 5 > infinitely) as we decrease to infinitely small. > > Musatov Inverse Counting Chain of Infinite > Proportions > (sample) > 0.0000473372528076171875 > 0.000094674505615234375 > 0.00018934901123046875 > 0.0003786980224609375 > 0.000757396044921875 > 0.00151479208984375 > 0.0030295841796875 > 0.006059168359375 > 0.01211833671875 > 0.0242366734375 > 0.048473346875 > 0.09694669375 > 0.1938933875 > 0.387786775 > 0.77557355 > 1.5511471 > 3.1022942 > 6.2035884 > 12.4071768 > 24.8143536 > 49.6287072 > 99.2574144 > 198.5148288 > 397.0296576 > 794.0593152 > 1588.1186304 > 3176.2372608 > 6352.4745216 > 12704.9490432 > 25409.8980864 > 50819.7961728 > 101639.5923456 > 203279.1846912 > 406558.3693824 > 813116.7387648 > 1626233.4775296 > 3252466.9550592 > 6504933.9101184 > 13009867.8202368 > 26019735.6404736 > 52039471.2809472 > 104078942.5618944 > 208157885.1237888 > 416315770.2475776 > 832631540.4951552 > 1665263080.9903104 > 3330526161.9806208 > 6661052323.9612416 > 13322104647.9224832 > 26644209295.8449664 > 53288418591.6899328 > 106576837183.3798656 > 213153674366.7597312 > 426307348733.5194624 > 852614697467.0389248 > 1705229394934.0778496 > 3410458789868.1556992 > 6820917579736.3113984 > 13641835159472.6227968 > 27283670318945.2455936 > 54567340637890.4911872 > 109134681275780.9823744 > 218269362551561.9647488 > 436538725103123.9294976 > 873077450206247.8589952 > 1746154900412495.7179904 > 3492309800824991.4359808 > 6984619601649982.8719616 > 13969239203299965.7439232 > 27938478406599931.4878464 > 55876956813199862.9756928 > 111753913626399725.9513856 > 223507827252799451.9027712 > 447015654505598903.8055424 > 894031309011197807.6110848 > 1788062618022395615.2221696 > 3576125236044791230.4443392 > 7152250472089582460.8886784 > 14304500944179164921.7773568 > 28609001888358329843.5547136 > 57218003776716659687.1094272 > 114436007553433319374.2188544 > 228872015106866638748.4377088 > 457744030213733277496.8754176 > 915488060427466554993.7508352 > 1830976120854933109987.5016704 > 3661952241709866219975.0033408 > 7323904483419732439950.0066816 > 14647808966839464879900.0133632 > 29295617933678929759800.0267264 > 58591235867357859519600.0534528 > 117182471734715719039200.1069056 > 234364943469431438078400.2138112 > 468729886938862876156800.4276224 > 937459773877725752313600.8552448 > 1874919547755451504627201.7104896 > 3749839095510903009254403.4209792 > 7499678191021806018508806.8419584 > (sample) > > If you would like to do the honors Mr. Plouffe, I > endear you to > calculate the angles and lengths of the number of > ways, it is quite > obviously possible to square a circle based on the > geometry and > numerical progression. > > As is self-evident to anyone with even a modest > understanding of > mathematics and geometry after visual inspection of > the drawn proof > (http://www.flickr.com/photos/24869933@N07/3619373444/ > sizes/l/) it is > quite easily accomplished to connect and triangulate > the outer points > of the rectangles to the outer points of the square > and circle and > even by dimension sphere. > > Yes, the drawing clearly by definition of inverse > symmetry (take the > drawing cut it in half at the equator and turn the > paper over and > rotate it and compare the top and bottom mirrored > symmetry and you > will see it is an exact inverse of symmetry itself) > the drawing > explains how to properly and mathematically soundly > represent a fourth > dimension on a two dimensional plane. > > In plane language if you take the picture and > separate the top half of > the sphere by the bottom half of the sphere they are > identical but > only if the sphere existed on a complete three > dimensional plane > through expansion. > > Since the paper is flat, this is an impossibility, > nonetheless it is > completely expressed and retractable and countable > infinitely. > > By the definition of our standard we have established > any k-digit > approximation to pi is rational. > > Which could have many practical scientific > applications. For instance, > search: > > ON THE THEOREM OF IVA$EV-MUSATOV IIIx e F(l) + F(2) + > ... + F(p). This > remark is the basis of our proof of Theorem .... -y(M > + 1), x' + y(M + > 1)] = 0 it is easy to check that Conditions (i), > ...... Now set i = > pi x. *jj, v. Since p, x is positive,. (i) fi is > positive, ... > plms.oxfordjournals.org/cgi/reprint/s3-53/1/143.pdf > > REPRESENTATION OF FUNCTIONSBY SERIES AND CLASSES > .83'(1)with measure ÄEÄ 2.83ë -- .83Ì and such that the trigonometric series for > g(x) converges > uniformly on [0, 2.83ë]. .... trigonometric series are > due to > Ivashev-Musatov [40] who has shown that there are > null series (in the > sense of ..... function .83'(.83ï) e C(0, 1) with .83'(0) = > .83'(1) = 0 such that > ...... <*i .pdf > by PL Ul'yanov - 1972 - Cited by 24 - Related > > Full text of Bulletin of the British Museum (Natural > History)X 20. > FIG. 5. Well preserved large fragment showing the > different aspect and > ...... 0-1-0-5 P i n height) so giving the shell > surface the > appearance of ...... 94-95, i pi. Maseru. BRINK, A. > S. 1963. Two > cynodonts from the Ntawere ...... MUSATOV, D. L, > NEMIROVSKAYA, V. N., > SHIROKOVA, E. V. & ZHURAVLEVA, I. T. 1961. ... > www.archive.org/stream/bulletinofbritis17geollond/bull > etinofbritis17geollond_djvu.txt > > Trigonometric series with rapidly decreasing > coefficients a i0, 5 > ...Nt,+i - Pi = n,. N tl+j. -pi =N tl+j. -i +pi +1. > ..... number 8 6 > (0,1) for the product (32) such that for every point > x 6 E there are > ... > _A01.pdf > versions > > Plouffe's formula for pi - sci.math | Google Groups3 > posts - 2 authors > - Last post: 41 minutes ago > x=pi e^(ipi)+1=0. Martin Musatov. Reply Forward. > Martin Musatov to > problem. ... > 65ba40fd9e5e8f46/ > bf6d1ee13d87270b?lnk=raot > - 41 minutes ago > > [PDF] .83j.83h.83h.83m .83j-.83kFile Format: PDF/Adobe Acrobat > Gli storici ci dicono che, nel X secolo, il principe > Vladi- ...... i > pi`u anziani e carismatici si evidenziavano i nomi di > ...... tin > Somov, Viktor Borisov-Musatov, Aleksandr Benois e Lev > Bakst ... 5/0; > Judovin 5/0; Korovin 2/1; Tukacov 1/0; Goli- cyn 1/1; > Zacharov 1/1; > Jurkunas 5/0; Krasauskas 2/0;. Bulaka 3/0. ... > www.esamizdat.it/eSamizdat_2005_(III)_2-3.pdf > > I am testing these identities: > > (1/2) * ( - k ) * ( k - 1/2 ) * ( - 3 ) / k! > > This equation produces > > M = k! ( - 3 ) * ( k - 1/2) * ( - k ) * (1/2) > > back! > > > Martin Musatov Reply Forward > > > ---------- Forwarded message ---------- === > Subject: CHALLENGE TO MATHEMATICIANS OF THE WORLD > > > On Apr 11, 11:40 am, Harris Moran > > > *plonk* > > message > > > muh brane i needs one- Hide quoted text - > > - Show quoted text - > > NOTE THE PROGRESSION OF NUMBERS PELLS PROPORTION > X^2=3 or 4 = 3 as > the numbers progression, WOW if this does not proof > this what will > > 1. Every odd number , number as Y value till > infinity is exactly > solvable with a proportionate result using the base > Pells equation i.e > 3,5,7,9,etc etc till infinty and the answer is as in > 2(2^2)+1=3^2 and > so on till infinity with a whole number integer. > > 2.Every even number , without exception, using Pells > equation, the > integer always is a .75 till infinity consistant > > as 63.75(2^4)=1=4^4 till infinity every integer > will end with .75. > is that the same proportion as 3 over 4, Y=3 for the > least with > numbers proportion > > There is a definite pattern by pells to numbers, and > I note that in > that pattern the number 19 proportion is strikingly > proprtionate, > and I clearly surmise that that circluar progression > is the same as > numbers for vector 19 progression. > Update from your neighborhood friendly Musatov. I am going to frame this first one!!! Exciting.... **************************************************************************** **************************************************************************** * **************************************************************************** * ** *Math Forum Discussions mathematics and geometry after visual inspection of the drawn proof ... approximation to pi is rational. Which could have many practical scientific applications. ... x=pi e^(ipi)+1=0. Martin* *Musatov. Reply Forward. Martin Musatov to ...* *http://mathforum.org/kb/message.jspa?messageID=6749244&tstart=0 - 4 hours ago * **************************************************************************** **************************************************************************** * ****************** And then this one is tease of the journey to come... ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++ Reactions to/against the Binary Tree - Page 6 - Docendi.org17 posts - Last post: May 28 But he claims that pi is the limit of the finite initial segments. These two claims suggest ... axiomatic basis. Are you an alias of Musatov? No I am Musatov. .... If Cantor's proof is correct, then an infinite path is the limit of ..... The real number sqrt(3) is the limit of a rational sequence. ... ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + +++++++++++++++++++++ === Subject: Re: futility of n-space posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > The restriction of n-space is reflected in the virtual impossibility of > finding roots to n-degree polynomials.If each coefficient is a component > to a vector of n dimensions, and each power of x is a component to a vector > of n dimensions, the two vectors are orthogonal since their dot product is > zero. I was just trying to study the ring quotient with the help of our > excellent Usenet community. As you use the term n-space the insistence > within the polynomials in use for ring theory are even worse than > this. They require an infinite space. Thus one is not even free to > consider an n-space. Some refute the usage of polynomials as dimensional and insist that > remaining in the limbo of an 'indeterminate value' that they are free > to speak of an X without ever describing X other than To say that >( a + X ) ( b + X ) = ab + (a + b) X + X X > and such simple operations. Yet here if these a and b are real then X > must have a superposition with a real. Somehow the indeterminate > stance is satisfactory to many. I do find it troubling. If we > ourselves validate our concern then what areas of mathematics fall? > These polynomials will still generate amazingly accurate sin() curves > in just ten steps with X real, but then this is the dimensionally > collapsed system whereby your concern goes away. This is no longer the > 'indeterminate' system. I guess another way to approach this is to make a simpler > indeterminate object and perform operations with it. It seems we might > do anything we like with an object that is indeterminate. I can't see > how an indeterminate object can be applied to mathematics. Well then > along comes the quotient form which supposedly will build say the > complex numbers as >R[X] / ( XX + 1 ) > yet here we are at a level of complexity in terms of the indeterminate > object. A composition of indeterminate objects ought to yield an > indeterminate object in my opinion, yet this one claims to make a > coherent composition out of them. Perhaps the crux is in the usage of > the word isomorphic. I'm not that impressed with isomorphism. I have > little doubt that a human is isomorphic to a coffee mug under some > math. Hmmm... I'll have to come up with some extensions on this... an > indeterminate form must represent everything isomorphically for if we > can compose one determinate object from an indeterminate object then > the bridge has been built. Voila. All of mathematics is now completed. - Tim > The liberty of n-space is that it opens up infinite possibilites. A > theorist can prove his brainchild in 4 dimensions if he can't do it in 3. > Just imagine.Associated with every point in (x1,x2,x3) is a point on the > x4 axis.That is, associated with every (x,y,z,t) is a point x4 hidden in > the system of points (x1,x2,x3,x4,t), not considering that furthermore t > (time) may be altered.The theorist becomes an artist producing > explanations at whim in equations that all come together perfectly. > Anything whatsoever can be proven using math of higher dimensions in E^n > where n>3. > This also delimits n-space math.On the one hand it is uselessly > constrictive and on the other uselessly diluted. > Consider a number of E^3 coordinate systems in synchrounous time with each > other separated by large distances.I think this is as best a model as can > be hoped for.One system is unbridled and actions in it mapped to form > reactions in the other systems.That way, associated with one system are > events in the other systems by cause and effect.When a tree is felled in > one system, a dam is caused to be broken in the other systems. This > meaningless association is diluted as well, but Master and Slave can swap > roles like a see-saw.This is in line with momentum that is channeled for > technological uses.No, there is no relationship. But can you find one? > The man's pulse was connected mechanically to regulate the speed of a > flywheel.After this went on for awhile, the technician altered the > flywheel and observed the man's pulse.He found out nothing.There are > some processes, however, that do swap Master/Slave relationships, like a > child's see-saw on a playground.What are they? These are good candidates > for multiple systems of E^3, particularly in distance relationships to > unknown cause and effect forces.The media may be EMR. The relatively > equipotential space-time around the planet may have anomalies, like > gradients or funnels.The ancient structures built in phylogeny are a > vortex.The path of least resistance is to delve and fall into them, > degenerating and destroying the foundations of everything that went in to > bringing the world up to the present. > Suppose one system was in the nucleus of a star and the other system in the > vortex of a Black Hole.Is there any cause and effect between the two? > No.- Hide quoted text - - Show quoted text - Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT - Docendi.org5 posts cannot be reduced to Null zero and n(2Pi^2-0.75) is constant curve for that value of n ... www.docendi.org/re-t241082.html Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANTJun 11, 2009 ... I have full irrefutable proof of this very imprtant constant and if there are any problems ... Inverse 19, Have a good rest and Godspeed. Best wishes, Musatov .... reply of N=NP what that means, and I will reply to you. ... Power . power is silent as is the space matrix, but it has great power and ... sci.tech-archive.net/Archive/sci.math/2009-06/msg01398.html Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT - Docendi.org6 posts - Last post: 15 hours ago constant, ... challenge you to provide *proof* of me mentioning any other film or ... www.docendi.org/re-t241056.html - 15 hours ago 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT...6 posts - Last post: 2 days ago *proof* of me mentioning any other film or ... www.groupsrv.com/science/post-3081440.html 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT - sci.physics | Google Groups4 posts - 3 authors - Last post: 16 hours ago the space matrix at 19. > Note: That this constant cannot be reduced to Null zero and n(2Pi^2-0.75) is constant curve for that value of n .... challenge you to provide *proof* of me mentioning any other film or ... 393d103f47a36a8c/e819a4171b52eb14?lnk=raot&fwc=1 - 16 hours ago 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT...10 posts - Last post: yesterday Dimension is silent, so is the space matrix at 19. Note: That this constant cannot be ... challenge you to provide *proof* of me mentioning any other film or ... The below text is everything I know regarding N=NP: Kindly, Musatov ... Inverse 19, Have a good rest and Godspeed. Best wishes, Musatov ... www.groupsrv.com/science/post-3083339.html Re: 2(3^2)+1=19 Good bye note to Sci math gurus!Jun 12, 2009 ... Since ... reply of N=NP what that means, and I will reply to you. I think much of ... Power . power is silent as is the space matrix, but it has great power ... sci.tech-archive.net/Archive/sci.math/2009-06/msg01417.html - 4 hours ago Higher order bifurcation of discrete map...10 posts - Last post: 2 days ago some higher space. ...http://scottaaronson.com/blog/%3Fp%3D387 + ... present ... author = {Martin M. Musatov}, title = {Proof of {Proof Net .... period-n region and we are plunging in there from the Shell- Thron region. .... On Jun 9, 10:13 pm, inverse 19 posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) On Jun 12, 6:33 am, The Sortov Institute Hello all, > Here's a nice identity: > (p+q)^4 + (r-s)^4 = (p-q)^4 + (r+s)^4 > where {p,q,r,s} = {a^7+a^5-2a^3+a, 3a^2, a^6-2a^4+a^2+1, 3a^5} > For similar stuff, you may be interested in A Collection of Algebraic > Identities: > with the basics with 2nd powers and goes up to 8th and higher powers. > Enjoy. > - Titus If > a = sqrt(mm + nn) > b = sqrt(mm ? 2mn + 2nn) > c = sqrt(2mm ? 2mn + nn) > then > (a+b+c)(a+b?c)(a?b+c)(?a+b+c) = 4(mm ? mn + nn)^2. http://sci.tech-archive.net/pdf/Archive/sci.math/2006-11/msg04721.pdf A typically egregious remark from the Sortov Institute. :D- Hide quoted text - - Show quoted text - On Jun 12, 6:33 am, The Sortov Institute Hello all, > Here's a nice identity: > (p+q)^4 + (r-s)^4 = (p-q)^4 + (r+s)^4 > where {p,q,r,s} = {a^7+a^5-2a^3+a, 3a^2, a^6-2a^4+a^2+1, 3a^5} > For similar stuff, you may be interested in A Collection of Algebraic > Identities: > with the basics with 2nd powers and goes up to 8th and higher powers. > Enjoy. > - Titus If > a = sqrt(mm + nn) > b = sqrt(mm ? 2mn + 2nn) > c = sqrt(2mm ? 2mn + nn) > then > (a+b+c)(a+b?c)(a?b+c)(?a+b+c) = 4(mm ? mn + nn)^2. http://sci.tech-archive.net/pdf/Archive/sci.math/2006-11/msg04721.pdf A typically egregious remark from the Sortov Institute. :D- Hide quoted text - - Show quoted text - Ahem! That is the Martin Michael Musatov institute to you, Good Sir! > On May 11, 5:37 pm, Robert Israel > The power series with prime coefficients > P(x) = 1 + 2x + 3x^2 + 5x^3 + 7x^4 + 11x^5 + 13x^6 + .... > and itsinverse > Q(x) = 1 / P(x) = 1 - 2x + x^2 - x^3 + 2x^4 - 3x^5 + 7x^6 - 10x^7 + > 13x^8 - 21x^9 + 26x^10 - 33x^11 + 53x^12 - .... > were studied in the mid-1990s, when Backhouse conjectured, and > Flajolet proved, that the limit of the absolute value of the ratio of > consecutive coefficients of Q(x) is equal to B=1.4560749...., > Backhouse's constant. (By comparison, the limit of the ratio of > consecutive primes is equal to 1.) > Consider instead the power series with prime power coefficients: > PP(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + 7x^5 + 8x^6 + 9x^7 + 11x^8 + > 13x^9 + 16x^10 + 17x^11 + 19x^12 + 23x^13 + 25x^14 + 27x^15 + 29x^16 + > 31x^17 + 32x^18 + 37x^19+ .... > I would expect itsinverseto be quite similar to the series Q(x) > above. But it appears to be quite different: > QQ(x) = 1 - 2x + x^2 - x^5 + 3x^6 - 3x^7 + 2x^9 - x^10 + x^15 - 4x^16 > + 8x^17 - 5x^18 - 13x^19+ 26x^20 - 9x^21 - 20x^22 + 25x^23 + 9x^24 - > 60x^25 + 65x^26 + .... > Note how many low terms are zero: x^3, x^4, x^8, and the four > consecutive terms x^11, x^12, x^13, x^14. I wonder if this is pure > coincidence or if it has some significance. > There seem to be no other zero coefficients up to the x^300 term. > Also, whereas the terms of Q(x) alternate regularly between positive > and negative, there is no apparent pattern to the signs of the terms > of QQ(x). > Finally, the terms of QQ(x) appear to grow much more slowly than the > terms of Q(x). Perhaps as the series continues and the prime powers > become sparser compared to the primes, QQ(x) will start to look and > behave more like Q(x). But I am curious: does the limit of the > absolute value of the ratio of consecutive coefficients of QQ(x) > exist? If so, what is it? > Looking at the first 300 or so terms, I see no pattern. > -- > Robert Israel isr...@math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada- Hide quoted text - > - Show quoted text - Remember me? Or are my results too trivial Mr. Moderator? Welcome to the real mathematics and not the fake Matrix movie > sorry excuse for chess you've been playing for the last 20 years. Musatov (Founder, Inverse 19 Mathematics) Not the only, but one without it > would not exist. QED > Efforts by mathematicians to determine the value of pi since have led > to the conclusion that it ... Unlike rational numbers, pi cannot be > represented by a common fraction, .... But what about an uncommon fraction? Will you look at my drawing and in particular how the twelve points > are the outside points of the sphere are all rationally reached by > decimal numbers? 6.2035884*2 and doubled an infinite number of times should yield a > constant decimal binary expansion (what I mean by this is the numbers > to the right of the decimal point should always end in 4, then 8, > then 6, then 2, then 4, then 8, then 6, then 2...repeating for > infinity). So in binary, by starting with this number we predict for n>2 the > infinite binary tree will end with decimal progression: 2, 4, 8, 6 For n<2 the infinite binary tree will begin and end (depends on if you > are increasing or decreasing the value) 1, 5, 5, 5 (repeating 5 > infinitely) as we decrease to infinitely small. Musatov Inverse Counting Chain of Infinite Proportions > (sample) > 0.0000473372528076171875 > 0.000094674505615234375 > 0.00018934901123046875 > 0.0003786980224609375 > 0.000757396044921875 > 0.00151479208984375 > 0.0030295841796875 > 0.006059168359375 > 0.01211833671875 > 0.0242366734375 > 0.048473346875 > 0.09694669375 > 0.1938933875 > 0.387786775 > 0.77557355 > 1.5511471 > 3.1022942 > 6.2035884 > 12.4071768 > 24.8143536 > 49.6287072 > 99.2574144 > 198.5148288 > 397.0296576 > 794.0593152 > 1588.1186304 > 3176.2372608 > 6352.4745216 > 12704.9490432 > 25409.8980864 > 50819.7961728 > 101639.5923456 > 203279.1846912 > 406558.3693824 > 813116.7387648 > 1626233.4775296 > 3252466.9550592 > 6504933.9101184 > 13009867.8202368 > 26019735.6404736 > 52039471.2809472 > 104078942.5618944 > 208157885.1237888 > 416315770.2475776 > 832631540.4951552 > 1665263080.9903104 > 3330526161.9806208 > 6661052323.9612416 > 13322104647.9224832 > 26644209295.8449664 > 53288418591.6899328 > 106576837183.3798656 > 213153674366.7597312 > 426307348733.5194624 > 852614697467.0389248 > 1705229394934.0778496 > 3410458789868.1556992 > 6820917579736.3113984 > 13641835159472.6227968 > 27283670318945.2455936 > 54567340637890.4911872 > 109134681275780.9823744 > 218269362551561.9647488 > 436538725103123.9294976 > 873077450206247.8589952 > 1746154900412495.7179904 > 3492309800824991.4359808 > 6984619601649982.8719616 > 13969239203299965.7439232 > 27938478406599931.4878464 > 55876956813199862.9756928 > 111753913626399725.9513856 > 223507827252799451.9027712 > 447015654505598903.8055424 > 894031309011197807.6110848 > 1788062618022395615.2221696 > 3576125236044791230.4443392 > 7152250472089582460.8886784 > 14304500944179164921.7773568 > 28609001888358329843.5547136 > 57218003776716659687.1094272 > 114436007553433319374.2188544 > 228872015106866638748.4377088 > 457744030213733277496.8754176 > 915488060427466554993.7508352 > 1830976120854933109987.5016704 > 3661952241709866219975.0033408 > 7323904483419732439950.0066816 > 14647808966839464879900.0133632 > 29295617933678929759800.0267264 > 58591235867357859519600.0534528 > 117182471734715719039200.1069056 > 234364943469431438078400.2138112 > 468729886938862876156800.4276224 > 937459773877725752313600.8552448 > 1874919547755451504627201.7104896 > 3749839095510903009254403.4209792 > 7499678191021806018508806.8419584 > (sample) If you would like to do the honors Mr. Plouffe, I endear you to > calculate the angles and lengths of the number of ways, it is quite > obviously possible to square a circle based on the geometry and > numerical progression. As is self-evident to anyone with even a modest understanding of > mathematics and geometry after visual inspection of the drawn proof > (http://www.flickr.com/photos/24869933@N07/3619373444/sizes/l/) it is > quite easily accomplished to connect and triangulate the outer points > of the rectangles to the outer points of the square and circle and > even by dimension sphere. Yes, the drawing clearly by definition of inverse symmetry (take the > drawing cut it in half at the equator and turn the paper over and > rotate it and compare the top and bottom mirrored symmetry and you > will see it is an exact inverse of symmetry itself) the drawing > explains how to properly and mathematically soundly represent a fourth > dimension on a two dimensional plane. In plane language if you take the picture and separate the top half of > the sphere by the bottom half of the sphere they are identical but > only if the sphere existed on a complete three dimensional plane > through expansion. Since the paper is flat, this is an impossibility, nonetheless it is > completely expressed and retractable and countable infinitely. By the definition of our standard we have established any k-digit > approximation to pi is rational. Which could have many practical scientific applications. For instance, > search: ON THE THEOREM OF IVA$EV-MUSATOV IIIx e F(l) + F(2) + ... + F(p). This > remark is the basis of our proof of Theorem .... -y(M + 1), x' + y(M + > 1)] = 0 it is easy to check that Conditions (i), ...... Now set i = > pi x. *jj, v. Since p, x is positive,. (i) fi is positive, ... > plms.oxfordjournals.org/cgi/reprint/s3-53/1/143.pdf REPRESENTATION OF FUNCTIONSBY SERIES AND CLASSES £r(1)with measure E 2£k -- £` and such that the trigonometric series for g(x) converges > uniformly on [0, 2£k]. .... trigonometric series are due to > Ivashev-Musatov [40] who has shown that there are null series (in the > sense of ..... function £r(£q) e C(0, 1) with £r(0) = £r(1) = 0 such that Full text of Bulletin of the British Museum (Natural History)X 20. > FIG. 5. Well preserved large fragment showing the different aspect and > ...... 0-1-0-5 P i n height) so giving the shell surface the > appearance of ...... 94-95, i pi. Maseru. BRINK, A. S. 1963. Two > cynodonts from the Ntawere ...... MUSATOV, D. L, NEMIROVSKAYA, V. N., > SHIROKOVA, E. V. & ZHURAVLEVA, I. T. 1961. ...www.archive.org/stream/bulletinofbritis17geollond/bulletinofbritis17g... Trigonometric series with rapidly decreasing coefficients a i0, 5 > ...Nt,+i - Pi = n,. N tl+j. -pi =N tl+j. -i +pi +1. ..... number 8 6 > (0,1) for the product (32) such that for every point x 6 E there are Plouffe's formula for pi - sci.math | Google Groups3 posts - 2 authors > - Last post: 41 minutes ago > x=pi e^(ipi)+1=0. Martin Musatov. Reply Forward. Martin Musatov to > bf6d1ee13d87270b?lnk=raot > - 41 minutes ago [PDF] ???? ?-?File Format: PDF/Adobe Acrobat > Gli storici ci dicono che, nel X secolo, il principe Vladi- ...... i > pi`u anziani e carismatici si evidenziavano i nomi di ...... tin > Somov, Viktor Borisov-Musatov, Aleksandr Benois e Lev Bakst ... 5/0; > Judovin 5/0; Korovin 2/1; Tukacov 1/0; Goli- cyn 1/1; Zacharov 1/1; > Jurkunas 5/0; Krasauskas 2/0;. Bulaka 3/0. ...www.esamizdat.it/eSamizdat 2005 (III) 2-3.pdf I am testing these identities: (1/2) * ( - k ) * ( k - 1/2 ) * ( - 3 ) / k! This equation produces M = k! ( - 3 ) * ( k - 1/2) * ( - k ) * (1/2) Martin Musatov Reply Forward read more >- Hide quoted text - - Show quoted text -... > The smallest number with phi (n) = 13! is n = 6,227,180,929 = 66529 * > 93601 > Number 100,000 is n = 18,817,814,916 = 673 * 1301 * 199 * 27 * 4 > The largest of 171,821 numbers is 37,020,293,310 = 19 * 31 * 23 * 169 > * 11 * 49 * 5 * 3 * 2 Wow -- how did you calculate that? I can see that there are two or > three classes of numbers which could give this initial result, and > from there it is a multiplicative problem....but how can I determine > all of the base cases, rather than the ones found via brute-force? Like this: I will explain to you what I am after. I am after a new theory, a new way of programming, a new way of measuring, a new way of communicating. I am talking about an end to the old tired way of 'P', if 'P' then 'Q' innefficiency in programming and language (and practical matters too). I assert my findings (all the following text up to (c) 2009 Martin Musatov (the 2nd time) it is read in this thread: 'X' if 'X' The idea is we create a thread. The first programming variable means nothing. It is an instruction to the system script. It is an open and I said> order of logic. 'X' alone simply means Open or I am talking or Listen up, it can be followed by anything, or an infinite number of things. And the idea of 'X' is arbitrary, as 'X' could be replaced by any other logical symbol or character in any language and the idea still stands. The operation still works. Let me explain: 'X' when I say 'X'. 'X' is the master open 'command' key or 'dialogue'. It sets up a system open script. We then can open any operation simply by repeating it. The system is not loyal to any language or frame. (or insert proprietary here) If I want to open up a program Y, I simply say, X, Y I want Y. The system knows that if a variable is repeated it means to execute it under the terms programmed. (or by pressing a key) Let me explain further (further): A formal illustration of how it would look in a system would be: X, Y, Y, X The first 'X' means open the root command, the first 'Y' means, I want 'Y', the second 'Y' means it is just 'Y' I seek and the 'X' closes to command. The platform solution requires complete symmetry in programming and logic threads. The programming implementation allows the setup of a system where repetition of a variable in symmetry order allows command. In football the command hut means pitch me the ball back. The Quarterback will communicate to his teammates, On the 2nd hut throw the ball. The first hut is to get set. Still 'hut' is the command to throw the ball. The real world variable and how it relates to the equivalency in programming is explained as follows: Instead of saying (1)Listen up....(2)I want to open my word processor.... X 1 1 X where (1)=the first command 1 and (2)=11 ***********Note in binary*: the machine literally must count the two ones and add them to get two.************) User inputs:(1)I want my word processor (2)I want my word processor and this means to the machine Open up the word processor... only when you read (1) I want my word processor (2) I want my word processor (forgive my terminology, I am simply being plain) but this inside/out programming illustrates perfectly my point. and forgive my for being so bold but this logic clear, simple, and plain, simply resolves P=NP and overcomes the halting problem irrefutably the solution to P=NP or P Versus NP, or N=NP or -1=1 or however you want to frame the problem is simply this statement: Do not do anything until I say it twice. Advanced Symmetry application: the solution to the machine is the overlooked symmetry. A complex command may look like: 'XYZZYX' which would mean, I want the YZ complex variable please Stephen Arthur Cook was right, the whole thing was simply due to lack of ingenuity on behalf of our programmer. (C)2009. Martin Musatov. All Rights Reserved in perpetuity*. http://www.MeAmI.org Search for the People *Applies to derviations of this work. Including software, programming, and other profit generating means extracted from this work. Work may be patent pending. === === >Subject: Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT Seriously, your time to write me has meant a great deal to me. >Strangers is fine, but I need not a name to recognize a kinship or >friendship or kindness. While I appreciate your fierce sentiment in my defense, I believe >people are basically good and only hope the ones who acted such a way >only did so because I was misunderstood. You do not need to speak >harshly of them on my account. I forgive them and wish them no ill >will. Truth is what I seek. Bertrand Russell said some brilliant things, one of them being, >Without God, life has no meaning. This floored me coming from an >atheist. I am religious and I take the value of your words to heart. May I ask >what if anything prompted you in this reference? Am I not still in my >soul? I assure I am. This last section in your thread: be still Musatov, find the mathematics in >the whistling wind , happiness in the garden, and watch the loon for >hours , >and be still and greater wisdom will come to you It puzzles me. I have seen so many statements like this at the end of >threads and they seem to follow intense debates by more senior level >posters. Can you tell me what purposes they serve? Keep in touch, (I hope) Martin Musatov >Sent from my Verizon Wireless BlackBerry -----Original Message----- > === >Subject: Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT > You have my respect and wishes and sincerely so. I will prefer to >remain >annonymous even though you choose other wise purely because I am very >independant and I do not have any regular allegiences but my family. >I just >want to see this current project through with a web site and then I am >done, >I have lots of other hobbies. I am very honest , I do not keep any >single >contact with no body, that is the way I live, life is less complex >that way. >Simplicity is genius, a quote from an English mathematician Bertrand >Russel.. You may have good ideas Musatov , and you seem to be decent, >but as >the Bible states , be still in the soul. If you succeed in your >ventures >let me know, otherwise we should stay as strangers, nothing personal , >I am >that way. If I succeed I will contact you,---- but those other >bastards at >the Forum, I gave them their due, be still Musatov, find the >mathematics in >the whistling wind , happiness in the garden, and watch the loon for >hours , >and be still and greater wisdom will come to you > Be still in the soul Musatov, be still, God bless you . Find success! > hope 9900 Musatov === Subject: Re: If Cantor set is uncountable and measure 0, then (Cantor)^c has measure 1 on [0, 1]? posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > I'm trying to get this straight. TheCantorset is uncountable and has > no interior; > Yes. > it's a collection of points. > Yes, but that's a strange comment. > So the complement of the >Cantorset is a collection of open sets. > I think you meant to say the complement of theCantorset is open. >He may have meant either and still been correct. > Er, no. The complement of theCantorset is an open set. > It's not a collection of open sets. My Bad. > Is the total set of points in > the open sets uncountable? It must be, it seems to me, since any open > set of real numbers contains uncountably many points. And the > complement of theCantorset on [0, 1] has to have measure 1, no? So > both theCantorset and its complement are uncountable? It would seem > a paradox if the complement of theCantorset on [0, 1] were > countable. > Another question: I've seen it claimed that theCantorset can be > modified so as to have measure 1 (on [0, 1], for example). Can someone > provide an example of a modifiedCantorset with measure 1 on [0, 1]? > There are Cantorsets that have measure 1, but they are not a subset > of [0,1].Cantorsets are compact, and a compact subset of [0, 1] of > measure 1 must be [0, 1] itself. > David C. Ullrich > Understanding Godel isn't about following his formal proof. > That would make a mockery of everything Godel was up to. > (John Jones, My talk about Godel to the post-grads. > in sci.logic.) -- > Virgil- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text - Exactly. > Hello there, > Anyone knows how to prove this? > Let k = 4m+3. Given the eqn, > v(pi^k) = 2^(k-1) Sum[f1, {n,1,inf}] - (2^(k-1)+1) > Sum[f2, {n,1,inf}] > + Sum[f3, {n,1,inf}] > where {1/f1, 1/f2, 1/f3} = {n^k(q^n-1), > n^k(q^(2n)-1), n^k(q^(4n)-1)} > and q = e^(pi), (with inf = infinity) > then v is a rational number. > Ex. > (1/180) pi^3 = 2^2 Sum[f1, {n,1,inf}] - (2^2+1) > Sum[f2, {n,1,inf}] + > Sum[f3, {n,1,inf}] > (13/14175) pi^7 = 2^6 Sum[f1, {n,1,inf}] - (2^6+1) > Sum[f2, {n,1,inf}] > + Sum[f3, {n,1,inf}] > and so on. See more at Identities inspired by > Ramanujan's Notebooks > (Part 2): > http://www.lacim.uqam.ca/~plouffe/inspired2.pdf > Actually, Plouffe's identities have the general > form, > pi^k = a*Sum[f1, {n,1,inf}] + b*Sum[f2, {n,1,inf}] > + c*Sum[f3, {n, > 1,inf}] > for some rational numbers {a,b,c} for all ODD k, > but, after some > tweaking, I noticed it has that simpler form when k > = 4m+3. > (He found these identities experimentally using > Maple and PSLQ.) > - Titus > Indeed for m=0, that is for k = 4*m+3 = 3 > using Maple in Integer Relation Algorithms option > of the inverse > symbolic calculator > gives exactly what Titus stated > 180*sum(1/n^3*(4/(exp(1)^(Pi*n)-1)-5/(exp(1)^(2*Pi*n)- > 1)+1/(exp(1)^ > (4*Pi*n)-1)) > ,n = 1 .. infinity) > Value to be looked up: > K = 31.006276680299820175476315067102 gave the > following > results: > K satisfies the following Z-linear combination : > - K + Pi**3 Hello there, Anyone knows how to prove this? Let k = 4m+3. Given the eqn, v(pi^k) = 2^(k-1) Sum[f1, {n,1,inf}] - (2^(k-1)+1) Sum[f2, {n,1,inf}] + Sum[f3, {n,1,inf}] where {1/f1, 1/f2, 1/f3} = {n^k(q^n-1), n^k(q^(2n)-1), n^k(q^(4n)-1)} and q = e^(pi), (with inf = infinity) then v is a rational number. Ex. (1/180) pi^3 = 2^2 Sum[f1, {n,1,inf}] - (2^2+1) Sum[f2, {n,1,inf}] + Sum[f3, {n,1,inf}] (13/14175) pi^7 = 2^6 Sum[f1, {n,1,inf}] - (2^6+1) Sum[f2, {n,1,inf}] + Sum[f3, {n,1,inf}] and so on. See more at Identities inspired by Ramanujan's Notebooks (Part 2): http://www.lacim.uqam.ca/~plouffe/inspired2.pdf Actually, Plouffe's identities have the general form, pi^k = a*Sum[f1, {n,1,inf}] + b*Sum[f2, {n,1,inf}] + c*Sum[f3, {n, 1,inf}] for some rational numbers {a,b,c} for all ODD k, but, after some tweaking, I noticed it has that simpler form when k = 4m+3. (He found these identities experimentally using Maple and PSLQ.) - Titus x=pi e^(ipi)+1=0 Martin Musatov - Show quoted text - Hello there, Anyone knows how to prove this? Let k = 4m+3. Given the eqn, v(pi^k) = 2^(k-1) Sum[f1, {n,1,inf}] - (2^(k-1)+1) Sum[f2, {n,1,inf}] + Sum[f3, {n,1,inf}] where {1/f1, 1/f2, 1/f3} = {n^k(q^n-1), n^k(q^(2n)-1), n^k(q^(4n)-1)} and q = e^(pi), (with inf = infinity) then v is a rational number. Ex. (1/180) pi^3 = 2^2 Sum[f1, {n,1,inf}] - (2^2+1) Sum[f2, {n,1,inf}] + Sum[f3, {n,1,inf}] (13/14175) pi^7 = 2^6 Sum[f1, {n,1,inf}] - (2^6+1) Sum[f2, {n,1,inf}] + Sum[f3, {n,1,inf}] and so on. See more at Identities inspired by Ramanujan's Notebooks (Part 2): http://www.lacim.uqam.ca/~plouffe/inspired2.pdf Actually, Plouffe's identities have the general form, pi^k = a*Sum[f1, {n,1,inf}] + b*Sum[f2, {n,1,inf}] + c*Sum[f3, {n, 1,inf}] for some rational numbers {a,b,c} for all ODD k, but, after some tweaking, I noticed it has that simpler form when k = 4m+3. (He found these identities experimentally using Maple and PSLQ.) - Titus x=pi e^(ipi)+1=0 Martin Musatov Simon Plouffe before...??? act... Martin Musatov to Simon the below text: Regarding what you said before, Mr. Plouffe, I must apologize and assert my responsibility to clear up any confusion you may have. You see it is not my intent with this writing to prove to you P=NP, I am more interested in geometry and the fact I cannot understand why a flat line to a circle to a ball cannot be drawn in logical lines on a flat piece of paper. I have a drawing to prove pi is not irrational. Do you agree with this statement after viewing the picture at the end of this message and reading the text before it? > The restriction of n-space is reflected in the > virtual impossibility of > finding roots to n-degree polynomials. If each > coefficient is a component > to a vector of n dimensions, and each power of x is > a > component to a vector > of n dimensions, the two vectors are orthogonal > since > their dot product is > zero. > The liberty of n-space is that it opens up > infinite > possibilites. A > theorist can prove his brainchild in 4 dimensions > if > he can't do it in 3. > Just imagine. Associated with every point in > (x1,x2,x3) is a point on the > x4 axis. That is, associated with every (x,y,z,t) > is > a point x4 hidden in > the system of points (x1,x2,x3,x4,t), not > considering > that furthermore t > (time) may be altered. The theorist becomes an > artist producing > explanations at whim in equations that all come > together perfectly. > Anything whatsoever can be proven using math of > higher dimensions in E^n > where n>3. > This also delimits n-space math. On the one hand > it > is uselessly > constrictive and on the other uselessly diluted. > Consider a number of E^3 coordinate systems in > synchrounous time with each > other separated by large distances. I think this > is > as best a model as can > be hoped for. One system is unbridled and actions > in > it mapped to form > reactions in the other systems. That way, > associated > with one system are > events in the other systems by cause and effect. > When a tree is felled in > one system, a dam is caused to be broken in the > other > systems. This > meaningless association is diluted as well, but > Master and Slave can swap > roles like a see-saw. This is in line with > momentum > that is channeled for > technological uses. No, there is no relationship. > But can you find one? > The man's pulse was connected mechanically to > regulate the speed of a > flywheel. After this went on for awhile, the > technician altered the > flywheel and observed the man's pulse. He found > out > nothing. There are > some processes, however, that do swap Master/Slave > relationships, like a > child's see-saw on a playground. What are they? > These are good candidates > for multiple systems of E^3, particularly in > distance > relationships to > unknown cause and effect forces. The media may be > EMR. The relatively > equipotential space-time around the planet may > have > anomalies, like > gradients or funnels. The ancient structures > built > in phylogeny are a > vortex. The path of least resistance is to delve > and > fall into them, > degenerating and destroying the foundations of > everything that went in to > bringing the world up to the present. > Suppose one system was in the nucleus of a star > and > the other system in the > vortex of a Black Hole. Is there any cause and > effect between the two? > No. > NOW here is a another great flaw in your current > mathematics that defies logic PROVE THIS!. etc. > things are what they are, 1 is a unit, what do you > need proof for it . I was just recently in my office > playing a mathematical game with a 6 year old child , > giving him difficult circles etc. The Child thinks a > little and realizes that I was giving him some thing > that he did not understand or know . So he devises > his own response, he drwas 67 lines one on top of > another in a mess, and asks me , how many lines are > these? (you smart alec answer me), and Prove this > he says. Now if this child is kept from being brain > washed from your Mathematics, he will develop into a > great mathematicain, because he has the logic to know > when he is asked why is the sky blue, prove ut , he > says Cause(you dummy some things are as they are). > Your mathematics is flawed and many of you are brain > washed by your mathematical dictums, your professors > and your journals are all Hobsonian --- come join > us all you slaves of current mathematics, learn to > think free of Einstein and your gods , join the > inverse 19 movement to change mathematical theory. > We will give you further expose in 15 days when the > monument to Lost matghematics is unvelied here in > Wisconsin Proof pi is not irrational: http://www.flickr.com/photos/24869933@N07/3619373444/sizes/l/ Musatov (Inverse 19 Mathematics Co-Creator): not the only but one without it would not exist. Simon, the above text is my work. Efforts by mathematicians to determine the value of pi since have led to the conclusion that it ... Unlike rational numbers, pi cannot be represented by a common fraction, .... But what about an uncommon fraction? Will you look at my drawing and in particular how the twelve points are the outside points of the sphere are all rationally reached by decimal numbers? 6.2035884*2 and doubled an infinite number of times should yield a constant decimal binary expansion (what I mean by this is the numbers to the right of the decimal point should always end in 4, then 8, then 6, then 2, then 4, then 8, then 6, then 2...repeating for infinity). So in binary, by starting with this number we predict for n>2 the infinite binary tree will end with decimal progression: 2, 4, 8, 6 For n<2 the infinite binary tree will begin and end (depends on if you are increasing or decreasing the value) 1, 5, 5, 5 (repeating 5 infinitely) as we decrease to infinitely small. Musatov Inverse Counting Chain of Infinite Proportions (sample) 0.0000473372528076171875 0.000094674505615234375 0.00018934901123046875 0.0003786980224609375 0.000757396044921875 0.00151479208984375 0.0030295841796875 0.006059168359375 0.01211833671875 0.0242366734375 0.048473346875 0.09694669375 0.1938933875 0.387786775 0.77557355 1.5511471 3.1022942 6.2035884 12.4071768 24.8143536 49.6287072 99.2574144 198.5148288 397.0296576 794.0593152 1588.1186304 3176.2372608 6352.4745216 12704.9490432 25409.8980864 50819.7961728 101639.5923456 203279.1846912 406558.3693824 813116.7387648 1626233.4775296 3252466.9550592 6504933.9101184 13009867.8202368 26019735.6404736 52039471.2809472 104078942.5618944 208157885.1237888 416315770.2475776 832631540.4951552 1665263080.9903104 3330526161.9806208 6661052323.9612416 13322104647.9224832 26644209295.8449664 53288418591.6899328 106576837183.3798656 213153674366.7597312 426307348733.5194624 852614697467.0389248 1705229394934.0778496 3410458789868.1556992 6820917579736.3113984 13641835159472.6227968 27283670318945.2455936 54567340637890.4911872 109134681275780.9823744 218269362551561.9647488 436538725103123.9294976 873077450206247.8589952 1746154900412495.7179904 3492309800824991.4359808 6984619601649982.8719616 13969239203299965.7439232 27938478406599931.4878464 55876956813199862.9756928 111753913626399725.9513856 223507827252799451.9027712 447015654505598903.8055424 894031309011197807.6110848 1788062618022395615.2221696 3576125236044791230.4443392 7152250472089582460.8886784 14304500944179164921.7773568 28609001888358329843.5547136 57218003776716659687.1094272 114436007553433319374.2188544 228872015106866638748.4377088 457744030213733277496.8754176 915488060427466554993.7508352 1830976120854933109987.5016704 3661952241709866219975.0033408 7323904483419732439950.0066816 14647808966839464879900.0133632 29295617933678929759800.0267264 58591235867357859519600.0534528 117182471734715719039200.1069056 234364943469431438078400.2138112 468729886938862876156800.4276224 937459773877725752313600.8552448 1874919547755451504627201.7104896 3749839095510903009254403.4209792 7499678191021806018508806.8419584 (sample) If you would like to do the honors Mr. Plouffe, I endear you to calculate the angles and lengths of the number of ways, it is quite obviously possible to square a circle based on the geometry and numerical progression. As is self-evident to anyone with even a modest understanding of mathematics and geometry after visual inspection of the drawn proof (http://www.flickr.com/photos/24869933@N07/3619373444/sizes/l/) it is quite easily accomplished to connect and triangulate the outer points of the rectangles to the outer points of the square and circle and even by dimension sphere. Yes, the drawing clearly by definition of inverse symmetry (take the drawing cut it in half at the equator and turn the paper over and rotate it and compare the top and bottom mirrored symmetry and you will see it is an exact inverse of symmetry itself) the drawing explains how to properly and mathematically soundly represent a fourth dimension on a two dimensional plane. In plane language if you take the picture and separate the top half of the sphere by the bottom half of the sphere they are identical but only if the sphere existed on a complete three dimensional plane through expansion. Since the paper is flat, this is an impossibility, nonetheless it is completely expressed and retractable and countable infinitely. By the definition of our standard we have established any k-digit approximation to pi is rational. Which could have many practical scientific applications. For instance, search: ON THE THEOREM OF IVA$EV-MUSATOV IIIx e F(l) + F(2) + ... + F(p). This remark is the basis of our proof of Theorem .... -y(M + 1), x' + y(M + 1)] = 0 it is easy to check that Conditions (i), ...... Now set i = pi x. *jj, v. Since p, x is positive,. (i) fi is positive, ... plms.oxfordjournals.org/cgi/reprint/s3-53/1/143.pdf REPRESENTATION OF FUNCTIONSBY SERIES AND CLASSES .83'(1)with measure ÄEÄ> 2.83ë -- .83Ì and such that the trigonometric series for g(x) converges uniformly on [0, 2.83ë]. .... trigonometric series are due to Ivashev-Musatov [40] who has shown that there are null series (in the sense of ..... function .83'(.83ï) e C(0, 1) with .83'(0) = .83'(1) = 0 such that ...... <*i Mr. Plouffe, (Musatov), I appreciate you keep > the discussion an open discussion and > please DO post my replies on sci.math. > This is entirely appropriate given the > of your theory. > Okay but, pi is rationnal > the process you describe was unknown, it is > a finite process, it has to go to > finite value to reach any value : that process > is equivalent to have an infinite series > for pi, this fact was discovered hundreds of > years ago by fran.8dois VI.8eTE. > Again, if you use an finite process of > a construction of polygons for example > it leads to an FINITE series, an FINITE > series of polygons can't reach a rational > value and at the same time reach a value > like PI it is possible. There is > only 1 type of rational fraction : an integer > over another integer, you can have a fraction > made up with radicals or integrals if you want > but these can be called rationals in general. THere is > a naming convention about that which is clear: > a fraction is made of a ratio of integers, > but this could change. > simon plouffe >Martin Musatov Results 1 - 10 for pi fraction uncommon. (0.22 seconds) === === Subject: Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT Seriously, your time to write me has meant a great deal to me. Strangers is fine, but I need not a name to recognize a kinship or friendship or kindness. While I appreciate your fierce sentiment in my defense, I believe people are basically good and only hope the ones who acted such a way only did so because I was misunderstood. You do not need to speak harshly of them on my account. I forgive them and wish them no ill will. Truth is what I seek. Bertrand Russell said some brilliant things, one of them being, Without God, life has no meaning. This floored me coming from an atheist. I am religious and I take the value of your words to heart. May I ask what if anything prompted you in this reference? Am I not still in my soul? I assure I am. This last section in your thread: be still Musatov, find the mathematics in the whistling wind , happiness in the garden, and watch the loon for hours , and be still and greater wisdom will come to you It puzzles me. I have seen so many statements like this at the end of threads and they seem to follow intense debates by more senior level posters. Can you tell me what purposes they serve? Keep in touch, (I hope) Martin Musatov Sent from my Verizon Wireless BlackBerry -----Original Message----- === Subject: Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT You have my respect and wishes and sincerely so. I will prefer to remain annonymous even though you choose other wise purely because I am very independant and I do not have any regular allegiences but my family. I just want to see this current project through with a web site and then I am done, I have lots of other hobbies. I am very honest , I do not keep any single contact with no body, that is the way I live, life is less complex that way. Simplicity is genius, a quote from an English mathematician Bertrand Russel.. You may have good ideas Musatov , and you seem to be decent, but as the Bible states , be still in the soul. If you succeed in your ventures let me know, otherwise we should stay as strangers, nothing personal , I am that way. If I succeed I will contact you,---- but those other bastards at the Forum, I gave them their due, be still Musatov, find the mathematics in the whistling wind , happiness in the garden, and watch the loon for hours , and be still and greater wisdom will come to you Be still in the soul Musatov, be still, God bless you . Find success! hope 9900 ----- Original Message ----- === === Subject: Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT Seriously, your time to write me has meant a great deal to me. Strangers is fine, but I need not a name to recognize a kinship or friendship or kindness. While I appreciate your fierce sentiment in my defense, I believe people are basically good and only hope the ones who acted such a way only did so because I was misunderstood. You do not need to speak harshly of them on my account. I forgive them and wish them no ill will. Truth is what I seek. Bertrand Russell said some brilliant things, one of them being, Without God, life has no meaning. This floored me coming from an atheist. I am religious and I take the value of your words to heart. May I ask what if anything prompted you in this reference? Am I not still in my soul? I assure I am. This last section in your thread: be still Musatov, find the mathematics in the whistling wind , happiness in the garden, and watch the loon for hours , and be still and greater wisdom will come to you It puzzles me. I have seen so many statements like this at the end of threads and they seem to follow intense debates by more senior level posters. Can you tell me what purposes they serve? Keep in touch, (I hope) Martin Musatov Sent from my Verizon Wireless BlackBerry -----Original Message----- === Subject: Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT You have my respect and wishes and sincerely so. I will prefer to remain annonymous even though you choose other wise purely because I am very independant and I do not have any regular allegiences but my family. I just want to see this current project through with a web site and then I am done, I have lots of other hobbies. I am very honest , I do not keep any single contact with no body, that is the way I live, life is less complex that way. Simplicity is genius, a quote from an English mathematician Bertrand Russel.. You may have good ideas Musatov , and you seem to be decent, but as the Bible states , be still in the soul. If you succeed in your ventures let me know, otherwise we should stay as strangers, nothing personal , I am that way. If I succeed I will contact you,---- but those other bastards at the Forum, I gave them their due, be still Musatov, find the mathematics in the whistling wind , happiness in the garden, and watch the loon for hours , and be still and greater wisdom will come to you Be still in the soul Musatov, be still, God bless you . Find success! hope 9900 Reply Forward New window Print all Turn on highlighting Turn off highlightingSponsored Links A Course in Miracles www.QuantumJumping.com USA Hockey Jerseys-Lowest Prices--With Sewn 0n Tackle Twill Letters And Numbers For Just 99.50! www.miraclehockeyjersey.com Attention Math Teachers Need Online Grad Math Courses? Offered by Converse College www.OnlineMathCourses.org Proven Math Interventions Address all tiers of student need with Math Navigator and Ramp-Up www.americaschoice.org/mathematics Computational Math MS Differentiate yourself in our integrated Math+CS+Stats program! www.compmath.duq.edu Get Spiritual Healing Start Your Spiritual Healing Path Download Spiritual Healing Audio www.SilvaLifeSystem.com Zero Carb Noodles Miracle Noodles Have Zero Carbs. The Way Japanese Women Stay Thin. www.MiracleNoodle.com Are You a Good Mom? Are you a Good Mom? 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DetailsGmail view: standard | turn off chat | basic HTML Learn more©2009 Google - Terms - Privacy Policy - Google Home === Subject: Re: There is either one universe or an infinite number posting-account=rsfFvAoAAADqHCqmunbjtmaKjfuP-pwP Trident/4.0; GTB6; Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; InfoPath.1; .NET CLR 1.1.4322; .NET CLR 3.5.30729; .NET CLR 3.0.30618; OfficeLiveConnector.1.3; OfficeLivePatch.0.0),gzip(gfe),gzip(gfe) > Assume only one universe.In such a case what are the odds that the > natural laws in that universe will allow intelligent life to evolve? > Just about nil. > No, that is not right. The odds of it happening somewhere in the > universe is quite large. The odds of it happening in a particular > place in the universe is quite small. Do you know how statistics work? > What about a huge, but finite, number of other universes with > differing laws? > It seems that if there are multiple universes then there must be an > infinite number of them---there cannot be a huge, but finite, number > of universes, each with differing laws? > Why?Because for there to be a limit to the number of universes there > would have to be one law, applicable in all cases, limiting the total > number of universes, while allowing a huge (but finite) number to > exist.But each of the universes has differing laws so there cannot > be this one overriding law. > Statistics are about odds and evens. have no audit and no means of taking one, no numbers to crunch! Anecdotes nothwithstanding, however, it is better to sell fridges to Arabians and CH to Eskimos, and perhaps there are no customers on say, Sol. On the other hand, the conditions that sustain life may be more diverse than we imagine - or even can imagine. 'Likely?' That's the problem. I put an underdog fiver on Andorra to win! Just thought it would be fun if somehow, perhaps a chapter of accidents [like life?], Andorra had won, being under the bottom of the heap. I got a good price and a ha-ha-ha! http://www.guardian.co.uk/football/2009/jun/10/england-andorra-live-world-cu p -- Yrs, UncleEnglish. === Subject: Re: Inequality with xy+yz+zx=1. > Hello teacher~ > > xy + yz + zx = 1 (x, y, z : positive real) > > Show that > [x / sqrt{(x^2) + 1)}] + [y / sqrt{(y^2) + 1}] + [z / > sqrt{(z^2) + 1}] <= > 3/2. > > ------------------------------------------------------ > ----------------------------------------------- > Hm... > > x^2 + 1 = x^2 + xy + yz + zx = (x+y)(x+z) > > y^2 + 1 = y^2 + xy + yz + zx = (y+z)(y+x) > > z^2 + 1 = z^2 + xy + yz + zx = (z+y)(z+x) > > I can't progress favorably. > so, I need your advice. > > Why do you call me teacher? You have only one teacher and that is the Christ. Here is the complext plane: (learn from it)http://www.flickr.com/photos/24869933@N07/3570546800/sizes/l/in/photostre a m/ Musatov === Subject: Re: Inequality with xy+yz+zx=1. > Hello teacher~ xy + yz + zx = 1 (x, y, z : positive real) Show that > [x / sqrt{(x^2) + 1}] + [y / sqrt{(y^2) + 1}] + [z / sqrt{(z^2) + 1}] <= > 3/2. ---------------------------------------------------------------------------- - ------------------------ > Hm... x^2 + 1 = x^2 + xy + yz + zx = (x+y)(x+z) y^2 + 1 = y^2 + xy + yz + zx = (y+z)(y+x) z^2 + 1 = z^2 + xy + yz + zx = (z+y)(z+x) I can't progress favorably. > so, I need your advice. Ok, let's go.. x^2 + 1 = x^2 + xy + yz + zx = (x+y)(x+z) y^2 + 1 = y^2 + xy + yz + zx = (y+z)(y+x) z^2 + 1 = z^2 + xy + yz + zx = (z+y)(z+x) so, [x / sqrt{(x^2) + 1}] = sqrt(x*x) / sqrt{(x+y)(x+z)} [y / sqrt{(y^2) + 1}] = sqrt(y*y) / sqrt{(y+z)(y+x)} [z / sqrt{(z^2) + 1}] = sqrt(z*z) / sqrt{(z+y)(z+x)} By Arithmetic-geometric mean, [x / sqrt{(x^2) + 1}] = sqrt(x*x) / sqrt{(x+y)(x+z)} <= [{x/(x+y)}+{x/(x+z)}] / 2 [y / sqrt{(y^2) + 1}] = sqrt(y*y) / sqrt{(y+z)(y+x)} <= [{y/(y+z)}+{y/(y+x)}] / 2 [z / sqrt{(z^2) + 1}] = sqrt(z*z) / sqrt{(z+y)(z+x)} <= [{z/(z+y)}+{z/(z+x)}] / 2 so, [x / sqrt{(x^2) + 1}] + [y / sqrt{(y^2) + 1}] + [z / sqrt{(z^2) + 1}] <= [{(x+y) / (x+y)} + {(x+z) / (x+z)} + {(y+z) / (y+z)}] / 2 = 3/2 === Subject: Re: Inequality with xy+yz+zx=1. posting-account=06BQLAoAAADoC7Y4z9FWcUwGvMa7xMG9 7.4),gzip(gfe),gzip(gfe) > Hello teacher~ > xy + yz + zx = 1(x, y, z : positive real) > Show that > [x / sqrt{(x^2) + 1}] + [y / sqrt{(y^2) + 1}] + [z / sqrt{(z^2) + 1}] <= > 3/2. > ---------------------------------------------------------------------------[ CapitalEth]-------------------------- > Hm... > x^2 + 1 = x^2 + xy + yz + zx = (x+y)(x+z) > y^2 + 1 = y^2 + xy + yz + zx = (y+z)(y+x) > z^2 + 1 = z^2 + xy + yz + zx = (z+y)(z+x) > I can't progress favorably. > so, I need your advice. Ok, let's go.. x^2 + 1 = x^2 + xy + yz + zx = (x+y)(x+z) y^2 + 1 = y^2 + xy + yz + zx = (y+z)(y+x) z^2 + 1 = z^2 + xy + yz + zx = (z+y)(z+x) so, > [x / sqrt{(x^2) + 1}] = sqrt(x*x) / sqrt{(x+y)(x+z)} > [y / sqrt{(y^2) + 1}] = sqrt(y*y) / sqrt{(y+z)(y+x)} > [z / sqrt{(z^2) + 1}] = sqrt(z*z) / sqrt{(z+y)(z+x)} By Arithmetic-geometric mean, > [x / sqrt{(x^2) + 1}] = sqrt(x*x) / sqrt{(x+y)(x+z)} <= > [{x/(x+y)}+{x/(x+z)}] / 2 > [y / sqrt{(y^2) + 1}] = sqrt(y*y) / sqrt{(y+z)(y+x)} <= > [{y/(y+z)}+{y/(y+x)}] / 2 > [z / sqrt{(z^2) + 1}] = sqrt(z*z) / sqrt{(z+y)(z+x)} <= > [{z/(z+y)}+{z/(z+x)}] / 2 so, > [x / sqrt{(x^2) + 1}] + [y / sqrt{(y^2) + 1}] + [z / sqrt{(z^2) + 1}] <= [{(x+y) / (x+y)} + {(x+z) / (x+z)} + {(y+z) / (y+z)}] / 2 = 3/2- Masquer le texte des messages pr.8ec.8edents - - Afficher le texte des messages pr.8ec.8edents - Bravo Mina! Comparisons by arithmetic-geometric mean seems a very useful tool while solving inequations, Wish you a nice week end, Alain === Subject: Re: Factoring a large number? > Hi all, > > could someone try to factor the following 1351 digit > large number > below? > I'm not 100% sure but there should be just 2 prime > factors. > > 1725073237205857947941058976975667133967120394002 > 9684505992358918550555677139461746853893933753466 > 3324516557318514017702376369458932632547335906243 > 4082891372765217842651075223855023307360880491775 > 3856442298683876128462251451917561739289186463033 > 4528305924022898503449773725676053169057058336217 > 3998344551379354992523133080009562978285271680807 > 7857202153188090923341503168045855143060178430084 > 8842399422446292885619604650271635139199715874316 > 6249545774204554432543412070289958510693650610447 > 6305663902358695943110225660265027554844049187469 > 9514113137552295755717908429386036094056921868492 > 9588333233322200116414929913714582094242085448843 > 9524458937344392220207261992727216424252007246532 > 5032666043798779148759791622022514866469373306815 > 9035038106710882360772044728893007882282428959869 > 5038864028210682514850556736397455671006628590617 > 1292571468948550229615122349988593207305772126201 > 5813505142130348354654341355595257604176115566233 > 7760677658198665401229108701128158628589993355922 > 1105711458385211338597647800333671798311700249370 > 4131890750911963299543818564505501830749682740718 > 2020435920283859037411266417359917460923787834769 > 9645078197551794635334020841500476920521122586462 > 1213415446332729016403692831468979349307996181633 > 4098404475482625626466031528939783050395972544355 > 2178677174089467788658949249 > > > Gerry Your search - gcd(x1,f1)=x1gcd(x2,f2)=x2gcd(x1351,f3)=x3gcd(x1351,f4)=x4Since i used a pretty simple definition for f1..f4 i wantedto find out if it is easy to factor a composte usingthe big factors of f1..f4 hence the number x1351. - did not match any documents. Suggestions: Make sure all words are spelled correctly. Try different keywords. Try more general keywords. Try fewer keywords. About ¡ Blog ¡ FreeWiki» ©2009 MeAmI.org Search for the People Powered by infinity: ¡ === Subject: Re: Factoring a large number? posting-account=PSzFRAoAAAARszS8zeFmxtqyivK9-1_f InfoPath.2; .NET CLR 2.0.50727; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Some make it sound as if it is easy to choose big primes. > Is it really? > It is not hard, if you have the right software and hardware, > to test, say, a 500-digit number for primality. > To find a 500-digit prime, take your favorite 500-digit number, > N, and test N, N + 1, N + 2, etc., until you find one that is > prime. Chances are you won't have to test too many (and there > are obvious savings to be made by not testing any even numbers, > etc.). > I don't mind explaining how i got the numbers and i will when the time > is right. > The time was right when you started this thread. > If you ask the group for help, why not level with us from the start? > -- > GM > Hi Gerry, > unfortunately its not the right time because i really don't have that > much time believe me. > I wish i had more time to write out how i can produce these numbers > but i have to go. > Maybe later... If you are producing the numbers by creating a few large primes and > then multiplying them together, then nobody will be impressed. > If someone gave you the original 1300 digit+ number and you factored > it then it is very impressive. > I am guessing the former rather than the latter. For instance, I can easily multiply these three numbers together to > get a very large composite that should be relatively difficult to > factor: > 111111111111111111111111111111111111111111111111111111111111111111111111111[ CapitalEth]11111111111111111111111111111111111111111111111111111111111111111 1 111111111[CapitalEth]1111111111111111111111111111111111111111111111111111111 129 > 222222222222222222222222222222222222222222222222222222222222222222222222222[ CapitalEth]22222222222222222222222222222222222222222222222222222222222222222 2 222222222[CapitalEth]2222222222222222222222222222222222222222222222222222222 22222222222222222222[CapitalEth]22222222222222222222222222222222222222222222 2222222222222222222222222222222[CapitalEth]222222222222222222222222222222222 222222222222222222222222222222222222222331 > 333333333333333333333333333333333333333333333333333333333333333333333333333[ CapitalEth]33333333333333333333333333333333333333333333333333333333333333333 3 333333333[CapitalEth]3333333333333333333333333333333333333333333333333333333 33333333333333333333[CapitalEth]33333333333333333333333333333333333333333333 3333333333333333333333333333333[CapitalEth]333333333333333333333333333333333 333333333333333333333333333333333333333333[CapitalEth]3333333333333333333333 33333333333333333333333333333333333333333333333333333[CapitalEth]33333333333 33333333333333333333333333333333333333333334903 But creating the factors is as easy as calling nextprime > (some big input) with pari. > Not impressive in the least. But if you are given some 1000+ digit composite with all of the > factors having more than 100 digits, then it will be an impressive > feat to factor it unless it has a special format (e.g. like a Mersenne > Prime).- Hide quoted text - - Show quoted text - Hi all, Okay here's how i defined the numbers: let x1 be the smallest prime i mentioned in reply 20 x2 the larger one. x3 and x4 are the factors of the x1351 number. so x1351=x3*x4; define : f1=11^(105)-19; f2=11^(210)-19; f3=11^(420)-19; f4=11^(889)-19; And f1..f4 factor very fast for some reason which i don't know. Now use gcd to verify that: gcd(x1,f1)=x1 gcd(x2,f2)=x2 gcd(x1351,f3)=x3 gcd(x1351,f4)=x4 Since i used a pretty simple definition for f1..f4 i wanted to find out if it is easy to factor a composte using the big factors of f1..f4 hence the number x1351. Thats all there is and i did not want to impress anyone. I'm just looking for answers. Gerry === Subject: Re: Factoring a large number? posting-account=PSzFRAoAAAARszS8zeFmxtqyivK9-1_f InfoPath.2; .NET CLR 2.0.50727; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Some make it sound as if it is easy to choose big primes. > Is it really? > It is not hard, if you have the right software and hardware, > to test, say, a 500-digit number for primality. > To find a 500-digit prime, take your favorite 500-digit number, > N, and test N, N + 1, N + 2, etc., until you find one that is > prime. Chances are you won't have to test too many (and there > are obvious savings to be made by not testing any even numbers, > etc.). > I don't mind explaining how i got the numbers and i will when the time > is right. > The time was right when you started this thread. > If you ask the group for help, why not level with us from the start? > -- > GM > Hi Gerry, > unfortunately its not the right time because i really don't have that > much time believe me. > I wish i had more time to write out how i can produce these numbers > but i have to go. > Maybe later... > If you are producing the numbers by creating a few large primes and > then multiplying them together, then nobody will be impressed. > If someone gave you the original 1300 digit+ number and you factored > it then it is very impressive. > I am guessing the former rather than the latter. > For instance, I can easily multiply these three numbers together to > get a very large composite that should be relatively difficult to > factor: > 111111111111111111111111111111111111111111111111111111111111111111111111111[ CapitalEth][CapitalEth]11111111111111111111111111111111111111111111111111111 111111111111111111111[CapitalEth]1[CapitalEth]11111111111111111111111111111 11111111111111111111111111129 > 222222222222222222222222222222222222222222222222222222222222222222222222222[ CapitalEth][CapitalEth]22222222222222222222222222222222222222222222222222222 222222222222222222222[CapitalEth]2[CapitalEth]22222222222222222222222222222 22222222222222222222222222222222222222222222[CapitalEth]22[CapitalEth]22222 2222222222222222222222222222222222222222222222222222222222222222222[CapitalE th]222[CapitalEth]2222222222222222222222222222222222222222222222222222222222 2222222222222[CapitalEth]2331 > 333333333333333333333333333333333333333333333333333333333333333333333333333[ CapitalEth][CapitalEth]33333333333333333333333333333333333333333333333333333 333333333333333333333[CapitalEth]3[CapitalEth]33333333333333333333333333333 33333333333333333333333333333333333333333333[CapitalEth]33[CapitalEth]33333 3333333333333333333333333333333333333333333333333333333333333333333[CapitalE th]333[CapitalEth]3333333333333333333333333333333333333333333333333333333333 3333333333333[CapitalEth]3333[CapitalEth]3333333333333333333333333333333333 333333333333333333333333333333333333[CapitalEth]33333[CapitalEth]3333333333 333333333333333333333333333333333333333333334903 > But creating the factors is as easy as calling nextprime > (some big input) with pari. > Not impressive in the least. > But if you are given some 1000+ digit composite with all of the > factors having more than 100 digits, then it will be an impressive > feat to factor it unless it has a special format (e.g. like a Mersenne > Prime).- Hide quoted text - > - Show quoted text - Hi all, Okay here's how i defined the numbers: let x1 be the smallest prime i mentioned in reply 20 > x2 the larger one. > x3 and x4 are the factors of the x1351 number. so x1351=x3*x4; define : f1=11^(105)-19; > f2=11^(210)-19; > f3=11^(420)-19; > f4=11^(889)-19; And f1..f4 factor very fast for some reason which i don't know. Now use gcd to verify that: gcd(x1,f1)=x1 > gcd(x2,f2)=x2 > gcd(x1351,f3)=x3 > gcd(x1351,f4)=x4 Since i used a pretty simple definition for f1..f4 i wanted > to find out if it is easy to factor a composte using > the big factors of f1..f4 hence the number x1351. Thats all there is and i did not want to impress anyone. I'm just looking for answers. > Gerry- Hide quoted text - - Show quoted text - So there should be an easy way to factor such composites by using log 11(x1351) right? === Subject: Re: Factoring a large number? posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2009060310 Ubuntu/8.10 (intrepid) Firefox/3.0.11,gzip(gfe),gzip(gfe) > Some make it sound as if it is easy to choose big primes. > Is it really? > It is not hard, if you have the right software and hardware, > to test, say, a 500-digit number for primality. > To find a 500-digit prime, take your favorite 500-digit number, > N, and test N, N + 1, N + 2, etc., until you find one that is > prime. Chances are you won't have to test too many (and there > are obvious savings to be made by not testing any even numbers, > etc.). > I don't mind explaining how i got the numbers and i will when the time > is right. > The time was right when you started this thread. > If you ask the group for help, why not level with us from the start? > -- > GM > Hi Gerry, > unfortunately its not the right time because i really don't have that > much time believe me. > I wish i had more time to write out how i can produce these numbers > but i have to go. > Maybe later... > If you are producing the numbers by creating a few large primes and > then multiplying them together, then nobody will be impressed. > If someone gave you the original 1300 digit+ number and you factored > it then it is very impressive. > I am guessing the former rather than the latter. > For instance, I can easily multiply these three numbers together to > get a very large composite that should be relatively difficult to > factor: > 111111111111111111111111111111111111111111111111111111111111111111111111111[ CapitalEth][CapitalEth]11111111111111111111111111111111111111111111111111111 111111111111111111111[CapitalEth]1[CapitalEth]11111111111111111111111111111 11111111111111111111111111129 > 222222222222222222222222222222222222222222222222222222222222222222222222222[ CapitalEth][CapitalEth]22222222222222222222222222222222222222222222222222222 222222222222222222222[CapitalEth]2[CapitalEth]22222222222222222222222222222 22222222222222222222222222222222222222222222[CapitalEth]22[CapitalEth]22222 2222222222222222222222222222222222222222222222222222222222222222222[CapitalE th]222[CapitalEth]2222222222222222222222222222222222222222222222222222222222 2222222222222[CapitalEth]2331 > 333333333333333333333333333333333333333333333333333333333333333333333333333[ CapitalEth][CapitalEth]33333333333333333333333333333333333333333333333333333 333333333333333333333[CapitalEth]3[CapitalEth]33333333333333333333333333333 33333333333333333333333333333333333333333333[CapitalEth]33[CapitalEth]33333 3333333333333333333333333333333333333333333333333333333333333333333[CapitalE th]333[CapitalEth]3333333333333333333333333333333333333333333333333333333333 3333333333333[CapitalEth]3333[CapitalEth]3333333333333333333333333333333333 333333333333333333333333333333333333[CapitalEth]33333[CapitalEth]3333333333 333333333333333333333333333333333333333333334903 > But creating the factors is as easy as calling nextprime > (some big input) with pari. > Not impressive in the least. > But if you are given some 1000+ digit composite with all of the > factors having more than 100 digits, then it will be an impressive > feat to factor it unless it has a special format (e.g. like a Mersenne > Prime).- Hide quoted text - > - Show quoted text - > Hi all, > Okay here's how i defined the numbers: > let > x1 be the smallest prime i mentioned in reply 20 > x2 the larger one. > x3 and x4 are the factors of the x1351 number. > so x1351=x3*x4; > define : > f1=11^(105)-19; > f2=11^(210)-19; > f3=11^(420)-19; > f4=11^(889)-19; > And f1..f4 factor very fast for some reason which i don't know. > Now use gcd to verify that: > gcd(x1,f1)=x1 > gcd(x2,f2)=x2 > gcd(x1351,f3)=x3 > gcd(x1351,f4)=x4 > Since i used a pretty simple definition for f1..f4 i wanted > to find out if it is easy to factor a composte using > the big factors of f1..f4 hence the number x1351. > Thats all there is and i did not want to impress anyone. > I'm just looking for answers. > Gerry- Hide quoted text - > - Show quoted text - I don't see why: > So there should be an easy way to factor such composites > by using log 11(x1351) right? Although f3 and f4 (to use your notations): > f3=11^(420)-19; > f4=11^(889)-19; could be recognized as near-powers-of 11 by way of such a logarithm, what you seem to have done is define x3,x4 as the largest prime factors of f3,f4 (resp.), then take the product of x3 and x4 (which you call x1351). But it seems unlikely that factors of x3 * x4 will be recognized via taking log base 11. A well-known criterion for primes p,q used in an RSA encryption scheme so that p*q is not easily factored is that p-1 and q-1 should each contain a large prime factor. Rivest further stated that gcd(p-1,q-1) should be small. The primes you've asked about are much larger than used in typical RSA applications. There's a practical trade- off in using larger prime factors for greater security and the speed of computation in carrying out encrypt/ decrypt operations. === Subject: Re: Help Stability ODE system > Hello. > I want to study the stability of the origin of the following problem: > dx/dt =3D -2y > dy/dt =3D x + 2y > dz/dt =3D -2z > So the eigenvalues of this 3 x 3 matrix are -2, 1 + i, 1-i. > The eigenvectors are (0,0,1) , (2,-1-i,0), (-2,-1+i,0). > The solution (confirmed with Mathematica) is given by: > x(t) =3D 2*exp(t)cos(t) * C_1 + 2*exp(t)*sin(t) * C_2 - > 2*exp(t)*sin(t) *C_3 > y(t) =3D C_1 * (-exp(t)*=A8sin(t) - exp(t)*cos(t)) + C_2 * > (exp(t)cos(t) - exp(t)sin(t)) + C_3 * (-exp(t)cos(t)+exp(t)sin(t) ) > z(t) =3D 2*exp(2*t) *C_3 > Where C_1,C_2,C_3 are constants. > How can I find (analytically, not by plotting) if the origin (0,0,0) > is stable, unstable, a node, a center? > I'm having trouble figuring this out. Can you please help? > The general rule is: > 1) if any eigenvalue has strictly positive real part, > then the critical point is unstable. > 2) if all eigenvalues have strictly negative real part, then the > critical point is asymptotically stable. > 3) if neither (1) nor (2) applies (i.e. no real parts are strictly > positive but at least one is 0, then the linear system is stable > but not asymptotically stable; the stability of a nonlinear system > with this linearization is not determined. Oops: the part about the linear system being stable in case (3) may not be true if the eigenvalues with real part 0 are not all simple. > In your case, 1-i and 1+i have real part 1. So... > > is 1) _always_ true, independant of the dimension of the system? Yes. See e.g. sections 7.l and 7.2 of Nonlinear differential equations and dynamical systems By Ferdinand Verhulst Edition: 2, illustrated Published by Springer, 1996 ISBN 3540609342, 9783540609346 > ...also, can you always get away with this type of analysis? or are there > systems where this type of linear stability analysis fails? The theorems apply to any system X' = F(X) where F has at least two continuous derivatives in a neighbourhood of the critical point. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: tensor inversion (matrix of matrices) posting-account=WOBCgAoAAACJuvQC2pYXyYd2HTLiKt9G Gecko/2009042513 Ubuntu/8.04 (hardy) Firefox/3.0.10,gzip(gfe),gzip(gfe) That was exactly the info I was looking for. I'm still amazed at all the linear algebra info and tricks out there. I've got a followup to that question: I have a system A*X=B that I'd like to solve. This would be a simpler matter if X and B were Nx1 vectors, and A were an NxN matrix, but in my case X and B are both Nx3 matrices, and A is NxN. To solve this, my method is to find the inverse of A and left multiply both sides to find the matrix X. I've implemented this using the LAPACK routines. Do you know of a generalized solver for these types of matrix equations that don't use Nx1 vectors? ~Eric > An NxN matrix, 'T', is a super-matrix whose elements are all 3x3 > scalar matrices. Thus, if I look at the element T(i,j) I get a 3x3 > matrix of scalars. If I look at T(i,j)(k,l) then I get a particular > scalar. > Is the proper way to look at this matrix as a rank-4 tensor? > how does one go about finding the inverse of such a matrix of > matrices? > ~Eric > As you have constructed T, it is equivalent for the sake of > matrix multiplication (and addition) to consider it simply > as a 3Nx3N matrix of scalars.In particular, inverting T > is equivalent (considered with 3x3 entries in an NxN array) > to inverting T as a 3Nx3N matrix (of scalars). T might well be called a block matrix, quite special in > that all blocks are the same (square) size.I don't think > tensor is a good term to use in this case. > === Subject: Re: tensor inversion (matrix of matrices) > > That was exactly the info I was looking for. I'm still amazed at all > the linear algebra info and tricks out there. > > I've got a followup to that question: > > I have a system A*X=3DB that I'd like to solve. This would be a simpler > matter if X and B were Nx1 vectors, and A were an NxN matrix, but in > my case X and B are both Nx3 matrices, and A is NxN. > > To solve this, my method is to find the inverse of A and left multiply > both sides to find the matrix X. I've implemented this using the > LAPACK routines. Do you know of a generalized solver for these types > of matrix equations that don't use Nx1 vectors? Note that A (column #j of X) = (column #j of B), so your problem is equivalent to solving Ax=b for three different b's. In general, inverse matrices are to be avoided whenever possible. You might use the LU decomposition of A. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: _Indices of subgroups > On Jun 11, 7:30pm, qsymmetry If G is a group and H is a subgroup of G of finite > index in G, and K is the core of H, then by the > correspondence theorem, > [G : H] = [G/K : H/K] > If we wanted to get an upper bound for [G : H], is > it enough to assume that G is finite? If so, why? > > Because the core of a subgroup of finite index is of > finite index. > > To see this, consider the action G by left > multiplication on the left > cosets of H; there are only finitely many of them, so > this gives you a > homomorphism from G to some S_n; the kernel is the > set of elements of > G that map every coset to themselves, and this is a > normal subgroup > contained in H. Thus, you can find N such that [G:N] > is finite, N and N normal. Since, as you note above, [G/N : H/N] = > [G:H], and G/N > is finite (it embeds in some S_n), you may work in > G/N instead of > working in G. So you may assume that G is finite. > So, the crux is: G/K is finite whether or not G is, so we might as well assume that G is finite? I feared that I might 'lose' something by assuming that G is finite. I thought that, since G/K is finite, work with G/K instead, get a bound on the index of H/K in G/K, and be done with it. Is there anything inherently wrong with this approach? In any case, I'm trying to understand why it's not (at all) necessary. === Subject: Re: _Indices of subgroups posting-account=_l4K0QkAAAC09JhOoK_ZfoJKXOmr_jZf Gecko/2009042513 Ubuntu/8.04 (hardy) Firefox/3.0.10,gzip(gfe),gzip(gfe) > On Jun 11, 7:30pm, qsymmetry If G is a group and H is a subgroup of G of finite > index in G, and K is the core of H, then by the > correspondence theorem, > [G : H] = [G/K : H/K] > If we wanted to get an upper bound for [G : H], is > it enough to assume that G is finite? If so, why? > Because the core of a subgroup of finite index is of > finite index. > To see this, consider the action G by left > multiplication on the left > cosets of H; there are only finitely many of them, so > this gives you a > homomorphism from G to some S n; the kernel is the > set of elements of > G that map every coset to themselves, and this is a > normal subgroup > contained in H. Thus, you can find N such that [G:N] >is finite, N and N normal. Since, as you note above, [G/N : H/N] = > [G:H], and G/N > is finite (it embeds in some S n), you may work in > G/N instead of > working in G. So you may assume that G is finite. > So, the crux is: G/K is finitewhether or not > G is, so we might as well assume that G is finite? The point is that you are going to want to work in G/K all the time rather than in G, and you will want to do that because (i) index results about H in G are the same as index results about H/K in G/K; and (ii) it's easier to work with H/K in G/K than with H in G because G/K is finite, so you have all sorts of counting arguments at your disposal [you also have them in the infinite case, but they are not usually as informative]. So, rather than drag the /K along all the time, you can just work in the new group overline(G) (which is equal/ isomorphic to G/K), and its subgroup overline(H) (which is equal/ isomorphic to H/K). This really amounts to going to the finite case. If all you are interested in are index results for a subgroup of finite index H, then saying we may assume that G is finite is really shorthand for H contains a subgroup K that is normal in G, and such that G/K is finite. Rather than working in G, we will work in G/K and lift the corresponding results for H/K to H. I feared that I might 'lose' something by assuming that G is finite. Certainly, there are things you lose; but if the only thing you are concerned about has to do with things that respect quotients (such as the index, since as you noted you have that [G:H] = [G/K : H/K]), then those things don't get lost because you are reducing to the finite case via a suitable quotient. Naturally, things that *do* (or may) get lost when going to quotients may be lost if you assume G is finite. > I thought that, since G/K is finite, work with G/K instead,get a bound on the index of H/K in G/K, and be done with it. That is, essentially, what you are implying by the shorthand we may assume that G is finite; it's just notationally easier. > Is there anything inherently wrong with this approach? No; in fact, it's what is really happening behind we may assume without loss of generality that G is finite. > In any case, I'm trying to understand why it's not (at all) necessary. Not necessary to do what? -- Arturo Magidin === Subject: Re: where is that pdf file about how to write mathematical proofs for dummies? posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) Re Jun 12, 4:40am, David C. Ullrich : David, I'll print your message and look it over this weekend. In the meantime, I'd like to share my own version that fills in a few details. My version is based more on Halmos, but it's basically the same as Kelley's (the even/odd thing is just a different way of saying the same thing). Note: By 'positive finite sequence' I mean a finite sequence starting at 1 instead of at 0. (Just because I like to start counting at 1 rather than 0.) Suppose X is dominated by Y, and Y is dominated by X. Without loss of generality we may assume X and Y are disjoint, by taking Xx{0} and Yx{1}. So we may as well consider X and Y as if disjoint. Let f be an injection from X into Y and g an injection from Y into X. Definition: v is a volley <-> v is a positive finite sequence & v(1) in XuY & An>0(n+1 in dom(v) -> ((v(n) in Y -> v(n+)=g(v(n))) & (v(n) in X -> v(n +)=f(v(n))))). Definition: p is an ancestor of d <-> Avn(v is a volley & n in dom(v) & v(1)=p & v(n)=d) Definition: p is a proper ancestor of d <-> p is an ancestor of d & ~p=d If v is a volley and v' is a volley and v(1)=v'(1), then v is a subset of v' or v' is a subset of v. (That is by induction on the length of v, as the length of v is a natural number not equal to 0.) Let X X = {j in X | Ez(z is an ancestor of j & z in X & ~Ep p is a proper ancestor of z)}. Let X Y = {j in X | Ez(z is an ancestor of j & z in Y & ~Ep p is a proper ancestor of z)}. Let X I = X(X X u X Y). Let Y X = {j in Y | Ez(z is an ancestor of j & z in X & ~Ep p is a proper ancestor of z)}. Let Y Y = {j in Y | Ez(z is an ancestor of j & z in Y & ~Ep p is a proper ancestor of z)}. Let Y I = Y(Y X u Y Y). It's easy to verify: X X and X Y are disjoint. X X and X I are disjoint. X Y and X I are disjoint. Y X and Y Y are disjoint. Y X and Y I are disjoint. Y Y and Y I are disjoint. X X u X Y u X I = X Y X u Y Y u Y I = Y f restricted to X X is a bijection from X X onto Y X. g^-1 restricted to X Y is a bijection from X Y onto Y Y. f restricted to X I is a bijection from X I onto Y I. So (f restricted to X X) u (g^-1 restricted to X Y) u (f restricted to X I) is a bijection from X onto Y. MoeBlee === Subject: Re: where is that pdf file about how to write mathematical proofs for dummies? posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) Re Jun 12, 11:24am, MoeBlee : That didn't format the line breaks as I wished. So I hope it's still readable despite some some awkard line breaks. MoeBlee === Subject: Re: where is that pdf file about how to write mathematical proofs for dummies? > Lamport, the author of that paper, claims there is an error in the > proof of Schroder-Bernstein in Kelley's 'General Topology'. That is > basically the same proof as in Halmos's 'Naive Set Theory'. I've > worked through those proofs. I had to fill in missing details, but > I've not seen any error. What error is Lamport referring to? > I have no idea, and have unfortunately lost my copy of _General > Topology_. Perhaps you could produce the proof here for our inspection? >Go to Google Books. Search 'Kelley Schroder Bernstein'. The first >result will be the book. Click on the book cover. You'll be at page 28 >of the book, where the proof appears. >[...] Ok, I looked it up on Google Books as you suggest. >For a second I said aha, that's just a simpler and >cleaner way to say what I just said. But it seems to me that what's in Kelly is _wrong_, >although it's easily fixed - what the author probably >meant, or should have meant, is ok. The problem >is indeed related to the fact that E_x can be finite >although the definition includes infinitely many >expressions: Suppose that there exist a in A and b in B with >f(a) = b and g(b) = a. Then f has exactly one >ancestor, namely b. Ok, two ancestors, a and b. So change every one to two below, and swap even and odd: > In particular a has an >odd number of ancestors. Kelley states that >this implies that f(a) has an even number of >ancestors, which is false; f(a) also has exactly >one ancestor. Error. But the problem is just that even/odd/infinite >number of ancestors is not exactly what we >meant or should have meant. Instead of >infinitely many ancestors consider the class >of a in A such that you can take the inverse >image, then the inverse image again, infinitely >many times. That would be the set of elements >with infinitely many ancestors and _also_ >the elements that are part of closed loops >as above. Subdivide the _other_ elements >by the parity of the number of ancestors. >Now things are fine. >MoeBlee David C. Ullrich Understanding Godel isn't about following his formal proof. >That would make a mockery of everything Godel was up to. >(John Jones, My talk about Godel to the post-grads. >in sci.logic.) === Subject: Re: where is that pdf file about how to write mathematical proofs for dummies? posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) On Jun 12, 11:16am, David C. Ullrich Suppose that there exist a in A and b in B with >f(a) = b and g(b) = a. Then a has exactly one >ancestor, namely b. a has two ancestors, viz. b and itself. a has one PROPER ancestor (if we wish to define 'proper ancestor' as 'an ancestor other than oneself'). >a has an >odd number of ancestors. Kelley states that >this implies that f(a) has an even number of >ancestors, which is false; f(a) also has exactly >one ancestor. Error. Where did Kelley say that? In any case, f(a) has two ancestors, viz. a and itself. By the way, here's Kelley's argument pretty much as he gives it: Suppose: A and B are disjoint f is an injection from A into B g is an injection from B into A Define: x is an ancestor of y iff you can get from x to y by successive alternate applications of f and g (or of g and f) Let: A E = the members of A that have an even number of ancestors A O = the members of A that have an odd number of ancestors A I = the members of A that have infinitely many ancestors B E = the members of B that have an even number of ancestors B O = the members of B that have an odd number of ancestors B I = the members of B that have infinitely many ancestors So: f maps A E onto B O f maps A I onto B I g^-1 maps A O onto B E So: (the restriction of f to (A E u A I)) u (the restriction of g^-1 to A O) maps A one-to-one onto B Then he gives some additional comments. I don't know whether those comments are being counted as part of the proof. But they're not needed. Indeed, I formalized his you can get from x to y by successive alternate applications of f and g (or of g and f) as a volley, without using his later mentioned E. And my version, aside from formalizing you can get from x to y by successive alternate applications of f and g (or of g and f) is essentially the same as Kelley, since my partitions of the two main sets are the same but using different wording (like Halmos, I dispense with the even/odd wording, though it amounts to the same thing). MoeBlee === posting-account=EI6PUAoAAAAazrTcDkROhitOfD8_tWFT .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) I'm developing an application of quantum field theory to quantum game theory, specifically to a model of a financial market, with bosonic and fermionic fields. And some of the field interactions do not fit exactly with the six basic interactions of QED, specifically, there is market equivalent of a boson. Does anyone here know if there any case in quantum field theory where this may occur, or is this specific to markets? My initial thought was that it is specific to markets but, just in case, maybe there is some interaction or some application of some theory within physics or in interdisciplinary applications, where this occurs? I'd really appreciate if anyone could help me on this matter. === Subject: Re: Understanding the quotient ring nomenclature posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2009042708 Fedora/3.0.10-1.fc10 Firefox/3.0.10,gzip(gfe),gzip(gfe) > How arbitrary a language is though ought > to be a concern of a mathemtatician. Fundamentals are important and so > the primitives in use and how they are used do weigh heavily in > mathematics. If by fundamentals you mean explicit constructions of objects then no, they are *not* important and they are *absolutely* irrelevant to the use of those objects. It is absolutely unimportant and irrelevant how you construct a complete totally ordered field. The only important thing is that such a construction can be carried out, and that all such objects are isomorphic, because then one can talk about the complete totally ordered field with the assurance that such a thing actually exists. Absolutely nothing in maths needs to know whether such a beast was constructed using Dedekind cuts, Cauchy sequences or through your pretty clumsy procedure involving magnitudes. -- m === Subject: Re: Understanding the quotient ring nomenclature posting-account=n26igQkAAACeF9xA2Ms8cKIdBH40qzwr Gecko/20070505 Iceape/1.0.9 (Debian-1.0.13~pre080614i-0etch1),gzip(gfe),gzip(gfe) On Jun 12, 2:43 pm, Mariano Su.87rez-Alvarez > How arbitrary a language is though ought > to be a concern of a mathemtatician. Fundamentals are important and so > the primitives in use and how they are used do weigh heavily in > mathematics. If by fundamentals you mean explicit constructions > of objects then no, they are *not* important and they > are *absolutely* irrelevant to the use of those objects. It is absolutely unimportant and irrelevant how you > construct a complete totally ordered field. The only > important thing is that such a construction can be > carried out, and that all such objects are isomorphic, > because then one can talk about the complete totally > ordered field with the assurance that such a thing > actually exists. Absolutely nothing in maths needs to > know whether such a beast was constructed using > Dedekind cuts, Cauchy sequences or through your pretty > clumsy procedure involving magnitudes. -- m I know you are an individual Mariano and others will no doubt study the presentation of a Dedekind cut versus a Cauchy sequence and attempt perhaps even another new method. Still, I do sense that your position is fairly equitable among mathematicians, which has been reinforced by the boundary between physics and mathematics, and philosophy too. How one resolves these as disparate or tied together and which one would place beneath another is an individual choice. It seems no matter what choices one makes the tighter these boundaries are laid then these do actually intersect each other. The topic of time is so fundamental yet has lacked a clean representative in mathematics. Clearly the real value which composes the 3D spatial representation cannot be symmetrically introduced as an additional dimension, for then time would be of the same type as space. The Minkowski metric answers this with a negative distance contribution- something that mathematicians ought to quibble over but don't since that is physics right? Rather than concern myself with mimicing these topics as perfectly as I can I prefer to try and cover new ground, mathematics nomenclature looks disfunctional. It is lost in uninstantiable abstraction. This subject of quotient ring does have this feature. The position of the physicist versus the mathematician is well known. This tangle could be allayed by admitting that we are not at any final theory in either subject. In other words the way we look at these ought to be taken with a more open stance- open to their being wrong, open to a better way, open perhaps to their being no finality in them. Your subject has a heavier burden than you are willing to admit to because you are caught in the accumulation. That is my simple criticism of both you and the topic of mathematics. It was to be a thing so fundamental that no quibbling over the construction should be possible without a direct contradiction somewhere. That was the axiomatic style. I feel comfortable saying that the ring quotient is far beyond this level. So far that perhaps it is fraudulent, but I'm not through it so I cannot say for certain. I have been able to make similar arguments on other topics such as bivectors where the bivector x ^ x = 0 becomes dubious through the construction of a unit bivector a ^ b whose angle and magnitudes vary but maintains constant value. Upon reaching the overlapped position the direction is undefined. Hence the unit bivector is undefined at this position. Thus assigning a zero value is as impossible as is assigning a sensible value to a division by zero. The usage of infinity here is in play as it is in Arturo's insistence on an infinite length polynomial, though in a completely different form. In this unit bivector form we see instantiable instances rather that a claim of specificity in terms of an indeterminate variable. Thus I use Grassman's bivector instance as support for my argument on instantiable mathematics. It seems that no such form will be permitted within the indeterminate polynomial representation that the quotient ring R[X] / ( X X + 1 ) requires. Thus no disproof by instance can ever exist. Hence the logic of the construction becomes dubious. I understand that the probablility of my argument being correct is slim. So I welcome a refutation. I must have made a mistake somewhere in my reasoning. It shouldn't be so hard to find this, especially for one who understands this subject. There are corollaries on human intelligence here which do have readily available instances. Unfortunately I am one of those instances. Call this an abstract Godel style if you like- the problems remain open universally. Sorry to be so long winded. Please feel free not to respond and I will do the same. - Tim === Subject: Re: Understanding the quotient ring nomenclature posting-account=_l4K0QkAAAC09JhOoK_ZfoJKXOmr_jZf Gecko/2009060215 Firefox/3.0.11 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) you have amply demonstrated again and again and again and again. You have consistently misunderstood it, misquoted it, and misrepresented it. As such, comments such as: > The usage of infinity here is in play as it is in Arturo's > insistence on an infinite length polynomial, though in a completely > different form. constitute nothing but rather pathetic (and dishonest) attempts on your part to try to gain some credibility by associating my name somehow with your ramblings and your ignorant assertions. Kindly, do *not* mention my name when you are being an ignoramus. As such, kindly do *not* mention my name at all whenever you post to sci.math, ever, because you have never shown an iota of an inkling of an idea of what I may or may not have said, let alone what I may have meant. -- Arturo Magidin === Subject: Re: Understanding the quotient ring nomenclature posting-account=n26igQkAAACeF9xA2Ms8cKIdBH40qzwr Gecko/20070505 Iceape/1.0.9 (Debian-1.0.13~pre080614i-0etch1),gzip(gfe),gzip(gfe) you have amply demonstrated again and again and again and again. You > have consistently misunderstood it, misquoted it, and misrepresented > it. As such, comments such as: > The usage of infinity here is in play as it is in Arturo's > insistence on an infinite length polynomial, though in a completely > different form. constitute nothing but rather pathetic (and dishonest) attempts on > your part to try to gain some credibility by associating my name > somehow with your ramblings and your ignorant assertions. You've presented a polynomial in real coefficients but insist that X is an indeterminate variable. The format you chose is ( a0, a1, a2, ... ) where a(n) are real. This is indistinguishable from an infinite dimensional cartesian coordinate. This is so simple that how you cannot concede the overlap is puzzling. Next, rather than attempt to remedy any nuance of interpretation you choose to dismiss my argument without discussing any detail, particularly the detail of this construction not allowing a standard representation ( a0, a1, ..., an ) due to the fact that products of this form require a result out to 2n-1. Thus under this form above the polynomials are not rings until the form is enforced to infinite length, whereupon the issue of a superinfinite form arises. This is very different from other usages of infinity. For instance within calculus we can typically choose a largest n and still get a result, with a larger choice yielding a more accurate result. Here what we see is that the math itself is broken under such choices. The usage of the phrase polynomial with real coefficients in X as a ring is not satisfactory, particularly in the construction of an instance of a quotient ring. Maintenance of X as an indeterminate variable helps to cloud this issue further. - Tim Kindly, do *not* mention my name when you are being an ignoramus. As > such, kindly do *not* mention my name at all whenever you post to > sci.math, ever, because you have never shown an iota of an inkling of > an idea of what I may or may not have said, let alone what I may have > meant. -- > Arturo Magidin === Subject: Re: Understanding the quotient ring nomenclature posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Trident/4.0; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > On Jun 12, 2:43 pm, Mariano Su.87rez-Alvarez > How arbitrary a language is though ought > to be a concern of a mathemtatician. Fundamentals are important and so > the primitives in use and how they are used do weigh heavily in > mathematics. > If by fundamentals you mean explicit constructions > of objects then no, they are *not* important and they > are *absolutely* irrelevant to the use of those objects. > It is absolutely unimportant and irrelevant how you > construct a complete totally ordered field. The only > important thing is that such a construction can be > carried out, and that all such objects are isomorphic, > because then one can talk about the complete totally > ordered field with the assurance that such a thing > actually exists. Absolutely nothing in maths needs to > know whether such a beast was constructed using > Dedekind cuts, Cauchy sequences or through your pretty > clumsy procedure involving magnitudes. > -- m I know you are an individual Mariano and others will no doubt study > the presentation of a Dedekind cut versus a Cauchy sequence and > attempt perhaps even another new method. Still, I do sense that your > position is fairly equitable among mathematicians, which has been > reinforced by the boundary between physics and mathematics, and > philosophy too. How one resolves these as disparate or tied together > and which one would place beneath another is an individual choice. It > seems no matter what choices one makes the tighter these boundaries > are laid then these do actually intersect each other. The topic of > time is so fundamental yet has lacked a clean representative in > mathematics. Clearly the real value which composes the 3D spatial > representation cannot be symmetrically introduced as an additional > dimension, for then time would be of the same type as space. The > Minkowski metric answers this with a negative distance contribution- > something that mathematicians ought to quibble over but don't since > that is physics right? Rather than concern myself with mimicing these > topics as perfectly as I can I prefer to try and cover new ground, > mathematics nomenclature looks disfunctional. It is lost in > uninstantiable abstraction. This subject of quotient ring does have > this feature. The position of the physicist versus the mathematician > is well known. This tangle could be allayed by admitting that we are > not at any final theory in either subject. In other words the way we > look at these ought to be taken with a more open stance- open to their > being wrong, open to a better way, open perhaps to their being no > finality in them. Your subject has a heavier burden than you are willing to admit to > because you are caught in the accumulation. That is my simple > criticism of both you and the topic of mathematics. It was to be a > thing so fundamental that no quibbling over the construction should be > possible without a direct contradiction somewhere. That was the > axiomatic style. I feel comfortable saying that the ring quotient is > far beyond this level. So far that perhaps it is fraudulent, but I'm > not through it so I cannot say for certain. I have been able to make > similar arguments on other topics such as bivectors where the bivector >x ^ x = 0 > becomes dubious through the construction of a unit bivector >a ^ b > whose angle and magnitudes vary but maintains constant value. Upon > reaching the overlapped position the direction is undefined. Hence the > unit bivector is undefined at this position. Thus assigning a zero > value is as impossible as is assigning a sensible value to a division > by zero. The usage of infinity here is in play as it is in Arturo's > insistence on an infinite length polynomial, though in a completely > different form. In this unit bivector form we see instantiable > instances rather that a claim of specificity in terms of an > indeterminate variable. Thus I use Grassman's bivector instance as > support for my argument on instantiable mathematics. It seems that no > such form will be permitted within the indeterminate polynomial > representation that the quotient ring >R[X] / ( X X + 1 ) > requires. Thus no disproof by instance can ever exist. Hence the logic > of the construction becomes dubious. I understand that the probablility of my argument being correct is > slim. So I welcome a refutation. I must have made a mistake somewhere > in my reasoning. It shouldn't be so hard to find this, especially for > one who understands this subject. [...] This impression of yours is one we see all the time with our students: they argue that the work they do in their exams cannot be *that* wrong since we cannot point to a specific mistake in it. And the thing is, it is impossible to point to a specific mistake neither in their work and in your reasoning, because both are not even wrong: they are meaningless. You plainly do not know the subject about which you have criticisms. And I do not have in mind the high and subtle points of any area of mathematics: you have shown quite clearly that you do not understand what a polynomial is. No matter how long a tirade you are able to write about your criticisms, using that obsrvation to put your writings in perspective it is aparent that those criticisms cannot possibly carry any weight, for they are firmly grounded on your ignorance. Ignorance is seldom a unsurmontable problem, for it can be resolved by learning. But it *is* an obstacle. -- m === Subject: Re: Understanding the quotient ring nomenclature Tim BandTech.com a .8ecrit : > On Jun 12, 2:43 pm, Mariano Su.87rez-Alvarez > How arbitrary a language is though ought > to be a concern of a mathemtatician. Fundamentals are important and so > the primitives in use and how they are used do weigh heavily in > mathematics. > If by fundamentals you mean explicit constructions > of objects then no, they are *not* important and they > are *absolutely* irrelevant to the use of those objects. > It is absolutely unimportant and irrelevant how you > construct a complete totally ordered field. The only > important thing is that such a construction can be > carried out, [cut a lot of nonsensical answer, then...] >Thus I use Grassman's bivector instance as > support for my argument on instantiable mathematics. It seems that no > such form will be permitted within the indeterminate polynomial > representation that the quotient ring > R[X] / ( X X + 1 ) > requires. Thus no disproof by instance can ever exist. Hence the logic > of the construction becomes dubious. > > I understand that the probablility of my argument being correct is > slim. So I welcome a refutation. Certainly not. You dont understand much, you would not welcome a refutation, but (lucky you) you wouldn't recognize one if it were slim chance you suspected it *was*, you would put your fingers in your ears and shout I cannot hear you I must have made a mistake somewhere > in my reasoning. Noo, you never made any reasoning (and dont know the meaning of the world) It shouldn't be so hard to find this, especially for > one who understands this subject. One typical mistake is saying things like Arturos 'dimensional arguyment ust be wrong, because 5<6, while Arturo never used the word dimension to begin with... There are corollaries on human > intelligence here which do have readily available instances. > Unfortunately I am one of those instances. You are drifting in insanity again... Call this an abstract Godel > style if you like- the problems remain open universally. > > Sorry to be so long winded. Please feel free not to respond and I will > do the same. Is this a promise? Oh, please, please, keep it... - Tim === Subject: Re: Understanding the quotient ring nomenclature Revelations 3:3 I am he. === Subject: Re: Understanding the quotient ring nomenclature A message from Martin Michael Musatov: This morning I picked up my bible and read two verses: (This is a real book. I bought it. It is in front of me. The back of it reads: Zondervan ISBN 0-310-92391-3) The first verse was Daniel 12:1 At that time Michael, the great prince who protects your people, will arise. There will be a time of distress such as has not happened from the begin- ning of nations unti8l then. But at that time your people--everyone whose name is found written in the book--will be deliv- ered. The second one was Revelations 12:9 The great drag- on was hurled down--that ancient serpent called the devil, or Satan, who leads the whole world astray. He was hurled to the earth, and his angels with him. Prince of Persia | Learn The BibleRather, Michael was the angel of Daniel's people. He was the prince of Israel. .... Similar Bible References. Jesus and Michael the Archangel ... www.learnthebible.org/prince-of-persia.html Daniel 10:13 But the prince of the kingdom of Persia was withstandingKing James Bible But the prince of the kingdom of Persia withstood me one and twenty days: but, lo, Michael, one of the chief princes, came to help me; ... bible.cc/daniel/10-13.htm Who is Michael the Archangel?What was Michael's purpose? Where is he discussed in the Bible and what mighty ... is] none that holdeth with me in these things, but Michael your prince. ... www.biblestudy.org/question/is-michael-the-archangel-jesus.html Daniel 10:21 However, I will tell you what is inscribed in theBut I will tell thee that which is inscribed in the writing of truth: and there is none that holdeth with me against these, but Michael your prince. Bible ... bible.cc/daniel/10-21.htm Seventh-Day Adventists: Moses Resurrected cont. and Jesus call ...Feb 23, 2008 ... I'll let the Bible speak for itself and you decide if Christ and Michael are one and the same. Daniel 10:13 states that the prince of the ... en.allexperts.com/q/Seventh-Day-Adventists-2318/2008/2/Moses-Resurrected-con t-Jesus-1.htm Daniel 10. The Holy Bible: King James Version.The Holy Bible: King James Version. Daniel 10. ... the Scripture of truth: and there is none that holdeth with me in these things, but Michael your prince. ... www.bartleby.com/108/27/10.html Results 1 - 10 for revelations michael prince. (0.17 seconds) Custom Search Prince Michael in scriptures ' AngelsMar 10, 2008 ... Revelation 12: 7 [CapitalEth] Apostle John saw Chief Prince/General Michael war against the dragon as, satan (also known as the Prince of the ... thelordsangels.wordpress.com/2008/03/10/prince-michael-in-scriptures/ Revelation 12:7 And there was war in heaven, Michael and his angelsDaniel 12:1 Now at that time Michael, the great prince who stands guard over the sons ... Revelation 12:3 Then another sign appeared in heaven: and behold, ... bible.cc/revelation/12-7.htm Saint Michael the Archangel A revelation was given to Daniel, who had been named Belteshazzar. ... I left him (Michael) there with the prince of the kings of Persia, and came to make ... http://www.medjugorjeusa.org/stmichael.htm Who is Michael the Archangel? It is the identity of Michael, the archangel and prince, mentioned in the last five ..... In Revelation, Michael is portrayed as leading the heavenly hosts, ... www.tofm.org/LIBRARY/CHRIST/book04-Who%20Is%20Michael.htm Michael definition of Michael in the Free Online Encyclopedia. In the Bible and early Jewish literature, Michael is one of the angels of God's presence. He is depicted as a warrior-prince leading the celestial armies ... http://encyclopedia2.thefreedictionary.com/Michael Daniel 12:1 Now at that time Michael, the great prince who stands King James Bible And at that time shall Michael stand up, the great prince which standeth for the children of thy people: and there shall be a time of ... http://bible.cc/daniel/12-1.htm Archangel Michael-The Prince-Messiah To Come Archangel Michael is written all over the Bible and in Dead Sea Scrolls, mentioned as the Prince of the Congregation, The Heavenly Prince Melchizedek(IIQ13) ... http://www.geocities.com/Athens/Academy/2203/index.html Who is Michael the Archangel in the Bible? Mar 27, 2009 ... Difficult Bible Passages About Michael Explained. ...... Here is Michael our prince, the great protector of the church, and the patron of ... http://www.creation-science-prophecy.com/michael.htm Amazon.com: APPROACHING THE QUR'AN: THE EARLY REVELATIONS ...Amazon.com: APPROACHING THE QUR'AN: THE EARLY REVELATIONS.: Michael Anthony (editor & translator). Sells: Books. http://www.amazon.com/APPROACHING-Michael-Anthony-editor-translator/dp/B000T 826MI Revelations Michael Martin | Metro.co.ukMay 12, 2009 ... Latest news about revelations michael martin. Breaking UK & world news from Metro, with celebrity gossip and weird new, and exclusive ... http://search.metro.co.uk/tag/revelations-michael-martin.html AFP: Two lawmakers forced out over expenses scandal By Michael Thurston [CapitalEth] May 21, 2009. LONDON (AFP) .84 Two lawmakers said Thursday they will quit after fresh embarrassing revelations over expenses, ... Iranian Revelations - Michael Ledeen - The Corner on National ...May 14, 2005 ... Iranian Revelations [Michael Ledeen]. I am still in Naples, finishing the research for my book virgil's golden egg and other neapolitan ... http://corner.nationalreview.com/post/?q=YzYzNzk5ZGQ4YmIwMjU4OWE0MTlmNzlkYTY yYTNkMGY= Google Directory - Society > Religion and Spirituality ...Sales of books by Derek Prince (no longer available through retail outlets. ... Back and Michael R. Back. Book claiming Islam is the Babylon of Revelation. ... Michael -- The Great Prince In the Book of Revelation, it is said that Michael and his angels cast the ... Even in the time of Daniel, Michael the prince is spoken of -- a prince ... Keeping It Catholic - The Blog!: St. Michael: Prince and Guardian ...Sep 29, 2008 ... That assistance came by the hand of St. Michael, Prince of the Heavenly ... 10) According to the apocryphal Revelation of Moses, St. Michael ... http://keepingitcatholic.blogspot.com/2008/09/st-michael-prince-and-guardian -of-gods.html Michael the Great Prince -- Who Is He? The spirit creature Michael ...Dec 15, 1984 ... And during that time Michael will stand up, the great prince who is .... Hence, the conquering Michael referred to in Revelation 12 must be ... http://www.textfiles.com/occult/CHRISTIAN/mike.txt Archangel Michael-The Prince-Messiah To Come(Hint: Michael was always at God's Right Hand and termed ñPrinceî and ñJudgeî of .... Jesus in many scripture verses calls himself the thief:Revelation 3:3 ... http://www.geocities.com/Athens/Academy/2203/index.html JSH: Negative Pell's Equation, consequences - alt.math.undergrad ...9 posts - 5 authors - Last post: May 31 problem with ring of algebraic integ. ...mymath.blogspot.com/2009/04/ ... Martin Musatov [P=NP, Just for the taste of it! Diet Coke! ... Pell's equation. ..... opinions concerning Jesus Christ ...Opera, p. 16. ... Extra Nos: Godel and Falsifiability of EvolutionI just love it when from time to time you would post something about philosophy of .... Yes, re:sci.math board, I know. Once I get more time which should be fairly soon, ... But this whole world is not about me but it is about Christ's payment for me. LPC. 11:25 PM .87 Martin Musatov said. ... Member Lutheran Ring ... extranos.blogspot.com/2008/12/godel-and-falsifiability-of-evolution.html Posts tagged Comicbook at CinematicalDon't be fooled by the fact that this one opens just like the teaser trailer .... Also, Moviehole.net spoke with the writing team of Marty Musatov and Ethan Erwin .... Fargus on 50 Cent and Chris Klein Will Be 'Caught in the Crossfire' ... Bells Ring .87 Gummi Bear Surgery -- a How-To .87 The Math All Girls Are Good At ... www.cinematical.com/tag/comicbook/page/2/ Shtetl-Optimized é Blog Archive é The complement of Atlas ShruggedAlas, the price of keeping out the likes of Martin Musatov (who at the height of .... fourteen-year old's life: The Lord of the Rings and Atlas Shrugged. ..... mathematics has became ñjustî theorem-proving, and political philosophy has .... day after day, about your alleged P=NP proof, Jesus Christ, your father, ... scottaaronson.com/blog/?p=386 Russian Poetry and the October Revolutionmathematics. Maiakovskii said that, having found theprinciple ofa poem, Khlebnikov would leave it ... the memory of a sweet,effeminate, pastoral Christ whom he adores but resists. ..... Borisov-Musatov, one of the Blue Rose painters renowned for ..... In his decrees the authoritative ring of an Abbot's exhortations ... www.informaworld.com/index/794192526.pdf by A Pyman - 1990 Result for query keyword(s)=theorem author= title=On a theorem of Ivasev-Musatov III, ...... Combining this with a theorem of Michael Christ cite ch, the authors prove; To prove Part 2 of Theorem in the case $1 ...... Small extensions of Witt rings [Pacific Journal of Mathematics] ...... The second part of the theorem is just Proposition 2.1, (iv) above. ... nyjm.albany.edu:8000/cgi-bin/aglimpse/19/nyjm/Http/search/j?firstyear=1994&j ournaldir=combined&lastyear=2001&query=theorem Wikipedia:Dead external links/404/m - Wikipedia, the free encyclopedia... United Church of Christ short course on the German Reformed Church 404 Not Found .... A paper on the mathematics of change ringing 404 Object Not Found ...... Middle-earth in popular culture, REC Studios' Fellowship of the Ring 404 ...... Students in the Classroom: Not Absent, Just Overlooked': 404 Not Found ... en.wikipedia.org/wiki/Wikipedia:Dead_external_links/404/m Result for query keyword(s)=theorem author= title=This property is proved in cite BKL, and we could have just; following theorem. the above theorem implies that the ...... On a theorem of Ivasev-Musatov III, ...... Combining this with a theorem of Michael Christ cite ch, the authors prove ...... Small extensions of Witt rings [Pacific Journal of Mathematics] ... nyjm.albany.edu:8000/cgi-bin/aglimpse/19/nyjm/Http/search/j?firstyear=1994&j ournaldir=combined&lastyear=2002&query=theorem www.questia.com/PM.qst?a=o&se=gglsc&d=77723673 Result for query keyword(s)=theorem author= title=The theorem just stated is a model result for the conjecture using; $b_ 2$ is to emphasize that the $b_ 2$ in the theorem ...... Rings with sovable quivers [Pacific Journal of Mathematics] ...... On a theorem of Ivasev-Musatov III, ...... Combining this with a theorem of Michael Christ cite ch, the authors prove ... nyjm.albany.edu:8000/cgi-bin/aglimpse/19/nyjm/Http/search/j?firstyear=1994&j ournaldir=combined&lastyear=2006&query=theorem === Subject: Re: Understanding the quotient ring nomenclature posting-account=_l4K0QkAAAC09JhOoK_ZfoJKXOmr_jZf Gecko/2009042513 Ubuntu/8.04 (hardy) Firefox/3.0.10,gzip(gfe),gzip(gfe) On Jun 12, 12:02am, cbr...@cbrownsystems.com The key is: POLYNOMIALS ARE NOT FUNCTIONS. POLYNOMIAL FUNCTIONS ARE What this interchange makes me reflect on is the pedantic value of > teaching group theory before approaching ring theory; it eases the > task of UN-learning certain things about the meaning of using a > notation. You don't really need the full abstraction of ring theory and group theory for the purpose of understanding the construction of the complex numbers as a quotient of the ring R[X]. One can talk about R [X], with R the reals, without really discussing much about rings other than the bare-bones definition; I learned it that way three courses before we actually got to 'abstract algebra', as an undergraduate. Once we did get to abstract algebra, we had this construction in hand as an instantiation of the more general concept. I would have suggested (and was attempting) that approach, leaving the abstract notions behind, but the OP is too busy not listening to make any headway. -- Arturo Magidin === Subject: Re: Understanding the quotient ring nomenclature posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > On Jun 11, 5:44 pm, Mariano Su.87rez-Alvarez > On Jun 11, 9:15 am, Tim Golden BandTech.com OK Tim. Jeeze, it's like I'm talking to myself... > I must say I did a double-take once some time ago when reading one of > your posts and coming to the last line where you signed your name in > exactly the same way I normally do.A moment of did I write that!? > Umm... I guess someone would quibble with you about leaving the zeroes > off the end of your instances of polynomials. > Quite possibly; I'm sure that on Usenet there is always someone who > can quibble about nearly anything. > Still, this the funny thing about the construction. For instance > with the polynomial ring quotient construction it is possible to > build n dimensional spaces consistent with the ring requirements, > I really don't know where you're getting this dimension thing. You > appear to be using it in the sense of dimension of a real vector > space.In this particular case the space of real polynomials does > satisfy the axioms of a real vector space, but in that case it has > infinite dimension, not any natural number n.It isn't a very useful > concept here. > There is an insistence on remaining in the 'indeterminate form' whereby > X is never evaluated. In effect X is undefined. But now that I've read > on and it's fairly obvious anyway the polynomial factors > a X + b > are evaluatable aren't they? Is it true that in polynomial of real > coefficients that X is real? This is the only way to evaluate a factor > isn't it? Yet this will then collapse any polynomial into a real value > for any X. We could then procede to simply study > y / ( x^2 + 1 ) > with x and y real couldn't we? We'd eventually get to a square route of > a negative value and be forced to concede i in the traditional manner. > This seems to be in the interpretations on the indeterminate form > anyway. At some point they introduce i. > yet the underlying components will not play out this way with a > limit like n since there will be terms up to 2n-1 in the product > domain so that to build a finite form is not possible until the > quotient is taken. > Yes, the particular quotient R[x]/(x^2+1) can also be interpreted as a > real vector space of dimension 2, but again dimensionality is not a > very useful concept in the context of quotient rings. > Well, this is troubling that you say this. I have worked out some math > which develops dimensional behaviors without the need of any cartesian > product. This math has been expressed in terms of ring theory by Hagen > von Eitzen through a ring quotient and nobody has found any fault with > that math. This is somewhat how I arrive at puzzling this all out. I'd > like to understand how dimension is developed through this math. So back > at my o.p. I ask specifically is this ring quotient considered an actual > construction or is it more like a hindsight type of thing where a given > construction fits into this form and behaves consistently with it? I > understand that the polysign numbers are rings. I have no problem with > that. But can the ring quotient form build the polysign numbers from a > more primitive ring? The example of the complex numbers as > R[X] / ( X X + 1 ) > is nearly an identical but more familiar instance. Whether the ring form > builds a number system or whether it merely reflects a number system's > behaviors is crucial to understanding this nomenclature. > This is all about dimension to me. Still, if you pick a domain for X > then I understand the dimension of the classes and elements being > discussed should collapse to that dimension assuming that the arithmetic > product XX is in the same domain as X. > Thus the instances you describe must already be quotient forms > within the formality and so their products would then wrap at some > dimension as well. > The polynomial instances I mentioned are two example members of a > single element of the quotient ring.The quotient does not have > polynomials as elements of the ring, it has infinite sets (equivalence > classes) of polynomials. > This double abstraction is not doing much for me. What seems to matter > is determining the domain X. If the quotient ring is capable of defining > higher complexity X from simpler complexity X then understanding this > feature would be useful to me. If it is merely describing behaviors > consistent with these domains regardless of their complexity as common > features then that is an important distinction. > Going back to the primitive operators of superposition and product it is > fairly clear that any math that follows these operators as their well > known behaviors will be similarly behaved. The construction of those > maths could come into question here. > Only in your imagination has anything been called > into question... and your imagination has but your > ignorance of elementary algebra to build upon--you'll > agree that that is not a very sound basis to build > anything. > Who constructed who as the most > primitive is then a problem and either may construct the other on its > own terms. For instance the polysign numbers construct the reals, yet > many insist upon constructing the polysign numbers from the reals. > I would say that what many keep doing is showing you, > by exhibiting a construction of your beloved > polysigned things using the most elementary algebra > available, that they do not provide anything new. > This quest for primitiveness of yours is of no real > interest: it may have had a certain charm a few > centuries ago, but nowadays mathematicians have > understood that the only important thing is that > objects actually exist, and not how they are shown to > exist: that the specific construction of an object > is irrelevant. > -- m I like whay you've said here Mariano. I do accept that I am back a few > centuries. > The polysign representation is only slightly different than the > polynomial representation, but it does already work in low n with no > need of requiring an infinite series. The modulo two behavior of the > real number is obvious but ignored as merely an inverse. But I am not trying to push polysign here. I would like to make it > through this ring quotient construction but it seems it's a number of > courses and as Leland says I am trying to make one meal out of it. I > accept that criticism. It is awfully difficult to get through this. I > can accept that it is my own failing too. But your stance here on > modern math not caring about fundamentals is really fascinating. I do > sometimes think of many of these math topics as cryptologies. People > are learning a language, yes. How arbitrary a language is though ought > to be a concern of a mathemtatician. Fundamentals are important and so > the primitives in use and how they are used do weigh heavily in > mathematics. - Tim- Hide quoted text - - Show quoted text - Results 1 - 10 for quotient ring inverse 19 musatov. (0.67 seconds) Custom Search Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANTJun 11, 2009 ... Inverse 19, Have a good rest and Godspeed. Best wishes, Musatov .... Next by sci.tech-archive.net/Archive/sci.math/2009-06/msg01398.html Discussions - sci.math | Google Groups10 posts - 8 authors Understanding the quotient ring nomenclature. 63 new of 63 - Jun 11 ... THIS for the record and for Dr Phil, Amy , Edgar,and the great mathematicians Musatov,and to the silence of the ... By inverse 19 - 8:31am - 2 new of 2 messages ... Re: .9 repeatingsummerschool2009/papers/musatov.pdf New Theory In PROVEN: Universe is inversely proportioned at inverse 19 ... sci.tech-archive.net/Archive/sci.math/2009-06/msg01419.html - 4 hours ago Discussions - sci.math | Google GroupsUnderstanding the quotient ring nomenclature, 67 new of 67, cbrown@cbrownsystems.com (16 authors) ... 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT, 10 new of 10, Musatov (5 authors), Jun 11 ... 2(3^2)+1=19 Good bye note to Sci math gurus! ... sci.tech-archive.net/Archive/sci.math/2009-06/msg01394.html Discussions - sci.math | Google Groups10 posts - 7 authors - Last post: yesterday 0.75(2^2)+1=2^2 (for Dr Phil,Prof Edgar, Musatov, amy , Mathematicians). 6 new of 6 - Jun 12 ... PROVEN: Universe is inversely proportioned at inverse 19 ... Understanding the quotient ring nomenclature ... sci.mathUnderstanding the quotient ring nomenclature, Tim BandTech.com ... Re: Methods to calculate integration numerically, Martin Musatov; Re: Methods to calculate ..... Re: JSH: Negative Pell's Equation existence conditions, inverse 19 ... sci.tech-archive.net/Archive/sci.math/2009-06/ Result for query keyword(s)=theorem author= title=Obviously the inverse Fourier transform is defined by replacing $mF$ with $mF^ast $ in eqref formula Fourier. ...... Theorem~ tells us that we need only to quotient the ring; Theorem~, or the purely combinatorial ...... On a theorem of Ivasev-Musatov III, ...... cite 19 (see also cite[Theorem 1.1] 22). ... nyjm.albany.edu:8000/cgi-bin/aglimpse/19/nyjm/Http/search/j? firstyear=2001&journaldir=combined&lastyear=2007&query=theorem [PDF] AUTHOR INDEX, VOLUMES 51[CapitalEth]75File Format: PDF/Adobe Acrobat - View as HTML The quotient of a free product of groups by a single high-powered relator. II. ..... T. W. On the theorem of Ivasev-Musatov III. 53 (1986) 143[CapitalEth]192 .... Inverse monoids with a natural semilat- tice ordering. 70 (1995) 146[CapitalEth]182 ...... of permutation groups, and group rings of locally finite ..... 52 (1986) 1[CapitalEth]19 ... www.lms.ac.uk/publications/proceedings/plmsindx.pdf Result for query keyword(s)=theorem author= title=Obviously the inverse Fourier transform is defined by replacing $mF$ with $mF^ast $ in eqref formula Fourier. ...... Theorem~ tells us that we need only to quotient the ring; Theorem~, or the purely ...... On a theorem of Ivasev-Musatov III, ...... The next corollary follows immediately from Theorem~thmref 19. ... nyjm.albany.edu:8000/cgi-bin/aglimpse/19/nyjm/Http/search/j? firstyear=1994&journaldir=combined&lastyear=2006&query=theorem === Subject: Working with quotient groups If G is a group and H, K are subgroups of G with K < H, and N is a normal subgroup of G, is the following equality true? [HN/N : KN/N] = [H : K] ? === Subject: Re: Working with quotient groups posting-account=-PngCgkAAAD2yUjosqWv1Nf1lkqWP4lp rv:1.8.1.20) Gecko/20081217 Firefox/2.0.0.20,gzip(gfe),gzip(gfe) > If G is a group and H, K are subgroups of G with K < H, and N is a normal subgroup of G, is the following equality true? [HN/N : KN/N] = [H : K] ? Not necessarily - what happens if N = G for example? In general [H:K] = [HN/N:KN/N][H^N:K^N]. Derek Holt. === Subject: Re: Working with quotient groups > On 12 June, 19:43, qsymmetry If G is a group and H, K are subgroups of G with K > < H, and N is a normal subgroup of G, is the > following equality true? > [HN/N : KN/N] = [H : K] ? > > Not necessarily - what happens if N = G for example? > > In general [H:K] = [HN/N:KN/N][H^N:K^N]. What's ^ ? Intersection? === Subject: Re: Working with quotient groups > On 12 June, 19:43, qsymmetry If G is a group and H, K are subgroups of G with K > < H, and N is a normal subgroup of G, is the > following equality true? > [HN/N : KN/N] = [H : K] ? > > Not necessarily - what happens if N = G for example? > > In general [H:K] = [HN/N:KN/N][H^N:K^N]. Incidentally, where can I find a proof of this result? I see why it follows when G itself is finite (or when H and are finite subgroups of G); but what is (or should be the) proof for arbitrary G? === Subject: Re: Working with quotient groups <6927038.9283.1244837961572.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=-PngCgkAAAD2yUjosqWv1Nf1lkqWP4lp Gecko/2009042513 Ubuntu/8.04 (hardy) Firefox/3.0.10,gzip(gfe),gzip(gfe) > On 12 June, 19:43, qsymmetry If G is a group and H, K are subgroups of G with K > < H,and N is a normalsubgroup of G, is the > following equality true? > [HN/N : KN/N] = [H : K] ? > Not necessarily - what happens if N = G for example? > In general [H:K] = [HN/N:KN/N][H^N:K^N]. Incidentally, where can I find a proof of this result? I see why it follows when G itself is finite (or when H and are finite subgroups of G);but what is (or should be the) proof for arbitrary G? [HN/N:KN/N] = [HN:KN] = [H:KN^H] and [H^N:K^N] = [KN^H:K] Derek Holt. === Subject: An schizophrenic called Musatov wants destroy Sci-math posting-account=Z9v8GgkAAAB54xGwWI4tZlTl3U6kju-g Alexa Toolbar; MEGAUPLOAD 2.0; .NET CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) A mentally sick named Musatov endeavor to destroy Sci-math with a deluge of filth as is showed in this sample: THE GENESIS OF THE YOUNG COSIMA: HENRY HANDEL RICHARDSON'S MOST ...result is a book almost devoid of imaginative and descriptive writing. Its matter is ..... During the proof stage she said: ... I wish Oh God I'd .... See letter to Nettie Palmer dated 6 May 1939, N. P., p. 201. 9. 24 November 1929. ...[23]http://www.informaworld.com/ index/795114923.pdf . We decent people never will open his posted ordure. Ludovicus === Subject: Re: A Warning of Civil Suit to any libel comments re: Musatov posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) If any of you dare change the subject of this heading your ISP will be subpoenad and I will seek injunction up to and including civil suit. I am very serious about this. Musatov Let the fool Ludovicus be an example of what not to do. The individual yahoo account this email is directed to (to which you are hereby copied) I am writing to report. TOS violation by the account holder. The acount holder is engaged in making defamatory slanderous and libel contacts regarding myself on various newsgroups and is doing so under the anonymity umbrella Yahoo has providem him/ her. I am writing to a) report this abuse, and b) notify you if any further defamatory comments are made by the account holder (in the event no action is taken by Yahoo to enforce the TOS) I will file for a subpoena for the account holders identity, as I am entitled by law to an injunction up to and including civlil suit. I am serious. Respectfully, Martin Musatov Please see the email below as evidence of this claim: > A mentally sick named Musatov endeavor to destroy Sci-math with a > deluge > of filth as is showed in this sample: THE GENESIS OF THE YOUNG COSIMA: HENRY HANDEL RICHARDSON'S > MOST ...result is a book almost devoid of imaginative and descriptive > writing. Its matter is ..... During the proof stage she said: ... I > wish Oh God I'd .... See letter to Nettie Palmer dated 6 May 1939, N. > P., p. 201. 9. 24 November 1929. ...[23]http://www.informaworld.com/ > index/795114923.pdf . We decent people never will open his posted ordure. > Ludovicus > A mentally sick named Musatov endeavor to destroy Sci-math with a > deluge > of filth as is showed in this sample: THE GENESIS OF THE YOUNG COSIMA: HENRY HANDEL RICHARDSON'S > MOST ...result is a book almost devoid of imaginative and descriptive > writing. Its matter is ..... During the proof stage she said: ... I > wish Oh God I'd .... See letter to Nettie Palmer dated 6 May 1939, N. > P., p. 201. 9. 24 November 1929. ...[23]http://www.informaworld.com/ > index/795114923.pdf . We decent people never will open his posted ordure. > Ludovicus === Subject: Re: An schizophrenic called Musatov wants destroy Sci-math We are going backwords to Russia/ Cambodia , the killing feilds, Solzenetzin etc as we stoop so low as to call some one mentally sick. We reserve these names for North Korean President , even who is quite functional and less mentally sick than Einstein, if you go by functionality. This fellow does not appear to be mentally sick, may be a little different , but YOU all what are judging him for? , you and your types voted for the Governor of illinois two times and read Ayn Rand books,listen to Rush Limbaugh, the former Governor of Illinois and Ayn Rand are proprtional to this this fellow( we are all created equal). Abe Lincoln was mentally ill too , but he did great function as President under stress of the Civil war that was started by such bigotry calling the blacks crazy and genetically inferior because of a science called eugenics, some stupid mathematics even backed eugenics . STOP IT, do not call such names to no body in Libel, unless the person carries that medical diagnosis asserted by several physician, in which case you must still show respect, discretion, because it could be youe son next. WE DECENT PEOPLE as they say in the post, who are you these We-- the same ones that call Obama Black and incompetent and think that Rush Limbaugh is any less disparaged than this Musatov. I Appologize to this man MUSATOV, on behalf of the really decent in Sci math and on behalf of all Americans, and in the name of my personal Lord Jesus Christ, who was called crazy by many. Inverse 19 Inverse 19 === Subject: Re: An schizophrenic called Musatov wants destroy Sci-math posting-account=n26igQkAAACeF9xA2Ms8cKIdBH40qzwr Gecko/20070505 Iceape/1.0.9 (Debian-1.0.13~pre080614i-0etch1),gzip(gfe),gzip(gfe) > We are going backwords to Russia/ Cambodia , the killing feilds, Solzenetzin etc as we stoop so low as to call some one mentally sick. We reserve these names for North Korean President , even who is quite functional and less mentally sick than Einstein, if you go by functionality. This fellow does not appear to be mentally sick, may be a little different , but YOU all what are judging him for? , you and your types voted for the Governor of illinois two times and read Ayn Rand books,listen to Rush Limbaugh, the former Governor of Illinois and Ayn Rand are proprtional to this this fellow( we are all created equal). Abe Lincoln was mentally ill too , but he did great function as President under stress of the Civil war that was started by such bigotry calling the blacks crazy and genetically inferior because of a science called eugenics, some stupid mathematics even backed eugenics . STOP IT, do not call such names to no body in Libel, unless the person carries that medical diagnosis asserted by several physician, in which case you must still show respect, discretion, because it could be youe son next. WE DECENT PEOPLE as they say in the post, who are you these We-- the same ones that call Obama Black and incompetent and think that Rush Limbaugh is any less disparaged than this Musatov. I Appologize to this man MUSATOV, on behalf of the really decent in Sci math and on behalf of all Americans, and in the name of my personal Lord Jesus Christ, who was called crazy by many. Inverse 19 Inverse 19 Great mathematicians are known to succumb to mental illness no different than others do. Somehow their logical abilities may even make them more susceptible to the deepest end. I was just reading recently that Godel died in a paranoid state, not eating because he believed someone would poison him. I can just see him double checking his logic and being forced to remain open to the supposition especially given some other evidence of events nearby in his life. How much one should trust one's own mind is a difficult subject. At the heart of mathematics I'm afraid we have to admit our humanity. Pattern recognition is a wonderful feature to develop and trust, but the fallibility of every individual human, brilliant or not, must be granted. This is a frustrating position but to shy away from the truth is not consistent with the mathematician. As a chicadee sits on the edge of a bird feeder looking left then right before each peck at a piece of food it may be that all of the animals in this world are subject to such a functional basis. For the mathematician to look at himself is nearly impossible to do accurately. Especially troubling is the social state of mimicry at the core of our ascension which can be construed as a fraud as much as a level of advancement. Within this context the academic metric is a matter of the level of complexity of mimicry attainable, for if one finds fault with the existent system and refuses it then a failing grade ensues. An attempt to purvey a sense of truth does come at this problem from quite a different angle. This pursuit of truth is very much an can be found. While it is extremely challenging to perfect this position it enlivens any subject since the problems studied must be left open in the ultimate pursuit. Thus sealing a system is not validated. If anything turning that system upside down and inside out is validated. Here is a medium that accomodates this pursuit without offending the professor/student paradigm since we are symmetrically positioned here. Still, abuse of the medium will silence one's own voice. Anyway for Musatov I extend a modicum of compassion in the hopes that he will substantiate his complaints and resolutions in a more convincing manner. It is OK to go off here in a tirade. You can do that. Do it well. - Tim === Subject: Re: An schizophrenic called Musatov wants destroy Sci-math > We decent people never will open his posted ordure. That's very nice of you decent people. May I suggest you also refrain from polluting sci.math with commentary on Musatov's polluting sci.math? -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: An schizophrenic called Musatov wants destroy Sci-math posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) > May I suggest you also refrain from polluting sci.math > with commentary on Musatov's polluting sci.math? :) I am not schizophrenic and the next person who comments as such their service provider will be served with a subpoena for their identification and they will meet civil suit for libel. I am serious. === Subject: Re: An ignorant cruel person Ludovicus wants to destroy Musatov > A mentally sick named Musatov endeavor to destroy > Sci-math with a > deluge > of filth as is showed in this sample: > > THE GENESIS OF THE YOUNG COSIMA: HENRY HANDEL > RICHARDSON'S > MOST ...result is a book almost devoid of imaginative > and descriptive > writing. Its matter is ..... During the proof stage > she said: ... I > wish Oh God I'd .... See letter to Nettie Palmer > dated 6 May 1939, N. > P., p. 201. 9. 24 November 1929. > ...[23]http://www.informaworld.com/ > index/795114923.pdf . > > We decent people never will open his posted ordure. > Ludovicus > If we can focust on the mathematics (and steer away from distracton please, I'm sure this is the purpose of Sci.Math) Here is my Galois connection proof: > How to prove that lower adjoint of a Galois connection determines > upper adjoint and vice versa? > > I want a direct proof (not the reference to the more general case of > adjoint functors in category theory). Math Forum DiscussionsJun 12, 2009 ... How to prove that lower adjoint of a Galois connection determines upper adjoint and vice versa? I want a direct proof (not the reference to ... mathforum.org/kb/thread.jspa?threadID=1954161&tstart=0 - 2 hours ago I am still very much experimenting but inverse 19 is opening doors very rapidly: Okay, here goes infinity: A similar Galois connection whose lower adjoint is given by the meet set of X. In the first Galois connection, G is the upper adjoint, of Galois connections is that one adjoint uniquely determines the other. ... Reference: http://en.wikipedia.org/wiki/Galois_connection As detailed above, each part of a Galois connection uniquely determines the other. The lower and upper adjoint: the situation is completely symmetrical. Reference: http://wapedia.mobi/en/Galois_connection In category-theoretic terminology Galois connections are adjoint situations: F is called a lower adjoint of G and G is called an upper adjoint of F. (b) Suppose (F, G) is a Galois connection. Then F and G determine each monotonic function (and) G.81üB.81©A is an upper adjoint in a Galois connection equivales G is inf-preserving. Dually, a monotonic function F.81üA.81©B is ... http://www.cs.nott.ac.uk/~rcb/G53PAL/slides/02Lect7.pdf A computer science applications for a Galois connection (f'PQ, g:Q-.P): (1). Upper and lower adjoints uniqly determine each other, that is, whenever g' is an upper adjoint to f, ... Reference: http://www.springerlink.com/index/y0t76w25x3186j3x.pdf by A Melton - Cited by 72 QED. Musatov === Subject: Re: An schizophrenic called Musatov wants destroy Sci-math posting-account=CtWhuAoAAAAZZ9vwdovdqB3NNaiUa20_ GTB5),gzip(gfe),gzip(gfe) > A mentally sick named Musatov endeavor to destroy Sci-math with a > deluge > of filth as is showed in this sample: THE GENESIS OF THE YOUNG COSIMA: HENRY HANDEL RICHARDSON'S > MOST ...result is a book almost devoid of imaginative and descriptive > writing. Its matter is ..... During the proof stage she said: ... I > wish Oh God I'd .... See letter to Nettie Palmer dated 6 May 1939, N. > P., p. 201. 9. 24 November 1929. ...[23]http://www.informaworld.com/ > index/795114923.pdf . We decent people never will open his posted ordure. > Ludovicus Nobody can destroy 'sci.math'! But he is the _only_ debater here I've put in my killfile. Quite an achievement. Han de Bruijn === Subject: Re: An schizophrenic called Musatov wants destroy Sci-math Well said by this last writer/poster, very enlightening to hear an intelligent response. Pi is still 3.1. Freedom is still in command, though some go commando as if it was war again.. The only war out there is against tyranny. === Subject: Re: An schizophrenic called Musatov wants destroy Sci-math > Nobody can destroy 'sci.math'! Sci.math is indestructible! > But he is the _only_ debater here I've put in my killfile. Quite an > achievement. That you haven't killfiled anyone else, or his managing to make it into your killfile? -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === === Subject: How to prove that lower adjoint determines the upper adjoint? How to prove that lower adjoint of a Galois connection determines upper adjoint and vice versa? I want a direct proof (not the reference to the more general case of adjoint functors in category theory). === Subject: Re: How to prove that lower adjoint determines the upper adjoint? > How to prove that lower adjoint of a Galois connection determines > upper adjoint and vice versa? > > I want a direct proof (not the reference to the more general case of > adjoint functors in category theory). The general case of categories is not really more complicated, but anyway: Let f: P --> Q and h,g: Q --> P be the maps with f -| g and f -| h. You have for all p in P and q in Q: p <= g(q) <==> f(p) <= q <==> p <= h(q) For p = g(q) you obtain g(q) <= h(q) and for p = h(q) you obtain h(q) <= g(q). -- Marc === Subject: Re: How to prove that lower adjoint determines the upper adjoint? > How to prove that lower adjoint of a Galois connection determines > upper adjoint and vice versa? I want a direct proof (not the reference to the more general case of > adjoint functors in category theory). An schizophrenic called Musatov wants destroy Sci-math - sci.math ...3 posts - 2 authors - Last post: yesterday How to prove that lower adjoint of a Galois connection determines upper adjoint and vice versa? I want a direct proof (not the reference to ... === === Subject: Re: Missing Link on the History Channel? You were supposed to stop replying. > Does your physics books mentioned the word audience refering > [sic] to stars? > No, which is precisely my point. > If your people physics books doesn't[sic] mentioned[sic] the word > audience, it doesn't[sic] understand how stars are made. > Books, of course, don't understand anything; they are just inert > chunks of paper. So this statement of yours is meaningless. > Although I'm curious to know why you think I have people physics > books, a sub-genre I've never heard of. > You never heard of the tree people? There are people who cut down > trees in Brazil, the natives call them the tree people. You are the > physics book people. You got the math people...wait I see this is > going to the science fiction people... And meanwhile, you are one of the incoherent people. > Children should not be reading sf books... You're welcome to your opinion. May you never breed. > It is the audience that created the universe, ...the audience > exist[sic] in the future. > Right. The universe was made by time travelers. > I'm traveling through time right now! Don't forget your lightning rod. You know, just in case you run out of plutonium for juicing up that flux capacitor. > People describe the universe with mathematics, but that is just an > illusion... > No, it is not. I can use calculus and the laws of motion to > describe the trajectory of a baseball bat conking you on the head. > Your ensuing headache and goose-egg will confirm handily for you > for several days that it was no illusion. > You should work for the department of defense! They need guys like > you! I already do, you ninny. Didn't I mention earlier my profession? > Mathematics does not describe the universe. Nor does physics. > Sure they do. > Scientism[sic]. > I spelled it right! Scientism. Oops, nope, and you cocked it up again. It's science. With an e. > You have limited intelligence... My intelligence greatly exceeds yours. > A dead universe is a universe > without an audience. No stars > will be born. > There is no evidence to support this contention. > You fail astrophysics forever. > I never heard of that word astrophysics...where did you get it from > An actual ing education. > Was it the 13th grade? > University, you jackass. Perhaps you didn't read the other thread, > but I'm a ing nuclear physicist. I know a lot more about a lot of > this than you do, as must be becoming quite obvious by now. > A nuclear physicist? You guys created an atom bomb to kill Germans and > you end up killing Japanese people with it. That was very smart. Yeah, imagine if the war of the Pacific had dragged on another five years. > What is your next big blunder? So far, I have had none, and I don't plan to. We let the Russians make the big blunders, then learn from their mistakes. :) > And why do they assassinate Nuclear Physics? Are you guys too > dangerous?? Are you threatening me? If so, I will be notifying the police. > Observers, the audience..makes the stars. > Not the celestial ones. > As long as you confined yourself to physics and math, you can > never understand...The Real World. > Wake up and smell the muons: the Real World is made of ... > physics and math. Go to university and get a bachelor of science > degree if you don't believe me. I know I did. > I never went to college...and I don't want to enter those > 'slave to corporations' you call university. > Well, that explains your utter failure to grasp astrophysics or > even some basic concepts of logic and reason. Although not your > near-illiteracy. Did you forget to mention that you dropped out of > school in sixth grade, or something like that? > I'll explain why you're able to grasp astrophysics, ....it's was an > accident. You are not making sense once again. Grammatically, or semantically. > Now, I'm going to answer the Big Question. > Why does the universe exist? > It exist for Informational and Entertainment purposes only. > There is no evidence to support this contention. > Do you believe the universe to be, ....informational? > There are theories along such lines, but they don't generally > claim that the universe has any a-priori purpose at all -- > particularly since there may not even be any a priori at all, > if time itself began at the Big Bang. > I didn't mentioned a-priori... Not using that exact word, no. But you stated that it was created for informational and entertainment purposes only, which implied it. > you seem to be stuck in the past. I keep talking about the future > and you keep talking about the past! Your problems with reading comprehension are exactly that: YOUR problems. Not mine. > Time itself may have began at the Big Bang, but the audience began > at the future! Observers created the Big Bang from the future. Did they send Michael J. Fox in his souped-up DeLorean, or was it that spinning brass-n-glass thingy from that movie with Guy Pearce? > Before the movie starts > the audience is sitting in the dark.. > waiting to give birth to a star. None of this is relevant to astrophysics. > There is no Big Bang without observers in the dark. You want proof, go > see a movie! This is completely irrational. > Turn on a TV...or just open a page in a magazine. > The Big Bang is the star of the show! Maybe if the magazine is a cosmology special issue of Scientific American. > The universe exists for Entertainment and Informational purposes only. Haven't I just gotten through debunking that claim? > Okay, now that you guys thrown away > your math and physics books... I have not. > when the movie ends > it is dark again.. > the star making process > is over. > > Same law applies > to the universe... > in the end > the inverse[sic] is completely dark. > The star making process is over. > > Why the universe exists? > For Entertainment purposes only. There is no evidence to support this claim. > You got a fish tank? > Little fishes inside... > eating, drinking, sleeping.. > for your Entertainment only. I'm sure they have other reasons. > To that fish > in his tank > that's his universe.. He can see out through the glass, you know. > he exists for entertainment > purposes only. > > Now, you can drop out of school. > Play the guitar.. > smoke some weeeed > and get stoned. If you don't mind being destitute and living on the streets for the rest of your life. > Or go back to school > make that atom bomb > and kill those Germans! Are you stuck in a time-warp or something? WWII is over, six decades ago. We won. > Are you smart enough to drop out of school? Smart enough not to, you mean? > The Starmaker...coming soon to a theater near you! Revenge Of The > Nitwits 2 We've already had Terminator 4 bomb. We don't need another box-office bomb this month. === Subject: Re: Missing Link on the History Channel? SHUT UP!! > Does your physics books mentioned the word audience refering > [sic] to stars? > No, which is precisely my point. > If your people physics books doesn't[sic] mentioned[sic] the word > audience, it doesn't[sic] understand how stars are made. > Books, of course, don't understand anything; they are just inert > chunks of paper. So this statement of yours is meaningless. > Although I'm curious to know why you think I have people physics > books, a sub-genre I've never heard of. > You never heard of the tree people? There are people who cut down > trees in Brazil, the natives call them the tree people. You are the > physics book people. You got the math people...wait I see this is > going to the science fiction people... > And meanwhile, you are one of the incoherent people. > > Much larger population than the coherent people, a dying breed... Due to nozzles like you. > Children should not be reading sf books... > You're welcome to your opinion. May you never breed. > > Not my opinion, ...Albert Einstein's opinion, he believed that people > should not read science fiction--that it distorts science, and gives > people the illusion of understanding. Even Einstein was occasionally wrong. > May he never breed? Or maybe, You?... The Bomb Maker?? Hey. My bombs kept the peace for decades, until the SU eventually collapsed under its own weight. Don't knock 'em. > It is the audience that created the universe, ...the audience > exist[sic] in the future. > Right. The universe was made by time travelers. > I'm traveling through time right now! > Don't forget your lightning rod. You know, just in case you run out of > plutonium for juicing up that flux capacitor. > > I can always buy plutonium from the Koreans! *shudder* > People describe the universe with mathematics, but that is just an > illusion... > No, it is not. I can use calculus and the laws of motion to > describe the trajectory of a baseball bat conking you on the head. > Your ensuing headache and goose-egg will confirm handily for you > for several days that it was no illusion. > You should work for the department of defense! They need guys like > you! > I already do, you ninny. Didn't I mention earlier my profession? > > You mention it, but you didn't say what country.... I thought it was obvious. > Mathematics does not describe the universe. Nor does physics. > Sure they do. > Scientism[sic]. > I spelled it right! Scientism. > Oops, nope, and you cocked it up again. It's science. With an e. > > There's no c. There certainly is. http://www.merriam-webster.com/dictionary/science Note spelling. > You have limited intelligence... > My intelligence greatly exceeds yours. > > It is still, limited. A meaningless statement. > My stupidity greatly exceeds yours, and it has no limits. That much has been obvious for quite some time now. > A dead universe is a universe > without an audience. No stars > will be born. > There is no evidence to support this contention. > You fail astrophysics forever. > I never heard of that word astrophysics...where did you get it from > An actual ing education. > Was it the 13th grade? > University, you jackass. Perhaps you didn't read the other thread, > but I'm a ing nuclear physicist. I know a lot more about a lot of > this than you do, as must be becoming quite obvious by now. > A nuclear physicist? You guys created an atom bomb to kill Germans and > you end up killing Japanese people with it. That was very smart. > Yeah, imagine if the war of the Pacific had dragged on another five years. > > I don't live in a if world... Oh, yeah, of course, you're too stupid to understand hypotheticals. Silly me. > Are you threatening me? > > Do you have anything against Iranian Nuclear Physics being > assassinated? I'm not Iranian, nutjob. > Observers, the audience..makes the stars. > Not the celestial ones. > As long as you confined yourself to physics and math, you can > never understand...The Real World. > Wake up and smell the muons: the Real World is made of ... > physics and math. Go to university and get a bachelor of science > degree if you don't believe me. I know I did. > I never went to college...and I don't want to enter those > 'slave to corporations' you call university. > Well, that explains your utter failure to grasp astrophysics or > even some basic concepts of logic and reason. Although not your > near-illiteracy. Did you forget to mention that you dropped out of > school in sixth grade, or something like that? > I'll explain why you're able to grasp astrophysics, ....it's was an > accident. > You are not making sense once again. Grammatically, or semantically. > > Accident! A mistake, or by chance.. There has been no mistake by me. I understand it by intelligence and reason, not by sheer dumb luck. > The truth is, the majority of people do *not* understand or have any > real grasp of science or arithmetic... Including you. Fortunately, I'm not in that majority. > if you do, it's a freak of nature, a mistake..and accident. No, it is not. It is called having had an education. You ought to try it sometime. > here is what an accident looks like: > http://www.telegraph.co.uk/telegraph/multimedia/archive/00441/news-graphics- 2 007-_441537a.jpg That's the result of disease, not an accident, wit. > There is no Big Bang without observers in the dark. You want proof, go > see a movie! > This is completely irrational. > > Have you tried learning without a textbook? I learn via all kinds of media, including but not limited to books. > To that fish > in his tank > that's his universe.. > He can see out through the glass, you know. > > Does he understands what he sees? Probably not. He's a fish, doofus. > Or go back to school > make that atom bomb > and kill those Germans! > Are you stuck in a time-warp or something? WWII is over, six decades > ago. We won. > > I've been watching too many Woody Allen movies... I'm more worried about the inhalants you use than the movies you watch, to be honest. > The Starmaker...coming soon to a theater near you! Revenge Of The > Nitwits 2 > We've already had Terminator 4 bomb. We don't need another box-office > bomb this month. > > You like bombs don't you? Not the box-office ones. > ..-^~~~^-.. > .~ ~. > (;: :;) > (: :) > ':._ _.:' > | | > | | > -O- | | > | | > / ((/ )) > What's that? A greenie fantasy about trees arming themselves and chasing people? Or maybe just failure to compose your ascii art in a nonproportional font... === Subject: Re: Missing Link on the History Channel? The Starmaker failed to respect the followup-to header: > Children should not be reading sf books... > You're welcome to your opinion. May you never breed. > Not my opinion, ...Albert Einstein's opinion, he believed that people > should not read science fiction--that it distorts science, and gives > people the illusion of understanding. > Even Einstein was occasionally wrong. > > You got kids out there believing in flying saucers, that's science > fiction, that's a distortion of science. Not according to Science Illustrated. We might soon be building our own. > Mathematics does not describe the universe. Nor does physics. > Sure they do. > Scientism[sic]. > I spelled it right! Scientism. > Oops, nope, and you cocked it up again. It's science. With an e. > There's no c. > There certainly is. > http://www.merriam-webster.com/dictionary/science > Note spelling. > > Scientism Sheesh! You're hopeless. > Just a belief, just a religion. No. Science self-corrects and is grounded in evidence and reproducible observations, unlike faith-based belief systems. But trying to educate you, about science or anything else, is like trying to educate a nozzle. > You have limited intelligence... > My intelligence greatly exceeds yours. > It is still, limited. > A meaningless statement. > > It means your intelligence is small in range. That is a vicious lie. > My stupidity greatly exceeds yours, and it has no limits. > That much has been obvious for quite some time now. > Are you threatening me? > Do you have anything against Iranian Nuclear Physics being > assassinated? > I'm not Iranian, nutjob. > > Do you have anything against any Nuclear Physicist from any country > being assassinated? Yes. Murder is murder, whatever the motive. > Observers, the audience..makes the stars. > Not the celestial ones. > As long as you confined yourself to physics and math, you can > never understand...The Real World. > Wake up and smell the muons: the Real World is made of ... > physics and math. Go to university and get a bachelor of science > degree if you don't believe me. I know I did. > I never went to college...and I don't want to enter those > 'slave to corporations' you call university. > Well, that explains your utter failure to grasp astrophysics or > even some basic concepts of logic and reason. Although not your > near-illiteracy. Did you forget to mention that you dropped out of > school in sixth grade, or something like that? > I'll explain why you're able to grasp astrophysics, ....it's was an > accident. > You are not making sense once again. Grammatically, or semantically. > Accident! A mistake, or by chance.. > There has been no mistake by me. I understand it by intelligence and > reason, not by sheer dumb luck. > > I bet your parents thought they were lucky. To be born intelligent, maybe. Since then I've made my own luck. > if you do, it's a freak of nature, a mistake..and accident. > No, it is not. It is called having had an education. You ought to try > it sometime. > > Anybody can walk into a college bookstore and buy a textbook. That's not quite the same thing as an education, but it's a start. Go right ahead. When you're done, post here again. > here is what an accident looks like: > http://www.telegraph.co.uk/telegraph/multimedia/archive/00441/news-graphics- 2 007-_441537a.jpg > That's the result of disease, not an accident, wit. > > I was discussing 'smart people' are a freak of nature No, not really. His illness has nothing to do with his intelligence; they're orthogonal. > It's no accident he got a disease, it comes with 'the smarts'. Nonsense. > There is no Big Bang without observers in the dark. You want proof, go > see a movie! > This is completely irrational. > Have you tried learning without a textbook? > I learn via all kinds of media, including but not limited to books. > > You mean like , watching a movie? Few of those are educational. > To that fish > in his tank > that's his universe.. > He can see out through the glass, you know. > Does he understands what he sees? > Probably not. He's a fish, doofus. > > Doofus? Yes, doofus. > Or go back to school > make that atom bomb > and kill those Germans! > Are you stuck in a time-warp or something? WWII is over, six decades > ago. We won. > I've been watching too many Woody Allen movies... > I'm more worried about the inhalants you use than the movies you watch, > to be honest. > > You mean like glue sniffing? You put glue in a paper bag and you > inhale it...do you got kids? They are probably glue sniffing right > now.. Fortunately, they have long since outgrown that, unlike you. > The Starmaker...coming soon to a theater near you! Revenge Of The > Nitwits 2 > We've already had Terminator 4 bomb. We don't need another box-office > bomb this month. > You like bombs don't you? > Not the box-office ones. > > Just the ones that kill people...and I guess you're going to breed > babies that grow up to be.. Weapons of Mass Destruction. Ridiculous. One is a doctor now. She *saves* people. > And they teach this stuff in school? > > Daddy, I want to take a How To Build A Nuclear Bomb class at the > university!!! I want to kill millions of people! Ridiculous. I always worked on the energy-technologies side anyway. You know, trying to save the planet from global warming, that sort of thing? === Subject: Re: Missing Link on the History Channel? > [death threat deleted] Police notified. Have a nice day. === Subject: Re: Missing Link on the History Channel? You were supposed to stop replying. > Does your physics books mentioned the word audience refering to stars? > No, which is precisely my point. > If your people physics books doesn't[sic] mentioned[sic] the word > audience, it doesn't[sic] understand how stars are made. > Books, of course, don't understand anything; they are just inert chunks > of paper. So this statement of yours is meaningless. > Although I'm curious to know why you think I have people physics > books, a sub-genre I've never heard of. > > You never heard of the tree people? There are people who cut down > trees in Brazil, the natives call them the tree people. You are the > physics book people. You got the math people...wait I see this is > going to the science fiction people... And meanwhile, you are one of the incoherent people. > Children should not be reading sf books... You're welcome to your opinion. May you never breed. > It is the audience that created the universe, ...the audience > exist[sic] in the future. > Right. The universe was made by time travelers. > > I'm traveling through time right now! Don't forget your lightning rod. You know, just in case you run out of plutonium for juicing up that flux capacitor. > People describe the universe with mathematics, but that is just an > illusion... > No, it is not. I can use calculus and the laws of motion to describe the > trajectory of a baseball bat conking you on the head. Your ensuing > headache and goose-egg will confirm handily for you for several days > that it was no illusion. > > You should work for the department of defense! They need guys like > you! I already do, you ninny. Didn't I mention earlier my profession? > Mathematics does not describe the universe. Nor does physics. > Sure they do. > Scientism[sic]. > > I spelled it right! Scientism. Oops, nope, and you cocked it up again. It's science. With an e. > You have limited intelligence... My intelligence greatly exceeds yours. > A dead universe is a universe > without an audience. No stars > will be born. > There is no evidence to support this contention. > You fail astrophysics forever. > I never heard of that word astrophysics...where did you get it from > An actual ing education. > Was it the 13th grade? > University, you jackass. Perhaps you didn't read the other thread, but > I'm a ing nuclear physicist. I know a lot more about a lot of this > than you do, as must be becoming quite obvious by now. > > A nuclear physicist? You guys created an atom bomb to kill Germans and > you end up killing Japanese people with it. That was very smart. Yeah, imagine if the war of the Pacific had dragged on another five years. > What is your next big blunder? So far, I have had none, and I don't plan to. We let the Russians make the big blunders, then learn from their mistakes. :) > And why do they assassinate Nuclear Physics? Are you guys too > dangerous?? Are you threatening me? If so, I will be notifying the police. > Observers, the audience..makes the stars. > Not the celestial ones. > As long as you confined yourself to physics and math, you can never understand...The Real World. > Wake up and smell the muons: the Real World is made of ... physics and > math. Go to university and get a bachelor of science degree if you don't > believe me. I know I did. > I never went to college...and I don't want to enter those > 'slave to corporations' you call university. > Well, that explains your utter failure to grasp astrophysics or even > some basic concepts of logic and reason. Although not your > near-illiteracy. Did you forget to mention that you dropped out of > school in sixth grade, or something like that? > > I'll explain why you're able to grasp astrophysics, ....it's was an > accident. You are not making sense once again. Grammatically, or semantically. > Now, I'm going to answer the Big Question. > Why does the universe exist? > It exist for Informational and Entertainment purposes only. > There is no evidence to support this contention. > Do you believe the universe to be, ....informational? > There are theories along such lines, but they don't generally claim that > the universe has any a-priori purpose at all -- particularly since there > may not even be any a priori at all, if time itself began at the Big Bang. > > I didn't mentioned a-priori... Not using that exact word, no. But you stated that it was created for informational and entertainment purposes only, which implied it. > you seem to be stuck in the past. I keep talking about the future > and you keep talking about the past! Your problems with reading comprehension are exactly that: YOUR problems. Not mine. > Time itself may have began at the Big Bang, but the audience began > at the future! Observers created the Big Bang from the future. Did they send Michael J. Fox in his souped-up DeLorean, or was it that spinning brass-n-glass thingy from that movie with Guy Pearce? > Before the movie starts > the audience is sitting in the dark.. > waiting to give birth to a star. None of this is relevant to astrophysics. > There is no Big Bang without observers in the dark. You want proof, go > see a movie! This is completely irrational. > Turn on a TV...or just open a page in a magazine. > > The Big Bang is the star of the show! Maybe if the magazine is a cosmology special issue of Scientific American. > The universe exists for Entertainment and Informational purposes only. Haven't I just gotten through debunking that claim? === Subject: Re: Missing Link on the History Channel? > http://www.nytimes.com/2009/05/19/business/media/19fos > sil.html?bl&ex=1242964800&en=03a8f7720cdc331a&ei=5087% > 0A > > They found a Missing Link and all they could get is > the History Channel? > > > I think someone in TV is missing a link! > > Math Forum Discussionssci.math. Topic: Missing Link on the History Channel? ... Paris Hilton, Britney Speares, The Beatles, Jesus Christ, Humphrey Bogart...they are your ... mathforum.org/kb/message.jspa?messageID=6724277&tstart=0 Missing Link on the History Channel?...15 posts - Last post: May 26 They found a Missing Link and all they could get is the History Channel? .... misconception of my handle The Starmaker to yous sci physics math people. ... Paris Hilton, Britney Speares, The Beatles, Jesus Christ, ... www.groupsrv.com/science/post-3049924.html Missing Link on the History Channel? - sci.physics | Google GroupsYou need to throw away your physics/math textbooks away, out the window. ... Paris Hilton, Britney Speares, The Beatles, Jesus Christ, ... Missing Link on the History Channel?...They found a Missing Link and all they could get is the History ... You need to throw away your physics/math textbooks away, out the window. ... Paris Hilton, Britney Speares, The Beatles, Jesus Christ, Humphrey Bogart...they are your ... www.groupsrv.com/science/about413919-0-asc-30.html Amazon.com: The Multiplying Church: The New Math for Starting New ...The missing link--pregnant mother churches * Antioch vs. .... The Church of Jesus Christ of Latter-day Saints: Truth Restored (4), 2431, 9 minutes ago ... www.amazon.com/Multiplying-Church-Math-Starting-Churches/dp/0310277167 Missing Link on the History Channel?...May 27, 2009 ... Also, the show keeps trumpeting a missing link to human evolution, but ... You need to throw away your physics/math textbooks away, out the window. ... Paris Hilton, Britney Speares, The Beatles, Jesus Christ, ... www.groupsrv.com/science/about413919-0-asc-45.html BREAKING NEWS: Missing Link Found. - Topic Powered by Eve For ...40 posts - 27 authors - Last post: May 20 Notice the lack of Missing link nonsense and other .... Jesus Christ could come down to Earth and tell all his followers ... Evolution is perhaps the only case in which Creationists suddenly grasp mathematics ' a line ... episteme.arstechnica.com/eve/forums/a/tpc/f/28609695/m/959009278931 Look Up Fellowship: Ida Is Not Missing LinkThe existence of Jesus Christ is an historical fact so why would we ignore .... So, Ida is not a missing link [CapitalEth] at least not between anthropoids and more ... www.lookupfellowship.com/2009/05/ida-is-not-missing-link.html Ida is not ñthe missing linkî ' How good is that?Tagged: Ida, missing link, not the. You can follow any responses to this entry ..... Mathematics is based upon Logic but how can logic just exist when its a concept in ones ...... In the ñDid Jesus Really Existî thread you posted a link to a book ...... It is not historians who propagate the 'Christ-myth' theories, ... howgoodisthat.wordpress.com/2009/05/20/ida-is-not-the-missing-link/ > > The Starmaker > > > What's the Missing Link, a big toe? > http://tinyurl.com/pm5gkg > > > You couldn't get ABC, or CBS or NBC...you got the > History Channel? === Subject: Re: a question about Radon-Nikodym derivative Just one more thing: what if mu_1 << mu_2? It seems not the case in your example. > Hi all, > > I have a question as follows: given mu_1 and mu_2 > two Borel probability measures on R, does there > always exist a Borel probability measure mu, such > that the Radon-Nikodym derivatives of mu_1 and mu_2 > w.r.t. mu are continuously differentiable? > > My feeling is yes: convolving a mixture of mu_i's > with a Gaussian distribution could probably work. But > I haven't been able to figure out the details... Any > suggestion will be appreciated! > > I believe you are asking for too much. Consider the > following: > > There is a probability density f in the Lebesgue > sense on the > unit interval, taking on only the values 0 and 2, > such that > each of the values is taken on a set of positive > measure in > every interval. Let dmu_1 = f*dx and dmu_2 = > (2-f)*dx. > > Since for almost all x in (0,1) {dmu_1/dx, > dmu_2/dx} = > {0, 2} and in every open interval each corresponds to > 2 almost > everywhere on sets of realtive measure close to one > in SOME > open subinterval, it does not seem possible to get > both of > the measures to have a continuous density. > > > -- > This address is for information only. I do not claim > that these views > are those of the Statistics Department or of Purdue > University. > Herman Rubin, Department of Statistics, Purdue > University > hrubin@stat.purdue.edu Phone: (765)494-6054 > FAX: (765)494-0558 === Subject: Your Radian and 2Pi value Reading about your Pi values ,I was amazed in current mathematics litreature of the use of 2Pi/radian value that is not time dimensional, 2 Pi value must be applicable to a dimension that is viable. Are all the each radians expansile in Time dimension , i.e is the circle expands in two Pi values at 180 degrees , I.E 180. Absolutly wrong! . Sorry to dissappoint you all and your gurus at Princeton, Stanford , U chicago , Ohio etc,, check this out. This current value of 2 Pi =radians in a circle is a dead value. 8 Pi ratio is likewise equal to an angle progresssion of around 2.1 degrees === Subject: Re: Your Radian and 2Pi value Enigma 1541 - Box clever...Jun 2, 2009 ... Hobby Forum Index é Puzzles é Enigma 1541 - Box clever... Page 1 of 1. Enigma 1541 - Box clever... Author, Message. Chappy. ... Mensanator. ... www.groupsrv.com/hobby/post-4040335.html philistine pig ignoranceWhat I wanted was the fame and glory of winning and having my clever ... Would it have killed the editors to at least mention my clever idea in their ... mensanator.com/mensanator/pig_ignorance.htm mail.python.org/pipermail/python-list/2006-September/573696.html The Quest for the Ultimate Cycle3n + 2147483647 seed = 1 attractor = 1 cycle length = 23091222 odd numbers and act all smug about how clever I am. ... For now. Back to The Mensanator. mensanator.com/mensanator/cycle/ultimate_cycle.htm pygame and python 2.5mensanator at aol.com mensanator at aol.com. Sun Feb 11 18:11:50 CET 2007 .... Because you want people to know how clever you are. ... mail.python.org/pipermail/python-list/2007-February/598079.html EIGHT PLANETS Mnemonics for the order of the planets. - nz.general ...10 posts - 7 authors - Last post: Mar 19 very clever but i donot get it please carl. 26/1/09. JANy > mad threE (letters.) below. .... Mensanator > Richard Heathfield ... implement random selection in Pythonmensanator at aol.com mensanator at aol.com ... Is there > some clever algorithm using Python standard random package to do > the trick? ... mail.python.org/pipermail/python-list/2007-November/637808.html number theory libraries / project euler | Python | Python8 posts - Last post: Feb 18 The idea of Project Euler is you are supposed to find clever enough ... mensanator at aol. Feb 18, 2009, 2:36 PM Post #5 of 5 (114 views) ... www.gossamer-threads.com/lists/python/python/722126 mail.python.org/pipermail/python-list/2007-February/598047.html === grahamkendall.net/Math/Math%20Newsgroups/mm-998.txt > Reading about your Pi values ,I was amazed in > current mathematics litreature of the use of > 2Pi/radian value that is not time dimensional, 2 Pi > value must be applicable to a dimension that is > viable. Are all the each radians expansile in Time > dimension , i.e is the circle expands in two Pi > values at 180 degrees , I.E 180. Absolutly wrong! . > Sorry to dissappoint you all and your gurus at > Princeton, Stanford , U chicago , Ohio etc,, check > this out. > > I disagree mathematically and can prove that a time > e dimensional Pi value of a circle is = 19 equal Pi > values ratios of 3.14 that are equal. Because the > time dimension fractionation of a Pi value is not > additive or fractional, because there are infinte Pi > values and ratios in a circle that are out of sink > Pi ratio that is proven , but 9.5 degrees expands at > 2Pi value ratio but half the time dimension > > This current value of 2 Pi =radians in a circle is a > dead value. > 8 Pi ratio is likewise equal to an angle progresssion > of around 2.1 degrees === Subject: Re: Your Radian and 2Pi value There is a better a clearer method of saying the value of the circumference that is time dimensional i.e PiR (2) or PiR value multiplied by 2 . 2PiR can be confusing mathematically but it is acceptable now that it is acceptable. Pi value is also the radial arc, and the linear progression of inverse 19 and definition more correct of a mathematicval definition than Circumference over the radius, because it is time dimensional, and predictive till infinity I truly apologize for being a little sarcastic because most mathematicians are decent, but I have found your mathematics to be so hierachial, as goes these people, as goes the rest of mathematics and physics, generally I have found these places to be arrogant , the little contact I have had with them. There might come a day when I may have a chance to meet these fellows on the blackboard with their complexities, or may be not, that is not my goal. Real authority is humble and then civil and respectful.. Inverse 19 === Subject: Re: Your Radian and 2Pi value <22102145.9437.1244839862150.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > There is a better a clearer method of saying the value of the circumference that is time dimensional i.e PiR(2) or PiR value multiplied by 2 . 2PiRcan be confusing mathematically but it is acceptable now that it is acceptable. Pi value is also the radial arc, and the linear progression of inverse 19 and definition more correct of a mathematicval definition than Circumference over the radius, because it is time dimensional, and predictive till infinity I truly apologize for being a little sarcastic because most mathematicians are decent, but I have found your mathematics to be so hierachial, as goes these people, as goes the rest of mathematics and physics, generally I have found these places to be arrogant , the little contact I have had with them. There might come a day when I may have a chance to meet these fellows on the blackboard with their complexities, or may be not, that is not my goal. Real authority is humble and then civil and respectful.. Inverse 19 Yes, I need to work on this. > Cantor's argument is erroneous and its adoption leads > to unsound > mathematics. The basic idea in the argument is that there is no > bijection between > the set of counting numbers and the set of infinite > binary strings. > But such a bijection exists, it can be expressed in > terms of limit > points, or by transfinite induction; informally, it > can be defined as > the correspondence between the paths and the leaf > (i.e. limit) nodes > in the infinite binary tree. This invalidates all > results relating to > Cantor's transfinite. In particular, it is invalid to state that the set of > infinite binary > strings is uncountable. It is countable, being in > bijection with a > subset of a countable set, the set of nodes in the > infinite binary > tree. The other option is that one drops the > countability of infinite > sets completely, but I can see no advantage in > banning the > transfinite. The problem is possibly much broader, and deeper, > because it is the > very soundness of archimedean arithmetic that seems > at stake here. I > say soundness because from the archimedean > framework a tension > results, between computability and tractability, that > manifests a > deeper tension between sound and unsound mathematics, > and then even > logic. -LV+Musatov QED https://share.acrobat.com/adc/adc.do?docid=80c126dd-8961-4a57-9351-22abcd68d e3b The proof 6P=NP is no longer so-called. It has been in fact established irrefutable and proven given the over Cantor and the rational argument for pi and programming, plus the disqualification of the Levy proof R is uncountable. We have also provided proof of discovery of hard to find easy to check. Signed with respect and well wishes to my peers, Martin Michael Musatov (all of it is right here in this thread) > Reading about your Pi values ,I was amazed in current mathematics litreature of the use of 2Pi/radian value that is not time dimensional, 2 Pi value must be applicable to a dimension that is viable. Are all the each radians expansile in Time dimension , i.e is the circle expands in two Pi values at 180 degrees , I.E 180. Absolutly wrong! . Sorry to dissappoint you all and your gurus at Princeton, Stanford , U chicago , Ohio etc,, check this out. > This current value of 2 Pi =radians in a circle is a dead value. > 8 Pi ratio is likewise equal to an angle progresssion of around 2.1 degrees Math Forum Discussions Efforts by mathematicians to determine the value of pi since have led ... Musatov Inverse Counting Chain of Infinite Proportions (sample) ... mathforum.org/kb/message.jspa?messageID=6749244&tstart=0 - 4 hours ago Math Forum Discussions Would life be better if pi = 3 exactly? > Huh? This is Musatov ... Pi is a CONSTANT. Its value is 4*ATAN(1). ... Martin Michael Musatov. 6/8/09 ... mathforum.org/kb/message.jspa?messageID=6743888&tstart=0 Discussions - comp.theory | Google Groups QED Plouffe. I own you. (kidding) Musatov Welcome, marty.musatov@gmail.com Logout Your Control Panel Watches FAQ Contact Us Math Forum Ask Dr. Math Discussions Internet Newsletter MathTools Problems of the Week Teacher2Teacher Teacher Exchange Workshops Search All of the Math Forum: Browse our Internet Mathematics Library Views expressed in these public forums are not endorsed by Drexel University or The Math Forum. Math Forum é Discussions é sci.math.* é sci.math Topic: inverse of power series with prime power coefficients Replies: 7 Last Post: Jun 12, 2009 4:11 PM Search Thread: Advanced Search Reply to this Topic Watch this Topic Back to Topic List Jump to Tree View Messages: [ Previous | Next ] Martin Michael Musatov Posts: 802 Registered: 4/19/09 Re: inverse of power series with prime power coefficients > On May 11, 5:37 pm, Robert Israel > The power series with prime coefficients > P(x) = 1 + 2x + 3x^2 + 5x^3 + 7x^4 + 11x^5 + > 13x^6 + .... > and itsinverse > Q(x) = 1 / P(x) = 1 - 2x + x^2 - x^3 + 2x^4 - > 3x^5 + 7x^6 - 10x^7 + > 13x^8 - 21x^9 + 26x^10 - 33x^11 + 53x^12 - .... > were studied in the mid-1990s, when Backhouse > conjectured, and > Flajolet proved, that the limit of the absolute > value of the ratio of > consecutive coefficients of Q(x) is equal to > B=1.4560749...., > Backhouse's constant. (By comparison, the limit > of the ratio of > consecutive primes is equal to 1.) > Consider instead the power series with prime > power coefficients: > PP(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + 7x^5 + 8x^6 > + 9x^7 + 11x^8 + > 13x^9 + 16x^10 + 17x^11 + 19x^12 + 23x^13 + > 25x^14 + 27x^15 + 29x^16 + > 31x^17 + 32x^18 + 37x^19+ .... > I would expect itsinverseto be quite similar to > the series Q(x) > above. But it appears to be quite different: > QQ(x) = 1 - 2x + x^2 - x^5 + 3x^6 - 3x^7 + 2x^9 - > x^10 + x^15 - 4x^16 > + 8x^17 - 5x^18 - 13x^19+ 26x^20 - 9x^21 - 20x^22 > + 25x^23 + 9x^24 - > 60x^25 + 65x^26 + .... > Note how many low terms are zero: x^3, x^4, x^8, > and the four > consecutive terms x^11, x^12, x^13, x^14. I > wonder if this is pure > coincidence or if it has some significance. > There seem to be no other zero coefficients up to > the x^300 term. > Also, whereas the terms of Q(x) alternate > regularly between positive > and negative, there is no apparent pattern to the > signs of the terms > of QQ(x). > Finally, the terms of QQ(x) appear to grow much > more slowly than the > terms of Q(x). Perhaps as the series continues > and the prime powers > become sparser compared to the primes, QQ(x) will > start to look and > behave more like Q(x). But I am curious: does the > limit of the > absolute value of the ratio of consecutive > coefficients of QQ(x) > exist? If so, what is it? > Looking at the first 300 or so terms, I see no > pattern. > -- > Robert Israel > isr...@math.MyUniversitysInitials.ca > Department of Mathematics > http://www.math.ubc.ca/~israel > University of British Columbia > Vancouver, BC, Canada- Hide quoted text - > - Show quoted text - Remember me? Or are my results too trivial Mr. > Moderator? Welcome to the real mathematics and not the fake > Matrix movie > excuse for chess you've been playing for the > 20 years. Musatov (Founder, Inverse 19 Mathematics) Not the only, but > one without it > would not exist. QED > Efforts by mathematicians to determine the value of > pi since have led > to the conclusion that it ... Unlike rational > numbers, pi cannot be > represented by a common fraction, .... But what about an uncommon fraction? Will you look at my drawing and in particular how the > twelve points > are the outside points of the sphere are all > rationally reached by > decimal numbers? 6.2035884*2 and doubled an infinite number of times > should yield a > constant decimal binary expansion (what I mean by > this is the numbers > to the right of the decimal point should always end > in 4, then 8, > then 6, then 2, then 4, then 8, then 6, then > 2...repeating for > infinity). So in binary, by starting with this number we predict > for n>2 the > infinite binary tree will end with decimal > progression: 2, 4, 8, 6 For n<2 the infinite binary tree will begin and end > (depends on if you > are increasing or decreasing the value) 1, 5, 5, 5 > (repeating 5 > infinitely) as we decrease to infinitely small. Musatov Inverse Counting Chain of Infinite > Proportions > (sample) > 0.0000473372528076171875 > 0.000094674505615234375 > 0.00018934901123046875 > 0.0003786980224609375 > 0.000757396044921875 > 0.00151479208984375 > 0.0030295841796875 > 0.006059168359375 > 0.01211833671875 > 0.0242366734375 > 0.048473346875 > 0.09694669375 > 0.1938933875 > 0.387786775 > 0.77557355 > 1.5511471 > 3.1022942 > 6.2035884 > 12.4071768 > 24.8143536 > 49.6287072 > 99.2574144 > 198.5148288 > 397.0296576 > 794.0593152 > 1588.1186304 > 3176.2372608 > 6352.4745216 > 12704.9490432 > 25409.8980864 > 50819.7961728 > 101639.5923456 > 203279.1846912 > 406558.3693824 > 813116.7387648 > 1626233.4775296 > 3252466.9550592 > 6504933.9101184 > 13009867.8202368 > 26019735.6404736 > 52039471.2809472 > 104078942.5618944 > 208157885.1237888 > 416315770.2475776 > 832631540.4951552 > 1665263080.9903104 > 3330526161.9806208 > 6661052323.9612416 > 13322104647.9224832 > 26644209295.8449664 > 53288418591.6899328 > 106576837183.3798656 > 213153674366.7597312 > 426307348733.5194624 > 852614697467.0389248 > 1705229394934.0778496 > 3410458789868.1556992 > 6820917579736.3113984 > 13641835159472.6227968 > 27283670318945.2455936 > 54567340637890.4911872 > 109134681275780.9823744 > 218269362551561.9647488 > 436538725103123.9294976 > 873077450206247.8589952 > 1746154900412495.7179904 > 3492309800824991.4359808 > 6984619601649982.8719616 > 13969239203299965.7439232 > 27938478406599931.4878464 > 55876956813199862.9756928 > 111753913626399725.9513856 > 223507827252799451.9027712 > 447015654505598903.8055424 > 894031309011197807.6110848 > 1788062618022395615.2221696 > 3576125236044791230.4443392 > 7152250472089582460.8886784 > 14304500944179164921.7773568 > 28609001888358329843.5547136 > 57218003776716659687.1094272 > 114436007553433319374.2188544 > 228872015106866638748.4377088 > 457744030213733277496.8754176 > 915488060427466554993.7508352 > 1830976120854933109987.5016704 > 3661952241709866219975.0033408 > 7323904483419732439950.0066816 > 14647808966839464879900.0133632 > 29295617933678929759800.0267264 > 58591235867357859519600.0534528 > 117182471734715719039200.1069056 > 234364943469431438078400.2138112 > 468729886938862876156800.4276224 > 937459773877725752313600.8552448 > 1874919547755451504627201.7104896 > 3749839095510903009254403.4209792 > 7499678191021806018508806.8419584 > (sample) If you would like to do the honors Mr. Plouffe, I > endear you to > calculate the angles and lengths of the number of > ways, it is quite > obviously possible to square a circle based on the > geometry and > numerical progression. As is self-evident to anyone with even a modest > understanding of > mathematics and geometry after visual inspection of > the drawn proof > (http://www.flickr.com/photos/24869933@N07/3619373444/ > sizes/l/) it is > quite easily accomplished to connect and triangulate > the outer points > of the rectangles to the outer points of the square > and circle and > even by dimension sphere. Yes, the drawing clearly by definition of inverse > symmetry (take the > drawing cut it in half at the equator and turn the > paper over and > rotate it and compare the top and bottom mirrored > symmetry and you > will see it is an exact inverse of symmetry itself) > the drawing > explains how to properly and mathematically soundly > represent a fourth > dimension on a two dimensional plane. In plane language if you take the picture and > separate the top half of > the sphere by the bottom half of the sphere they are > identical but > only if the sphere existed on a complete three > dimensional plane > through expansion. Since the paper is flat, this is an impossibility, > nonetheless it is > completely expressed and retractable and countable > infinitely. By the definition of our standard we have established > any k-digit > approximation to pi is rational. Which could have many practical scientific > applications. For instance, > search: ON THE THEOREM OF IVA$EV-MUSATOV IIIx e F(l) + F(2) + > ... + F(p). This > remark is the basis of our proof of Theorem .... -y(M > + 1), x' + y(M + > 1)] = 0 it is easy to check that Conditions (i), > ...... Now set i = > pi x. *jj, v. Since p, x is positive,. (i) fi is > positive, ... > plms.oxfordjournals.org/cgi/reprint/s3-53/1/143.pdf REPRESENTATION OF FUNCTIONSBY SERIES AND CLASSES > ?(1)with measure E 2? -- ? and such that the trigonometric series for > g(x) converges > uniformly on [0, 2?]. .... trigonometric series are > due to > Ivashev-Musatov [40] who has shown that there are > null series (in the > sense of ..... function ?(?) e C(0, 1) with ?(0) = > ?(1) = 0 such that > ...... <*i .pdf > by PL Ul'yanov - 1972 - Cited by 24 - Related Full text of Bulletin of the British Museum (Natural > History)X 20. > FIG. 5. Well preserved large fragment showing the > different aspect and > ...... 0-1-0-5 P i n height) so giving the shell > surface the > appearance of ...... 94-95, i pi. Maseru. BRINK, A. > S. 1963. Two > cynodonts from the Ntawere ...... MUSATOV, D. L, > NEMIROVSKAYA, V. N., > SHIROKOVA, E. V. & ZHURAVLEVA, I. T. 1961. ... > www.archive.org/stream/bulletinofbritis17geollond/bull > etinofbritis17geollond djvu.txt Trigonometric series with rapidly decreasing > coefficients a i0, 5 > ...Nt,+i - Pi = n,. N tl+j. -pi =N tl+j. -i +pi +1. > ..... number 8 6 > (0,1) for the product (32) such that for every point > x 6 E there are > ... > A01.pdf > versions Plouffe's formula for pi - sci.math | Google Groups3 > posts - 2 authors > - Last post: 41 minutes ago > x=pi e^(ipi)+1=0. Martin Musatov. Reply Forward. > Martin Musatov to > problem. ... > 65ba40fd9e5e8f46/ > bf6d1ee13d87270b?lnk=raot > - 41 minutes ago [PDF] ???? ?-?File Format: PDF/Adobe Acrobat > Gli storici ci dicono che, nel X secolo, il principe > Vladi- ...... i > pi`u anziani e carismatici si evidenziavano i nomi di > ...... tin > Somov, Viktor Borisov-Musatov, Aleksandr Benois e Lev > Bakst ... 5/0; > Judovin 5/0; Korovin 2/1; Tukacov 1/0; Goli- cyn 1/1; > Zacharov 1/1; > Jurkunas 5/0; Krasauskas 2/0;. Bulaka 3/0. ... > www.esamizdat.it/eSamizdat 2005 (III) 2-3.pdf I am testing these identities: (1/2) * ( - k ) * ( k - 1/2 ) * ( - 3 ) / k! This equation produces M = k! ( - 3 ) * ( k - 1/2) * ( - k ) * (1/2) back! > Martin Musatov Reply Forward > ---------- Forwarded message ---------- === > Subject: CHALLENGE TO MATHEMATICIANS OF THE WORLD > On Apr 11, 11:40 am, Harris Moran > *plonk* message > muh brane i needs one- Hide quoted text - > - Show quoted text - NOTE THE PROGRESSION OF NUMBERS PELLS PROPORTION > X^2=3 or 4 = 3 as > the numbers progression, WOW if this does not proof > this what will 1. Every odd number , number as Y value till > infinity is exactly > solvable with a proportionate result using the base > Pells equation i.e > 3,5,7,9,etc etc till infinty and the answer is as in > 2(2^2)+1=3^2 and > so on till infinity with a whole number integer. 2.Every even number , without exception, using Pells > equation, the > integer always is a .75 till infinity consistant as 63.75(2^4)=1=4^4 till infinity every integer > will end with .75. > is that the same proportion as 3 over 4, Y=3 for the > least with > numbers proportion There is a definite pattern by pells to numbers, and > I note that in > that pattern the number 19 proportion is strikingly > proprtionate, > and I clearly surmise that that circluar progression > is the same as > numbers for vector 19 progression. > Update from your neighborhood friendly Musatov. I am going to frame this first one!!! Exciting.... **************************************************************************** **************************************************************************** * **************************************************************************** * ** *Math Forum Discussions mathematics and geometry after visual inspection of the drawn proof ... approximation to pi is rational. Which could have many practical scientific applications. ... x=pi e^ (ipi)+1=0. Martin* *Musatov. Reply Forward. Martin Musatov to ...* *http://mathforum.org/kb/message.jspa?messageID=6749244&tstart=0 - 4 hours ago * **************************************************************************** **************************************************************************** * ****************** And then this one is tease of the journey to come... +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++ Reactions to/against the Binary Tree - Page 6 - Docendi.org17 posts - Last post: May 28 But he claims that pi is the limit of the finite initial segments. These two claims suggest ... axiomatic basis. Are you an alias of Musatov? No I am Musatov. .... If Cantor's proof is correct, then an infinite path is the limit of ..... The real number sqrt(3) is the limit of a rational sequence. ... +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++ 5/11/09 inverse of power series with prime power coefficients rokirovka@gmail.com 5/11/09 Re: inverse of power series with prime power coefficients Gerry Myerson 5/11/09 Re: inverse of power series with prime power coefficients Robert Israel 5/12/09 Re: inverse of power series with prime power coefficients hagman 5/13/09 small tommy prize ! amy666 5/14/09 Re: small tommy prize ! amy666 6/12/09 Re: inverse of power series with prime power coefficients Martin Michael Musatov 6/12/09 Re: inverse of power series with prime power coefficients Martin Michael Musatov [Privacy Policy] [Terms of Use] © Drexel University 1994-2009. All Rights Reserved. The Math Forum is a research and educational enterprise of the Goodwin College of Professional Studies. 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Math Forum é Discussions é sci.math.* é sci.math Topic: The Beast (seeing is believing) Replies: 0 Search Thread: Advanced Search Reply to this Topic Watch this Topic Back to Topic List THEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEB EASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEAST T HEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBE A STTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTH E BEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEAS T THEBEASTTHEBEASTTHEBEASTTHEBEASTVTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHE B EASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEAST T HEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBE A STTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTH E BEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEAS T THEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEB E ASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTT H EBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTVTHEBEASTTHEBEASTTHEBE A STTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTH E BEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEAST Martin Michael Musatov Posts: 805 Registered: 4/19/09 The Beast (seeing is believing) THEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEB EASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEAST T HEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBE A STTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTH E BEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEAS T THEBEASTTHEBEASTTHEBEASTTHEBEASTVTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHE B EASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEAST T HEBEASTTHEBEASTTHEBEAST In[170]:= 0.0000473372528076171875*2 0.000094674505615234375 0.00018934901123046875 0.0003786980224609375 0.000757396044921875 0.00151479208984375 0.0030295841796875 0.006059168359375 0.01211833671875 0.0242366734375 0.048473346875 0.09694669375 0.1938933875 0.387786775 0.77557355 1.5511471 3.1022942 6.2035884 12.4071768 24.8143536 49.6287072 99.2574144 198.5148288 397.0296576 794.0593152 1588.1186304 3176.2372608 6352.4745216 12704.9490432 25409.8980864 50819.7961728 101639.5923456 203279.1846912 406558.3693824 813116.7387648 1626233.4775296 3252466.9550592 6504933.9101184 13009867.8202368 26019735.6404736 52039471.2809472 104078942.5618944 208157885.1237888 416315770.2475776 832631540.4951552 1665263080.9903104 3330526161.9806208 6661052323.9612416 13322104647.9224832 26644209295.8449664 53288418591.6899328 106576837183.3798656 213153674366.7597312 426307348733.5194624 852614697467.0389248 1705229394934.0778496 3410458789868.1556992 6820917579736.3113984 13641835159472.6227968 27283670318945.2455936 54567340637890.4911872 109134681275780.9823744 218269362551561.9647488 436538725103123.9294976 873077450206247.8589952 1746154900412495.7179904 3492309800824991.4359808 6984619601649982.8719616 13969239203299965.7439232 27938478406599931.4878464 55876956813199862.9756928 111753913626399725.9513856 223507827252799451.9027712 447015654505598903.8055424 894031309011197807.6110848 1788062618022395615.2221696 3576125236044791230.4443392 7152250472089582460.8886784 14304500944179164921.7773568 28609001888358329843.5547136 57218003776716659687.1094272 114436007553433319374.2188544 228872015106866638748.4377088 457744030213733277496.8754176 915488060427466554993.7508352 1830976120854933109987.5016704 3661952241709866219975.0033408 7323904483419732439950.0066816 14647808966839464879900.0133632 29295617933678929759800.0267264 58591235867357859519600.0534528 117182471734715719039200.1069056 234364943469431438078400.2138112 468729886938862876156800.4276224 937459773877725752313600.8552448 1874919547755451504627201.7104896 3749839095510903009254403.4209792 7499678191021806018508806.8419584 Out[170]= 0.000094674505615234375 Out[171]= 0.0000946745 Out[172]= 0.000189349 Out[173]= 0.000378698 Out[174]= 0.000757396 Out[175]= 0.00151479 Out[176]= 0.00302958 Out[177]= 0.00605917 Out[178]= 0.0121183 Out[179]= 0.0242367 Out[180]= 0.0484733 Out[181]= 0.0969467 Out[182]= 0.193893 Out[183]= 0.387787 Out[184]= 0.775574 Out[185]= 1.55115 Out[186]= 3.10229 Out[187]= 6.20359 Out[188]= 12.4072 Out[189]= 24.8144 Out[190]= 49.6287 Out[191]= 99.2574 Out[192]= 198.515 Out[193]= 397.03 Out[194]= 794.059 Out[195]= 1588.12 Out[196]= 3176.24 Out[197]= 6352.47 Out[198]= 12704.9 Out[199]= 25409.9 Out[200]= 50819.8 Out[201]= 101640. Out[202]= 203279. Out[203]= 406558. Out[204]= 813117. Out[205]= 1.62623*10^6 Out[206]= 3.25247*10^6 Out[207]= 6.50493*10^6 Out[208]= 1.30099*10^7 Out[209]= 2.60197*10^7 Out[210]= 5.20395*10^7 Out[211]= 1.04079*10^8 Out[212]= 2.08158*10^8 Out[213]= 4.16316*10^8 Out[214]= 8.32632*10^8 Out[215]= 1.66526*10^9 Out[216]= 3.33053*10^9 Out[217]= 6.66105*10^9 Out[218]= 1.3322104647922483*10^10 Out[219]= 2.6644209295844966*10^10 Out[220]= 5.3288418591689933*10^10 Out[221]= 1.06576837183379866*10^11 Out[222]= 2.13153674366759731*10^11 Out[223]= 4.26307348733519462*10^11 Out[224]= 8.52614697467038925*10^11 Out[225]= 1.705229394934077850*10^12 Out[226]= 3.410458789868155699*10^12 Out[227]= 6.820917579736311398*10^12 Out[228]= 1.36418351594726227968*10^13 Out[229]= 2.7283670318945245594*10^13 Out[230]= 5.4567340637890491187*10^13 Out[231]= 1.091346812757809823744*10^14 Out[232]= 2.18269362551561964749*10^14 Out[233]= 4.36538725103123929498*10^14 Out[234]= 8.73077450206247858995*10^14 Out[235]= 1.746154900412495717990*10^15 Out[236]= 3.492309800824991435981*10^15 Out[237]= 6.984619601649982871962*10^15 Out[238]= 1.3969239203299965743923*10^16 Out[239]= 2.7938478406599931487846*10^16 Out[240]= 5.5876956813199862975693*10^16 Out[241]= 1.11753913626399725951386*10^17 Out[242]= 2.23507827252799451902771*10^17 Out[243]= 4.47015654505598903805542*10^17 Out[244]= 8.94031309011197807611085*10^17 Out[245]= 1.788062618022395615222170*10^18 Out[246]= 3.576125236044791230444339*10^18 Out[247]= 7.152250472089582460888678*10^18 Out[248]= 1.4304500944179164921777357*10^19 Out[249]= 2.8609001888358329843554714*10^19 Out[250]= 5.7218003776716659687109427*10^19 Out[251]= 1.14436007553433319374218854*10^20 Out[252]= 2.28872015106866638748437709*10^20 Out[253]= 4.57744030213733277496875418*10^20 Out[254]= 9.15488060427466554993750835*10^20 Out[255]= 1.830976120854933109987501670*10^21 Out[256]= 3.661952241709866219975003341*10^21 Out[257]= 7.323904483419732439950006682*10^21 Out[258]= 1.4647808966839464879900013363*10^22 Out[259]= 2.9295617933678929759800026726*10^22 Out[260]= 5.85912358673578595196000534528*10^22 Out[261]= 1.171824717347157190392001069056*10^23 Out[262]= 2.34364943469431438078400213811*10^23 Out[263]= 4.687298869388628761568004276224*10^23 Out[264]= 9.374597738777257523136008552448*10^23 Out[265]= 1.874919547755451504627201710490*10^24 Out[266]= 3.749839095510903009254403420979*10^24 Out[267]= 7.499678191021806018508806841958*10^24 In[72]:= 0.0000473372528076171875/2 0.000094674505615234375 0.00018934901123046875 0.0003786980224609375 0.000757396044921875 0.00151479208984375 0.0030295841796875 0.006059168359375 0.01211833671875 0.0242366734375 0.048473346875 0.09694669375 0.1938933875 0.387786775 0.77557355 1.5511471 3.1022942 6.2035884 12.4071768 24.8143536 49.6287072 99.2574144 198.5148288 397.0296576 794.0593152 1588.1186304 3176.2372608 6352.4745216 12704.9490432 25409.8980864 50819.7961728 101639.5923456 203279.1846912 406558.3693824 813116.7387648 1626233.4775296 3252466.9550592 6504933.9101184 13009867.8202368 26019735.6404736 52039471.2809472 104078942.5618944 208157885.1237888 416315770.2475776 832631540.4951552 1665263080.9903104 3330526161.9806208 6661052323.9612416 13322104647.9224832 26644209295.8449664 53288418591.6899328 106576837183.3798656 213153674366.7597312 426307348733.5194624 852614697467.0389248 1705229394934.0778496 3410458789868.1556992 6820917579736.3113984 13641835159472.6227968 27283670318945.2455936 54567340637890.4911872 109134681275780.9823744 218269362551561.9647488 436538725103123.9294976 873077450206247.8589952 1746154900412495.7179904 3492309800824991.4359808 6984619601649982.8719616 13969239203299965.7439232 27938478406599931.4878464 55876956813199862.9756928 111753913626399725.9513856 223507827252799451.9027712 447015654505598903.8055424 894031309011197807.6110848 1788062618022395615.2221696 3576125236044791230.4443392 7152250472089582460.8886784 14304500944179164921.7773568 28609001888358329843.5547136 57218003776716659687.1094272 114436007553433319374.2188544 228872015106866638748.4377088 457744030213733277496.8754176 915488060427466554993.7508352 1830976120854933109987.5016704 3661952241709866219975.0033408 7323904483419732439950.0066816 14647808966839464879900.0133632 29295617933678929759800.0267264 58591235867357859519600.0534528 117182471734715719039200.1069056 234364943469431438078400.2138112 468729886938862876156800.4276224 937459773877725752313600.8552448 1874919547755451504627201.7104896 3749839095510903009254403.4209792 7499678191021806018508806.8419584 Out[72]= 0.0000236686264038085938 Out[73]= 0.0000946745 Out[74]= 0.000189349 Out[75]= 0.000378698 Out[76]= 0.000757396 Out[77]= 0.00151479 Out[78]= 0.00302958 Out[79]= 0.00605917 Out[80]= 0.0121183 Out[81]= 0.0242367 Out[82]= 0.0484733 Out[83]= 0.0969467 Out[84]= 0.193893 Out[85]= 0.387787 Out[86]= 0.775574 Out[87]= 1.55115 Out[88]= 3.10229 Out[89]= 6.20359 Out[90]= 12.4072 Out[91]= 24.8144 Out[92]= 49.6287 Out[93]= 99.2574 Out[94]= 198.515 Out[95]= 397.03 Out[96]= 794.059 Out[97]= 1588.12 Out[98]= 3176.24 Out[99]= 6352.47 Out[100]= 12704.9 Out[101]= 25409.9 Out[102]= 50819.8 Out[103]= 101640. Out[104]= 203279. Out[105]= 406558. Out[106]= 813117. Out[107]= 1.62623*10^6 Out[108]= 3.25247*10^6 Out[109]= 6.50493*10^6 Out[110]= 1.30099*10^7 Out[111]= 2.60197*10^7 Out[112]= 5.20395*10^7 Out[113]= 1.04079*10^8 Out[114]= 2.08158*10^8 Out[115]= 4.16316*10^8 Out[116]= 8.32632*10^8 Out[117]= 1.66526*10^9 Out[118]= 3.33053*10^9 Out[119]= 6.66105*10^9 Out[120]= 1.3322104647922483*10^10 Out[121]= 2.6644209295844966*10^10 Out[122]= 5.3288418591689933*10^10 Out[123]= 1.06576837183379866*10^11 Out[124]= 2.13153674366759731*10^11 Out[125]= 4.26307348733519462*10^11 Out[126]= 8.52614697467038925*10^11 Out[127]= 1.705229394934077850*10^12 Out[128]= 3.410458789868155699*10^12 Out[129]= 6.820917579736311398*10^12 Out[130]= 1.36418351594726227968*10^13 Out[131]= 2.7283670318945245594*10^13 Out[132]= 5.4567340637890491187*10^13 Out[133]= 1.091346812757809823744*10^14 Out[134]= 2.18269362551561964749*10^14 Out[135]= 4.36538725103123929498*10^14 Out[136]= 8.73077450206247858995*10^14 Out[137]= 1.746154900412495717990*10^15 Out[138]= 3.492309800824991435981*10^15 Out[139]= 6.984619601649982871962*10^15 Out[140]= 1.3969239203299965743923*10^16 Out[141]= 2.7938478406599931487846*10^16 Out[142]= 5.5876956813199862975693*10^16 Out[143]= 1.11753913626399725951386*10^17 Out[144]= 2.23507827252799451902771*10^17 Out[145]= 4.47015654505598903805542*10^17 Out[146]= 8.94031309011197807611085*10^17 Out[147]= 1.788062618022395615222170*10^18 Out[148]= 3.576125236044791230444339*10^18 Out[149]= 7.152250472089582460888678*10^18 Out[150]= 1.4304500944179164921777357*10^19 Out[151]= 2.8609001888358329843554714*10^19 Out[152]= 5.7218003776716659687109427*10^19 Out[153]= 1.14436007553433319374218854*10^20 Out[154]= 2.28872015106866638748437709*10^20 Out[155]= 4.57744030213733277496875418*10^20 Out[156]= 9.15488060427466554993750835*10^20 Out[157]= 1.830976120854933109987501670*10^21 Out[158]= 3.661952241709866219975003341*10^21 Out[159]= 7.323904483419732439950006682*10^21 Out[160]= 1.4647808966839464879900013363*10^22 Out[161]= 2.9295617933678929759800026726*10^22 Out[162]= 5.85912358673578595196000534528*10^22 Out[163]= 1.171824717347157190392001069056*10^23 Out[164]= 2.34364943469431438078400213811*10^23 Out[165]= 4.687298869388628761568004276224*10^23 Out[166]= 9.374597738777257523136008552448*10^23 Out[167]= 1.874919547755451504627201710490*10^24 Out[168]= 3.749839095510903009254403420979*10^24 Out[169]= 7.499678191021806018508806841958*10^24 JEREMIAH 19:13THEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEAS TTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHE B EASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEAST T HEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBE A STTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTH E BEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTVTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEA S TTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHE B EASTTHEBEASTTHEBEASTTHEBEAST revelations 13:18 THEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEB EASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEAST T HEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBE A STTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTH E BEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEAS T THEBEASTTHEBEASTTHEBEASTTHEBEASTVTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHE B EASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEASTTHEBEAST T HEBEASTTHEBEASTTHEBEAST No replies to this message. Never can one reply to this message [Privacy Policy] [Terms of Use] © Drexel University 1994-2009. All Rights Reserved. The Math Forum is a research and educational enterprise of the Goodwin College of Professional Studies. > I'm trying to get this straight. TheCantorset is uncountable and has > no interior; > Yes. > it's a collection of points. > Yes, but that's a strange comment. > So the complement of the >Cantorset is a collection of open sets. > I think you meant to say the complement of theCantorset is open. >He may have meant either and still been correct. > Er, no. The complement of theCantorset is an open set. > It's not a collection of open sets. > My Bad. > Is the total set of points in > the open sets uncountable? It must be, it seems to me, since any open > set of real numbers contains uncountably many points. And the > complement of theCantorset on [0, 1] has to have measure 1, no? So > both theCantorset and its complement are uncountable? It would seem > a paradox if the complement of theCantorset on [0, 1] were > countable. > Another question: I've seen it claimed that theCantorset can be > modified so as to have measure 1 (on [0, 1], for example). Can someone > provide an example of a modifiedCantorset with measure 1 on [0, 1]? > There are Cantorsets that have measure 1, but they are not a subset > of [0,1].Cantorsets are compact, and a compact subset of [0, 1] of > measure 1 must be [0, 1] itself. > David C. Ullrich > Understanding Godel isn't about following his formal proof. > That would make a mockery of everything Godel was up to. > (John Jones, My talk about Godel to the post-grads. > in sci.logic.) > -- > Virgil- Hide quoted text - > - Show quoted text -- Hide quoted text - > - Show quoted text - Exactly. > Hello there, > Anyone knows how to prove this? > Let k = 4m+3. Given the eqn, > v(pi^k) = 2^(k-1) Sum[f1, {n,1,inf}] - (2^(k-1)+1) > Sum[f2, {n,1,inf}] > + Sum[f3, {n,1,inf}] > where {1/f1, 1/f2, 1/f3} = {n^k(q^n-1), > n^k(q^(2n)-1), n^k(q^(4n)-1)} > and q = e^(pi), (with inf = infinity) > then v is a rational number. > Ex. > (1/180) pi^3 = 2^2 Sum[f1, {n,1,inf}] - (2^2+1) > Sum[f2, {n,1,inf}] + > Sum[f3, {n,1,inf}] > (13/14175) pi^7 = 2^6 Sum[f1, {n,1,inf}] - (2^6+1) > Sum[f2, {n,1,inf}] > + Sum[f3, {n,1,inf}] > and so on. See more at Identities inspired by > Ramanujan's Notebooks > (Part 2): >http://www.lacim.uqam.ca/~plouffe/inspired2.pdf > Actually, Plouffe's identities have the general > form, > pi^k = a*Sum[f1, {n,1,inf}] + b*Sum[f2, {n,1,inf}] > + c*Sum[f3, {n, > 1,inf}] > for some rational numbers {a,b,c} for all ODD k, > but, after some > tweaking, I noticed it has that simpler form when k > = 4m+3. > (He found these identities experimentally using > Maple and PSLQ.) > - Titus > Indeed for m=0, that is for k = 4*m+3 = 3 > using Maple in Integer Relation Algorithms option > of the inverse > symbolic calculator > gives exactly what Titus stated > 180*sum(1/n^3*(4/(exp(1)^(Pi*n)-1)-5/(exp(1)^(2*Pi*n)- > 1)+1/(exp(1)^ > (4*Pi*n)-1)) > ,n = 1 .. infinity) > Value to be looked up: > K = 31.006276680299820175476315067102 gave the > following > results: > K satisfies the following Z-linear combination : > - K + Pi**3 > Hello there, Anyone knows how to prove this? Let k = 4m+3. Given the eqn, v(pi^k) = 2^(k-1) Sum[f1, {n,1,inf}] - (2^(k-1)+1) Sum[f2, {n,1,inf}] > + Sum[f3, {n,1,inf}] where {1/f1, 1/f2, 1/f3} = {n^k(q^n-1), n^k(q^(2n)-1), n^k(q^(4n)-1)} > and q = e^(pi), (with inf = infinity) then v is a rational number. Ex. (1/180) pi^3 = 2^2 Sum[f1, {n,1,inf}] - (2^2+1) Sum[f2, {n,1,inf}] + > Sum[f3, {n,1,inf}] (13/14175) pi^7 = 2^6 Sum[f1, {n,1,inf}] - (2^6+1) Sum[f2, {n,1,inf}] > + Sum[f3, {n,1,inf}] and so on. See more at Identities inspired by Ramanujan's Notebooks > (Part 2): http://www.lacim.uqam.ca/~plouffe/inspired2.pdf Actually, Plouffe's identities have the general form, pi^k = a*Sum[f1, {n,1,inf}] + b*Sum[f2, {n,1,inf}] + c*Sum[f3, {n, > 1,inf}] for some rational numbers {a,b,c} for all ODD k, but, after some > tweaking, I noticed it has that simpler form when k = 4m+3. (He found these identities experimentally using Maple and PSLQ.) - Titus x=pi > e^(ipi)+1=0 Martin Musatov - Show quoted text - Hello there, Anyone knows how to prove this? Let k = 4m+3. Given the eqn, v(pi^k) = 2^(k-1) Sum[f1, {n,1,inf}] - (2^(k-1)+1) Sum[f2, {n,1,inf}] > + Sum[f3, {n,1,inf}] where {1/f1, 1/f2, 1/f3} = {n^k(q^n-1), n^k(q^(2n)-1), n^k(q^(4n)-1)} > and q = e^(pi), (with inf = infinity) then v is a rational number. Ex. (1/180) pi^3 = 2^2 Sum[f1, {n,1,inf}] - (2^2+1) Sum[f2, {n,1,inf}] + > Sum[f3, {n,1,inf}] (13/14175) pi^7 = 2^6 Sum[f1, {n,1,inf}] - (2^6+1) Sum[f2, {n,1,inf}] > + Sum[f3, {n,1,inf}] and so on. See more at Identities inspired by Ramanujan's Notebooks > (Part 2): http://www.lacim.uqam.ca/~plouffe/inspired2.pdf Actually, Plouffe's identities have the general form, pi^k = a*Sum[f1, {n,1,inf}] + b*Sum[f2, {n,1,inf}] + c*Sum[f3, {n, > 1,inf}] for some rational numbers {a,b,c} for all ODD k, but, after some > tweaking, I noticed it has that simpler form when k = 4m+3. (He found these identities experimentally using Maple and PSLQ.) - Titus x=pi > e^(ipi)+1=0 Martin Musatov Simon Plouffe > before...??? act... Martin Musatov to Simon > the > below text: Regarding what you said before, Mr. Plouffe, I must apologize and > assert my responsibility to clear up any confusion you may have. You see it is not my intent with this writing to prove to you P=NP, I > am more interested in geometry and the fact I cannot understand why a > flat line to a circle to a ball cannot be drawn in logical lines on a > flat piece of paper. I have a drawing to prove pi is not irrational. Do you agree with > this statement after viewing the picture at the end of this message > and reading the text before it? > The restriction of n-space is reflected in the > virtual impossibility of > finding roots to n-degree polynomials. If each > coefficient is a component > to a vector of n dimensions, and each power of x is > a > component to a vector > of n dimensions, the two vectors are orthogonal > since > their dot product is > zero. > The liberty of n-space is that it opens up > infinite > possibilites. A > theorist can prove his brainchild in 4 dimensions > if > he can't do it in 3. > Just imagine. Associated with every point in > (x1,x2,x3) is a point on the > x4 axis. That is, associated with every (x,y,z,t) > is > a point x4 hidden in > the system of points (x1,x2,x3,x4,t), not > considering > that furthermore t > (time) may be altered. The theorist becomes an > artist producing > explanations at whim in equations that all come > together perfectly. > Anything whatsoever can be proven using math of > higher dimensions in E^n > where n>3. > This also delimits n-space math. On the one hand > it > is uselessly > constrictive and on the other uselessly diluted. > Consider a number of E^3 coordinate systems in > synchrounous time with each > other separated by large distances. I think this > is > as best a model as can > be hoped for. One system is unbridled and actions > in > it mapped to form > reactions in the other systems. That way, > associated > with one system are > events in the other systems by cause and effect. > When a tree is felled in > one system, a dam is caused to be broken in the > other > systems. This > meaningless association is diluted as well, but > Master and Slave can swap > roles like a see-saw. This is in line with > momentum > that is channeled for > technological uses. No, there is no relationship. > But can you find one? > The man's pulse was connected mechanically to > regulate the speed of a > flywheel. After this went on for awhile, the > technician altered the > flywheel and observed the man's pulse. He found > out > nothing. There are > some processes, however, that do swap Master/Slave > relationships, like a > child's see-saw on a playground. What are they? > These are good candidates > for multiple systems of E^3, particularly in > distance > relationships to > unknown cause and effect forces. The media may be > EMR. The relatively > equipotential space-time around the planet may > have > anomalies, like > gradients or funnels. The ancient structures > built > in phylogeny are a > vortex. The path of least resistance is to delve > and > fall into them, > degenerating and destroying the foundations of > everything that went in to > bringing the world up to the present. > Suppose one system was in the nucleus of a star > and > the other system in the > vortex of a Black Hole. Is there any cause and > effect between the two? > No. > NOW here is a another great flaw in your current > mathematics that defies logic PROVE THIS!. etc. > things are what they are, 1 is a unit, what do you > need proof for it . I was just recently in my office > playing a mathematical game with a 6 year old child , > giving him difficult circles etc. The Child thinks a > little and realizes that I was giving him some thing > that he did not understand or know . So he devises > his own response, he drwas 67 lines one on top of > another in a mess, and asks me , how many lines are > these? (you smart alec answer me), and Prove this > he says. Now if this child is kept from being brain > washed from your Mathematics, he will develop into a > great mathematicain, because he has the logic to know > when he is asked why is the sky blue, prove ut , he > says Cause(you dummy some things are as they are). > Your mathematics is flawed and many of you are brain > washed by your mathematical dictums, your professors > and your journals are all Hobsonian --- come join > us all you slaves of current mathematics, learn to > think free of Einstein and your gods , join the > inverse 19 movement to change mathematical theory. > We will give you further expose in 15 days when the > monument to Lost matghematics is unvelied here in > Wisconsin Proof pi is not irrational: http://www.flickr.com/photos/24869933@N07/3619373444/sizes/l/ Musatov (Inverse 19 Mathematics Co-Creator): not the only but one > without it would not exist. Simon, the above text is my work. > Efforts by mathematicians to determine the value of pi since have led > to the conclusion that it ... Unlike rational numbers, pi cannot be > represented by a common fraction, .... But what about an uncommon fraction? Will you look at my drawing and in particular how the twelve points > are the outside points of the sphere are all rationally reached by > decimal numbers? 6.2035884*2 and doubled an infinite number of times should yield a > constant decimal binary expansion (what I mean by this is the numbers > to the right of the decimal point should always end in 4, then 8, > then 6, then 2, then 4, then 8, then 6, then 2...repeating for > infinity). So in binary, by starting with this number we predict for n>2 the > infinite binary tree will end with decimal progression: 2, 4, 8, 6 For n<2 the infinite binary tree will begin and end (depends on if you > are increasing or decreasing the value) 1, 5, 5, 5 (repeating 5 > infinitely) as we decrease to infinitely small. Musatov Inverse Counting Chain of Infinite Proportions > (sample) > 0.0000473372528076171875 > 0.000094674505615234375 > 0.00018934901123046875 > 0.0003786980224609375 > 0.000757396044921875 > 0.00151479208984375 > 0.0030295841796875 > 0.006059168359375 > 0.01211833671875 > 0.0242366734375 > 0.048473346875 > 0.09694669375 > 0.1938933875 > 0.387786775 > 0.77557355 > 1.5511471 > 3.1022942 > I will explain to you what I am after. I am after a new theory, a new > way of programming, a new way of measuring, a new way of > communicating. I am talking about an end to the old tired way of 'P', > if 'P' then 'Q' innefficiency in programming and language (and > practical matters too). > When I was a teenager I noticed that lots of Propositional Calculus > axioms are derivable from the fact that AND and OR (as well as ADD and > MUL) are what I call Set Functions. You can apply them to any > finite set - so the order cannot matter. A&B = B&A , A&(B&C)=(A&B)&C > etc. > As a young adult I utilize that principle in my automated theorem- > proving software by representing A&B&C not as 2 applications of & but > as a set {A,B,C} with the & Set Function applied to it. > The result is that my programs generate theorems orders of magnitude > faster. > Yes, standard Logic is poorly designed, with shameful duplication of > principles between different branches of Mathematics e.g. Logic and > Set Theory both have DeMorgan rules etc. Researchers keep looking for > ways to prove the axioms, but overlook the fact that they have the > wrong model for that. It should be as Set Functions. Just like they > use Predicate Calculus for MetaMathematics (e.g.; ZFC axioms) when a > higher level of abstraction is much more efficient and yields many > more theorems in practice (finite world). > Always work at the highest level of abstraction you can. > C-B > I propose the following: > No need. I have implemented it and the Set Function new idea solves > the problem nicely. > 'X' if 'X' > The idea is we create a thread. The first programming variable means > nothing. It is an instruction to the system script. It is an open and > said> order of logic. > 'X' alone simply means Open or I am talking or Listen up, it can > be followed by anything, or an infinite number of things. > And the idea of 'X' is arbitrary, as 'X' could be replaced by any > other logical symbol or character in any language and the idea still > stands. The operation still works. > Let me explain: > 'X' when I say 'X'. > 'X' is the master open 'command' key or 'dialogue'. It sets up a > system open script. > We then can open any operation simply by repeating it. The system is > not loyal to any language or frame. (or insert proprietary here) > If I want to open up a program Y, I simply say, X, Y I want Y. > The system knows that if a variable is repeated it means to execute it > under the terms programmed. (or by pressing a key) Let me explain > further (further): > A formal illustration of how it would look in a system would be: > X, Y, Y, X > The first 'X' means open the root command, the first 'Y' means, I want > 'Y', the second 'Y' means it is just 'Y' I seek and the 'X' closes to > command. > The platform solution requires complete symmetry in programming and > logic threads. > The programming implementation allows the setup of a system where > repetition of a variable in symmetry order allows command. > In football the command hut means pitch me the ball back. The > Quarterback will communicate to his teammates, On the 2nd hut throw > the ball. The first hut is to get set. Still 'hut' is the command to > throw the ball. > The real world variable and how it relates to the equivalency in > programming is explained as follows: > Instead of saying (1)Listen up....(2)I want to open my word > processor.... > X 1 1 X > where (1)=the first command 1 > and (2)=11 > ***********Note in binary*: the machine literally must count the two > ones and add them to get two.************) > User inputs:(1)I want my word processor (2)I want my word > processor > and this means to the machine Open up the word processor... only > when you read (1) I want my word processor (2) I want my word > processor > (forgive my terminology, I am simply being plain) but this inside/out > programming > illustrates perfectly my point. > and forgive my for being so bold but this logic clear, simple, and > plain, simply resolves P=NP > and overcomes the halting problem irrefutably > the solution to P=NP or P Versus NP, or N=NP or -1=1 or however you > want to frame > the problem is simply this statement: Do not do anything until I say > it twice. > Advanced Symmetry application: the solution to the machine is the > overlooked symmetry. A complex command may look like: 'XYZZYX' which > would mean, I want the YZ complex variable please > Stephen Arthur Cook was right, the whole thing was simply due to lack > of ingenuity on behalf of our programmer. > (C)2009. Martin Musatov. All Rights Reserved in perpetuity*. > http://www.MeAmI.orgSearch for the People > *Applies to derviations of this work. Including software, programming, > and other profit generating means extracted from this work. Work may > be patent pending.- Hide quoted text - > - Show quoted text -- Hide quoted text - > - Show quoted text - So it is agreed we have indeed overcome the P Versus NP problem and/or > halting problem and/or proven 6P=NP, correct? (Yes if any or all of the > above apply, No, if none) = this will do to begin with ;)- Hide quoted text - - Show quoted text - >Yes. > 12.4071768 > 24.8143536 > 49.6287072 > 99.2574144 > 198.5148288 > 397.0296576 > 794.0593152 > 1588.1186304 > 3176.2372608 > 6352.4745216 > 12704.9490432 > 25409.8980864 > 50819.7961728 > 101639.5923456 > 203279.1846912 > 406558.3693824 > 813116.7387648 > 1626233.4775296 > 3252466.9550592 > 6504933.9101184 > 13009867.8202368 > 26019735.6404736 > 52039471.2809472 > 104078942.5618944 > 208157885.1237888 > 416315770.2475776 > 832631540.4951552 > 1665263080.9903104 > 3330526161.9806208 > 6661052323.9612416 > 13322104647.9224832 > 26644209295.8449664 > 53288418591.6899328 > 106576837183.3798656 > 213153674366.7597312 > 426307348733.5194624 > 852614697467.0389248 > 1705229394934.0778496 > 3410458789868.1556992 > 6820917579736.3113984 > 13641835159472.6227968 > 27283670318945.2455936 > 54567340637890.4911872 > 109134681275780.9823744 > 218269362551561.9647488 > 436538725103123.9294976 > 873077450206247.8589952 > 1746154900412495.7179904 > 3492309800824991.4359808 > 6984619601649982.8719616 > 13969239203299965.7439232 > 27938478406599931.4878464 > 55876956813199862.9756928 > 111753913626399725.9513856 > 223507827252799451.9027712 > 447015654505598903.8055424 > 894031309011197807.6110848 > 1788062618022395615.2221696 > 3576125236044791230.4443392 > 7152250472089582460.8886784 > 14304500944179164921.7773568 > 28609001888358329843.5547136 > 57218003776716659687.1094272 > 114436007553433319374.2188544 > 228872015106866638748.4377088 > 457744030213733277496.8754176 > 915488060427466554993.7508352 > 1830976120854933109987.5016704 > 3661952241709866219975.0033408 > 7323904483419732439950.0066816 > 14647808966839464879900.0133632 > 29295617933678929759800.0267264 > 58591235867357859519600.0534528 > 117182471734715719039200.1069056 > 234364943469431438078400.2138112 > 468729886938862876156800.4276224 > 937459773877725752313600.8552448 > 1874919547755451504627201.7104896 > 3749839095510903009254403.4209792 > 7499678191021806018508806.8419584 > (sample) If you would like to do the honors Mr. Plouffe, I endear you to > calculate the angles and lengths of the number of ways, it is quite > obviously possible to square a circle based on the geometry and > numerical progression. As is self-evident to anyone with even a modest understanding of > mathematics and geometry after visual inspection of the drawn proof > (http://www.flickr.com/photos/24869933@N07/3619373444/sizes/l/) it is > quite easily accomplished to connect and triangulate the outer points > of the rectangles to the outer points of the square and circle and > even by dimension sphere. Yes, the drawing clearly by definition of inverse symmetry (take the > drawing cut it in half at the equator and turn the paper over and > rotate it and compare the top and bottom mirrored symmetry and you > will see it is an exact inverse of symmetry itself) the drawing > explains how to properly and mathematically soundly represent a fourth > dimension on a two dimensional plane. In plane language if you take the picture and separate the top half of > the sphere by the bottom half of the sphere they are identical but > only if the sphere existed on a complete three dimensional plane > through expansion. Since the paper is flat, this is an impossibility, nonetheless it is > completely expressed and retractable and countable infinitely. By the definition of our standard we have established any k-digit > approximation to pi is rational. Which could have many practical scientific applications. For instance, > search: ON THE THEOREM OF IVA$EV-MUSATOV IIIx e F(l) + F(2) + ... + F(p). This > remark is the basis of our proof of Theorem .... -y(M + 1), x' + y(M + > 1)] = 0 it is easy to check that Conditions (i), ...... Now set i = > pi x. *jj, v. Since p, x is positive,. (i) fi is positive, ... > plms.oxfordjournals.org/cgi/reprint/s3-53/1/143.pdf REPRESENTATION OF FUNCTIONSBY SERIES AND CLASSES .83'(1)with measure ÄEÄ 2.83ë -- .83Ì and such that the trigonometric series for g(x) converges > uniformly on [0, 2.83ë]. .... trigonometric series are due to > Ivashev-Musatov [40] who has shown that there are null series (in the > sense of ..... function .83'(.83ï) e C(0, 1) with .83'(0) = .83'(1) = 0 such that Full text of Bulletin of the British Museum (Natural History)X 20. > FIG. 5. Well preserved large fragment showing the different aspect and > ...... 0-1-0-5 P i n height) so giving the shell surface the > appearance of ...... 94-95, i pi. Maseru. BRINK, A. S. 1963. Two > cynodonts from the Ntawere ...... MUSATOV, D. L, NEMIROVSKAYA, V. N., > SHIROKOVA, E. V. & ZHURAVLEVA, I. T. 1961. ...www.archive.org/stream/bulletinofbritis17geollond/bulletinofbritis17g... Trigonometric series with rapidly decreasing coefficients a i0, 5 > ...Nt,+i - Pi = n,. N tl+j. -pi =N tl+j. -i +pi +1. ..... number 8 6 > (0,1) for the product (32) such that for every point x 6 E there are Plouffe's formula for pi - sci.math | Google Groups3 posts - 2 authors > - Last post: 41 minutes ago > x=pi e^(ipi)+1=0. Martin Musatov. Reply Forward. Martin Musatov to > bf6d1ee13d87270b?lnk=raot > - 41 minutes ago [PDF] ???? ?-?File Format: PDF/Adobe Acrobat > Gli storici ci dicono che, nel X secolo, il principe Vladi- ...... i > pi`u anziani e carismatici si evidenziavano i nomi di ...... tin > Somov, Viktor Borisov-Musatov, Aleksandr Benois e Lev Bakst ... 5/0; > Judovin 5/0; Korovin 2/1; Tukacov 1/0; Goli- cyn 1/1; Zacharov 1/1; > Jurkunas 5/0; Krasauskas 2/0;. Bulaka 3/0. ...www.esamizdat.it/eSamizdat 2005 (III) 2-3.pdf I am testing these identities: (1/2) * ( - k ) * ( k - 1/2 ) * ( - 3 ) / k! This equation produces M = k! ( - 3 ) * ( k - 1/2) * ( - k ) * (1/2) Martin Musatov- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text - === Subject: Re: Your Radian and 2Pi value > There is a better a clearer method of saying the value of the > circumference that is time dimensional i.e PiR(2) or PiR value > multiplied by 2 . 2PiRcan be confusing mathematically but it is > acceptable now that it is acceptable. Pi value is also the radial arc, > and the linear progression of inverse 19 and definition more correct of > a mathematicval definition than Circumference over the radius, because > it is time dimensional, and predictive till infinity >I truly apologize for being a little sarcastic because most > mathematicians are decent, but I have found your mathematics to be so > hierachial, as goes these people, as goes the rest of mathematics and > physics, generally I have found these places to be arrogant , the > little contact I have had with them. There might come a day when I may > have a chance to meet these fellows on the blackboard with their > complexities, or may be not, that is not my goal. > Real authority is humble and then civil and respectful.. >Inverse 19 Yes, I need to work on this. <*> Wow, this debate between kooks might be entertaining if not for the massive quoting in the posts! 8( === Subject: Re: Your Radian and 2Pi value <22102145.9437.1244839862150.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > There is a better a clearer method of saying the value of the > circumference that is time dimensional i.e PiR(2) or PiR value > multiplied by 2 . 2PiRcan be confusing mathematically but it is > acceptable now that it is acceptable. Pi value is also the radial arc, > and the linear progression of inverse 19 and definition more correct of > a mathematicval definition than Circumference over the radius, because > it is time dimensional, and predictive till infinity >I truly apologize for being a little sarcastic because most > mathematicians are decent, but I have found your mathematics to be so > hierachial, as goes these people, as goes the rest of mathematics and > physics, generally I have found these places to be arrogant , the > little contact I have had with them. There might come a day when I may > have a chance to meet these fellows on the blackboard with their > complexities, or may be not, that is not my goal. > Real authority is humble and then civil and respectful.. >Inverse 19 > Yes, I need to work on this. <* > Wow, this debate between kooks might be entertaining if not for the > massive quoting in the posts! 8(- Hide quoted text - - Show quoted text - Results 1 - 10 for !8(. (0.28 seconds) Custom Search Download Internet Explorer 8 optimized for GoogleUpgrade now to Internet Explorer 8 optimized for Google. We're sorry, but Internet Explorer 8 optimized for Google is only available for Windows XP and ... 8 (number) - Wikipedia, the free encyclopedia8 (eight) is the natural number following 7 and preceding 9. The SI prefix for 10008 is yotta (Y), and for its reciprocal yocto (y). ... en.wikipedia.org/wiki/8 (number) Internet Explorer 8 doesw not allow attached files to Gmail. - Web ... 2 posts - 1 author - Last post: 3 days ago Internet Explorer 8 doesw not allow attached files to Gmail. ... FYI - Since downloading and installing Internet Explorer 8 (using Microsoft ... Webmaster Tools: Improve your site's visibility in Google SearchApr 15, 2006 ... We require your Sitemap file to be UTF-8 encoded (you can generally do ... As with Sitemaps, your Sitemap index file must be UTF-8 encoded. ... 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(This version CORRECTS the number of victims killed to 8, not 10, based on updated official information.) Copyright © 2009 The Associated ... Web Search - Web Search HelpJun 10, 2009 ... 3 4 5 6 7 8 9 10 11 12 a Scam? 10 replies ... === Subject: Re: Your Radian and 2Pi value <22102145.9437.1244839862150.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > There is a better a clearer method of saying the value of the > circumference that is time dimensional i.e PiR(2) or PiR value > multiplied by 2 . 2PiRcan be confusing mathematically but it is > acceptable now that it is acceptable. Pi value is also the radial arc, > and the linear progression of inverse 19 and definition more correct of > a mathematicval definition than Circumference over the radius, because > it is time dimensional, and predictive till infinity >I truly apologize for being a little sarcastic because most > mathematicians are decent, but I have found your mathematics to be so > hierachial, as goes these people, as goes the rest of mathematics and > physics, generally I have found these places to be arrogant , the > little contact I have had with them. There might come a day when I may > have a chance to meet these fellows on the blackboard with their > complexities, or may be not, that is not my goal. > Real authority is humble and then civil and respectful.. >Inverse 19 > Yes, I need to work on this. <* > Wow, this debate between kooks might be entertaining if not for the > massive quoting in the posts! 8(- Hide quoted text - - Show quoted text - posts - 2 authors - Last post: May 29, 2008 forum.iz3d.com/viewtopic.php? t=267&sid=58e66705be9ff9decb49626baa7838d0 Google Finance: Genta Incorporated3 posts - Last post: yesterday To post a message you must first join this group. ... Are they just flooding it so they can do a massive RS and then > dilute again? I think I am waiting for a while ... Local: Thurs, Jun 11 2009 8:15 am. === Subject: Re: More Dillution? ... http://cxa.marketwatch.com/tsx/en/market/quote.aspx?symbol=nri ... Remote file access in php causes massive page load delay ...2 posts - 1 author Remote file access in php causes massive page load delay Coding. ... Posts: 8. Almost forgot: Here's one of my error ... Posts: 8. Quote: ... www.phpbuilder.com/board/showthread.php?t=10365901 Massive Windows Me Problem - Suggest A Fix PC Support Forums8 posts - 5 authors - Last post: May 14 work ... www.suggestafix.com/index.php?showtopic=32985&view=getnewpost View: Jen Harmon Has Massive Cleavage - Page 8 - Two Plus Two ...Page 8 of 18, ' First .87 < .87 3 .87 4 .87 5 .87 6 .87 7, 8, 9 .87 10 .87 11 .87 12 .87 13 .87 > .87 Last é ... Quote: ... Posts: 2092. Re: View: Jen Harmon Has Massive massive cleavage ... forumserver.twoplustwo.com/29/news-views-gossip/view-jen-harmon-has- massive-cleavage-482230/index8.html DAZ 3D Forums :: View topic - Massive Bryce content sale bundle ...tengoku1. Joined: 19 May 2009. Posts: 8 ... Quote: So now do for a nice reply ... forum.daz3d.com/viewtopic.php? p=1838107&sid=4280938e853cf2d27c0e0ebd80763e80 Massive 8/9 Inch Clown Loach For Sale - Tropical Fish Forums20 posts - 17 authors - Last post: Mar 27 So for £100 you get 1 x massive loach ( gotta be 9 inch!) ... This post has been edited by drowning fish: Mar 17 2009, 09:30 PM ... post Mar 18 2009, 07:42 PM. Post #9. QUOTE (Nick16 @ Mar 18 2009, 07:44 PM) * ... www.fishforums.net/index.php?showtopic=278545 Sims 3 leaks early, still guaranteed massive sales | GameRiot: The ...None 7 replies to Godsmak (show). Permalink Quote Comment Post Reply .... Epic Drop 5/27/09: Challenging US-born Illegals, Prop 8 goes down, ... wowriot.gameriot.com/blogs/GameRiot-The-Blog/Sims-3-leaks-early-still- guaranteed-massive-sales Remote file access in php causes massive page load delay ...Remote file access in php causes massive page load delay Coding. ... Old 8. Quote: ... www.phpbuilder.com/board/showthread.php?p=10917034 Sims 3 leaks early, still guaranteed massive sales | GameRiot: The ... 7 replies to Godsmak (show). Permalink Quote Comment Post Reply .... Lusitania May 18, 2009 at 8:12 pm unhide comment ... conqueragon.gameriot.com/blogs/GameRiot-The-Blog/Sims-3-leaks-early- still-guaranteed-massive-sales === Subject: Re: FOL and AST <87r5xrlyh1.fsf@alatheia.truth.invalid> <4A31ED5F.3C60089@gmail.com> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > As for uncountable languages, I don't see why we should be much > interested in such highly theoretical objects. > As a possible answer, Shoenfield defines validity in a structure by > adding to the language a constant for each individual. > I do not know if it is possible to avoid this expansion of the > language in that definition. But if it was not, then the need for > uncountable languages would not be of any high theoricity, but a quite > basic need. > We don't have to use that method. We can define satisfaction and truth > in a structure (what Shoenfield calls 'validity', as opposed to the > more ordinary sense of 'validity' as 'satisfied or true in all > structures') with an uncountable universe without having to use a > method of adding constants. > That there are other uses for uncountable languages though is another > matter. > MoeBlee The more this discussion goes on, the more I think I am reading the > wrong book. I feel the need to see the big picture before diving into > the technical details.- Hide quoted text - - Show quoted text - Diagonal wanderings (incongruent by construction) - comp.theory ...May 29, 2009 ... where aristotle was concerned with ontological term calculi .... a given proof has a physical representation -- though ... Discussion subject changed to Pure Variable Assignment by Martin Michael Musatov ... 18dcda439d5b806f Accepted papers - CSR-2009Canonical Calculi: Invertibility, Axiom expansion and (Non)-determinism ... Characterizing the Existence of Optimal Proof Systems and Complete Sets for Promise Classes ... Daniil Musatov, Andrei Romashchenko and Alexander Shen. ... math.nsc.ru/conference/csr2009/accepted.shtml Uniqueness of trigonometric series and descriptive set theory ...simultaneously ALEKSANDROV'S proof of the continuum hypothesis for BOREL ...... 1754 Subsidium calculi sinuum,. Novi Commentarii Academice Scientiarum lmperialis ... IVASHEV-MUSATOV, O. S. (I/IBAIIIEB-MYCATOB, O. C.) ... www.springerlink.com/index/K1V8323323830662.pdf Martin Musatov ... Useless Stoic verbalism proof. Nature, miracle and sin: a study of ... a39b7c033e56f148 Ë.9cËò : Mu Mu 1 $(J,J')$-Lossless Factorization for Descriptor ...100, A Constructive Proof of a Fixed Point Theorem of Soardi(233-237 ...... 1933, On Continuous Functions with No Derivatives(53-56 Page), O.S. Ivashev-Musatov ...... 2487, Quantum groups, differential calculi and the eigenvalues of the ... mathnet.kaist.ac.kr/mathnet/thesis author.php?author=Mu%20Mu In order to show you the most relevant results, we have omitted By nothing. If you have something of value to exchange in honest trade or partnership contact me: marty.musatov@gmail.com Musatov P.S. It is clear I have spent a lot of time on inverse 19 and P=NP foundation work to the point that may very last name is ground into reality with computational complexity. I have the techniques I can make it work. *IBM, Microsoft, Apple, whoever wants innovation the most come to me and I will lead the way. === Subject: Re: If you could change one fundamental thing about the universe? > On Jun 9, 5:33 am, Dirk Bruere at NeoPax > Just for fun, what would you most like to see as > different about the > way the universe works. > Not sure, if I'd actually do this: > The universe, being a process in the almighty > computer should fork into > a new universe process anytime someone wishes the > universe to change, > granting the wish in the child process and make > the person forget the > wish in the parent process. The child process > shall terminate if the > person regrets the wish, making the original > person remember the wish > and be aware of why he regretted it. > Easy to > do.http://en.wikipedia.org/wiki/Quantum_suicide > > As I read Berhard's proposal, it is for a kind of > Timmy Turner (*) > universe. Rather than have a universe within which > fatal alternatives > are created and pruned away quietly, his proposal > is for a universe > where alternatives are consciously wished for and > then conciously > pruned. And the key point is that memory of the > failed branches is > retained in the principle branch. > > (*) Timmy Turner is the protagonist in Fairly > Odd-Parents, a > children's cartoon where the episodes tend to be > cautionary tales > about wish outcomes and un-wishing is routine. > > A similar plot device has used in at least two > movies, both of whose > names escape me. In one, the main character had an > ability to forsee > possible futures out to a horizon of about 10 > minutes and then conduct > himself so as to select a favorable outcome. In > the other the main > character experiences the present but has the > ability to influence his > past actions, thus putting himself into an > alternate timeline. > > Cautionary tales on immortality abound. If quantum > suicide ends up > pruning away the cases where you die in a house > fire in favor of the > cases where you survive in excruciating pain with > permanent > disabilities, it may not be something to be hoped > for. > > True. > Still, if it is the correct interpretation we will > all get to discover > it eventually. It's my explanation as to why the > world is getting > rapidly weirder. It has to in order to support my > aging life. And it > will speed up exponentially as I approach my death. > > -- > Dirk > > http://www.transcendence.me.uk/ - Transcendence UK > http://www.theconsensus.org/ - A UK political party > http://www.onetribe.me.uk/wordpress/?cat=5 - Our > podcasts on weird stuff > Reading about your Pi values ,I was amazed in current mathematics litreature of the use of 2Pi/radian value that is not time dimensional, 2 Pi value must be applicable to a dimension that is viable. Are all the each radians expansile in Time dimension , i.e is the circle expands in two Pi values at 180 degrees , I.E 180. Absolutly wrong! . Sorry to dissappoint you all and your gurus at Princeton, Stanford , U chicago , Ohio etc,, check this out. > > > This current value of 2 Pi =radians in a circle is a dead value. > 8 Pi ratio is likewise equal to an angle progresssion of around 2.1 degrees Math Forum Discussions Efforts by mathematicians to determine the value of pi since have led ... Musatov Inverse Counting Chain of Infinite Proportions (sample) ... mathforum.org/kb/message.jspa?messageID=6749244&tstart=0 - 4 hours ago Math Forum Discussions Would life be better if pi = 3 exactly? > Huh? This is Musatov ... Pi is a CONSTANT. Its value is 4*ATAN(1). ... Martin Michael Musatov. 6/8/09 ... mathforum.org/kb/message.jspa?messageID=6743888&tstart=0 Discussions - comp.theory | Google Groups QED Plouffe. I own you. Musatov === Subject: Re: Pi Is Rational Re: If you could change one fundamental thing about the universe? > On Jun 9, 5:33 am, Dirk Bruere at NeoPax > Just for fun, what would you most like to see as > different about the > way the universe works. > Not sure, if I'd actually do this: > The universe, being a process in the almighty > computer should fork into > a new universe process anytime someone wishes the > universe to change, > granting the wish in the child process and make > the person forget the > wish in the parent process. The child process > shall terminate if the > person regrets the wish, making the original > person remember the wish > and be aware of why he regretted it. > Easy to > do.http://en.wikipedia.org/wiki/Quantum_suicide > > As I read Berhard's proposal, it is for a kind of > Timmy Turner (*) > universe. Rather than have a universe within which > fatal alternatives > are created and pruned away quietly, his proposal > is for a universe > where alternatives are consciously wished for and > then conciously > pruned. And the key point is that memory of the > failed branches is > retained in the principle branch. > > (*) Timmy Turner is the protagonist in Fairly > Odd-Parents, a > children's cartoon where the episodes tend to be > cautionary tales > about wish outcomes and un-wishing is routine. > > A similar plot device has used in at least two > movies, both of whose > names escape me. In one, the main character had an > ability to forsee > possible futures out to a horizon of about 10 > minutes and then conduct > himself so as to select a favorable outcome. In > the other the main > character experiences the present but has the > ability to influence his > past actions, thus putting himself into an > alternate timeline. > > Cautionary tales on immortality abound. If quantum > suicide ends up > pruning away the cases where you die in a house > fire in favor of the > cases where you survive in excruciating pain with > permanent > disabilities, it may not be something to be hoped > for. > > True. > Still, if it is the correct interpretation we will > all get to discover > it eventually. It's my explanation as to why the > world is getting > rapidly weirder. It has to in order to support my > aging life. And it > will speed up exponentially as I approach my death. > > -- > Dirk > > http://www.transcendence.me.uk/ - Transcendence UK > http://www.theconsensus.org/ - A UK political party > http://www.onetribe.me.uk/wordpress/?cat=5 - Our > podcasts on weird stuff > Reading about your Pi values ,I was amazed in current mathematics litreature of the use of 2Pi/radian value that is not time dimensional, 2 Pi value must be applicable to a dimension that is viable. Are all the each radians expansile in Time dimension , i.e is the circle expands in two Pi values at 180 degrees , I.E 180. Absolutly wrong! . Sorry to dissappoint you all and your gurus at Princeton, Stanford , U chicago , Ohio etc,, check this out. > > > This current value of 2 Pi =radians in a circle is a dead value. > 8 Pi ratio is likewise equal to an angle progresssion of around 2.1 degrees Math Forum Discussions Efforts by mathematicians to determine the value of pi since have led ... Musatov Inverse Counting Chain of Infinite Proportions (sample) ... mathforum.org/kb/message.jspa?messageID=6749244&tstart=0 - 4 hours ago Math Forum Discussions Would life be better if pi = 3 exactly? > Huh? This is Musatov ... Pi is a CONSTANT. Its value is 4*ATAN(1). ... Martin Michael Musatov. 6/8/09 ... mathforum.org/kb/message.jspa?messageID=6743888&tstart=0 Discussions - comp.theory | Google Groups QED Plouffe. I own you. Musatov > > Well, you need the correct rational value of pi, > obviously. > I've found a web page which gives it: > http://www.correctpi.com/ > Clearly, the Oracle isn't all-knowning. > Most-knowing, at best. > > > I have investigated and researched on Pi for 35 > 35 years and > it has taken me this long to come to these > ese conclusions. > To share my knowledge and discuss the contents of > of my book > with fellow colleagues I have put a few words > rds together on > this site. > > Old methods of identifying and calculating Pi have > ve not been > mentioned, discussed or used in this book. History > ory of Pi has > not been mentioned either. This book only > nly considers a new idea > and new formulas. To understand and get a complete > ete view of the > new formulas and ideas one has to study the book. > > The second paragraph does not surprise me in the > least. In > matters of basic history and literature awareness, > the vast > majority of cranks put forth virtually no effort at > all. > They'll spend years, sometimes decades, working on > something > and the thought never seems to occur to them that > they might > benefit by visiting a college library, or even just a > public > library. > > Dave L. Renfro > Re: If you could change one fundamental thing about the universe? > On Jun 9, 5:33 am, Dirk Bruere at NeoPax > Just for fun, what would you most like to see as > different about the > way the universe works. > Not sure, if I'd actually do this: > The universe, being a process in the almighty > computer should fork into > a new universe process anytime someone wishes the > universe to change, > granting the wish in the child process and make > the person forget the > wish in the parent process. The child process > shall terminate if the > person regrets the wish, making the original > person remember the wish > and be aware of why he regretted it. > Easy to > do.http://en.wikipedia.org/wiki/Quantum_suicide > > As I read Berhard's proposal, it is for a kind of > Timmy Turner (*) > universe. Rather than have a universe within which > fatal alternatives > are created and pruned away quietly, his proposal > is for a universe > where alternatives are consciously wished for and > then conciously > pruned. And the key point is that memory of the > failed branches is > retained in the principle branch. > > (*) Timmy Turner is the protagonist in Fairly > Odd-Parents, a > children's cartoon where the episodes tend to be > cautionary tales > about wish outcomes and un-wishing is routine. > > A similar plot device has used in at least two > movies, both of whose > names escape me. In one, the main character had an > ability to forsee > possible futures out to a horizon of about 10 > minutes and then conduct > himself so as to select a favorable outcome. In > the other the main > character experiences the present but has the > ability to influence his > past actions, thus putting himself into an > alternate timeline. > > Cautionary tales on immortality abound. If quantum > suicide ends up > pruning away the cases where you die in a house > fire in favor of the > cases where you survive in excruciating pain with > permanent > disabilities, it may not be something to be hoped > for. > > True. > Still, if it is the correct interpretation we will > all get to discover > it eventually. It's my explanation as to why the > world is getting > rapidly weirder. It has to in order to support my > aging life. And it > will speed up exponentially as I approach my death. > > -- > Dirk > > http://www.transcendence.me.uk/ - Transcendence UK > http://www.theconsensus.org/ - A UK political party > http://www.onetribe.me.uk/wordpress/?cat=5 - Our > podcasts on weird stuff > Reading about your Pi values ,I was amazed in current mathematics litreature of the use of 2Pi/radian value that is not time dimensional, 2 Pi value must be applicable to a dimension that is viable. Are all the each radians expansile in Time dimension , i.e is the circle expands in two Pi values at 180 degrees , I.E 180. Absolutly wrong! . Sorry to dissappoint you all and your gurus at Princeton, Stanford , U chicago , Ohio etc,, check this out. > > > This current value of 2 Pi =radians in a circle is a dead value. > 8 Pi ratio is likewise equal to an angle progresssion of around 2.1 degrees Math Forum Discussions Efforts by mathematicians to determine the value of pi since have led ... Musatov Inverse Counting Chain of Infinite Proportions (sample) ... mathforum.org/kb/message.jspa?messageID=6749244&tstart=0 - 4 hours ago Math Forum Discussions Would life be better if pi = 3 exactly? > Huh? This is Musatov ... Pi is a CONSTANT. Its value is 4*ATAN(1). ... Martin Michael Musatov. 6/8/09 ... mathforum.org/kb/message.jspa?messageID=6743888&tstart=0 Discussions - comp.theory | Google Groups QED Plouffe. I own you. Musatov === Subject: The Beast (seeing is believing) In[170]:= 0.0000473372528076171875*2 0.000094674505615234375 0.00018934901123046875 0.0003786980224609375 0.000757396044921875 0.00151479208984375 0.0030295841796875 0.006059168359375 0.01211833671875 0.0242366734375 0.048473346875 0.09694669375 0.1938933875 0.387786775 0.77557355 1.5511471 3.1022942 6.2035884 12.4071768 24.8143536 49.6287072 99.2574144 198.5148288 397.0296576 794.0593152 1588.1186304 3176.2372608 6352.4745216 12704.9490432 25409.8980864 50819.7961728 101639.5923456 203279.1846912 406558.3693824 813116.7387648 1626233.4775296 3252466.9550592 6504933.9101184 13009867.8202368 26019735.6404736 52039471.2809472 104078942.5618944 208157885.1237888 416315770.2475776 832631540.4951552 1665263080.9903104 3330526161.9806208 6661052323.9612416 13322104647.9224832 26644209295.8449664 53288418591.6899328 106576837183.3798656 213153674366.7597312 426307348733.5194624 852614697467.0389248 1705229394934.0778496 3410458789868.1556992 6820917579736.3113984 13641835159472.6227968 27283670318945.2455936 54567340637890.4911872 109134681275780.9823744 218269362551561.9647488 436538725103123.9294976 873077450206247.8589952 1746154900412495.7179904 3492309800824991.4359808 6984619601649982.8719616 13969239203299965.7439232 27938478406599931.4878464 55876956813199862.9756928 111753913626399725.9513856 223507827252799451.9027712 447015654505598903.8055424 894031309011197807.6110848 1788062618022395615.2221696 3576125236044791230.4443392 7152250472089582460.8886784 14304500944179164921.7773568 28609001888358329843.5547136 57218003776716659687.1094272 114436007553433319374.2188544 228872015106866638748.4377088 457744030213733277496.8754176 915488060427466554993.7508352 1830976120854933109987.5016704 3661952241709866219975.0033408 7323904483419732439950.0066816 14647808966839464879900.0133632 29295617933678929759800.0267264 58591235867357859519600.0534528 117182471734715719039200.1069056 234364943469431438078400.2138112 468729886938862876156800.4276224 937459773877725752313600.8552448 1874919547755451504627201.7104896 3749839095510903009254403.4209792 7499678191021806018508806.8419584 Out[170]= 0.000094674505615234375 Out[171]= 0.0000946745 Out[172]= 0.000189349 Out[173]= 0.000378698 Out[174]= 0.000757396 Out[175]= 0.00151479 Out[176]= 0.00302958 Out[177]= 0.00605917 Out[178]= 0.0121183 Out[179]= 0.0242367 Out[180]= 0.0484733 Out[181]= 0.0969467 Out[182]= 0.193893 Out[183]= 0.387787 Out[184]= 0.775574 Out[185]= 1.55115 Out[186]= 3.10229 Out[187]= 6.20359 Out[188]= 12.4072 Out[189]= 24.8144 Out[190]= 49.6287 Out[191]= 99.2574 Out[192]= 198.515 Out[193]= 397.03 Out[194]= 794.059 Out[195]= 1588.12 Out[196]= 3176.24 Out[197]= 6352.47 Out[198]= 12704.9 Out[199]= 25409.9 Out[200]= 50819.8 Out[201]= 101640. Out[202]= 203279. Out[203]= 406558. Out[204]= 813117. Out[205]= 1.62623*10^6 Out[206]= 3.25247*10^6 Out[207]= 6.50493*10^6 Out[208]= 1.30099*10^7 Out[209]= 2.60197*10^7 Out[210]= 5.20395*10^7 Out[211]= 1.04079*10^8 Out[212]= 2.08158*10^8 Out[213]= 4.16316*10^8 Out[214]= 8.32632*10^8 Out[215]= 1.66526*10^9 Out[216]= 3.33053*10^9 Out[217]= 6.66105*10^9 Out[218]= 1.3322104647922483*10^10 Out[219]= 2.6644209295844966*10^10 Out[220]= 5.3288418591689933*10^10 Out[221]= 1.06576837183379866*10^11 Out[222]= 2.13153674366759731*10^11 Out[223]= 4.26307348733519462*10^11 Out[224]= 8.52614697467038925*10^11 Out[225]= 1.705229394934077850*10^12 Out[226]= 3.410458789868155699*10^12 Out[227]= 6.820917579736311398*10^12 Out[228]= 1.36418351594726227968*10^13 Out[229]= 2.7283670318945245594*10^13 Out[230]= 5.4567340637890491187*10^13 Out[231]= 1.091346812757809823744*10^14 Out[232]= 2.18269362551561964749*10^14 Out[233]= 4.36538725103123929498*10^14 Out[234]= 8.73077450206247858995*10^14 Out[235]= 1.746154900412495717990*10^15 Out[236]= 3.492309800824991435981*10^15 Out[237]= 6.984619601649982871962*10^15 Out[238]= 1.3969239203299965743923*10^16 Out[239]= 2.7938478406599931487846*10^16 Out[240]= 5.5876956813199862975693*10^16 Out[241]= 1.11753913626399725951386*10^17 Out[242]= 2.23507827252799451902771*10^17 Out[243]= 4.47015654505598903805542*10^17 Out[244]= 8.94031309011197807611085*10^17 Out[245]= 1.788062618022395615222170*10^18 Out[246]= 3.576125236044791230444339*10^18 Out[247]= 7.152250472089582460888678*10^18 Out[248]= 1.4304500944179164921777357*10^19 Out[249]= 2.8609001888358329843554714*10^19 Out[250]= 5.7218003776716659687109427*10^19 Out[251]= 1.14436007553433319374218854*10^20 Out[252]= 2.28872015106866638748437709*10^20 Out[253]= 4.57744030213733277496875418*10^20 Out[254]= 9.15488060427466554993750835*10^20 Out[255]= 1.830976120854933109987501670*10^21 Out[256]= 3.661952241709866219975003341*10^21 Out[257]= 7.323904483419732439950006682*10^21 Out[258]= 1.4647808966839464879900013363*10^22 Out[259]= 2.9295617933678929759800026726*10^22 Out[260]= 5.85912358673578595196000534528*10^22 Out[261]= 1.171824717347157190392001069056*10^23 Out[262]= 2.34364943469431438078400213811*10^23 Out[263]= 4.687298869388628761568004276224*10^23 Out[264]= 9.374597738777257523136008552448*10^23 Out[265]= 1.874919547755451504627201710490*10^24 Out[266]= 3.749839095510903009254403420979*10^24 Out[267]= 7.499678191021806018508806841958*10^24 In[72]:= 0.0000473372528076171875/2 0.000094674505615234375 0.00018934901123046875 0.0003786980224609375 0.000757396044921875 0.00151479208984375 0.0030295841796875 0.006059168359375 0.01211833671875 0.0242366734375 0.048473346875 0.09694669375 0.1938933875 0.387786775 0.77557355 1.5511471 3.1022942 6.2035884 12.4071768 24.8143536 49.6287072 99.2574144 198.5148288 397.0296576 794.0593152 1588.1186304 3176.2372608 6352.4745216 12704.9490432 25409.8980864 50819.7961728 101639.5923456 203279.1846912 406558.3693824 813116.7387648 1626233.4775296 3252466.9550592 6504933.9101184 13009867.8202368 26019735.6404736 52039471.2809472 104078942.5618944 208157885.1237888 416315770.2475776 832631540.4951552 1665263080.9903104 3330526161.9806208 6661052323.9612416 13322104647.9224832 26644209295.8449664 53288418591.6899328 106576837183.3798656 213153674366.7597312 426307348733.5194624 852614697467.0389248 1705229394934.0778496 3410458789868.1556992 6820917579736.3113984 13641835159472.6227968 27283670318945.2455936 54567340637890.4911872 109134681275780.9823744 218269362551561.9647488 436538725103123.9294976 873077450206247.8589952 1746154900412495.7179904 3492309800824991.4359808 6984619601649982.8719616 13969239203299965.7439232 27938478406599931.4878464 55876956813199862.9756928 111753913626399725.9513856 223507827252799451.9027712 447015654505598903.8055424 894031309011197807.6110848 1788062618022395615.2221696 3576125236044791230.4443392 7152250472089582460.8886784 14304500944179164921.7773568 28609001888358329843.5547136 57218003776716659687.1094272 114436007553433319374.2188544 228872015106866638748.4377088 457744030213733277496.8754176 915488060427466554993.7508352 1830976120854933109987.5016704 3661952241709866219975.0033408 7323904483419732439950.0066816 14647808966839464879900.0133632 29295617933678929759800.0267264 58591235867357859519600.0534528 117182471734715719039200.1069056 234364943469431438078400.2138112 468729886938862876156800.4276224 937459773877725752313600.8552448 1874919547755451504627201.7104896 3749839095510903009254403.4209792 7499678191021806018508806.8419584 Out[72]= 0.0000236686264038085938 Out[73]= 0.0000946745 Out[74]= 0.000189349 Out[75]= 0.000378698 Out[76]= 0.000757396 Out[77]= 0.00151479 Out[78]= 0.00302958 Out[79]= 0.00605917 Out[80]= 0.0121183 Out[81]= 0.0242367 Out[82]= 0.0484733 Out[83]= 0.0969467 Out[84]= 0.193893 Out[85]= 0.387787 Out[86]= 0.775574 Out[87]= 1.55115 Out[88]= 3.10229 Out[89]= 6.20359 Out[90]= 12.4072 Out[91]= 24.8144 Out[92]= 49.6287 Out[93]= 99.2574 Out[94]= 198.515 Out[95]= 397.03 Out[96]= 794.059 Out[97]= 1588.12 Out[98]= 3176.24 Out[99]= 6352.47 Out[100]= 12704.9 Out[101]= 25409.9 Out[102]= 50819.8 Out[103]= 101640. Out[104]= 203279. Out[105]= 406558. Out[106]= 813117. Out[107]= 1.62623*10^6 Out[108]= 3.25247*10^6 Out[109]= 6.50493*10^6 Out[110]= 1.30099*10^7 Out[111]= 2.60197*10^7 Out[112]= 5.20395*10^7 Out[113]= 1.04079*10^8 Out[114]= 2.08158*10^8 Out[115]= 4.16316*10^8 Out[116]= 8.32632*10^8 Out[117]= 1.66526*10^9 Out[118]= 3.33053*10^9 Out[119]= 6.66105*10^9 Out[120]= 1.3322104647922483*10^10 Out[121]= 2.6644209295844966*10^10 Out[122]= 5.3288418591689933*10^10 Out[123]= 1.06576837183379866*10^11 Out[124]= 2.13153674366759731*10^11 Out[125]= 4.26307348733519462*10^11 Out[126]= 8.52614697467038925*10^11 Out[127]= 1.705229394934077850*10^12 Out[128]= 3.410458789868155699*10^12 Out[129]= 6.820917579736311398*10^12 Out[130]= 1.36418351594726227968*10^13 Out[131]= 2.7283670318945245594*10^13 Out[132]= 5.4567340637890491187*10^13 Out[133]= 1.091346812757809823744*10^14 Out[134]= 2.18269362551561964749*10^14 Out[135]= 4.36538725103123929498*10^14 Out[136]= 8.73077450206247858995*10^14 Out[137]= 1.746154900412495717990*10^15 Out[138]= 3.492309800824991435981*10^15 Out[139]= 6.984619601649982871962*10^15 Out[140]= 1.3969239203299965743923*10^16 Out[141]= 2.7938478406599931487846*10^16 Out[142]= 5.5876956813199862975693*10^16 Out[143]= 1.11753913626399725951386*10^17 Out[144]= 2.23507827252799451902771*10^17 Out[145]= 4.47015654505598903805542*10^17 Out[146]= 8.94031309011197807611085*10^17 Out[147]= 1.788062618022395615222170*10^18 Out[148]= 3.576125236044791230444339*10^18 Out[149]= 7.152250472089582460888678*10^18 Out[150]= 1.4304500944179164921777357*10^19 Out[151]= 2.8609001888358329843554714*10^19 Out[152]= 5.7218003776716659687109427*10^19 Out[153]= 1.14436007553433319374218854*10^20 Out[154]= 2.28872015106866638748437709*10^20 Out[155]= 4.57744030213733277496875418*10^20 Out[156]= 9.15488060427466554993750835*10^20 Out[157]= 1.830976120854933109987501670*10^21 Out[158]= 3.661952241709866219975003341*10^21 Out[159]= 7.323904483419732439950006682*10^21 Out[160]= 1.4647808966839464879900013363*10^22 Out[161]= 2.9295617933678929759800026726*10^22 Out[162]= 5.85912358673578595196000534528*10^22 Out[163]= 1.171824717347157190392001069056*10^23 Out[164]= 2.34364943469431438078400213811*10^23 Out[165]= 4.687298869388628761568004276224*10^23 Out[166]= 9.374597738777257523136008552448*10^23 Out[167]= 1.874919547755451504627201710490*10^24 Out[168]= 3.749839095510903009254403420979*10^24 Out[169]= 7.499678191021806018508806841958*10^24 === Subject: Norms Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) Suppos I have a vector X=(x1,x2,x3) then the Euclidian norm is the square root of the sum of squares. What is the L infinity norm? is it just the element which has largest absolute value? ie suppose X=(1,2,-3) is L inf =3? also what is the difference bewteen H infinity and L infinity - are they the same? Hardy === Subject: Re: Norms > Suppos I have a vector > > X=(x1,x2,x3) then the Euclidian norm is the square > root of the sum of > squares. > > What is the L infinity norm? is it just the element > which has largest > absolute value? > > ie suppose X=(1,2,-3) is L inf =3? > Discussions - sci.math | Google Groups10 posts - 8 authors Suppos I have a vector X=(x1,x2,x3) then the Euclidian norm is the ... has largest absolute value? ie suppose X=(1,2,-3) is L inf =3? also what is the difference bewteen H infinity and L ... Are all the each radians expansile in Time dimension , i.e is the circle expands in two Pi values at 180 degrees , I.E 180. ... On the Relationships between Induced L-Infinity Gain and Various ...This difference between (2) and (3) is a primary reason .... results from [1], [4] (see also [7], [8] for discrete-time .... For the special case X = Y and h(x,w) .81§ x, having l ... Suppose X is a normed linear space; then we say .... in which the input is unconstrained (i.e. w k can take on any values in W) as was ... http://ieeexplore.ieee.org/iel5/9774/30838/01429239.pdf?arnumber=1429239 Math Forum DiscussionsJun 12, 2009 ... absolute value? ie suppose X=(1,2,-3) is L inf =3? also what is the difference bewteen H infinity and L infinity - are they the same? ... http://mathforum.org/kb/thread.jspa?threadID=1954224&tstart=0 - 1 hour ago Methods of solving a polynomial equation for an H/sub infinity ...H. = (BoB+A,Y)-l(AoA+A,X) where X and Yare the solutions of. (2),(3),and(4)withX=hIanddeg(X)=n,+ .... is zero at several values of A, the largest absolute value of A .... solved for a 6 x. 6 matrix. The relation between the optimality index E ... by Step 1) is - 11.459147 and the optimal X is - 11.482921. This also ... http://ieeexplore.ieee.org/iel4/9/860/00021088.pdf?arnumber=21088 Relations between Common Lyapunov Functions of Quadratic and ...vector infinity-norm have also received a lot of attentions. .... in Definition 3. Now we suppose that the set of systems (1) has a. CQLF V(x) = x ... really strict, i.e., L. Q. .81.bc L. I . (QED). Theorem 1 shows that the strict inclusion relation be- .... tained from A by taking the absolute value of all entries, ... http://ietfec.oxfordjournals.org/cgi/reprint/E89-A/6/1794.pdf Adaptive FIR filtering based on minimum L/sub infinity /-norm ...sumption, we then derive a set of nonlinear difference equations that .... (0,l). Substituting. (3) in. (5) yields pe(n). = AT(n)X(n). .... that the mean-absolute value of a Gaussian random variable with zero mean and variance ... tions in (19) and (20) as well as Assumptions 1 and 2. Notice also ... http://ieeexplore.ieee.org/iel2/518/4206/00160821.pdf?arnumber=160821 Testing for Polynomial Regression Using Nonparametric Regression ...Also, under assumption A, Nychka (1989) showed that [absolute value of [nh.sub.ij]] [less than] .... 2]) for i = 1, 2, 3 as n [approaches] [infinity]. ... Hall, P., and Hart, J. D. (1990), Bootstrap Test for Difference Between Means in ... Holst, L., and Rao, J. S. (1980), Asymptotic Theory for Some Families of ... http://www.questia.com/PM.qst?a=o&se=gglsc&d=5002277810 Fast iterative computation of optimal two-block H/sup infinity ...6[@3(jw)l. (1-2) where. 6[+(jw)] denotes the largest singular value of ... which solves (1-5) will also be an optimal solution of ..... that Theorem 2.4 and Corollary 2.1 are so alike that the only difference lies between yI and y2,. i.e., yl. < yo. < yz. Suppose ys is a point such that yI ... http://ieeexplore.ieee.org/iel4/9/1253/00029402.pdf?arnumber=29402 Points of difference: Relative infinity in the Euclidean planevalue is necessary on two counts= the choice between p and q. (since Ap(A, B)=-zq(A, .... Suppose now we have two observers with distinct antipodes e 1 and e 2. AS ...... /irst observer's coordinates, la llb l = x 2 = la llb l, .... What happens in Euclidean 3-space with a relative infinity? (One ... Infinity-norm torque minimization for redundant manipulators using ...J(e)H(B)-l. Recall that the infinity-norm of a vector x ... denotes the absolute value of the component, is defined as: llllw. = m {l 1I l 21,...,1 nl}, (7) .... is the squared duality gap; i.e., the squared difference between the objective functions ..... driving joint torques for joint 1, joint 2 and joint 3 are ... http://ieeexplore.ieee.org/iel5/6713/18008/00831241.pdf?arnumber=831241 > also what is the difference bewteen H infinity and L > infinity - are > they the same? > > > Hardy > Google Base: 8 Glass Door Reach-in Freezer Infinity 8 Glass Door Reach-in Freezer Infinity ... 4'w x 16'-10 L x 7'-7 H (STD). Shelving Extra-Deep 36. T-8 Lighting, 75 Light on either side of all glass ... ON THE USE OF JPEG 2000 TO ACHIEVE MINIMUM L-INFINITY ERROR WHEN ...the quantized residuals (the difference between the original .... They also include a column where the number of bits used to quantize the residuals is shown. ... of the same answer as far as the total achieved rate can also be seen for this variable. .... with a JPEG2000 Framework for Controlling L-Infinity Error ... http://ieeexplore.ieee.org/iel5/4378863/4379065/04379167.pdf?arnumber=437916 7 Google Base: 5 Glass Door Reach-in Cooler InfinityDec 30, 2008 ... Infinity Design. Assistance Call (M-F 8am-5pm EST) 516-342-9322 or To buy another model Click the Link ... Actual Out side Dimensions: 36Wx125L x95 inch H, ... Floor can Handle 650 lbs/sq ft Distributed. Options ... [PDF] RELATION BETWEEN QUASIASYMPTOTICS, EQUIVALENCE AT INFINITY ANDFile Format: PDF/Adobe Acrobat - View as HTML function L G Lioc is a slowly varying function at infinity if L(x) > 0, x > .... 5-asymptotics related to haL(h) and that C0 .84.86 0. We can also suppose that i is .... where L'ka_1(x)/L(x) .81> a~k, x .81>.81Á oo. Now, using the same method of ... the difference between these two definitions of asymptotic behaviour follows ... http://www.emis.de/journals/NSJOM/Papers/23_1/NSJOM_23_1_001_011.pdf H-infinity Optimal Control and Related Minimax Design ProblemsThis is the second edition of our 1991 book with the same title, which, ... and at places under more refined conditions, also contains substantial new ... Urbana-Champaign ; Ecole Superieure des Sciences de l'Informatique, ... A Relationship Between H-infinity Optimal Control and LQ Zero-Sum Dynamic Games (p. ... http://decision.csl.uiuc.edu/~tbasar/birkhauserbook.html http://mathforum.org/library/drmath/view/52393.html L-Infinity Bounded Robust Control for Hybrid SystemsThe difference between autonomous discrete transition set. A and the control one C is in the fact that the ... That is to say, the autonomous discrete dynamics are also ..... time for the other hybrid control and they have the same .... [2] Basar T. and Bernhard (1995) H-infinity Optimal Control and Related ... http://ieeexplore.ieee.org/iel5/4176992/4176993/04177275.pdf?isnumber=417699 3&prod=CNF&arnumber=4177275&arSt=4801...M... Projective Infinity from Interactive Mathematics Miscellany and ...According to [H. Eves, p. 59], infinity has been introduced into geometry by Johann ... initial position, the smaller becomes the angle between the lines l and m. ... so that, when parallel, they intersect at the same point at infinity. ... http://www.cut-the-knot.org/WhatIs/Infinity/Projective.shtml Robust H/sub infinity / control for linear systems with norm ...Also, the relationship between. H , control and quadratic stabilization has been established in ... results are restricted to time-invariant systems and hence they .... also quadratically stabilizes the system. (X. )with the same levcl .... L. Xie and C.E. de Souza, .81gRobust H, control for linear ... http://ieeexplore.ieee.org/iel5/164/5238/00203757.pdf [PPT] Objects at infinity used in calibrationFile Format: Microsoft Powerpoint - View as HTML once again, we shall be fascinated by objects at infinity ... identifies the excess: the difference between an affinity (12dof) and a projectivity (15dof) is exactly the 3dof .... they are h1 +- i h2, where h1 and h2 are first 2 columns of H ... interesting: the pair of circular points encode the same information, ... http://www.cis.uab.edu/jj/sfm/lecture11-infinity+3squarealg.ppt ++ Musatov === Subject: Re: Norms > Suppos I have a vector > > X=(x1,x2,x3) then the Euclidian norm is the square root of the sum of > squares. > > What is the L infinity norm? is it just the element which has largest > absolute value? > > ie suppose X=(1,2,-3) is L inf =3? yes > > also what is the difference bewteen H infinity and L infinity - are > they the same? H_infinity is a norm for spaces of analytic functions > > > Hardy > === Subject: Re: Norms <120620091810212796%anniel@nym.alias.net.invalid> Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Suppos I have a vector > X=(x1,x2,x3) then the Euclidian norm is the square root of the sum of > squares. > What is the L infinity norm? is it just the element which has largest > absolutevalue? > ie suppose X=(1,2,-3) is L inf =3? yes > also what is the difference bewteen H infinity and L infinity - are > they the same? H infinity is a norm for spaces of analytic functions > Hardy f(x)=x^2+3x+2 or does it have to be a vector of (say) polynomials. Hardy === Subject: Re: Norms > Suppos I have a vector > X=(x1,x2,x3) then the Euclidian norm is the square root of the sum of > squares. > What is the L infinity norm? is it just the element which has largest > absolutevalue? > ie suppose X=(1,2,-3) is L inf =3? > yes > > also what is the difference bewteen H infinity and L infinity - are > they the same? > H_infinity is a norm for spaces of analytic functions > > Hardy > > > f(x)=x^2+3x+2 or does it have to be a vector of (say) polynomials. > > Hardy So x is a complex variable that ranges over the disk |x|<1 in the complex plane. For the H_infinity norm, take the supremum of all values |f(x)| ... For your function, the maximum value (on the closed disk) is at x=1, namely 6, so that the H_infinity norm in this case is 6. One can also consider H_infinity norm on domains other than the unit disk. More here http://en.wikipedia.org/wiki/Hardy_space === Subject: Counting combinations of ordered pairs of items without repetition posting-account=lGXbOwoAAADQIxuV27e3GJ5gffEagGrr Gecko/2009060215 Firefox/3.0.11 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) Assume we have lists of ordered pairs of items, such that each list has the same pairs. For example say we have 2 lists: {(1,2),(3,4), (4,5),(6,7)} and {(1,2),(3,4),(4,5),(6,7)}. Is there a way, using combinatorial math, not brute force, to count the number of ways of choosing a single pair from each list such that no single item is duplicated? For example, if you choose (3,4) from the first list, you cannot choose (3,4) or (4,5) from the second list. === Subject: Re: Counting combinations of ordered pairs of items without repetition posting-account=lGXbOwoAAADQIxuV27e3GJ5gffEagGrr Gecko/2009060215 Firefox/3.0.11 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Assume we have lists of ordered pairs of items, such that each list > has the same pairs. For example say we have 2 lists: {(1,2),(3,4), > (4,5),(6,7)} and {(1,2),(3,4),(4,5),(6,7)}. Is there a way, using > combinatorial math, not brute force, to count the number of ways of > choosing a single pair from each list such that no single item is > duplicated? For example, if you choose (3,4) from the first list, you > cannot choose (3,4) or (4,5) from the second list. You can't calculate it without knowing how many overlapping pairs > there are. Assuming you know that, then: The Cartesian Product of a list of N items with itself gives N * N or 4 + 4 + 4 + 4 when N=4 But since an item can't be paired with itself, you get N * (N-1) or 3 + 3 + 3 + 3 But, per your final constraint, reduce by one each > the overlapping pairs 3 + 2 + 2 + 3 So there should be 10 possibilities. Table: > s t u > -------------------- > aba b > cdc d > ded e > fgf g Query: > SELECT t.s, t 1.s > FROM t, t AS t 1 > WHERE ((([t].[t]<>[t 1].[t] And [t].[t]<>[t 1].[u])=True) AND >(([t].[u]<>[t 1].[t] And [t].[u]<>[t 1].[u])=True)); t.s t 1.s > --------------- > cdab <----- 3 > deab | > fgab | > abcd <----- 2 > fgcd | > abde <----- 2 > fgde | > abfg <----- 3 > cdfg | > defg | 10 records lists of pairs? === Subject: Re: Counting combinations of ordered pairs of items without repetition posting-account=lGXbOwoAAADQIxuV27e3GJ5gffEagGrr Gecko/2009060215 Firefox/3.0.11 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Assume we have lists of ordered pairs of items, such that each list > has the same pairs. For example say we have 2 lists: {(1,2),(3,4), > (4,5),(6,7)} and {(1,2),(3,4),(4,5),(6,7)}. Is there a way, using > combinatorial math, not brute force, to count the number of ways of > choosing a single pair from each list such that no single item is > duplicated? For example, if you choose (3,4) from the first list, you > cannot choose (3,4) or (4,5) from the second list. You can't calculate it without knowing how many overlapping pairs > there are. Assuming you know that, then: The Cartesian Product of a list of N items with itself gives N * N or 4 + 4 + 4 + 4 when N=4 But since an item can't be paired with itself, you get N * (N-1) or 3 + 3 + 3 + 3 But, per your final constraint, reduce by one each > the overlapping pairs 3 + 2 + 2 + 3 So there should be 10 possibilities. Table: > s t u > -------------------- > aba b > cdc d > ded e > fgf g Query: > SELECT t.s, t 1.s > FROM t, t AS t 1 > WHERE ((([t].[t]<>[t 1].[t] And [t].[t]<>[t 1].[u])=True) AND >(([t].[u]<>[t 1].[t] And [t].[u]<>[t 1].[u])=True)); t.s t 1.s > --------------- > cdab <----- 3 > deab | > fgab | > abcd <----- 2 > fgcd | > abde <----- 2 > fgde | > abfg <----- 3 > cdfg | > defg | 10 records lists of pairs? === Subject: Re: Counting combinations of ordered pairs of items without repetition posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > Assume we have lists of ordered pairs of items, such that each list > has the same pairs. For example say we have 2 lists: {(1,2),(3,4), > (4,5),(6,7)} and {(1,2),(3,4),(4,5),(6,7)}. Is there a way, using > combinatorial math, not brute force, to count the number of ways of > choosing a single pair from each list such that no single item is > duplicated? For example, if you choose (3,4) from the first list, you > cannot choose (3,4) or (4,5) from the second list. Isn't this just the same as choosing a pair (i,j) from the first list and then choosing another pair (p,q) with p and/or q different from i and/or j? In other words, we just need a *single* list. We can interpret this graphically: say we have a bipartite graph with two sets of nodes U = {1,2,...,n} and V = {1,2,...,n}. There is an arc from i in U to j in V if the pair (i,j) is in your given set. See, eg., http://en.wikipedia.org/wiki/Bipartite graph . Now, when you choose a pair (i,j) and then want to count all the other pairs (p,q) that satisfy your conditions, you are really choosing an arc (i,j) in the graph and are then counting all the other arcs in the graph whose nodes are different from i and j in either set U, V. This does not actually solve your problem, but it does give you another way of thinking about it that you may find helpful. As Mensanator says, you need to know more about the set of available pairs = set of arcs in the graph. R.G. Vickson === Subject: Re: Counting combinations of ordered pairs of items without repetition > Assume we have lists of ordered pairs of items, such > that each list > has the same pairs. For example say we have 2 lists: > {(1,2),(3,4), > (4,5),(6,7)} and {(1,2),(3,4),(4,5),(6,7)}. Is there > a way, using > combinatorial math, not brute force, to count the > number of ways of > choosing a single pair from each list such that no > single item is > duplicated? For example, if you choose (3,4) from the > first list, you Did you mean: 2 lights: {(1,2),(3,4),(4,5),(6,7)} and {(1,2),(3,4),(4,5),(6,7)}.? Android Market - Android Market HelpApplication promotion: How to get on the featured list? 4:35 PM. Question from Burrotech in 'Appy' ... Answered (2 replies). can i watch some other than youtube 3:06 PM ... 1 reply. I'm from Colombia. How can I purchase the Dev Phone? 6/7/09 ... Answered (5 replies). Wifi Shows Connected but I do not get any data ... [PDF] THe BUsINessWeek BesT seLLer LIsTFile Format: PDF/Adobe Acrobat - View as HTML 1. 4. 4. 2. 8. 7. 9. 3. 11. 1. 3. 5. 11. 6. 3. 9. 4. .84. 1. 10. 12. .84. 1. 13. 2. 14. 15. .84. 1. HarDCOver BUsINess BOOks. LasT. MONTH. MONTHs. ON LIsT ... http://www.businessweek.com/pdfs/2008/0806_bestseller.pdf What is Level 5? - AdSense Help100+ posts - 23 authors Level 2 5/13/09 ... So, do you also have a progress bar, sriraj, to level 3? ... There are total of Levels 1 through 10. And since I am level 4, .... Level 5 Awesome. Mine bar is 25% blue. I hope there is no Level 6,7,8 . .... Mine is 7/93 (I just reached Level 4). I wish I know more about Google's algorithms for ... Doodle 4 Google Region 1. Johnny Zuk. Region 2. Sinceraty Alexander. Region 3. Grace Para. Region 4. Anagha Kangovi. Region 5. Alaina Beaver. Region 6. Jackie Tian. Region 7 ... Region 2. Courtney Bodine. Region 3. Austin Gage Druid. Region 4 ... Google Friend Connect - Google Friend Connect Help Jun 5, 2009 ... Lots of inconsistencies in member list and replies lists 6/7/09 ... 2 replies. Sign In Confusion 6/5/09 ... 1 reply. 1 2 3 4 5 6 7 8 9 10 11 ... Next. Popular discussions ... 1 reply. recommendation gadget. 3 replies ... > cannot choose (3,4) or (4,5) from the second list. === Subject: Re: The Truman Proof - update > And everyone around me hears it and comments on it. > Except your doctors, it seems. > Sylvia. > My GP is quite a hottie and I used to calculate my odds of having sex in > her office, I put it at 1 in a million. Anyway, at this clinic you sit > just outside the doctors office and she can hear my thoughts while she's > with another patient. So ofcourse my mind slipped and I was > broadcasting about the fantasy and she hollers out OH YEAH. Is this Kirby? Kirby is even hotter. But I don't see her anymore. Herc === === Subject: Re: Complex Made Simple: Update > If you have a copy you can now find a list of typos online - > go to > www.ams.org/bookstore?fn=20&arg1=gsmseries&item=GSM-97 > and click on the Supplementary Material. (The one that > really bothers me is this: The f in Theorem B in Chapter > 20 should be u.) > I suppose that you have better things to do than correcting the typos >from your list of typos, > > > but here it goes: > 1) The full stops at the end of the corrections concerning the pages 273 > and 373 are misplaced. > > Probably everyone but me would agree with that, but the placement > is intentional. Those entries read: > > p. n, l. m: display_1 > > should be display_2 > > . > > Here I'm using $$...$$ as means of _quotation_. The period > in question is not part of what display_1 should be changed > to, so it doesn't belong in display_2. Cute. Now that I understood it it turns out that I agree with you. Here's another remark, this time about the book itself. On page 417 you It is relatively easy to show that the trivial zeroes are the only zeroes outside the ``critical strip'' defined by 0 < Re(z) < 1. Well, the meaning of relatively easy vague. I would certainly agree that it applies to the assertion that the trivial zeroes are the only zeroes _z_ such that Re(z) < 0 or that Re(z) > 1. On the other hand, I would not say that it is relatively easy to prove that there are no zeroes _z_ of the zeta function such that Re(z) is 0 or 1. It's a hard (IMHO) theorem due to Hadamard and de la Vall.8ee Poussin. Jose Carlos Santos === Subject: Re: Complex Made Simple: Update <79g2s9F1pdlcqU1@mid.individual.net> posting-account=n26igQkAAACeF9xA2Ms8cKIdBH40qzwr Gecko/20070505 Iceape/1.0.9 (Debian-1.0.13~pre080614i-0etch1),gzip(gfe),gzip(gfe) I keep catching the title of this thread and thought I'd throw three cents at it. The disc or plane is easily built by extending the real behavior - 1 + 1 = 0 to a third level - 1 + 1 * 1 = 0 where '*' is a new third sign. What were modulo two sign mechanics on the reals now become modulo three mechanics on these three-signed numbers. Geometry, superposition, and product are all present through the generalzation of sign. These three-signed mumbers are the complex numbers in a more elemental form. Hence they do extend naturally from the reals, as do other number systems. Physics is nearby. http://BandTechnology.com/Polysigned This is now old hat to some and I do not mean to take away from the discussion but merely wish to point out this option on an appropriately titled thread. - Tim > [...] >Here's another remark, this time about the book itself. Comments on the book are certainly more interesting, > at least to me, than comments on the typesetting of > the errata page. >On page 417 you > It is relatively easy to show that the trivial zeroes are the only > zeroes outside the ``critical strip'' defined by 0 < Re(z) < 1. >Well, the meaning of relatively easy vague. I would certainly agree >that it applies to the assertion that the trivial zeroes are the only >zeroes z such that Re(z) < 0 or that Re(z) > 1. On the other hand, I >would not say that it is relatively easy to prove that there are no >zeroes z of the zeta function such that Re(z) is 0 or 1. It's a hard >(IMHO) theorem due to Hadamard and de la Vall.8ee Poussin. You may have a point. Of course it is relatively easy, by comparison > with proving RH, which is what relatively was referring to > (what else could I have been comparing it to?). And I do come > very close to saying in the next paragraph that showing > zeta(1+iy) <> 0 is nontrivial. >Jose Carlos Santos David C. Ullrich Understanding Godel isn't about following his formal proof. > That would make a mockery of everything Godel was up to. > (John Jones, My talk about Godel to the post-grads. > in sci.logic.) === Subject: Re: Complex Made Simple: Update Tim BandTech.com a .8ecrit : > > I keep catching the title of this thread and thought I'd throw three > cents at it. PLease send more money > The disc or plane is easily built by extending the real behavior > - 1 + 1 = 0 > to a third level > - 1 + 1 * 1 = 0 > where '*' is a new third sign. What were modulo two sign mechanics on > the reals now become modulo three mechanics on these three-signed > numbers. Geometry, superposition, and product are all present through > the generalzation of sign. These three-signed mumbers are the complex > numbers in a more elemental form. Hence they do extend naturally from > the reals, as do other number systems. Physics is nearby. > http://BandTechnology.com/Polysigned > > This is now old hat to some and I do not mean to take away from the > discussion but merely wish to point out this option on an > appropriately titled thread. It was not titled polysign made simple, was it? === Subject: Re: Complex Made Simple: Update <87ws7hfv20.fsf@phiwumbda.org> posting-account=CtWhuAoAAAAZZ9vwdovdqB3NNaiUa20_ GTB5),gzip(gfe),gzip(gfe) There's a certain logic to your placement, but it's nonetheless > awkward and unusual.Why not just re-word the sentence? Replace > should be with, say, should be as follows. Yeah, there's a certain logic in the whole book. But somehow Dave cultivates the unusual. For example on page 37 of Complex Made Simple (which is in itself an oxymoron) he deems the Fundamental Theorem of Algebra to be a Corollary (3.7) instead of a famous theorem. And he calls it just an amusing digression on page 38. Trivial theorem (HdB): A polynomial of degree n assumes every complex value n times. Proof: polynomial - (a complex value) = another polynomial with n zeroes. Now say that an entier function is a polynomial of degree infinity and we're almost there for the Big Picard Theorem (20.1). Not so? Han de Bruijn === Subject: Re: Complex Made Simple: Update <87ws7hfv20.fsf@phiwumbda.org> posting-account=CtWhuAoAAAAZZ9vwdovdqB3NNaiUa20_ GTB5),gzip(gfe),gzip(gfe) > There's a certain logic to your placement, but it's nonetheless > awkward and unusual.Why not just re-word the sentence? Replace > should be with, say, should be as follows. Yeah, there's a certain logic in the whole book. But somehow Dave > cultivates the unusual. For example on page 37 of Complex Made > Simple (which is in itself an oxymoron) he deems the Fundamental > Theorem of Algebra to be a Corollary (3.7) instead of a famous > theorem. And he calls it just an amusing digression on page 38. More awkward writing. On page 51 of CMS it reads: Chapter 4. Logarithms, Winding Numbers and Cauchy's Theorem. Versions of Cauchy's Theorem already appeared in previous chapters. Thus, as an indication of its content, the title of this chapter 4 is at least a bit misleading, but leave that aside. Suddenly, in the middle of the same Chapter 4, on page 71, we read the following: Our next big topic is the Residue Theorem. So it is clear that Ullrich has never followed a course in decent essay writing. It's incredible that a big topic is not even mentioned in the title of a chapter where it is treated. CMS is full of these amateur designs. I'm talking about form here, not about (mathematical) content. Guess it's can be the editor of a book. Han de Bruijn === Subject: Re: Complex Made Simple: Update >Giggle. There was no LaTeX involved in writing the book. I hate to interrupt the lively flow of this thread with a question that may not even be on topic for this group, but what typesetting system did you use to produce CMS? I had thought that LaTeX was the format that most publishers wanted to receive. Rob Johnson take out the trash before replying === Subject: Re: Complex Made Simple: Update > Giggle. There was no LaTeX involved in writing the book. > > I hate to interrupt the lively flow of this thread with a question > that may not even be on topic for this group, but what typesetting > system did you use to produce CMS? I had thought that LaTeX was > the format that most publishers wanted to receive. Jose Carlos Santos === Subject: Re: Complex Made Simple: Update > Giggle. There was no LaTeX involved in writing the book. > I hate to interrupt the lively flow of this thread with a question > that may not even be on topic for this group, but what typesetting > system did you use to produce CMS? I had thought that LaTeX was > the format that most publishers wanted to receive. > > How'd you know that? Because we had this talk: five years ago. Jose Carlos Santos === Subject: Re: Complex Made Simple: Update <87ws7hfv20.fsf@phiwumbda.org> posting-account=CtWhuAoAAAAZZ9vwdovdqB3NNaiUa20_ GTB5),gzip(gfe),gzip(gfe) > There's a certain logic to your placement, but it's nonetheless > awkward and unusual.Why not just re-word the sentence? Replace > should be with, say, should be as follows. >Yeah, there's a certain logic in the whole book. But somehow Dave > cultivates the unusual. For example on page 37 of Complex Made >Simple (which is in itself an oxymoron) >he deems the Fundamental >Theorem of Algebra to be a Corollary (3.7) instead of a famous >theorem. Huh? It is a corollary. I don't say that it's not a famous theorem. >And he calls it just an amusing digression on page 38. I say that it's just an amusing digression for our purposes , > at least at present . That's all precisely correct - it has no > particular application to any of the complex analysis we're > studying there. >Trivial theorem (HdB): >A polynomial of degree n assumes every complex value n times. >Proof: >polynomial - (a complex value) = another polynomial with n zeroes. >Now say that an entier function is a polynomial of degree infinity >and we're almost there for the Big Picard Theorem (20.1). Not so? Not so. Correction (not very serious but want to get some feeling for this) Now say that an entier function (not a polynomial of finite degree) is like a polynomial of infinite degree and we're almost there for the Big Picard Theorem (20.1) ( not the Little one, Tonio!). Then it's (intuitively) clear that such an entier function attains every complex value infinitely many times. A minor detail being that we are missing some exceptional values. But the Where are the zeroes thread shows that the latter are out of reach at infinity. Still not a chance, I suppose? Han de Bruijn === Subject: Re: Complex Made Simple: Update <79apu8F1m4gtlU1@mid.individual.net> <87ws7hfv20.fsf@phiwumbda.org> <87ws7hfv20.fsf@phiwumbda.org> <87ws7hfv20.fsf@phiwumbda.org> posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e 2.0.50727),gzip(gfe),gzip(gfe) > There's a certain logic to your placement, but it's nonetheless > awkward and unusual.Why not just re-word the sentence? Replace > should be with, say, should be as follows. Yeah, there's a certain logic in the whole book. But somehow Dave > cultivates the unusual. For example on page 37 of Complex Made > Simple (which is in itself an oxymoron) you? So you think David called his book the way he did just to be...witty? And so nice of you agreeing there's A certain logic in the whole book...reviews like this from people like you are any decent mathematician's dream. he deems the Fundamental > Theorem of Algebra to be a Corollary (3.7) instead of a famous > theorem. And he calls it just an amusing digression on page 38. > It indeed is a corollary, and in fact a rather trivial one, of Liouville's Theorem, and this is one reason more that makes complex analysis so fascinating and absorbing...for mathematicians, of course, and within the general context of the stuff being taught in David's book, and in practically any other complex analysis book I've ever met (Ahlfors', Bak & Newman's, Freitag and Busam's, etc.) it indeed is just a marginal result, very impressive for the previous knowledge one assumes students have of it, but still just a disgression within the context of the stuff being studied. Why do you sound so deeply hurt? > Trivial theorem (HdB): > A polynomial of degree n assumes every complex value n times. > Proof: > polynomial - (a complex value) = another polynomial with n zeroes. Now say that an entier function is a polynomial of degree infinity > and we're almost there for the Big Picard Theorem (20.1). Not so? > Well, YOU can say that, undoubtedly, though you'd probably mean the Little Picard theorem for entire functions (since you talk about polynomials...), but hey, it never minds: as long as polynomials of degree infinit [sic] is our theme, we can say lots of things. Tonio > Han de Bruijn === Subject: Re: Complex Made Simple: Update > Probably everyone but me would agree with that, but the placement > is intentional. Those entries read: > p. n, l. m: > display_1 > should be > display_2 > . > Here I'm using $$...$$ as means of _quotation_. The period > in question is not part of what display_1 should be changed > to, so it doesn't belong in display_2. There's a certain logic to your placement, but it's nonetheless >awkward and unusual. Why not just re-word the sentence? Replace >should be with, say, should be as follows. I know I know this one, just gimme a second... Probably because I spent _so_ much friggin time getting all this sort of thing just right for the book, and here it doesn't really matter? I _think_ that's it, anyway. === Subject: Re: Complex Made Simple: Update posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > If you have a copy you can now find a list of typos online - > go to > www.ams.org/bookstore?fn=20&arg1=gsmseries&item=GSM-97 > and click on the Supplementary Material. (The one that > really bothers me is this: The f in Theorem B in Chapter > 20 should be u.) >I suppose that you have better things to do than correcting the typos >from your list of typos, >but here it goes: >1) The full stops at the end of the corrections concerning the pages 273 >and 373 are misplaced. Probably everyone but me would agree with that, but the placement > is intentional. Those entries read: p. n, l. m: display 1 should be display 2 . Here I'm using $$...$$ as means of quotation . The period > in question is not part of what display 1 should be changed > to, so it doesn't belong in display 2. >My guess is that you meant to write just Errata. Heh. I never typed . Errata - I was formatting the thing as a > chapter using the tex things they provided, in order to get > the horizontal bars at the top of the pages, etc, and the period > appeared automatically (because there was no chapter number; > with a chapter number it would have been n. Errata.) There must be a way to fix that, but I couldn't motivate > myself to spend enough time looking for it. (Hmm, maybe > I could just use XXX or something for the chapter number...) >3) The minuses signs concerning line numbers (as in p. 239, l. -1) are >not actually minuses signs (they are hyphens) but they should be. I suppose... >Jose Carlos Santos David C. Ullrich Understanding Godel isn't about following his formal proof. > That would make a mockery of everything Godel was up to. > (John Jones, My talk about Godel to the post-grads. > in sci.logic.) Welcome to the real mathematics and not the fake Matrix movie sorry excuse for chess you've been playing for the last 20 years. Musatov (Founder, Inverse 19 Mathematics) Not the only, but one without it would not exist. QED Efforts by mathematicians to determine the value of pi since have led to the conclusion that it ... Unlike rational numbers, pi cannot be represented by a common fraction, .... But what about an uncommon fraction? Will you look at my drawing and in particular how the twelve points are the outside points of the sphere are all rationally reached by decimal numbers? 6.2035884*2 and doubled an infinite number of times should yield a constant decimal binary expansion (what I mean by this is the numbers to the right of the decimal point should always end in 4, then 8, then 6, then 2, then 4, then 8, then 6, then 2...repeating for infinity). So in binary, by starting with this number we predict for n>2 the infinite binary tree will end with decimal progression: 2, 4, 8, 6 For n<2 the infinite binary tree will begin and end (depends on if you are increasing or decreasing the value) 1, 5, 5, 5 (repeating 5 infinitely) as we decrease to infinitely small. Musatov Inverse Counting Chain of Infinite Proportions (sample) 0.0000473372528076171875 0.000094674505615234375 0.00018934901123046875 0.0003786980224609375 0.000757396044921875 0.00151479208984375 0.0030295841796875 0.006059168359375 0.01211833671875 0.0242366734375 0.048473346875 0.09694669375 0.1938933875 0.387786775 0.77557355 1.5511471 3.1022942 6.2035884 12.4071768 24.8143536 49.6287072 99.2574144 198.5148288 397.0296576 794.0593152 1588.1186304 3176.2372608 6352.4745216 12704.9490432 25409.8980864 50819.7961728 101639.5923456 203279.1846912 406558.3693824 813116.7387648 1626233.4775296 3252466.9550592 6504933.9101184 13009867.8202368 26019735.6404736 52039471.2809472 104078942.5618944 208157885.1237888 416315770.2475776 832631540.4951552 1665263080.9903104 3330526161.9806208 6661052323.9612416 13322104647.9224832 26644209295.8449664 53288418591.6899328 106576837183.3798656 213153674366.7597312 426307348733.5194624 852614697467.0389248 1705229394934.0778496 3410458789868.1556992 6820917579736.3113984 13641835159472.6227968 27283670318945.2455936 54567340637890.4911872 109134681275780.9823744 218269362551561.9647488 436538725103123.9294976 873077450206247.8589952 1746154900412495.7179904 3492309800824991.4359808 6984619601649982.8719616 13969239203299965.7439232 27938478406599931.4878464 55876956813199862.9756928 111753913626399725.9513856 223507827252799451.9027712 447015654505598903.8055424 894031309011197807.6110848 1788062618022395615.2221696 3576125236044791230.4443392 7152250472089582460.8886784 14304500944179164921.7773568 28609001888358329843.5547136 57218003776716659687.1094272 114436007553433319374.2188544 228872015106866638748.4377088 457744030213733277496.8754176 915488060427466554993.7508352 1830976120854933109987.5016704 3661952241709866219975.0033408 7323904483419732439950.0066816 14647808966839464879900.0133632 29295617933678929759800.0267264 58591235867357859519600.0534528 117182471734715719039200.1069056 234364943469431438078400.2138112 468729886938862876156800.4276224 937459773877725752313600.8552448 1874919547755451504627201.7104896 3749839095510903009254403.4209792 7499678191021806018508806.8419584 (sample) If you would like to do the honors Mr. Plouffe, I endear you to calculate the angles and lengths of the number of ways, it is quite obviously possible to square a circle based on the geometry and numerical progression. As is self-evident to anyone with even a modest understanding of mathematics and geometry after visual inspection of the drawn proof (http://www.flickr.com/photos/24869933@N07/3619373444/sizes/l/) it is quite easily accomplished to connect and triangulate the outer points of the rectangles to the outer points of the square and circle and even by dimension sphere. Yes, the drawing clearly by definition of inverse symmetry (take the drawing cut it in half at the equator and turn the paper over and rotate it and compare the top and bottom mirrored symmetry and you will see it is an exact inverse of symmetry itself) the drawing explains how to properly and mathematically soundly represent a fourth dimension on a two dimensional plane. In plane language if you take the picture and separate the top half of the sphere by the bottom half of the sphere they are identical but only if the sphere existed on a complete three dimensional plane through expansion. Since the paper is flat, this is an impossibility, nonetheless it is completely expressed and retractable and countable infinitely. By the definition of our standard we have established any k-digit approximation to pi is rational. Which could have many practical scientific applications. For instance, search: ON THE THEOREM OF IVA$EV-MUSATOV IIIx e F(l) + F(2) + ... + F(p). This remark is the basis of our proof of Theorem .... -y(M + 1), x' + y(M + 1)] = 0 it is easy to check that Conditions (i), ...... Now set i = pi x. *jj, v. Since p, x is positive,. (i) fi is positive, ... plms.oxfordjournals.org/cgi/reprint/s3-53/1/143.pdf REPRESENTATION OF FUNCTIONSBY SERIES AND CLASSES .83'(1)with measure ÄEÄ> 2.83ë -- .83Ì and such that the trigonometric series for g(x) converges uniformly on [0, 2.83ë]. .... trigonometric series are due to Ivashev-Musatov [40] who has shown that there are null series (in the sense of ..... function .83'(.83ï) e C(0, 1) with .83'(0) = .83'(1) = 0 such that ...... <*i *plonk* > muh brane i needs one- Hide quoted text - > - Show quoted text - NOTE THE PROGRESSION OF NUMBERS PELLS PROPORTION X^2=3 or 4 = 3 as the numbers progression, WOW if this does not proof this what will 1. Every odd number , number as Y value till infinity is exactly solvable with a proportionate result using the base Pells equation i.e 3,5,7,9,etc etc till infinty and the answer is as in 2(2^2)+1=3^2 and so on till infinity with a whole number integer. 2.Every even number , without exception, using Pells equation, the integer always is a .75 till infinity consistant as 63.75(2^4)=1=4^4 till infinity every integer will end with .75. is that the same proportion as 3 over 4, Y=3 for the least with numbers proportion There is a definite pattern by pells to numbers, and I note that in that pattern the number 19 proportion is strikingly proprtionate, and I clearly surmise that that circluar progression is the same as numbers for vector 19 progression. === Subject: Re: INVERSE zero versus VOID(NULL)zero > BE CIVIL if you are so secure in your null zero, Null > zero or Vouid zero, is an adaptaion into mathematical > theory from the east where there was a belief that > fire(agni) reduces everything into nothing, and > cleanses all and everything, purification. > > Since mathematics is a continuum in time dimension > in our clear opinion at inverse 19 research > > 1-1=-1(inverse zero), (-1)(1)=1, (-1)(0.5)=-1 , > 1-500=-1 > > (a)Inverse zero=-1= no time progression > > (b)+1 value is a prime C-constant of unit time > dimension progression by proportion. > > (c)(1)(1) is the basis for variable > expansion(multiplication) and its proportion is 19 > degrees proprtion exact (X^2 +1 is proportionate to > 19^2). > > (d) a circle has 19 equal proportions that are > proportionate to numbers, and the number 19 is > exactly proprtional to 19^2, proprtionate to 1. > > (e) There is a value for the maximamal proportionate > value, that is being developed and most probably is > (19^2)^19 which will define maximal speed proportion > and mass density, and maximum dilation per unit time, > with relation to inverse zero( this value plus > inverse zero is a prime discovery of inverse 19 > research) > > > > EQUALIZATION of numbers and degrees BY PELLS > EQUATION > > > 360(1^2)+1=19^2=361 > > 90(2^2)+1=19^2=361 > > We have by a rough draft paper proved to the journal > 19^2 value of Y absolute(i.e most definitly x= 2 , > and Y is =19 in the universe mathematical continuum > matrix). We have solved Pells 9 different ways . We > have no time now for a sophisticated challenge to > current mathematical theory ,since a 12 ton rock > monument to lost mathematics is being installed in > n Wisconsin to record this challenge > > The final equations and supporting diagrams as they > y appear on the monument will be posted in this > Mathematics forum in about 22 days from today, and > then we will answer all challenges to the inverse 19 > mathematical theory > > > Inverse 19 Proof: Did you mean: Topic: Your Radian and 2Pi value Replies: 2 Last Post: June 12, 2009 4:50 PM Search Thread: Advanced Search Reply to this Topic Watch this Topic Back to Topic List Jump to Tree View Messages: [ Previous | Next ] Topics: [ Previous | Next ]mensan No standard web pages containing all your search terms were found. Your search - Topic: Your Radian and 2Pi valueReplies: 2 Last Post: Jun 12, 2009 4:50 PM Search Thread: Advanced Search Reply to this Topic Watch this Topic Back to Topic List Jump to Tree View Messages: [ Previous | Next ] Topics: [ Previous | Next ]mensan - did not match any documents. Suggestions: Make sure all words are spelled correctly. Try different keywords. Try more general keywords. Try fewer keywords. Musatov (co-founder Inverse 19 Mathematics) not the only but without it would not exist === Subject: Re: Project Euler Problem 248 posting-account=DSjMzAoAAAAISwlGnTY2c6bnmUVmdFNf AppleWebKit/530.17 (KHTML, like Gecko) Version/4.0 Safari/530.17,gzip(gfe),gzip(gfe) > The smallest number with phi (n) = 13! is n = 6,227,180,929 = 66529 * > 93601 > Number 100,000 is n = 18,817,814,916 = 673 * 1301 * 199 * 27 * 4 > The largest of 171,821 numbers is 37,020,293,310 = 19 * 31 * 23 * 169 > * 11 * 49 * 5 * 3 * 2 Wow -- how did you calculate that?I can see that there are two or > three classes of numbers which could give this initial result, and > from there it is a multiplicative problem....but how can I determine > all of the base cases, rather than the ones found via brute-force? Ok, here goes: We want to find all the numbers x such that phi (x) = n, in this case for n = 13!. Each x is made up of the product of distinct primes raised to some power p^k, and the values phi (p^k) = (p-1)*p^(k-1), multiplied together, must give n. First step is finding all the values p^k where phi (p^k) divides n. We first find all the prime factors of n, here p = 2, 3, 5, 7, 11, 13. If p^k divides n, and (p-1) divides n, then p, p^2 etc. up to p^(k+1) are candidates for factors of x. That's one class of possible factors. For the other class of factors, we take all numbers d>1 dividing n, and if (d+1) is a prime and (d+1) is not one of the prime factors of n, then d+1 is also a possible factor of x. That catches all possible factors of x. In the case of n = 13!, there are about 500-600 primes which are one larger than a number dividing 13!. We split these numbers d into sets depending on their largest prime factor. Now we systematically multiply factors together to get phi (x) = n. We start with x = 1, phi (x) = 1. In the case of n = 13!, a prime factor of 13 is missing. So we have the following choices: We can use any of the primes d+1 where d divides n and the largest prime factor of d is 13, multiply x by d+1 and multiply phi by d. Or we can use the factor 13^2, multiplying x by 13*13 and phi by 13*12. If we used one of the factors d, we can in addition use the factor 13 (multiply x by 13, phi by 12) because once phi has a factor 13 the higher powers of 13 are not used anymore. For each of these cases we do exactly the same thing for the prime factor 11, except when we look at a prime d+1 where d has the largest prime factor 11, we have to check that phi*d still divides n. Then the same for the lower primes. If phi has a factor p^2 or a higher power of p missing, then we can use multiple different values d, until we either have the necessary powers of p, or we can finally add another factor p^2, p^3 etc. Eventually we end up with phi = n. In addition, we ignored that x could have a factor 2 which doesn't affect phi, so if we end up with an odd x where phi (x) = n, then phi (2x) = n as well. Given n > 1: Let m be the number of distinct prime factors of n. Let p1, p2, ..., pm be the distinct prime factors of m in ascending order. Let D by the set of factors d > 1 of n such that d+1 is a prime that doesn't divide n. Let D(k) be the set of elements d of D where the largest prime factor of d is p(k), for 1 <= k <= m. Recursively perform the function Enumerate, starting with Enumerate (1, 1, m, card (D (m)). Function Enumerate with arguments x, phi, m, c: If m = 0: Output x. If x is odd then output 2x. Exit the function. If p (m) divides n / phi: For all 1 <= i <= c: Let d = D (m, i). If d divides n / phi then Enumerate (x * (d+1), phi * d, m, i-1). If p (m) does not divide n / phi then Enumerate (x, phi, m-1, card (D (m-1)). If p (m-1) divides n / phi: Let x = x * p (m), let phi = phi * (p (m) - 1). While p (m) divides n / phi let x = x * p (m) and phi = phi * p (m). Enumerate (x, phi, m-1, card (D (m-1)). === Subject: Re: Project Euler Problem 248 posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) On Jun 12, 4:38pm, christian.bau The smallest number with phi (n) = 13! is n = 6,227,180,929 = 66529 * > 93601 > Number 100,000 is n = 18,817,814,916 = 673 * 1301 * 199 * 27 * 4 > The largest of 171,821 numbers is 37,020,293,310 = 19 * 31 * 23 * 169 > * 11 * 49 * 5 * 3 * 2 > Wow -- how did you calculate that?I can see that there are two or > three classes of numbers which could give this initial result, and > from there it is a multiplicative problem....but how can I determine > all of the base cases, rather than the ones found via brute-force? Ok, here goes: We want to find all the numbers x such that phi (x) = > n, in this case for n = 13!. Each x is made up of the product of > distinct primes raised to some power p^k, and the values phi (p^k) = > (p-1)*p^(k-1), multiplied together, must give n. First step is finding > all the values p^k where phi (p^k) divides n. We first find all the prime factors of n, here p = 2, 3, 5, 7, 11, 13. > If p^k divides n, and (p-1) divides n, then p, p^2 etc. up to p^(k+1) > are candidates for factors of x. That's one class of possible factors. > For the other class of factors, we take all numbers d>1 dividing n, > and if (d+1) is a prime and (d+1) is not one of the prime factors of > n, then d+1 is also a possible factor of x. That catches all possible > factors of x. In the case of n = 13!, there are about 500-600 primes > which are one larger than a number dividing 13!. We split these numbers d into sets depending on their largest prime > factor. Now we systematically multiply factors together to get phi (x) = n. We > start with x = 1, phi (x) = 1. In the case of n = 13!, a prime factor > of 13 is missing. So we have the following choices: We can use any of > the primes d+1 where d divides n and the largest prime factor of d is > 13, multiply x by d+1 and multiply phi by d. Or we can use the factor > 13^2, multiplying x by 13*13 and phi by 13*12. If we used one of the > factors d, we can in addition use the factor 13 (multiply x by 13, phi > by 12) because once phi has a factor 13 the higher powers of 13 are > not used anymore. For each of these cases we do exactly the same thing for the prime > factor 11, except when we look at a prime d+1 where d has the largest > prime factor 11, we have to check that phi*d still divides n. Then the > same for the lower primes. If phi has a factor p^2 or a higher power of p missing, then we can > use multiple different values d, until we either have the necessary > powers of p, or we can finally add another factor p^2, p^3 etc. > Eventually we end up with phi = n. In addition, we ignored that x > could have a factor 2 which doesn't affect phi, so if we end up with > an odd x where phi (x) = n, then phi (2x) = n as well. Given n > 1: Let m be the number of distinct prime factors of n. > Let p1, p2, ..., pm be the distinct prime factors of m in ascending > order. Let D by the set of factors d > 1 of n such that d+1 is a prime that > doesn't divide n. > Let D(k) be the set of elements d of D where the largest prime factor > of d is p(k), for 1 <= k <= m. > Recursively perform the function Enumerate, starting with Enumerate > (1, 1, m, card (D (m)). Function Enumerate with arguments x, phi, m, c: >If m = 0: Output x. If x is odd then output 2x. Exit the function. If p (m) divides n / phi: > For all 1 <= i <= c: > Let d = D (m, i). > If d divides n / phi then Enumerate (x * (d+1), phi * d, m, > i-1). If p (m) does not divide n / phi then Enumerate (x, phi, m-1, card > (D (m-1)). If p (m-1) divides n / phi: > Let x = x * p (m), let phi = phi * (p (m) - 1). > While p (m) divides n / phi let x = x * p (m) and phi = phi * p > (m). > Enumerate (x, phi, m-1, card (D (m-1)). usepackage{amsmath, amssymb, graphics} newcommand{mathsym}[1]{{}} newcommand{unicode}{{}} begin{document} noindent(pmb{13!}) noindent(pmb{text{FactorInteger}[6227020800]}) noindent({{2,10},{3,5},{5,2},{7,1},{11,1},{13,1 }}) end{document} === Subject: Re: Project Euler Problem 248 > Could some one please help me how to solve this problem. > The first number n for which .9a(n)=13! is 6227180929. > Find the 100,000^(th) such number. > This is project euler problem and I have no idea how to solve > this.http://projecteuler.net/index.php?section=problems&id=248 The smallest number with phi (n) = 13! is n = 6,227,180,929 = 66529 * > 93601 > Number 100,000 is n = 18,817,814,916 = 673 * 1301 * 199 * 27 * 4 Hummm... Are you sure? -- Ignacio Larrosa Ca.96estro A Coru.96a (Espa.96a) ilarrosaQUITARMAYUSCULAS@mundo-r.com > The largest of 171,821 numbers is 37,020,293,310 = 19 * 31 * 23 * 169 > * 11 * 49 * 5 * 3 * 2 === Subject: Re: Project Euler Problem 248 posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > The smallest number with phi (n) = 13! is n = 6,227,180,929 = 66529 * > 93601 > Number 100,000 is n = 18,817,814,916 = 673 * 1301 * 199 * 27 * 4 > The largest of 171,821 numbers is 37,020,293,310 = 19 * 31 * 23 * 169 > * 11 * 49 * 5 * 3 * 2 Wow -- how did you calculate that?I can see that there are two or > three classes of numbers which could give this initial result, and > from there it is a multiplicative problem....but how can I determine > all of the base cases, rather than the ones found via brute-force? > The smallest number with phi (n) = 13! is n = 6,227,180,929 = 66529 * > 93601 > Number 100,000 is n = 18,817,814,916 = 673 * 1301 * 199 * 27 * 4 > The largest of 171,821 numbers is 37,020,293,310 = 19 * 31 * 23 * 169 > * 11 * 49 * 5 * 3 * 2 Wow -- how did you calculate that? I can see that there are two or > three classes of numbers which could give this initial result, and > from there it is a multiplicative problem....but how can I determine > all of the base cases, rather than the ones found via brute-force? Like this: I will explain to you what I am after. I am after a new theory, a new way of programming, a new way of measuring, a new way of communicating. I am talking about an end to the old tired way of 'P', if 'P' then 'Q' innefficiency in programming and language (and practical matters too). I assert my findings (all the following text up to (c) 2009 Martin Musatov (the 2nd time) it is read in this thread: 'X' if 'X' The idea is we create a thread. The first programming variable means nothing. It is an instruction to the system script. It is an open and I said> order of logic. 'X' alone simply means Open or I am talking or Listen up, it can be followed by anything, or an infinite number of things. And the idea of 'X' is arbitrary, as 'X' could be replaced by any other logical symbol or character in any language and the idea still stands. The operation still works. Let me explain: 'X' when I say 'X'. 'X' is the master open 'command' key or 'dialogue'. It sets up a system open script. We then can open any operation simply by repeating it. The system is not loyal to any language or frame. (or insert proprietary here) If I want to open up a program Y, I simply say, X, Y I want Y. The system knows that if a variable is repeated it means to execute it under the terms programmed. (or by pressing a key) Let me explain further (further): A formal illustration of how it would look in a system would be: X, Y, Y, X The first 'X' means open the root command, the first 'Y' means, I want 'Y', the second 'Y' means it is just 'Y' I seek and the 'X' closes to command. The platform solution requires complete symmetry in programming and logic threads. The programming implementation allows the setup of a system where repetition of a variable in symmetry order allows command. In football the command hut means pitch me the ball back. The Quarterback will communicate to his teammates, On the 2nd hut throw the ball. The first hut is to get set. Still 'hut' is the command to throw the ball. The real world variable and how it relates to the equivalency in programming is explained as follows: Instead of saying (1)Listen up....(2)I want to open my word processor.... X 1 1 X where (1)=the first command 1 and (2)=11 ***********Note in binary*: the machine literally must count the two ones and add them to get two.************) User inputs:(1)I want my word processor (2)I want my word processor and this means to the machine Open up the word processor... only when you read (1) I want my word processor (2) I want my word processor (forgive my terminology, I am simply being plain) but this inside/out programming illustrates perfectly my point. and forgive my for being so bold but this logic clear, simple, and plain, simply resolves P=NP and overcomes the halting problem irrefutably the solution to P=NP or P Versus NP, or N=NP or -1=1 or however you want to frame the problem is simply this statement: Do not do anything until I say it twice. Advanced Symmetry application: the solution to the machine is the overlooked symmetry. A complex command may look like: 'XYZZYX' which would mean, I want the YZ complex variable please Stephen Arthur Cook was right, the whole thing was simply due to lack of ingenuity on behalf of our programmer. (C)2009. Martin Musatov. All Rights Reserved in perpetuity*. http://www.MeAmI.org Search for the People *Applies to derviations of this work. Including software, programming, and other profit generating means extracted from this work. Work may be patent pending. Reply Reply to author Forward === Subject: Injection and surjection If X and Y are nonempty sets, and f : X -> Y is surjective, is there an injective map Y --> X ? If so, how does one construct such an injection? === Subject: Re: Injection and surjection Illustration proof http://farm4.static.flickr.com/3496/3256947403_398e26dde8_o.jpg === Subject: Re: Injection and surjection posting-account=_l4K0QkAAAC09JhOoK_ZfoJKXOmr_jZf Gecko/2009060215 Firefox/3.0.11 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > If X and Y are nonempty sets, andf : X -> Y is surjective, is there an injective mapY --> X ? If so, how does one construct such an injection? Assuming the Axiom of Choice, for each y in Y you select one x in f^ {-1}(y) and define g(y) to be that x. surjectivity of f guarantees that f^{-1}{y} is nonempty, and the fact that f is a function guarantees that f^{-1}(y) is disjoint from f^{-1}(z) whenever y=/=z, thus establishing that g is an injection. This construction does more than simply show there is an injective map Y->X: it shows that there is a left inverse for f; that every surjective function has a left inverse is in fact equivalent to the Axiom of Choice, so your question may have a less-heavy-handed solution. -- Arturo Magidin === Subject: Re: Injection and surjection > On Jun 12, 6:53pm, qsymmetry If X and Y are nonempty sets, andf : X -> Y is > surjective,is there an injective mapY --> X ? > If so, how does one construct such an injection? > > Assuming the Axiom of Choice, for each y in Y you > select one x in f^ > {-1}(y) and define g(y) to be that x. I am not sure why we need the axiom of choice here: if f is surjective, then f^{-1}(y) is nonempty, by definition. Perhaps I've overlooked something? === Subject: Re: Injection and surjection <6366016.10725.1244870290553.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=_l4K0QkAAAC09JhOoK_ZfoJKXOmr_jZf Gecko/2009060215 Firefox/3.0.11 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > On Jun 12, 6:53pm, qsymmetry If X and Y are nonempty sets, andf : X -> Y is > surjective,is there an injective mapY --> X ? > If so, how does one construct such an injection? > Assuming the Axiom of Choice, for each y in Y you > select one x in f^ > {-1}(y) and define g(y) to be that x. I am not sure why we need the axiom of choice here: > iff is surjective,then f^{-1}(y) is nonempty, by definition. You need AC in order to define g for all y; that is, you need to make one choice for *each* y in Y, and that may involve making infinitely many arbitrary choices. Choosing a single x in f^{-1}(y) for a single y does not, as you note, require AC; but picking a single x from possibly infinitely many such y does. For example, if f were the function from the infinitely many pairs of socks into the natural numbers Y, and you g were meant to be choose a single sock from each pair; picking a single sock from any particular pair does not require choice, but making a choice of a sock for *all* pairs does. You may avoid the use of AC under some circumstances: if Y is finite, or if X if well-ordered (or if you can establish it is well-orderable without invoking choice). But in the arbitrary case, Choice is needed to guarantee that you can define g on all of Y. Also, don't overlook that my g is more than you are asking for; I honestly do not remember whether the existence of an *arbitrary* injection Y->X given a surjection X->Y requires AC or not; I know that AC is equivalent to the existence of a *right inverse* to the surjection X->Y (and a right inverse is necessarily 1 to 1), but that is more than you were asking for. -- Arturo Magidin === Subject: Re: Injection and surjection <6366016.10725.1244870290553.JavaMail.jakarta@nitrogen.mathforum.org I am not sure why we need the axiom of choice here: > if f is surjective, then f^{-1}(y) is nonempty, by definition. Perhaps I've overlooked something? Nonemptiness is not enough. That gives you a proof that for any given y in Y, there exists a function with a suitable value for that particular y. What you need to prove is that a suitable function exists that has suitable values for all y in Y at once. If Y has some special properties you can do it without the axiom of choice - e.g. if there is some predicate that distinguishes some value in each f^-1(y) set from the others. Well-ordering is one such property, so you can do without AC if Y is well-orderable. Countable is a special case of well-orderable, so that would suffice. - Tim === Subject: Re: Injection and surjection <6366016.10725.1244870290553.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=_l4K0QkAAAC09JhOoK_ZfoJKXOmr_jZf Gecko/2009060215 Firefox/3.0.11 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > I am not sure why we need the axiom of choice here: > iff is surjective,then f^{-1}(y) is nonempty, by definition. > Perhaps I've overlooked something? Nonemptiness is not enough.That gives you a proof that for any given > y in Y, there exists a function with a suitable value for that > particular y.What you need to prove is that a suitable function > exists that has suitable values for all y in Y at once. If Y has some special properties you can do it without the axiom of > choice - e.g. if there is some predicate that distinguishes some value > in each f^-1(y) set from the others.Well-ordering is one such > property, so you can do without AC if Y is well-orderable. Having Y well-ordered does you no good; if X were well-orderable, however, then you'd be in business. Your choices are being taken from the family {f^{-1}(y) : y in Y} of subsets of X; so it is X (or the sets in the family) that need to have nice properties in order for the choices to be possible. One such condition would be finitely many sets (that is, Y is finite). But if Y is infinite, I do not think there is any condition on Y that would guarantee a left inverse for an arbitrary surjection into Y. -- Arturo Magidin === Subject: Re: Injection and surjection > I am not sure why we need the axiom of choice here: > if f is surjective, then f^{-1}(y) is nonempty, by definition. > Perhaps I've overlooked something? Nonemptiness is not enough. That gives you a proof that for any given >y in Y, there exists a function with a suitable value for that >particular y. What you need to prove is that a suitable function >exists that has suitable values for all y in Y at once. I think that by there exists a function with a suitable value you mean that we can choose an x in X. That is, for each y in Y, there is at least one x in X so that f(x) = y, and we could choose one to define f^{-1}(y) = x. We can do this for any finite subset of Y, but if Y is infinite, and if we don't have some prior way of specifying exactly which x in X to choose for each y in Y, we need the Axiom of Choice. >If Y has some special properties you can do it without the axiom of >choice - e.g. if there is some predicate that distinguishes some value >in each f^-1(y) set from the others. Well-ordering is one such >property, so you can do without AC if Y is well-orderable. Countable >is a special case of well-orderable, so that would suffice. Isn't it X that needs a special property, such as a well-order, so that we don't need the axiom of choice? That way, given the set of x in X so that f(x) = y, we can choose the least such x and say that f^{-1}(y) is that least x. Being well-orderable means that at least one such order exists, but to define f in this way, we need to choose one of those orders. For example, suppose we have a countable set of marbles, two marked 1, two marked 2, two marked 3, and so on, so that all marbles are so marked, but otherwise identical. We have a simple surjective map from this set of marbles to the positive integers. However, even though there is a countable set of marbles, and only two marbles for each integer, we still need the axiom of choice to choose one marble for each positive integer in order to construct the inverse function from the the positive integers to this set of marbles. Rob Johnson take out the trash before replying === Subject: Re: Injection and surjection > On Jun 12, 6:53pm, qsymmetry If X and Y are nonempty sets, and f : X -> Y is > surjective, is there an injective map Y --> X ? > If so, how does one construct such an injection? > > Assuming the Axiom of Choice, for each y in Y you > select one x in f^ > {-1}(y) and define g(y) to be that x. > > > > I am not sure why we need the axiom of choice here: > if f is surjective, then f^{-1}(y) is nonempty, by definition. That's true, but you need the axiom of choice to put all the choices together (one for each y) to construct the function g. Ted === Subject: Re: Injection and surjection <6366016.10725.1244870290553.JavaMail.jakarta@nitrogen.mathforum.org>, > On Jun 12, 6:53.8cpm, qsymmetry If X and Y are nonempty sets, and .8cf : X -> Y is > surjective, .8cis there an injective map .8cY --> X ? > If so, how does one construct such an injection? > > Assuming the Axiom of Choice, for each y in Y you > select one x in f^{-1}(y) and define g(y) to be that x. > > > > I am not sure why we need the axiom of choice here: > if f is surjective, then f^{-1}(y) is nonempty, by definition. > > Perhaps I've overlooked something? There is the problem of picking a single element from the possibly very large set f^{-1}(y). Picking an arbitrary element from a arbitrary set of more than one element in the absence of an axiom of choice is not a trivial problem. Now if Y were already well-ordered... -- Virgil === Subject: Re: Injection and surjection <6366016.10725.1244870290553.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=_l4K0QkAAAC09JhOoK_ZfoJKXOmr_jZf Gecko/2009060215 Firefox/3.0.11 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > <6366016.10725.1244870290553.JavaMail.jaka...@nitrogen.mathforum.org>, > On Jun 12, 6:53pm, qsymmetry If X and Y are nonempty sets, andf : X -> Y is > surjective,is there an injective mapY --> X ? > If so, how does one construct such an injection? > Assuming the Axiom of Choice, for each y in Y you > select one x in f^{-1}(y) and define g(y) to be that x. > I am not sure why we need the axiom of choice here: > iff is surjective,then f^{-1}(y) is nonempty, by definition. > Perhaps I've overlooked something? There is the problem of picking a single element from the possibly very > large set f^{-1}(y). No, that's not it. Picking a single element form an arbitrary nonempty set does not require the Axiom of Choice, no matter how large the set is. The reason you need the axiom of choice here is that if the set Y is large, then you need to make *many* simultaneous choices, one for each y in Y. > Picking an arbitrary element from a arbitrary set of more than one > element in the absence of an axiom of choice is not a trivial problem. No; picking an arbitrary element from an arbitrary nonempty set does not require AC. > Now if Y were already well-ordered... That is also no good; if X were well ordered, however, then you would be fine, since you could define g(y) to be the least element of f^{-1} (y). -- Arturo Magidin === Subject: Re: Injection and surjection <6366016.10725.1244870290553.JavaMail.jakarta@nitrogen.mathforum.org> element in the absence of an axiom of choice is not a trivial problem. It is a trivial problem, solvable essentially by an application of E-elimination. I suspect you didn't quite say what you intended. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Injection and surjection posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > If X and Y are nonempty sets, andf : X -> Y is surjective,is there an injective mapY --> X ? > If so, how does one construct such an injection? Assuming the Axiom of Choice, for each y in Y you select one x in f^ > {-1}(y) and define g(y) to be that x. surjectivity of f guarantees > that f^{-1}{y} is nonempty, and the fact that f is a function > guarantees that f^{-1}(y) is disjoint from f^{-1}(z) whenever y=/=z, > thus establishing that g is an injection. This construction does more than simply show there is an injective map > Y->X: it shows that there is a left inverse for f; that every > surjective function has a left inverse is in fact equivalent to the > Axiom of Choice, so your question may have a less-heavy-handed > solution. -- > Arturo Magidin > If X and Y are nonempty sets, and f : X -> Y is surjective, is there an injective map Y --> X ? > If so, how does one construct such an injection? Assuming the Axiom of Choice, for each y in Y you select one x in f^ > {-1}(y) and define g(y) to be that x. surjectivity of f guarantees > that f^{-1}{y} is nonempty, and the fact that f is a function > guarantees that f^{-1}(y) is disjoint from f^{-1}(z) whenever y=/=z, > thus establishing that g is an injection. This construction does more than simply show there is an injective map > Y->X: it shows that there is a left inverse for f; that every > surjective function has a left inverse is in fact equivalent to the > Axiom of Choice, so your question may have a less-heavy-handed > solution. -- > Arturo Magidin Q: If X and Y are nonempty sets, and f : X -> Y is surjective, is there an injective map Y --> X ? If so, how does one construct such an injection? A: Countable sets Text - Physics Forums Library13 posts Well, for any set X, the empty map is an injection of emptyset into that ... so we biject a set X onto a subset Z of Y to show that cardX<=cardY ... if there exists a bijection, i.e. an injective and surjective function, from A to B. [. ... B and forall a in A there is exactly one b in B such that (a,b) in F. ... http://www.physicsforums.com/archive/index.php/t-115630.html ++ Musatov (Inverse 19) === Subject: Re: Injection and surjection posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > If X and Y are nonempty sets, andf : X -> Y is surjective, is there an injective mapY --> X ? If so, how does one construct such an injection? Like this: On Aug 17 1990, 5:15 am, n...@ccncsu.ColoState.EDU (USENET news) > An extraordinary recursive factoring algorithm has been discovered (?) by > Risto Lankinen (ri...@yj.data.nokia.fi) based on mathematical properties he > which finds all natural numbers u, v such that u * v = n for an odd n, is > as follows: 1. select an odd n > 1. > 2. u <- 1, v <- 1, b <- 1, n <- (n - 1). > 3. n <- (n / 2). > 4. if n = 0, (then) u * v = n; return. > 5. if n < 0, (then) u * v <> n; return. > 6. b <- (b * 2). > 7. if n is odd, (then) > 7a. restart at 3 with n <- (n - u), v <- (v + b), > 7b. and again with n <- (n - v), u <- (u + b). > 8. (otherwise) for even n, > 8a. restart at 3, > 8b. and again with n <- (n - u - v - b), u <- (u + b), v <- (v + b). Execution time is comparable to the technique of division by all odd numbers > less than the square root of n. > - - - Following is an implementation of Lankinen's algorithm, with embellishments, > in the C language: DIVISOR.C 1: #include 3: void factor(long remain, > 4: long left, > 5: long right, > 6: long bit) > 7: > 8: { > 9: if (remain >= 1) > 10: { > 11: if (remain > 0) > 12: { > 13: bit <<= 1; > 14: if (remain & 1) > 15: { > 16: factor(remain - left, left, right | bit, bit); > 17: if (left != right) > 18: factor(remain - right, right, left | bit, bit); > 19: > 20: } > 21: else > 22: { > 23: factor(remain, left, right, bit); > 24: factor(remain - left - right - bit, left | bit, right | bit, bit); > 25: } > 26: } > 27: } > 28: else > 29: printf(%li %lin, left, right); > 30: > 31: } > 32: > 33: void main(void) > 34: > 35: { > 36: long number; > 37: > 38: scanf(%li, &number); > 39: while (!(number & 1)) > 40: number >= 1; > 41: factor(number, 1, 1, 1); > 42: } The correspondence is as follows: step -> lines | variables > ---- ----- | --------- > 1 38-40 | n -> number, remain > 2 41,9 | u left > 3 9 | v right > 4 9,28-29 | b bit > 5 11,21-25 | > 6 13 | > 7 14,15-20 | > 7a 16 | > 7b 18* | > 8 14,21 | > 8a 23 | > 8b 24 | * Embellishment, see notes below. The C language is ideal for implementing Lankinen's algorithm because both are > optimized for binary machine languages. Hence many arithmetical operations in > the algorithm can be replaced with C's bitwise operators for maximized > computational efficiency: This function at [line] replaces at [step] (respectively). > ---- -------- ------ -------- ------ > `|' (binary OR) 16,18,24 addition 7a,7b,8b > `>' (shift right) 9, 40 division 3 (1 implicit) > '<<' (shift left) 13 multiplication 6 Moreover, the parity test at steps 7 and 8 can be reduced to simply branching > on the state of the rightmost bit of the variable at lines 14 and 21. Recall > the existence of the implicit nonzero test in C, utilized in lines 14 and 39, > which can even be combined with assignment as at 9 (here some compilers may > warn of a possibly unintentional assignment instead of comparison). This C routine embellishes steps 7 and 8 in Lankinen's algorithm at lines 16-18 > and 23-24: * The conditional statement at line 17 decreases total processing time roughly > by a factor of two by avoiding a recursive call that results in finding > reversed (duplicate) pairs. Hence at line 29 the pair (left,right) is > unique; for no (u,v) will the program also report (v,u). (If the trial > factors are equivalent to some point, the recursive branches originating > there would be symmetric with respect to each other.) * Lankinen left unspecified the order of the two recursive calls in steps 7 > and 8 (they were assigned definite order above for exposition), but when 7 is > encoded as above, in concert with the test at 17 and the implied interchange > of trial factors at 18 (compare with step 7b), an ordering is established > such that for all divisor pairs (left,right) found at line 29, left <= right. - - - Lankinen's basic algorithm can be modified slightly to determine the complete > prime decomposition of a number: 1: #include 3: #define factor(what) find(what, 1, 1, 1, what) > 4: > 5: long find(long remain, > 6: long left, > 7: long right, > 8: long bit, > 9: long now) > 10: > 11: { > 12: if (right > now) > 13: return now; > 14: > 15: if (remain >= 1) > 16: { > 17: if (remain > 0) > 18: { > 19: bit <<= 1; > 20: if (remain & 1) > 21: { > 22: if (left != right) > 23: now = find(remain - right, right, left | bit, bit, now); > 24: return find(remain - left, left, right | bit, bit, now); > 25: > 26: } > 27: else > 28: return find(remain, > 29: left, > 30: right, > 31: bit, > 32: find(remain - left - right - bit, > 33: left | bit, > 34: right | bit, > 35: bit, > 36: now)); > 37: } > 38: else > 39: return now; > 40: } > 41: else > 42: if (now > 1) > 43: if (left == 1) > 44: printf(%li , right); > 45: else > 46: { > 47: factor(left); > 48: factor(right); > 49: now /= right; > 50: } > 51: > 52: return now; > 53: } > 54: > 55: void main(void) > 56: > 57: { > 58: long number; > 59: > 60: scanf(%li, &number); > 61: while (!(number & 1)) > 62: { > 63: printf(2 ); > 64: number >= 1; > 65: } > 66: if (number > 1) > 67: factor(number); > 68: } The function `factor' is replaced by a macro that calls on the new function > `find' that decomposes an odd number into its prime factors. This program > utilizes `intercommunication' between the branches of the recursive tree. > Whenever `factor' is started, the macro (line 3) places the original n in > the `now' variable (line 9). This variable represents the current unfactored > part of n (the original number divided by all actual factors found at some > point). It is passed from `parent' to `child' process as a parameter and as > the function result in the opposite direction. Then if the tentative factor under consideration is ever greater than `now', > it is impossibly large. The comparison for this test occurs at line 12. > At that point both `left' and `right' are trial factors but again right >= left > by ordering so only `right' need be tested. The division occurs at line 49. With adjustment to the unspecified ordering in the algorithm at lines 22-24 > and 28-36, there is an implicit guarantee that the divisor that is the number > itself (left = 1, right = n) will be found last. Hence, if an unfactored part > remains (line 42), either it is composite and all factors will be found > recursively at lines 47-49, or it is prime and reported at line 44. The `find' routine can be rewritten in several other forms. In this version > the `else' keywords at lines 38 and 41 are redundant and should be removed > if they improve the efficiency of the compilation (they were included here > for clarity). Also, the routine is readily modified to use a single exit point > at line 52 by eliminating the other three at lines 13, 24, and 28 with block > scoping and assignment to `now' (the version above attempts to maximize speed). - - - For a more rigorous mathematical treatment of Lankinen's approach to factoring, > see his preliminary findings in <6...@tuura.UUCP>. The description that follows > is intended to be more intuitively appealing than logically precise. Lankinen's algorithm can be thought of as the `inverse' of binary > multiplication (otherwise known as the `Russian Peasant's Method') > in the special case where both the multiplier and multiplicand are odd > (and hence the product also). The algorithm simply explores all possibilities in u and v that could match n > by focusing on the 2nd bit from the right and advancing left one bit at a time. > (The rightmost bit in n is matched immediately because both factors are odd.) > The fundamental rule is employed that if the next bit in n is 1, then it is > matched only when the next bits in u and v are different from each other, > and if the digit is 0 the bits must match. In either case, there are only two > possibilities, each of which is pursued recursively. As the trial divisors accumulate, so does a carry that has to be accounted for > in the subsequent comparisons. This is accomplished exactly and conveniently > by subtracting the newly accumulated amount from n prior to entering each > recursive branch. If n < 0, the product of the trial factors has accrued a > carry that exceeds the original n to be factored, thereby eliminating them. > Otherwise, if n = 0, they are actual factors. ld231...@longs.LANCE.ColoState.EDU Musatov === Subject: Re: .9 repeating <29753597.71734.1241733955686.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > <29753597.71734.1241733955686.JavaMail.jaka...@nitrogen.mathforum.org>, > Please help me why this proof is incorrect: > let x = 0.9... > 10x = 9.9... > 10x -x = 9x > 9x = 9.9... - 0.9... = > 9 > and 9x / 9 = 1 > so > 1 = 0.9.... > It is correct (in the real numger system). Of course all the steps > involved should have been previously explained. Such as the meaning > and convergence of the series represented by those dots. > -- > G. A. Edgar http://www.math.ohio-state.edu/~edgar/ But it was not. Base logic defies numerical characteristic. Here are your infinite strings of 9's: (in long division) Math Forum Discussions - Re: A Formula For Constant PrimesMusatov's > theorem: > 2 and 3 are relatively and constantly prime to > infinity Áfi > 2*2=4 > 3*3=9 > 4*9=36+1=37 is > prime. ...http://mathforum.org/kb/thread.jspa%3FforumID%3D13%26threadID > %3D1933132%26messageID%3D6709333 > Math Forum Discussionsinfinite amount of mass exists. ANYTHING > infinitely massive cannot ... Re: P=NP Proof Published at CERNOn 9 > mayo, 05:18, Martin Musatov. thread.jspa%3FthreadID%3D1938110%26tstart%3D135 > Stable jurisdiction partitions under monotonically decreasing ...live > on the segment [0,F] and transportation costs go to infinity on > its ..... 9 . If C is not blocking then population of either A or B is > greater than. 9 ...http://www.ecares.org/ecaresdocuments/ws/ pm ...http://www.groupsrv.com/science/about413926.htmlvalue of second- > order logic - sci.math | Google Groups for symbols=the initials of > Martin Michael Musatov=the initials of an infinite numbers of people > with first middle and last names all ... series...(Infinite variable > assignment) ... Local: Thurs, Jun 11 2009 9:34 pm ...http:// > Google Groups This is a method from which to calculate infinite sets > of prime numbers, ... ending in 1,3,5,7,9 that were multiplied by > numbers ending in ... Martin M. Musatov. How is this in any way an > group/alt.math/browse thread/thread/ > 7d5b2f468335ef52/4084bcf09ac40121%3Flnk%3Draot pravila-07.vp We assume > that an infinite curved regular surface divides the space into and > its ... Analytical integration in the second term of the series (9) > gives ... Klem-Musatov K. D. Theory of Edge Waves. -- Novosibirsk: > Nauka Press, 1980. 296 p. ...http://www.sscc.ru/Conf/mmg2008/papers/ > Aizenberg.doc > Re: Artinian non-noetherian module 10 Jun 2009 ... started with: the p- Hosting and Rack Housing from 9 Net Avenue Deutschland. ...http:// > sci.tech-archive.net/Archive/sci.math/2009-06/msg01261.html > Phys. Rev. D 56, 2713 (1997): Musatov and Radyushkin - Transverse ... > 26 is equivalent to incorporating the infinite momentum frame. ...... > 4.12 in k 2 /Q 2 one would get an infinite series of power corrections > under the x integral. ..... 2735 9 D. J. Gross and S. B. Treiman, > Phys. Rev. D 4, 2105 1971 . ...http://link.aps.org/doi/10.1103/ > PhysRevD.56.2713 Shtetl-Optimized > Blog Archive > The LEGO Turing > machine Comment #3 March 2nd, 2009 at 9:43 pm. Scott, if blogging mode > is on, what do you think ... Martin M. Musatov Says: Comment #6 March > 2nd, 2009 at 11:24 pm .... Scott says: going to infinite-dimensional > Hilbert spaces ... takes you even ...http://scottaaronson.com/blog/%3Fp > %3D387 123456789 Inverse19 the key is Hope | > | > |Please do not cease contact. I apologize if | > |my repost of our correspondence betrayed | > |your trust or impacted your assessment of | > |my character. I stand by my actions and can > |only hope it was the right decision to make. I > |thought generally of the mentality of groups > |and group behavior in sociology and. > |psychology and specifically a TED > |(Technology, Education, and Design) video. > |had seen on the manifests of evil in > |anonymity and groups. [Link: http:// > |www.ted.com/talk/ > philip zimbardo on the psychology of evil.html] Based on the textbooks > I have read and ||this testimony of expert consulted in the | > aforementioned video, the best chance at > |ending this manifest (or digital holocost if > |left unchecked) is to immediately tear down |the walls between the > decent bystanders and > | the intervening. (You) Being one brave > | enough to speak to the defense of one being | mistreated is such a > rare occurence in front > |of large groups and you are brilliant > |combination of logic, decency, intelligence, > | sensitivity, and kindness. My heart rejoices > | and my faith in the fundamental goodness > | of people has been reaffirmed in you. By > | your actions you set a fine example and > |deserve praise, (minus the accusations, > | insults and negative consequence I can only |assume exist since you > referred to me asnot |your leper but their leper and feeling > |the need to assert you were not unclean for |touching me with > respect in your words. > | > | I must ask, why do you refer to me as a > | leper? I do not understand. Based on the | situation you > sparked in me a curiosity I can > | only identify akin to Sherlock Holmes > |worthy in analysis. Indulge me this, and offer > | me logic (if you must even in a hypothetical |sense--given the > circumstanced I will take |||||what I can get). > | > |1)Leprocy does not exist (except for > |extremely rare cases, which I only assert > |based on an episode of House) and... > | > | > |2) You have never met me. > | > |3) And I do not have leprocy. > | > |Conclusion: I can only rationally assume this |was an expression you > made for some other |situation or context you were not openly > |referring to. > | > |Further analysis and consideration reveals: > | > |A)In the context you directed your comment > |to Pharisees (your words in quotes). > | > |B) Pharisees mostly refer to ancient Biblical > |text and groups. > | > |Conclusioon: > | > |Since I do not have leprocy and the disease > |is extremely rare and you have never met > |me and 'Pharisees' almost always in modern |language refers to an > ancient religious sect, I |can only rationally assume you were not > |referring to literal Pharisees like you could > |not have been rationally referring to literal |leprocy. > | > |So my questions for you: > | > |1) What parallel dynamics (the physical > |persons and the relationship) between me > |personally and the community in context > |warrant the use of the term? How does it > |apply and why? > | > |2) Who are the real counterparts to the |Pharisees? > | > |3) What physical conditions exist on USENET |to make one a leper? > (Since the physical > |conditions for real leprocy do not exist there > |has to be other physical conditions present > |which define a leper in this context. Please |list them. > | > |4) Since I am not physically unclean (in the |rational ommission/ > absence of leprocy, how > |could you possibly be unclean simply be |touching me with > respect (your words) on > |the Internet? Well since a) no physical > |leprocy is or can be rationally present and b) > | it is impossible to physically catch an absent |disease by > electronically typing a kind > |message on behalf of one being mistreated, > |there is more going on here. The question is > |what conditions in the context of your > |comments and the physical reality define |clean and unclean? In > other words how > |could and why would one be considered |unclean for speaking out in > defense of one |being ridiculed by electronic messages? > | > | > |5) Define clean and unclean as physical > |conditions as they exist on USENET. > | > |6) What about me (Martin Michael Musatov, > |Caucasian male, D.O.B. 9/23/1978, Unity > |Hospital, Fridley, MN, 55432) have I done to > |take on the characteristics, or what physical > |characteristics or conditions existed or do > |exist at the time of your comments to > |rationally fullfill the requirements of a |leper? > | > |7) What does the group (the USENET > |community -- or a portion of them) gain by > |not only mistreating a leper (in this > |context) but inspiring a defender of a leper |to assert they are > different than me? > | > |THE TIPPING POINT: (To quote Malcolm > |Gladwell, an author I am fond of) > | > [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[- [ [[[[[[[[******* > $50,000 > |Question********What logical conditions > |exist to completely explain in full context the |underlying anaolgy > of the leper and the |pharisees?***************************** > | > |Answer here: > | > |[Please answer $50,000 (?) above this > |comment] > | > ***************************************************************************- * ********** > [[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[- [ [[[[[[[[<<<<<<<<<<<<<<<<| > <<< | > |You said it so you must have the answer to > |this question and it would be entirely > |irrational for you to not have the answer to > |this question since you are indeed the one > |who made it. > | > | I do not see the literal reference or abstract > |comparison here, so I beg you kindly to > | please explain all these puzzling words and > | place them and your reference in clear > | simple terms. Will you do this for me? Can > |you do this? > | > | > |::::::::::END STRICTLY LOGIC QUESTIONS TO > |||BE > |ANSWERED:::::::::::::: > | > |::::::::::BEGIN THEOLOGICAL > ||ARGIUMNENT::::::::THE LAW:::::::::::: (Atheists, | > |please look away if you are easily annoyed > |by illogical Christians*) > | > |*I am not saying anything just sparing > | *some* read more >- Hide quoted text - - Show quoted text -... On Aug 17 1990, 5:15 am, n...@ccncsu.ColoState.EDU (USENET news) > An extraordinary recursive factoring algorithm has been discovered (?) by > Risto Lankinen (ri...@yj.data.nokia.fi) based on mathematical properties he > which finds all natural numbers u, v such that u * v = n for an odd n, is > as follows: 1. select an odd n > 1. > 2. u <- 1, v <- 1, b <- 1, n <- (n - 1). > 3. n <- (n / 2). > 4. if n = 0, (then) u * v = n; return. > 5. if n < 0, (then) u * v <> n; return. > 6. b <- (b * 2). > 7. if n is odd, (then) > 7a. restart at 3 with n <- (n - u), v <- (v + b), > 7b. and again with n <- (n - v), u <- (u + b). > 8. (otherwise) for even n, > 8a. restart at 3, > 8b. and again with n <- (n - u - v - b), u <- (u + b), v <- (v + b). Execution time is comparable to the technique of division by all odd numbers > less than the square root of n. > - - - Following is an implementation of Lankinen's algorithm, with embellishments, > in the C language: DIVISOR.C 1: #include 3: void factor(long remain, > 4: long left, > 5: long right, > 6: long bit) > 7: > 8: { > 9: if (remain >= 1) > 10: { > 11: if (remain > 0) > 12: { > 13: bit <<= 1; > 14: if (remain & 1) > 15: { > 16: factor(remain - left, left, right | bit, bit); > 17: if (left != right) > 18: factor(remain - right, right, left | bit, bit); > 19: > 20: } > 21: else > 22: { > 23: factor(remain, left, right, bit); > 24: factor(remain - left - right - bit, left | bit, right | bit, bit); > 25: } > 26: } > 27: } > 28: else > 29: printf(%li %lin, left, right); > 30: > 31: } > 32: > 33: void main(void) > 34: > 35: { > 36: long number; > 37: > 38: scanf(%li, &number); > 39: while (!(number & 1)) > 40: number >= 1; > 41: factor(number, 1, 1, 1); > 42: } The correspondence is as follows: step -> lines | variables > ---- ----- | --------- > 1 38-40 | n -> number, remain > 2 41,9 | u left > 3 9 | v right > 4 9,28-29 | b bit > 5 11,21-25 | > 6 13 | > 7 14,15-20 | > 7a 16 | > 7b 18* | > 8 14,21 | > 8a 23 | > 8b 24 | * Embellishment, see notes below. The C language is ideal for implementing Lankinen's algorithm because both are > optimized for binary machine languages. Hence many arithmetical operations in > the algorithm can be replaced with C's bitwise operators for maximized > computational efficiency: This function at [line] replaces at [step] (respectively). > ---- -------- ------ -------- ------ > `|' (binary OR) 16,18,24 addition 7a,7b,8b > `>' (shift right) 9, 40 division 3 (1 implicit) > '<<' (shift left) 13 multiplication 6 Moreover, the parity test at steps 7 and 8 can be reduced to simply branching > on the state of the rightmost bit of the variable at lines 14 and 21. Recall > the existence of the implicit nonzero test in C, utilized in lines 14 and 39, > which can even be combined with assignment as at 9 (here some compilers may > warn of a possibly unintentional assignment instead of comparison). This C routine embellishes steps 7 and 8 in Lankinen's algorithm at lines 16-18 > and 23-24: * The conditional statement at line 17 decreases total processing time roughly > by a factor of two by avoiding a recursive call that results in finding > reversed (duplicate) pairs. Hence at line 29 the pair (left,right) is > unique; for no (u,v) will the program also report (v,u). (If the trial > factors are equivalent to some point, the recursive branches originating > there would be symmetric with respect to each other.) * Lankinen left unspecified the order of the two recursive calls in steps 7 > and 8 (they were assigned definite order above for exposition), but when 7 is > encoded as above, in concert with the test at 17 and the implied interchange > of trial factors at 18 (compare with step 7b), an ordering is established > such that for all divisor pairs (left,right) found at line 29, left <= right. - - - Lankinen's basic algorithm can be modified slightly to determine the complete > prime decomposition of a number: 1: #include 3: #define factor(what) find(what, 1, 1, 1, what) > 4: > 5: long find(long remain, > 6: long left, > 7: long right, > 8: long bit, > 9: long now) > 10: > 11: { > 12: if (right > now) > 13: return now; > 14: > 15: if (remain >= 1) > 16: { > 17: if (remain > 0) > 18: { > 19: bit <<= 1; > 20: if (remain & 1) > 21: { > 22: if (left != right) > 23: now = find(remain - right, right, left | bit, bit, now); > 24: return find(remain - left, left, right | bit, bit, now); > 25: > 26: } > 27: else > 28: return find(remain, > 29: left, > 30: right, > 31: bit, > 32: find(remain - left - right - bit, > 33: left | bit, > 34: right | bit, > 35: bit, > 36: now)); > 37: } > 38: else > 39: return now; > 40: } > 41: else > 42: if (now > 1) > 43: if (left == 1) > 44: printf(%li , right); > 45: else > 46: { > 47: factor(left); > 48: factor(right); > 49: now /= right; > 50: } > 51: > 52: return now; > 53: } > 54: > 55: void main(void) > 56: > 57: { > 58: long number; > 59: > 60: scanf(%li, &number); > 61: while (!(number & 1)) > 62: { > 63: printf(2 ); > 64: number >= 1; > 65: } > 66: if (number > 1) > 67: factor(number); > 68: } The function `factor' is replaced by a macro that calls on the new function > `find' that decomposes an odd number into its prime factors. This program > utilizes `intercommunication' between the branches of the recursive tree. > Whenever `factor' is started, the macro (line 3) places the original n in > the `now' variable (line 9). This variable represents the current unfactored > part of n (the original number divided by all actual factors found at some > point). It is passed from `parent' to `child' process as a parameter and as > the function result in the opposite direction. Then if the tentative factor under consideration is ever greater than `now', > it is impossibly large. The comparison for this test occurs at line 12. > At that point both `left' and `right' are trial factors but again right >= left > by ordering so only `right' need be tested. The division occurs at line 49. With adjustment to the unspecified ordering in the algorithm at lines 22-24 > and 28-36, there is an implicit guarantee that the divisor that is the number > itself (left = 1, right = n) will be found last. Hence, if an unfactored part > remains (line 42), either it is composite and all factors will be found > recursively at lines 47-49, or it is prime and reported at line 44. The `find' routine can be rewritten in several other forms. In this version > the `else' keywords at lines 38 and 41 are redundant and should be removed > if they improve the efficiency of the compilation (they were included here > for clarity). Also, the routine is readily modified to use a single exit point > at line 52 by eliminating the other three at lines 13, 24, and 28 with block > scoping and assignment to `now' (the version above attempts to maximize speed). - - - For a more rigorous mathematical treatment of Lankinen's approach to factoring, > see his preliminary findings in <6...@tuura.UUCP>. The description that follows > is intended to be more intuitively appealing than logically precise. Lankinen's algorithm can be thought of as the `inverse' of binary > multiplication (otherwise known as the `Russian Peasant's Method') > in the special case where both the multiplier and multiplicand are odd > (and hence the product also). The algorithm simply explores all possibilities in u and v that could match n > by focusing on the 2nd bit from the right and advancing left one bit at a time. > (The rightmost bit in n is matched immediately because both factors are odd.) > The fundamental rule is employed that if the next bit in n is 1, then it is > matched only when the next bits in u and v are different from each other, > and if the digit is 0 the bits must match. In either case, there are only two > possibilities, each of which is pursued recursively. As the trial divisors accumulate, so does a carry that has to be accounted for > in the subsequent comparisons. This is accomplished exactly and conveniently > by subtracting the newly accumulated amount from n prior to entering each > recursive branch. If n < 0, the product of the trial factors has accrued a > carry that exceeds the original n to be factored, thereby eliminating them. > Otherwise, if n = 0, they are actual factors. ld231...@longs.LANCE.ColoState.EDU === Subject: Re: Sum of two squares QED: Musatov > On Jun 1, 1:05 pm, Jos.8e Carlos Santos > There is a well-known way of determining whether > or not a natural > number can be written as a sum of two squares. Is > anyone aware of > some way of knowing in *how many* ways that can > be done? > Yes. The analysis can be found, for example, in > An Introduction to > the Theory of Numbers, 5th Edition, by Ivan > Niven, Herbert > Zuckerman, and Hugh Montgomery, Section 3.6 (pp. > 163-169). > I feel you are all using a slegehammer to crack a > nut. > I take it the number, N(n) say, > of ways of expressing n as a sum of two squares, > is multiplicative in the number-theoretic sense, ie > gcd(m,n)=1 => N(mn)=N(m)N(n). > If that is true, can it be proved easily in an > elementary way? > > The number of representations of n as sum of two > squares, essentially > differents, is > > N(m) = (1/2)(1 + r1)(1 + r2)(1 + r3)... > > where r1, r2, r3, ... are the exponents of its prime > factors equals to 1 > (mod 4), assuming that m isn`t a perfect square nor a > double or a perfect > square, and all the exponents of the factors equals > to 3 (mod 4) are even. > Evidently, N() isn't multiplicative. > > By counterexample, the first: > > 5 = 2^2 + 1^2, 13 = 3^2 + 2^2, 65 = 8^2 + 1^2 = > 7^2 + 4^2 == > N(65) = 2 =/= N(13)*N(5) = 1*1 = 1 > > > -- > > Ignacio Larrosa Ca.96estro > A Coru.96a (Espa.96a) > ilarrosaQUITARMAYUSCULAS@mundo-r.com N(5) = 1, N(13) = 1, N(65) > > Math Forum DiscussionsMartin Michael Musatov Posts: 641. Registered: 4/19/09 ... from Wolfram MathWorldFermat's 4n+1 theorem, sometimes called Fermat's ... mathforum.org/kb/thread.jspa?messageID=6738341&tstart=0 mathforum.org/kb/message.jspa?messageID=6746768&tstart=75 Math Forum DiscussionsMartin Michael Musatov Posts: 734. Registered: 4/19/09 ... MathWorld Headline News ... mathforum.org/kb/message.jspa?messageID=6747368&tstart=75 mathforum.org/kb/message.jspa?messageID=6736479&tstart=0 mathforum.org/kb/message.jspa?messageID=6749035&tstart=60 - 12 hours ago Math Forum DiscussionsJun 11, 2009 ... The Math Forum Donate to the Math Forum ... ... Martin Michael Musatov ... mathforum.org/kb/message.jspa?messageID=6747287&tstart=75 Math Forum DiscussionsJun 10, 2009 ... The Math Forum Donate to the Math Forum ... http://functions.wolfram.com/Polynomials/LegendreP2/21/02/01/ ... Martin Michael Musatov ... mathforum.org/kb/message.jspa?messageID=6746671&tstart=75 Math Forum DiscussionsThe Math Forum Donate to the Math Forum ... Wolfram Research The idea is correct, but your formula mixes vectors and scalars. ... Martin Michael Musatov ... mathforum.org/kb/message.jspa?messageID=6732833&tstart=60 - 12 hours ago Math Forum DiscussionsThe Math Forum Donate to the Math Forum ... ... Martin Michael Musatov. 5/31/09 ... mathforum.org/kb/thread.jspa?messageID=6732837&tstart=0 Math Forum DiscussionsThe Math Forum Donate to the Math Forum ... would be: || (A-B) - (A-B).(C-B)/||C-B|| || Daniel Lichtblau Wolfram Research ... Martin Michael Musatov ... mathforum.org/kb/message.jspa?messageID=6729970&tstart=60 - 12 hours ago QED: Musato v === Subject: Re: Irrationality of sqrt (n^2 - 1) > On May 18, 9:48pm, cbr...@cbrownsystems.com > As Bill Dubuque pointed out, though, you don't > even need all of that; > you only need GCD's and Euclid's Lemma: a|xy and > (a,x)=1, then x|y. > > > -- > Arturo Magidin V Musatov V Forwarded conversation === Subject: Re: Injection and surjection ------------------------ I have neither interest nor desire to exchange any messages with you. Please do not send me any e-mail again; I will consider future messages to constitute harassment, and will report them as such to your service provider. ---------- Yes, Sir. I am sorry if my mathematics upset you. Sent from my Verizon Wireless BlackBerry +++++++++++++++++++++++++++++++++++++++++++++++++++++ === Subject: Re: Diagonal wanderings (inorder) CLICK THIS LINK:::::: http://www.flickr.com/photos/24869933@N07/3619373444/siz YOU SHOULD SEE A SET OF PICTURES THEY ARE ALL MINE AND OR THEY MIGHT BE IF IT DOES NOT MAKE YOU SIGN IN TO VIEW THEM. I GET THIS MESSAGE WHEN I GO THERE: note: these pictures are my proof of pi and my activation of the complex plane my splitting an electric signal between two monitors and running a forced P=NP Rimann Proof: This is not the page you're looking for. It looks like you're trying to see something in scriber77's photostream. Here are their most recent uploads... Did you click on a broken link? Please let us know. | Deutsch | English | Espa.96ol | Fran.8dais | | Italiano | Portugu.90s Copyright © 2009 MeAmI.org! Inc. All rights reserved. http://www.flickr.com/photos/24869933@N07/3619373444/siz > Diagonal wanderings (inorder) > ----------------------------- > > --- Introduction: > > A complete list of infinite binary expansions is > given, by walking > inorder the infinite binary tree, and selecting > branches > appropriately. The selection logic is here > illustrated: > > The infinite binary tree: R > / > 0 1 > / / > / > 0 1 0 > 0 1 > / / / / > / > 0 1 0 1 0 1 0 > 0 1 > / / / / / / / > / > 0 1 0 1 0 1 0 1 0 1 0 1 0 > 0 1 0 1 > / / / / / / / / / / / / / / > / / > . . . . . . . . . . . . . . . . . . . . . . . . . . . > . . . . . > > 1st entry: (0) R > / > [0] 1 > / / > / > [0] 1 0 > 0 1 > / / / / > / > [0] 1 0 1 0 1 0 > 0 1 > / / / / / / / > / > [0] 1 0 1 0 1 0 1 0 1 0 1 0 > 1 0 1 > / / / / / / / / / / / / / / > / / > . . . . . . . . . . . . . . . . . . . . . . . . . . . > . . . . . > > 2nd entry: (1) R > / > (0) [1] > / / > / > (0) 1 0 > 0 [1] > / / / / > / > (0) 1 0 1 0 1 0 > 0 [1] > / / / / / / / > / > (0) 1 0 1 0 1 0 1 0 1 0 1 0 > 1 0 [1] > / / / / / / / / / / / / / / > / / > . . . . . . . . . . . . . . . . . . . . . . . . . . . > . . . . . > > 3rd entry: 0(1) R > / > (0) (1) > / / > / > (0) [1] 0 > 0 (1) > / / / / > / > (0) 1 0 [1] 0 1 0 > 0 (1) > / / / / / / / > / > (0) 1 0 1 0 1 0 [1] 0 1 0 1 0 > 1 0 (1) > / / / / / / / / / / / / / / > / / > . . . . . . . . . . . . . . . . . . . . . . . . . . . > . . . . . > > 4th entry: 1(0) R > / > (0) (1) > / / > / > (0) (1) [0] > [0] (1) > / / / / > / > (0) 1 0 (1) [0] 1 0 > 0 (1) > / / / / / / / > / > (0) 1 0 1 0 1 0 (1) [0] 1 0 1 0 > 1 0 (1) > / / / / / / / / / / / / / / > / / > . . . . . . . . . . . . . . . . . . . . . . . . . . . > . . . . . > > 5th entry: 00(1) R > / > (0) (1) > / / > / > (0) (1) (0) > (0) (1) > / / / / > / > (0) [1] 0 (1) (0) 1 0 > 0 (1) > / / / / / / / > / > (0) 1 0 [1] 0 1 0 (1) (0) 1 0 1 0 > 1 0 (1) > / / / / / / / / / / / / / / > / / > . . . . . . . . . . . . . . . . . . . . . . . . . . . > . . . . . > > 6th entry: 01(0) R > / > (0) (1) > / / > / > (0) (1) (0) > (0) (1) > / / / / > / > (0) (1) [0] (1) (0) 1 0 > 0 (1) > / / / / / / / > / > (0) 1 0 (1) [0] 1 0 (1) (0) 1 0 1 0 > 1 0 (1) > / / / / / / / / / / / / / / > / / > . . . . . . . . . . . . . . . . . . . . . . . . . . . > . . . . . > > 7th entry: 10(1) R > / > (0) (1) > / / > / > (0) (1) (0) > (0) (1) > / / / / > / > (0) (1) (0) (1) (0) [1] 0 > 0 (1) > / / / / / / / > / > (0) 1 0 (1) (0) 1 0 (1) (0) 1 0 [1] 0 > 1 0 (1) > / / / / / / / / / / / / / / > / / > . . . . . . . . . . . . . . . . . . . . . . . . . . . > . . . . . > > And so on... > > (NOTE: In this construction we consider 0(1) == > 1(0). Incidentally, > this way we can work with a binary alphabet with no > special handling.) > > --- Construction: > > I have coded this logic into a set of inductive > definitions, then > implemented it in Prolog [*]. The program can answer > questions like, > which expansion is at a specific index in the list, > or viceversa, at > which index is a given expansion. > > [*] http://julio.diegidio.name/Share/RealTree.zip > (Note that the program is optimised w.r.t. the naive > inductive > definitions.) > > Here is some output: first the list (first 32 > entries), then the > corresponding diagonal sequence, finally the > anti-diagonal sequence. > > These are the first 32 expansions (the reals). > The format is: '[index] expansion ~ approx.value'. > > LIST: > [0] (0) ~ 0 > [1] (1) ~ 1 > [2] 0(1) ~ 0.5 > [3] 1(0) ~ 0.5 > [4] 00(1) ~ 0.25 > [5] 01(0) ~ 0.25 > [6] 10(1) ~ 0.75 > [7] 11(0) ~ 0.75 > [8] 000(1) ~ 0.125 > [9] 001(0) ~ 0.125 > [10] 010(1) ~ 0.375 > [11] 011(0) ~ 0.375 > [12] 100(1) ~ 0.625 > [13] 101(0) ~ 0.625 > [14] 110(1) ~ 0.875 > [15] 111(0) ~ 0.875 > [16] 0000(1) ~ 0.0625 > [17] 0001(0) ~ 0.0625 > [18] 0010(1) ~ 0.1875 > [19] 0011(0) ~ 0.1875 > [20] 0100(1) ~ 0.3125 > [21] 0101(0) ~ 0.3125 > [22] 0110(1) ~ 0.4375 > [23] 0111(0) ~ 0.4375 > [24] 1000(1) ~ 0.5625 > [25] 1001(0) ~ 0.5625 > [26] 1010(1) ~ 0.6875 > [27] 1011(0) ~ 0.6875 > [28] 1100(1) ~ 0.8125 > [29] 1101(0) ~ 0.8125 > [30] 1110(1) ~ 0.9375 > [31] 1111(0) ~ 0.9375 > > These are the first 32 diagonal expansions. > The format is: '[index] [diag-index] expansion ~ > approx.value', where > 'diag-index' is the index of the corresponding > diagonal in the initial > list. > > DIAGONAL: > [0] -> [2] 0(1) ~ 0.5 > [1] -> [5] 01(0) ~ 0.25 > [2] -> [11] 011(0) ~ 0.375 > [3] -> [22] 0110(1) ~ 0.4375 > [4] -> [45] 01101(0) ~ 0.40625 > [5] -> [90] 011010(1) ~ 0.421875 > [6] -> [181] 0110101(0) ~ 0.414063 > [7] -> [362] 01101010(1) ~ 0.417969 > [8] -> [725] 011010101(0) ~ 0.416016 > [9] -> [1450] 0110101010(1) ~ 0.416992 > [10] -> [2901] 01101010101(0) ~ 0.416504 > [11] -> [5802] 011010101010(1) ~ 0.416748 > [12] -> [11605] 0110101010101(0) ~ 0.416626 > [13] -> [23210] 01101010101010(1) ~ 0.416687 > [14] -> [46421] 011010101010101(0) ~ 0.416656 > [15] -> [92842] 0110101010101010(1) ~ 0.416672 > [16] -> [185685] 01101010101010101(0) ~ 0.416664 > [17] -> [371370] 011010101010101010(1) ~ 0.416668 > [18] -> [742741] 0110101010101010101(0) ~ 0.416666 > [19] -> [1485482] 01101010101010101010(1) ~ 0.416667 > [20] -> [2970965] 011010101010101010101(0) ~ 0.416667 > [21] -> [5941930] 0110101010101010101010(1) ~ > 0.416667 > [22] -> [11883861] 01101010101010101010101(0) ~ > 0.416667 > [23] -> [23767722] 011010101010101010101010(1) ~ > 0.416667 > [24] -> [47535445] 0110101010101010101010101(0) ~ > 0.416667 > [25] -> [95070890] 01101010101010101010101010(1) ~ > 0.416667 > [26] -> [190141781] 011010101010101010101010101(0) ~ > 0.416667 > [27] -> [380283562] 0110101010101010101010101010(1) ~ > 0.416667 > [28] -> [760567125] 01101010101010101010101010101(0) > ~ 0.416667 > [29] -> [1521134250] > 011010101010101010101010101010(1) ~ 0.416667 > [30] -> [3042268501] > 0110101010101010101010101010101(0) ~ 0.416667 > [31] -> [6084537002] > 01101010101010101010101010101010(1) ~ 0.416667 > > These are the first 32 anti-diagonal expansions. > The format is: '[index] [anti-index] expansion ~ > approx.value', where > 'anti-index' is the index of the corresponding > anti-diagonal in the > initial list. > > ANTI-DIAGONAL: > [0] -> [3] 1(0) ~ 0.5 > [1] -> [6] 10(1) ~ 0.75 > [2] -> [12] 100(1) ~ 0.625 > [3] -> [25] 1001(0) ~ 0.5625 > [4] -> [50] 10010(1) ~ 0.59375 > [5] -> [101] 100101(0) ~ 0.578125 > [6] -> [202] 1001010(1) ~ 0.585938 > [7] -> [405] 10010101(0) ~ 0.582031 > [8] -> [810] 100101010(1) ~ 0.583984 > [9] -> [1621] 1001010101(0) ~ 0.583008 > [10] -> [3242] 10010101010(1) ~ 0.583496 > [11] -> [6485] 100101010101(0) ~ 0.583252 > [12] -> [12970] 1001010101010(1) ~ 0.583374 > [13] -> [25941] 10010101010101(0) ~ 0.583313 > [14] -> [51882] 100101010101010(1) ~ 0.583344 > [15] -> [103765] 1001010101010101(0) ~ 0.583328 > [16] -> [207530] 10010101010101010(1) ~ 0.583336 > [17] -> [415061] 100101010101010101(0) ~ 0.583332 > [18] -> [830122] 1001010101010101010(1) ~ 0.583334 > [19] -> [1660245] 10010101010101010101(0) ~ 0.583333 > [20] -> [3320490] 100101010101010101010(1) ~ 0.583333 > [21] -> [6640981] 1001010101010101010101(0) ~ > 0.583333 > [22] -> [13281962] 10010101010101010101010(1) ~ > 0.583333 > [23] -> [26563925] 100101010101010101010101(0) ~ > 0.583333 > [24] -> [53127850] 1001010101010101010101010(1) ~ > 0.583333 > [25] -> [106255701] 10010101010101010101010101(0) ~ > 0.583333 > [26] -> [212511402] 100101010101010101010101010(1) ~ > 0.583333 > [27] -> [425022805] 1001010101010101010101010101(0) ~ > 0.583333 > [28] -> [850045610] 10010101010101010101010101010(1) > ~ 0.583333 > [29] -> [1700091221] > 100101010101010101010101010101(0) ~ 0.583333 > [30] -> [3400182442] > 1001010101010101010101010101010(1) ~ 0.583333 > [31] -> [6800364885] > 10010101010101010101010101010101(0) ~ 0.583333 > > --- The diagonal argument: > > By ordinary induction, the diagonal (just as the > anti-diagonal) is not (is) in > the list. This represents a perfect example to the > diagonal argument. Ergo, the diagonal argument is valid. > > -MM > === Subject: Re: Diagonal wanderings (inorder) <4A124C4F.C32F82CC@gmail.com> <4A1338A5.35ED7B97@gmail.com> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Each real is potentially > infinite, in the sense that we can compute arbitrarily large > approximations. > Ok, that's a locution that makes some sense. > Although, it seems a little strange to say that sqrt(2) is > potentially infinite since it's less than 3. > Right; > it's rather the computational reading of the binary expansion > that can be described this way. > -- > hz- Hide quoted text - - Show quoted text - Got it folks. Musatov (Inverse 19) represent: Efforts by mathematicians to determine the value of pi since have led to the conclusion that it ... Unlike rational numbers, pi cannot be represented by a common fraction, .... But what about an uncommon fraction? Will you look at my drawing and in particular how the twelve points are the outside points of the sphere are all rationally reached by decimal numbers? 6.2035884*2 and doubled an infinite number of times should yield a constant decimal binary expansion (what I mean by this is the numbers to the right of the decimal point should always end in 4, then 8, then 6, then 2, then 4, then 8, then 6, then 2...repeating for infinity). So in binary, by starting with this number we predict for n>2 the infinite binary tree will end with decimal progression: 2, 4, 8, 6 For n<2 the infinite binary tree will begin and end (depends on if you are increasing or decreasing the value) 1, 5, 5, 5 (repeating 5 infinitely) as we decrease to infinitely small. Musatov Inverse Counting Chain of Infinite Proportions (sample) 0.0000473372528076171875 0.000094674505615234375 0.00018934901123046875 0.0003786980224609375 0.000757396044921875 0.00151479208984375 0.0030295841796875 0.006059168359375 0.01211833671875 0.0242366734375 0.048473346875 0.09694669375 0.1938933875 0.387786775 0.77557355 1.5511471 3.1022942 6.2035884 12.4071768 24.8143536 49.6287072 99.2574144 198.5148288 397.0296576 794.0593152 1588.1186304 3176.2372608 6352.4745216 12704.9490432 25409.8980864 50819.7961728 101639.5923456 203279.1846912 406558.3693824 813116.7387648 1626233.4775296 3252466.9550592 6504933.9101184 13009867.8202368 26019735.6404736 52039471.2809472 104078942.5618944 208157885.1237888 416315770.2475776 832631540.4951552 1665263080.9903104 3330526161.9806208 6661052323.9612416 13322104647.9224832 26644209295.8449664 53288418591.6899328 106576837183.3798656 213153674366.7597312 426307348733.5194624 852614697467.0389248 1705229394934.0778496 3410458789868.1556992 6820917579736.3113984 13641835159472.6227968 27283670318945.2455936 54567340637890.4911872 109134681275780.9823744 218269362551561.9647488 436538725103123.9294976 873077450206247.8589952 1746154900412495.7179904 3492309800824991.4359808 6984619601649982.8719616 13969239203299965.7439232 27938478406599931.4878464 55876956813199862.9756928 111753913626399725.9513856 223507827252799451.9027712 447015654505598903.8055424 894031309011197807.6110848 1788062618022395615.2221696 3576125236044791230.4443392 7152250472089582460.8886784 14304500944179164921.7773568 28609001888358329843.5547136 57218003776716659687.1094272 114436007553433319374.2188544 228872015106866638748.4377088 457744030213733277496.8754176 915488060427466554993.7508352 1830976120854933109987.5016704 3661952241709866219975.0033408 7323904483419732439950.0066816 14647808966839464879900.0133632 29295617933678929759800.0267264 58591235867357859519600.0534528 117182471734715719039200.1069056 234364943469431438078400.2138112 468729886938862876156800.4276224 937459773877725752313600.8552448 1874919547755451504627201.7104896 3749839095510903009254403.4209792 7499678191021806018508806.8419584 (sample) If you would like to do the honors Mr. Plouffe, I endear you to calculate the angles and lengths of the number of ways, it is quite obviously possible to square a circle based on the geometry and numerical progression. As is self-evident to anyone with even a modest understanding of mathematics and geometry after visual inspection of the drawn proof (http://www.flickr.com/photos/24869933@N07/3619373444/sizes/l/) it is quite easily accomplished to connect and triangulate the outer points of the rectangles to the outer points of the square and circle and even by dimension sphere. Yes, the drawing clearly by definition of inverse symmetry (take the drawing cut it in half at the equator and turn the paper over and rotate it and compare the top and bottom mirrored symmetry and you will see it is an exact inverse of symmetry itself) the drawing explains how to properly and mathematically soundly represent a fourth dimension on a two dimensional plane. In plane language if you take the picture and separate the top half of the sphere by the bottom half of the sphere they are identical but only if the sphere existed on a complete three dimensional plane through expansion. Since the paper is flat, this is an impossibility, nonetheless it is completely expressed and retractable and countable infinitely. By the definition of our standard we have established any k-digit approximation to pi is rational. Which could have many practical scientific applications. For instance, search: ON THE THEOREM OF IVA$EV-MUSATOV IIIx e F(l) + F(2) + ... + F(p). This remark is the basis of our proof of Theorem .... -y(M + 1), x' + y(M + 1)] = 0 it is easy to check that Conditions (i), ...... Now set i = pi x. *jj, v. Since p, x is positive,. (i) fi is positive, ... plms.oxfordjournals.org/cgi/reprint/s3-53/1/143.pdf REPRESENTATION OF FUNCTIONSBY SERIES AND CLASSES £r(1)with measure E> 2£k -- £` and such that the trigonometric series for g(x) converges uniformly on [0, 2£k]. .... trigonometric series are due to Ivashev-Musatov [40] who has shown that there are null series (in the sense of ..... function £r(£q) e C(0, 1) with £r(0) = £r(1) = 0 such that ...... <*i After a lot of time and a some progress in my Riemann > Zeta Function > studies, I'm stuck again. > Firstly, I want to thank everyone helped me in the > past (Raymond > Manzoni above all, more than one year ago, phew...). > I'll explain my problem shortly: I'm still following > Edwards' book in Seeing is believing. --Musatov http://farm4.static.flickr.com/3496/3256947403_398e26dde8_o.jpg > my study and I got to the point where he studies the > rate of growth > of the quotient (p. 43 - 45). > I understood quite everything in the proof but the > point where he > analyzes the function v_r(s). In fact I can't > understand why this > function changes from the first definition he gives > at the top of p. > 44 and the following one at the bottom: > > v_r(s) = Re{ log{ 1 / ( prod(1 - (s - 1/2)/(p - > p - 1/2) ), |p - 1/2| 2R )} } first def > > v_r(s) = Re{ log{ 1 / ( prod(1 - (s - 1/2)^2/(p - > p - 1/2)^2 ), |p - > 1/2| > 2R )} } second def > > The rest of the proof is pretty clear to me... > Hope to get some answers... === Subject: Re: Q on Riemann domains > A riemann domain is similar to a Riemann surface > except that it can be > higher-dimensional. > > Also because of the global projection map P (which > makes it a complex > http://farm4.static.flickr.com/3496/3256947403_398e26dde8_o.jpg See it. ++Musatov > I don't know why they want to exclude compact > surfaces like CP^n. It > would seem as relevant as Riemann surfaces that are > compact. > > Laura > === Subject: Re: #259 Chapter 7, the COOL Riemann Hypothesis and its proof ; new book 2nd edition: New True Mathematics > I do not often use the word cool in my lexicon but > I should, even > though it is an > inflated term. > > But the Old Riemann Hypothesis wanted to know if all > the zeroes, the > nontrivial > zeroes lie on the 1/2 Real strip in Complex Plane. > They wanted to know > this: http://farm4.static.flickr.com/3496/3256947403_398e26dde8_o.jpg > because it brings some order to how the Prime > numbers occurr > embedded > within all the Counting Numbers. That was why there > was so much fuss > about > proving RH for so many other propositions hinged on > whether RH was > true or > false. > > Trouble with the Old RH was that mathematicians did > not have the full > deck > of Numbers. They had only the numbers 0, 1, 2, 3, 4, > ..... out to > 00000xyz... > That is why they could never prove RH, for it is like > playing bridge > in cardgames > with only 5 cards as a deck. > > What they needed for RH is the full deck of numbers > from 0 to > 9999....99999 > > So what is the COOL Riemann Hypothesis? Well, we do > not confine > ourselves > to the simpleton Real 1/2 strip and confine ourselves > to only the > Primes, but > we prove a RH that is vastly more general. We prove a > Universal RH > which looks > at Primes and Composites. Where we ask does the > Primes follow x/Ln(x)? > Do > the 2-prime composites such as 3x5 = 15 follow > x/Ln(x)? Do the 3-prime > composites etc etc follow x/Ln(x) ? > > That is the Cool Riemann Hypothesis for it no longer > narrowly > restricts itself to > just the Primes on a 1/2 Real strip but asks about > all the numbers > that make up > the Counting Number from 0 to 9999....99999 > > So what is the Cool Riemann Hypothesis Proposition? > Well the > Proposition is also > Cool, and so very Cool indeed. > > Cool Riemann Hypothesis: There exists one and only > one function, a > unique function > amoung all other functions and it is a Self Similar > Function for it > behaves the same > whether going forward in counting or counting > backwards as to the > distribution of > all the Primes, and of all the 2-prime Composites and > of all the 3- > prime Composites > etc etc. So the Cool Riemann Hypothesis looks for one > thing-- looks > for a function > that serves all the Counting Numbers whether we count > them forwards or > backwards > and that they have to be distributed amoungst 0 to > 99999.....99999 and > obey this > one formula. > > Proof of the Cool Riemann Hypothesis: x/Ln(x) > > Marvellous. Did anyone ever imagine that the worst > mathematical > problem > of modern day science is proved by just seven > letters? The worst math > problem becomes the shortest proof. > > There is only one mathematical function that can be > Self Similar to > all the Primes > and all the Categories of Composites. It is a > Logarithmic function. It > is what the > Ancients knew from the nautilus shell that as it > grows bigger it still > stays the same > in shape. The size varies but the shape remains the > same. > > So the proof of the Old Riemann Hypothesis and the > Cool New Riemann > Hypothesis > are both the same proofs. > > Proof: x/Ln(x) > > Explanation: there exists one and only one function > in mathematics > that can arrange > All Possible Digit Arrangements so that the primes > and the composites > distribution > are all the same and that Sameness is the logarithmic > function x/Ln(x) > > Reflection: forget about pondering nontrivial zeroes > on the 1/2 Real > strip because > you neglect and overlook all the other numbers which > the primes are > immersed. It is > like a picture that has no surroundings so the > picture cannot exist. > True, the Primes > follow x/Ln(x) but the RH of old forgot to look at > the composites and > that they > have to follow some formula. > > Archimedes Plutonium > www.iw.net/~a_plutonium > whole entire Universe is just one big atom > where dots of the electron-dot-cloud are galaxies === Subject: Re: Riemann-Stieljes > I try to find a continuous function f from [0,1/Pi] > to R so that f is > not Riemann-Stieljes with > respect to the function g:[0,1/Pi]-->R defined by > g(x)=x^(1/2)sin(1/x) > if x>0 and g(x)=0 if x=0. Illustrates: http://farm4.static.flickr.com/3496/3256947403_398e26dde8_o.jpg === Subject: Re: Riemann Integral Illustraton: proof http://farm4.static.flickr.com/3496/3256947403_398e26dde8_o.jpg === Subject: Re: Riemann Integral > pubkeybreaker > >On Apr 28, 8:38pm, Linear1983 > Let f be a bounded function defined on the > interval [a,b]. If f is integrable on [a,b], then > given any e > 0 there exists a partition P of [a, b] > such that L(f, P) > U(f) - e. > Where L is the lower Riemann integral and U is the > upper. > I am trying to figure out if this is true or > false. However, I cannot find anything like it in my > notes or textbook. Any help would be great. >I am obviously being stupid here, but how can this > be a strict >inequality for e > 0? >For a constant function, L and U are the same..... > > ..so you have L > L - e, no? Have another beer! :-) > > --Lynn > > http://math.asu.edu/~kurtz > What it looks like on paper: http://farm4.static.flickr.com/3496/3256947403_398e26dde8_o.jpg Musatov === Subject: Re: Cubic polynomial conjectures (?) > I don't know how much of this is already known > (probably most of it), > and it is mainly conjectural, but here goes anyway. > > Suppose that P = x^3 + a*x^2 + b*x + c is an > irreducible cubic > polynomial with coefficients a,b,c in Z. Let D be its > discriminant: > > D = -4*b^3 + a^2*b^2 + 18*c*a*b + -4*c*a^3 - 27*c^2 > > and define E as > > E = a^2 - 3*b > > Define Q as the sextic polynomial > > Q = x^6 - 2*E*x^4 + E^2*x^2 - D = (x^3 - E*x)^2 - D > > I have some conjectures about the relationship > between P and Q. > > 1. If the Galois group of P is of order 6, then Q is > irreducible and > generates the splitting field of P. > > 2. If the Galois group of P is of order 3, then a) D > is a square in Z, > say D = d^2 with d in Z; b) Q is the product of two > cubic factors, > namely > > Q = (x^3 - E*x + d)*(x^3 - E*x - d) > > I can see how some of this implies other parts, e.g. > if D = d^2 with d > in Z, then Q indeed factors as shown above. > > I've verified these conjectures for all values of the > coefficients of > P in the range -10..10. Yes, see it http://farm4.static.flickr.com/3496/3256947403_398e26dde8_o.jpg musatov (please) === Subject: Re: Cubic polynomial conjectures (?) posting-account=oX5FAgkAAAD2tWLfJT1nYaPn9C8kADIl Gecko/20090423 Firefox/3.5b4,gzip(gfe),gzip(gfe) On Jun 11, 8:45pm, Gerry Myerson On Jun 11, 2:17am, Gerry Myerson Just as a sanity check, is it true that the order of the splitting > field of an irreducible quartic is either 24, 8 or 4? I haven't come > across any of order 12 yet. > It is conjectured that every finite group is G(K/Q) for some extension K > of the rationals. I'm sure there are A 4 quartics. Again I suggest you > find a good text on Field Theory (or Galois Theory, or perhaps even > Algenraic Number Theory), and you may find all your questions > answered. > -- > Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email) > I actually read them, but my memory has unfortunately only a finite > capacity. I suppose I could break down and use them for the purpose > for which they were intended. > Actually, the nonexistence of A4 quartics would not disprove the > conjecture, since A4 is also a subgroup of Sn for all n > 4. > Oh, never mind. I've already come across a few A4 quartics, so they do > indeed exist. You may be interested in this paper: MR0564533 (81c:12006) > Baily, Andrew Marc > On the density of discriminants of quartic fields. > J. Reine Angew. Math. 315 (1980), 190--210. > 12A30 (12A50) For a group $G$, let $N n(X,G)$, $N n^+(X,G)$, $N n^-(X,G)$, > respectively, denote the number of (nonconjugate) algebraic number > fields of degree $n$ whose normal closure has Galois group $G$ and > whose discriminant $d$ satisfies, respectively, $|d|leq X$, > $0dgeq-X$. In this paper the author considers the case > $n=4$. > Using lemmas on class field theory together with the results of H. Cohn > [Proc. Amer. Math. Soc. 5 (1954), 476--477; MR0064076 (16,222a)] and H. > Davenport and H. Heilbronn [Proc. Roy. Soc. London Ser. A 322 (1971), > no. 1551, 405--420; MR0491593 (58 #10816)] on cubic fields, he obtains > lower bounds and upper bounds for $N 4^+(X,S 4)$, $N 4^-(X,S 4)$, > $N 4(X,A 4)$ and $N 4(X,D 4)$, where $S 4,A 4$ and $D 4$ denote the > symmetric group, the alternating group, and the dihedral group (of > order eight), respectively. Let $C 4$ denote the cyclic group of order > four, and $V 4$ the Klein four-group; for these abelian cases he shows > $N 4(X,C 4)sim a CX^{1/2}$, and $N 4(X,V 4)sim a VX^{1/2}log^2X$, > where $a C$ and $a V$ are constants. > Reviewed by Kenz.99 Komatsu A more recent paper on similar topics: MR2183288 (2006m:11163) > Bhargava, Manjul > The density of discriminants of quartic rings and fields. > Ann. of Math. (2) 162 (2005), no. 2, 1031--1063. > 11R45 (11R16 11R29) Let $k$ be a number field of degree 4 over $Bbb Q$ and $d k$ its > discriminant. One says that $k$ is an $S 4$-extension if the Galois > group of its Galois closure (in a fixed algebraic closure > $overline {Bbb Q}$ of $Bbb Q$) is the symmetric group $S 4$. Let $scr F(S 4)$ be the set of isomorphism classes of > $S 4$-extensions. The author seeks to evaluate the counting function > $$ > N 4(S 4,X)=#{kinscr F(S 4):, |d k|leq X}, > $$ > as well as to determine the asymptotic behavior of $N 4(S 4,X)$ as $X$ > tends to infinity. Since there exist $S 4$-extensions for each of the signatures $(r,s)$, > where $r=0,2,4$ is the number of real embeddings of $k$ into $Bbb C$ > and $r+2s=4$, one can consider the more precise counting functions > $$ > N 4^{(s)}(S 4,X)=#{kinscr F(S 4):, |d k|leq X, (r,s)=(4-2s,s)}. > $$ The function $N 4(S 4,X)$ was studied by A. Yukie [Surikaisekikenkyusho > Kokyuroku No. 1173 (2000), 171--183; MR1840077], who proved that > $N 4(S 4,X)=O(Xlog^2X)$. He conjectured that, in fact, > $N 4(S 4,X)sim c.87X$ for a constant $c$ whose Euler product expression > is explicitly known and that this could be proved under some very > reasonable convergence conditions. The main result of the paper under review (Theorem 1) is a proof of > this conjecture confirming the value of the constant predicted by > Yukie, although the Archimedean factor had to be corrected slightly. In > fact, the author proves that $N 4^{(s)}(S 4,X)sim c s X$ and that > $$ > c 0=frac{c 1}6=frac{c 2}3=frac1{48}prod p(p^4+p^2-p-1)/p^4. > $$ The proof relies essentially on results obtained by the author in > [``Higher composition laws'', Ph.D. thesis, Princeton Univ., Princeton, > 3, 1329--1360; MR2113024 (2005k:11214)]), where he established a > parametrization of quartic orders by the orbits of pairs of entire > ternary quadratic forms under the explicit action of > ${rm GL} s(Bbb Z)times {rm SL} 3(Bbb Z)$ and then counted the > number of integer points in certain fundamental domains. This method > should be compared with that used by H. Davenport and H. Heilbronn > [Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405--420; > MR0491593 (58 #10816)] to determine the density of the $S 3$-extensions > of $Bbb Q$ using a correspondence between cubic orders and equivalence > classes of binary cubic forms. The paper under review also contains a number of other results (some of > a different nature) that the author considers as deriving from the main > theorem. First, let us cite the determination (Theorem 2), quite > natural in light of the method of proof that is employed, of the > functions $M^{(s)} 4(S 4,X)$ counting the number of quartic orders > $scr O$ contained in $S 4$-extensions of $Bbb Q$ of signature $(r,s)$ > and discriminant $d(scr O)={rm disc}(scr O)$ with > $0<|d(scr O)|leq X$. One has $M^{(s)} 4(S 4,X)sim b sX$, with > $$ > b 0=frac{b 1}6=frac {b 2}3=frac{zeta(2)^2zeta(3)}{48zeta(5)}. > $$ Now, consider the field $k {24}$ of degree 24 over $Bbb Q$, which is > the Galois closure of an $S 4$-extension $k$. To every nonramified > prime $p$ in $k/Bbb Q$ the author associates the conjugacy class of > the Artin symbol $(tfrac{k {24}}p)$; the various conjugacy classes can > be represented by the permutations of $S 4$: $e$, $(12)$, $(123)$, > $(1234)$, $(12)(34)$, and, according to the Chebotarev density theorem, > if $k$ is a fixed $S 4$-extension and $p$ runs through the nonramified > primes in $k/Bbb Q$, the conjugacy classes of the corresponding Artin > symbols appear with the relative frequencies > $1colon 6colon 8colon 6colon 3$. The author proves (Theorem 3) that > if $p$ is fixed and $k$ runs through the $S 4$-extensions in which $p$ > does not ramify (they are ordered by the absolute value of the > discriminant), the conjugacy classes of the symbols > $(tfrac {k {24}}p)$ again appear with the relative frequencies > $1:6:8colon 6:3$. A. M. Baily [J. Reine Angew. Math. 315 (1980), 190--210; MR0564533 > (81c:12006)] proved that, aside from the $S 4$-extensions, the only > extensions of degree 4 of $Bbb Q$ with positive density are the > $D 4$-extensions. The value of the constant that arises in the counting > of these $D 4$-extensions has been determined explicitly by H. Cohen, > F. Diaz y Diaz and M. Olivier [Compositio Math. 133 (2002), no. 1, > exactly determine (Theorem 4) the percentage of $D 4$-extensions and of > $S 4$-extensions among the extensions of $Bbb Q$ of degree 4, > confirming the fact that, unlike in the case of polynomials, for which > the Hilbert irreducibility theorem shows that all polynomials of degree > $n$ have $S n$ as Galois group, only $82.889%$ of the quartic fields > are $S 4$-extensions. Finally, the author shows (Theorem 5) that for cubic fields whose > absolute value of the discriminant is bounded by $X$, the mean value of > the number of ideal classes with exponent 2 tends, as $X$ tends to > infinity, to $tfrac 54$ for totally really fields and to $tfrac 32$ > for complex cubic fields; in particular, this proves that at least > $75%$ of totally real cubic fields have an odd number of classes, as do > at least $50%$ of complex cubic fields. This theorem proves the > validity of the heuristics of Cohen and J. Martinet [J. Reine Angew. > Math. 404 (1990), 39--76; MR1037430 (91k:11097)] for $p=2$. A remarkable introduction, very clear in its description of the results > obtained, which it puts within the context of the works of other > researchers, precedes the more technical part of the paper containing > the proofs. Reviewed by F. Diaz y Diaz -- > Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email) These look interesting, but I'm afraid they are only peripherally related to what I am trying to do, which I can describe in greater detail. I only asked about the orders of the Galois groups of quartics because if it turned out that a certain n is not the order of such a Galois group, then I could simplify my work (no point really in reinventing the wheel, after all, unless it provides greater insight into the underlying process.) generates the splitting field of a given polynomial. Obviously, this will only be of limited utility, because the degrees of the polynomials increase too rapidly, which makes it very time- and space- consuming to derive the splitting field in this way. I noticed that the coefficients of the particular polynomials I was coming up with appeared to bear some relationship to the coefficients of the polynomial that I started with, so I decided to construct a heuristic examination of the splitting polynomials to see if I could derive an exact relationship. (Of course, this only has a chance to succeed if the PARI functions I used to derive the splitting polynomial were based on algorithms that had some unknown symmetry built in -- these are mainly the PARI functions nffactor and rnfequation.) I was able to derive a conjectural relationship fairly quickly for trying the same thing for quartics right now, but I expect it to take me at least a few days to get a result. This is of course far from proving anything, since my results are purely heuristic, but maybe someone would like to take a shot at it. To restate the original conjecture (with simplifications resulting from your comments): Let P be an irreducible (in Z) cubic x^3 + a*x^2 + b*x + c with coefficients in Z. Let D be its discriminant, D = -4*b^3 + a^2*b^2 + 18*c*a*b + -4*c*a^3 - 27*c^2 and let E be the leading coefficient of its Hessian covariant, E = 3*b - a^2 Let Q be the sextic, Q = (x^3 + E*x)^2 - D (Actually, Q can also be described as a cubic in x^2.) Conjecture: If the Galois group of P is of order 6 (i.e. P is not 'cyclic'), then Q is irreducible and generates the splitting field of P. Note that if P is cyclic, then D is a square in Z, say D = d^2, in which case Q is the difference of two squares over Z, and can thus be factored into two cubic components, namely x^3 + E*x + d and x^3 + E*x - d. === Subject: Re: Identifying possible expressions for numeric approximations of numbers. have a look http://farm4.static.flickr.com/3496/3256947403_398e26dde8_o.jpg qed === Subject: Re: Identifying possible expressions for numeric approximations of numbers. Math Forum DiscussionsMartin Michael Musatov Posts: 641. Registered: 4/19/09 ... from Wolfram MathWorldFermat's 4n+1 theorem, sometimes called Fermat's ... mathforum.org/kb/thread.jspa?messageID=6738341&tstart=0 mathforum.org/kb/message.jspa?messageID=6746768&tstart=75 Math Forum DiscussionsMartin Michael Musatov Posts: 734. Registered: 4/19/09 ... MathWorld Headline News ... mathforum.org/kb/message.jspa?messageID=6747368&tstart=75 mathforum.org/kb/message.jspa?messageID=6736479&tstart=0 mathforum.org/kb/message.jspa?messageID=6749035&tstart=60 - 12 hours ago Math Forum DiscussionsJun 11, 2009 ... The Math Forum Donate to the Math Forum ... ... Martin Michael Musatov ... mathforum.org/kb/message.jspa?messageID=6747287&tstart=75 Math Forum DiscussionsJun 10, 2009 ... The Math Forum Donate to the Math Forum ... http://functions.wolfram.com/Polynomials/LegendreP2/21/02/01/ ... Martin Michael Musatov ... mathforum.org/kb/message.jspa?messageID=6746671&tstart=75 Math Forum DiscussionsThe Math Forum Donate to the Math Forum ... Wolfram Research The idea is correct, but your formula mixes vectors and scalars. ... Martin Michael Musatov ... mathforum.org/kb/message.jspa?messageID=6732833&tstart=60 - 12 hours ago Math Forum DiscussionsThe Math Forum Donate to the Math Forum ... ... Martin Michael Musatov. 5/31/09 ... mathforum.org/kb/thread.jspa?messageID=6732837&tstart=0 Math Forum DiscussionsThe Math Forum Donate to the Math Forum ... would be: || (A-B) - (A-B).(C-B)/||C-B|| || Daniel Lichtblau Wolfram Research ... Martin Michael Musatov ... mathforum.org/kb/message.jspa?messageID=6729970&tstart=60 - 12 hours ago === Subject: Can FOL formalize all of math? posting-account=XD7nMAkAAAAUcvFxxCbDAfgYkTA2oJDz Gecko/2009042513 Ubuntu/8.04 (hardy) Firefox/3.0.10,gzip(gfe),gzip(gfe) Can first-order logic formalize all of mathematics, i.e. express all proven axioms, theorems so that the latter could be proven in it as well? If not, what logic can do that? In an earlier thread, I believe someone claimed that second-order logic can express more than first-order logic, but can not prove more. Doesn't this at least mean that second-order logic is useless for formalizing mathematics? === Subject: Re: Can FOL formalize all of math? posting-account=OWfgwwgAAADQpH2XgMDMe2wuQ7OFPXlE Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > Can first-order logic formalize all of mathematics, i.e. express all > proven axioms, theorems so that the latter could be proven in it as > well? > As an amateur, I have spent many years pondering this question. Without some kind of 2nd-order-like constructs (e.g. sets), it seems you would not be able to do things like proofs by induction. Dan Download my DC Proof software at http://www.dcproof.com === Subject: Re: Can FOL formalize all of math? posting-account=XD7nMAkAAAAUcvFxxCbDAfgYkTA2oJDz Gecko/2009042513 Ubuntu/8.04 (hardy) Firefox/3.0.10,gzip(gfe),gzip(gfe) > proven axioms, theorems I meant axioms and proven theorems === Subject: Is Boolean algebra a finite field? posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) Is Boolean algebra a finite field? === Subject: Re: Is Boolean algebra a finite field? > Is Boolean algebra a finite field? Take each thread and read by each color each pass, how many coherent threads are there? http://mathforum.org/kb/message.jspa?messageID=6751446&tstart=0 === Subject: Re: Is Boolean algebra a finite field? Is Boolean algebra a finite field? The question doesn't make any sense as posed since you don't specify what the operations are supposed to be on the finite field. Any finite set of the same cardinality as a finite field can be given the structure of a finite field - that is trivial. Every boolean algebra is, in a certain sense, equivalent to a boolean ring (i.e. a ring where x^2 = x) where product = intersection, sum = symmetric difference i.e. xy = x/y, x+y = (x'/y)/(x/y') But x^2 = x => x = 0 or x = 1 in a field. Hence the only such field is the field of two elements F_2. --Bill Dubuque === Subject: Re: Is Boolean algebra a finite field? > Is Boolean algebra a finite field? I would say yes in the two valued case, with + defined as xor. The additive inverse is identity function. a+(-a) = a+a = a xor a = 0 The multiplicative inverse is identify function. a*(a^-1) = a*a = a and a = 1 for a!=0. The complement is neither the additive nor the multiplicative inverse, it can be viewed as the shift by 1: ~a = 1 + a = 1 xor a In the multivalued case (take the variable to range over sets), with + define as symmetric difference we have: The additive inverse is again the identity function: a + (-a) = a + a = a symdiff a = (a a) u (a a) = 0 But a multiplicative inverse does not exist, there- fore in the general case: A boolean algebra is only a boolean ring, but not a field. === Subject: Re: Is Boolean algebra a finite field? > therefore in the general case: A boolean algebra is only a boolean ring, but not a field. > Or as William Elliot has already put it: > No. > Every element has a complement but > not every element has a reciprocal. Yes in the multi-valued case the complement is again a shift, but neither the multiplicative nor the additive inverse. Namely: ~a = U + a = U simdiff a = (U a) u (a U) = U a u {} = U a Where U is the top element. Bye === Subject: Re: Is Boolean algebra a finite field? > therefore in the general case: > A boolean algebra is only a boolean ring, but not a field. > > Or as William Elliot has already put it: > > No. > Every element has a complement but > not every element has a reciprocal. > > Yes in the multi-valued case the complement is > again a shift, but neither the multiplicative > nor the additive inverse. Namely: ~a = U + a = U simdiff a = (U a) u (a U) = U a u {} = U a > > Where U is the top element. > > Bye And there is still the infinite case. Bye === Subject: Re: Is Boolean algebra a finite field? > Is Boolean algebra a finite field? or is equivalent to max(a,b) and is equivalent to min(a,b) where a, b are in {0,1} and 'and' is also equal to 1 - max(a,b). It should be obvious by several methods there is no unique inverses 0 1 0 0 1 1 1 1 0 1 0 0 0 1 0 1 But for xor we do have a inverse(this is why it's used in encoding/encryption) we have 0 1 0 0 1 1 1 0 and maybe we can have someting like xnd which gives 0 1 0 1 0 1 0 1 of course xnd = neg xor and not all that useful and there isn't anyway to recover or and and from them. === Subject: Re: Is Boolean algebra a finite field? posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Is Boolean algebra a finite field? or is equivalent to max(a,b) and is equivalent to min(a,b) where a, b are in > {0,1} and 'and' is also equal to 1 - max(a,b). It should be obvious by several methods there is no unique inverses Here is a definition from Wikipedia Additive and multiplicative inverses For every a in F, there exists an element -a in F, such that a + (-a) = 0. Similarly, for any a in F other than 0, there exists an element a -1 in F, such that a .87 a-1 = 1. (The elements a + (-b) and a .87 b-1 are also denoted a - b and a/b, respectively.) In other words, subtraction and division operations exist. It says there exists an element -a. It does not say that -a must be unique. 0 1 > 0 0 1 > 1 1 1 0 1 > 0 0 0 > 1 0 1 But for xor we do have a inverse(this is why it's used in > encoding/encryption) we have 0 1 > 0 0 1 > 1 1 0 and maybe we can have someting like xnd which gives 0 1 > 0 1 0 > 1 0 1 of course xnd = neg xor and not all that useful and there isn't anyway to > recover or and and from them. === Subject: Re: Is Boolean algebra a finite field? > It says there exists an element -a. It does not say that -a must > be unique. You can prove that it is. Just repeat these steps: (1) Assume s and t are additive inverses of a. (2) Then a + s = 0 = a + t. (3) Then s + (a + s) = s + (a + t). (4) Then (s + a) + s = (s + a) + t. (5) Then 0 + s = 0 + t. (6) Then s = t. Its uniqueness, incidentally, is why the inverse can be notated -a without ambiguity. === Subject: Re: Is Boolean algebra a finite field? unique. What Wikipedia says or leaves unsaid is immaterial. That -a is unique follows from the field axioms. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Is Boolean algebra a finite field? > Is Boolean algebra a finite field? or is equivalent to max(a,b) and is equivalent to min(a,b) where a, b are > in > {0,1} and 'and' is also equal to 1 - max(a,b). It should be obvious by several methods there is no unique inverses Here is a definition from Wikipedia Additive and multiplicative inverses For every a in F, there exists an element -a in F, such that a + (-a) = 0. Similarly, for any a in F other than 0, there exists an element a -1 in F, such that a .87 a-1 = 1. (The elements a + (-b) and a .87 b-1 are also denoted a - b and a/b, respectively.) In other words, subtraction and division operations exist. It says there exists an element -a. It does not say that -a must be unique. --- If the element exists it must be unique. This is a basic proof in group theory 101. Suppose you have two inverses a' and a'' to a. By the definition of inverses, a'a = e = a''a. Right multiplying by a' one === Subject: Re: Is Boolean algebra a finite field? posting-account=U44YcwkAAAAbGXB70Qr7gA3kornmKE4i Gecko/20080922 Ubuntu/7.10 (gutsy) Firefox/2.0.0.17,gzip(gfe),gzip(gfe) > Is Boolean algebra a finite field? (I didn't understand the other answer, but anyway...) No. Boolean algebra (at least according to Wikipedia) consists of a (finite) set of Truth values, typically {T, F}, and logical operations, numbering at least three, viz AND, OR, NOT. A finite field consists of a finite set of entities, and *two* operations, typically called + and x. So they are different. But Boolean algebra under AND and XOR is a finite set of entities and two operations, and you only need a piece of A7 paper to write it all down, so why not check for yourself if it meets the full set of field axioms. Does that answer your question? Brian Chandler === Subject: Re: Is Boolean algebra a finite field? posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Is Boolean algebra a finite field? (I didn't understand the other answer, but anyway...) No. Boolean algebra (at least according to Wikipedia) consists of a > (finite) set of Truth values, typically {T, F}, and logical > operations, numbering at least three, viz AND, OR, NOT. A finite field consists of a finite set of entities, and *two* > operations, typically called + and x. So they are different. But Boolean algebra under AND and XOR is a > finite set of entities and two operations, and you only need a piece > of A7 paper to write it all down, so why not check for yourself if it > meets the full set of field axioms. Does that answer your question? No. Brian Chandler === Subject: Re: Is Boolean algebra a finite field? posting-account=U44YcwkAAAAbGXB70Qr7gA3kornmKE4i Gecko/20080922 Ubuntu/7.10 (gutsy) Firefox/2.0.0.17,gzip(gfe),gzip(gfe) > Is Boolean algebra a finite field? > (I didn't understand the other answer, but anyway...) > No. Boolean algebra (at least according to Wikipedia) consists of a > (finite) set of Truth values, typically {T, F}, and logical > operations, numbering at least three, viz AND, OR, NOT. > A finite field consists of a finite set of entities, and *two* > operations, typically called + and x. > So they are different. But Boolean algebra under AND and XOR is a > finite set of entities and two operations, and you only need a piece > of A7 paper to write it all down, so why not check for yourself if it > meets the full set of field axioms. > Does that answer your question? No. It might help if you clarified just what your question was. I interpreted Boolean algebra as meaning simply the algebra of truth values and operations on them which is what one of the Wikipedia entries refers to. One reason for thinking this is that the other meaning (and WP entry) is for *a* Boolean algebra, which is one of a class of abstract structures; if you had meant this, you might have been expected to write Is a Boolean algebra a finite field? -- except that this is an odd thing to ask, rather than simply Is a Boolean algebra the same as a finite field? (to which the answer is fairly obviously No. described here: http://en.wikipedia.org/wiki/Boolean_algebra_(introduction) In this case my answer stands. Obviously, since a finite field is a structure with two operators, general truth value arithmetic can't be a finite field as such, because it has more operators. But write out the tables for AND(&) and XOR(x) on T and F (should look OK in fixed font): & F T x F T F F F F F T T F T T T F Compare with the tables for the field of order 2. HTH Brian Chandler === Subject: Re: Is Boolean algebra a finite field? posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Is Boolean algebra a finite field? > (I didn't understand the other answer, but anyway...) > No. Boolean algebra (at least according to Wikipedia) consists of a > (finite) set of Truth values, typically {T, F}, and logical > operations, numbering at least three, viz AND, OR, NOT. > A finite field consists of a finite set of entities, and *two* > operations, typically called + and x. > So they are different. But Boolean algebra under AND and XOR is a > finite set of entities and two operations, and you only need a piece > of A7 paper to write it all down, so why not check for yourself if it > meets the full set of field axioms. > Does that answer your question? > No. It might help if you clarified just what your question was. I > interpreted Boolean algebra as meaning simply the algebra of truth > values and operations on them which is what one of the Wikipedia > entries refers to. One reason for thinking this is that the other > meaning (and WP entry) is for *a* Boolean algebra, which is one of a > class of abstract structures; if you had meant this, you might have > been expected to write Is a Boolean algebra a finite field? -- > except that this is an odd thing to ask, rather than simply Is a > Boolean algebra the same as a finite field? (to which the answer is > fairly obviously No. described here:http://en.wikipedia.org/wiki/Boolean_algebra_(introduction) In this case my answer stands. Obviously, since a finite field is a > structure with two operators, general truth value arithmetic can't > be a finite field as such, because it has more operators. But write > out the tables for AND(&) and XOR(x) on T and F (should look OK in > fixed font): & F T x F T > F F F F F T > T F T T T F Compare with the tables for the field of order 2. HTH > Brian Chandler- Hide quoted text - This is the answer I was looking for: Based on this: For every a in F, there exists an element -a in F, such that a + (-a) = 0 If a = 1 there is no -a such that 1 + (-a) = 0. I got somehow mixed up, that is why I asked the question. So what kind of algebra is Boolean algebra? Is it a category on its own? === Subject: Re: Is Boolean algebra a finite field? posting-account=U44YcwkAAAAbGXB70Qr7gA3kornmKE4i Gecko/20080922 Ubuntu/7.10 (gutsy) Firefox/2.0.0.17,gzip(gfe),gzip(gfe) > Is Boolean algebra a finite field? > No. Boolean algebra (at least according to Wikipedia) consists of a > (finite) set of Truth values, typically {T, F}, and logical > operations, numbering at least three, viz AND, OR, NOT. ... the tables for AND(&) and XOR(x) on T and F (should look OK in > fixed font): > & F T x F T > F F F F F T > T F T T T F > Compare with the tables for the field of order 2. > This is the answer I was looking for: Good! > Based on this: For every a in F, there exists an element -a in F, such that a + (-a) > = 0 If a = 1 there is no -a such that 1 + (-a) = 0. Yes there is (in the above context), because F is the 0 of the field, T is the 1 of the field, x (XOR) is the + of the field, and & is the * of the field. So translating to canonical terminology: > For every a in F, there exists an element -a in F, such that a + (-a) > = 0 Means (remembering that F is now False, not Field) for any (either) of {F, T}, there is an inverse under x (XOR) -- and of course everything is its own inverse under XOR. Thus: If a = T, T x T = F, so -T = T > I got somehow mixed up, that is why I asked the question. So what kind of algebra is Boolean algebra? Is it a category on its > own? Ouch! Say that and someone will start ranting about category theory. Like all other noun phrases in mathematics, Boolean algebra is rather ambiguous. The meaning I took from your question was the more elementary one simply meaning arithmetic with truth values. But -- a Boolean algebra -- which refers to the structure of a power set with union and intersect. This is not a field. Brian Chandler === Subject: Re: Is Boolean algebra a finite field? posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Is Boolean algebra a finite field? > No. Boolean algebra (at least according to Wikipedia) consists of a > (finite) set of Truth values, typically {T, F}, and logical > operations, numbering at least three, viz AND, OR, NOT. ... the tables for AND(&) and XOR(x) on T and F (should look OK in > fixed font): > & F Tx F T > F F FF F T > T F TT T F > Compare with the tables for the field of order 2. > This is the answer I was looking for: Good! > Based on this: > For every a in F, there exists an element -a in F, such that a + (-a) > = 0 > If a = 1 there is no -a such that 1 + (-a) = 0. Yes there is (in the above context), because F is the 0 of the > field, T is the 1 of the field, x (XOR) is the + of the field, and > & is the * of the field. So translating to canonical terminology: > For every a in F, there exists an element -a in F, such that a + (-a) > = 0 Means (remembering that F is now False, not Field) for any > (either) of {F, T}, there is an inverse under x (XOR) -- and of course > everything is its own inverse under XOR. Thus: > If a = T, T x T = F,so -T = T > I got somehow mixed up, that is why I asked the question. > So what kind of algebra is Boolean algebra? Is it a category on its > own? Ouch! Say that and someone will start ranting about category theory. > Like all other noun phrases in mathematics, Boolean algebra is > rather ambiguous. The meaning I took from your question was the more > elementary one simply meaning arithmetic with truth values. But > -- a Boolean algebra -- which refers to the structure of a power set > with union and intersect. So where did xor come from? > This is not a field. So what is it? > Brian Chandler === Subject: Re: Is Boolean algebra a finite field? > No. Boolean algebra (at least according to Wikipedia) consists of a > (finite) set of Truth values, typically {T, F}, and logical > operations, numbering at least three, viz AND, OR, NOT. This is an idiosyncratic definition of Boolean algebra -- or, rather, sounds more like a description, not of a class of mathematical structures, but of Boolean algerba as a field of mathematics, the algebraic study of propositional logic. The more usual definition has it that a Boolean algebra is a complemented distributive bounded lattice, or something to that effect, i.e. a partially ordered set such that infima and suprema of any two elements exist, the distributivity holds for join and meet, there is a least element and a greatest element, and every element has a complement. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Is Boolean algebra a finite field? <87zlccfsge.fsf@alatheia.truth.invalid> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > The more usual definition has it > that a Boolean algebra is a complemented distributive bounded lattice, > or something to that effect, i.e. a partially ordered set such that > infima and suprema of any two elements exist, the distributivity holds > for join and meet, there is a least element and a greatest element, and > every element has a complement. Which is the more usual definition is a matter of convention, I suppose. But the above, while pleasingly algebraic, assumes a specific signature, despite the fact that others are possible. What seems most significant to me about the various signatures of boolean algebras is that they completely axiomatize the two-element model. This immediately leads me to wonder if the three element model has ever been completely axiomatized. Anyone know? Here's a smallish but complete set of axioms for boolean algebra, with the and/or/not signature (^/2 v/2 -/1). x ^ 1 = x. x v 0 = x. x ^ -x = 0. x v -x = 1. x ^ y = y ^ x. x ^ (y v z) = (x ^ y) v (x ^ z). x v (y ^ z) = (x v y) ^ (x v z). Its little fillip of asymmetry amuses me. Marshall === Subject: Re: Is Boolean algebra a finite field? > Here's a smallish but complete set of axioms for boolean > algebra, with the and/or/not signature (^/2 v/2 -/1). > > x ^ 1 = x. > x v 0 = x. > > x ^ -x = 0. > x v -x = 1. > > x ^ y = y ^ x. > > x ^ (y v z) = (x ^ y) v (x ^ z). > x v (y ^ z) = (x v y) ^ (x v z). Are you sure we don't need x v y = y ^ x ? Hm, since each pair of formula above are dual pairs (swap '^' and 'v', and '0' and '1') I wonder if we can get away with just one formula from each pair ... -- hz === Subject: Re: Is Boolean algebra a finite field? <4A3420B6.BC0E59F0@gmail.com> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > Here's a smallish but complete set of axioms for boolean > algebra, with the and/or/not signature (^/2 v/2 -/1). > x ^ 1 = x. > x v 0 = x. > x ^ -x = 0. > x v -x = 1. > x ^ y = y ^ x. > x ^ (y v z) = (x ^ y) v (x ^ z). > x v (y ^ z) = (x v y) ^ (x v z). Are you sure we don't need x v y = y ^ x ? [You meant to say Are you sure we don't need x v y = y v x ?] Yes. Although it is of course a theorem. Alternatively, we can include the commutativity of 'or' as an axiom and take the commutativity of 'and' as a theorem. That's the amusing fillip of asymmetry. Are you not entertained? http://www.youtube.com/watch?v=rbg5AAm2 XM > Hm, since each pair of formula above are dual pairs > (swap '^' and 'v', and '0' and '1') I wonder if we > can get away with just one formula from each pair ... Alas, no. If we omit any of the seven we don't have a basis any more. ---------------- At the complete other end of the spectrum, we can get the whole issue completely solved with just this: (((x v y)' v z)' v (x v (z' v (z v u)')')')' = z for the signature {or, not}. (That is, the above is a complete axiomatization of the boolean algebra.) Or again: (x | ((y | x) | x)) | (y | (z | x)) = y for the signature {nand}. Any equational axiomatization of the boolean algebra must have at least one equation of three variables. I think it might still be an open question whether there exist {or, not} axiomatizations with three variables, or with size smaller than the above one, which is 22. (Counting operators, including equals, and variables, but not parentheses.) Starting with the -------- the results are all from the paper Short Single Axioms for the Boolean Algebra by McCune et al. (The earlier stuff I figgered out m'self.) Bill McCune is one cool cat. Marshall === Subject: Re: Is Boolean algebra a finite field? > > The more usual definition has it > that a Boolean algebra is a complemented distributive bounded lattice, > or something to that effect, i.e. a partially ordered set such that > infima and suprema of any two elements exist, the distributivity holds > for join and meet, there is a least element and a greatest element, and > every element has a complement. > > Which is the more usual definition is a matter of convention, > I suppose. But the above, while pleasingly algebraic, assumes > a specific signature, despite the fact that others are possible. > > What seems most significant to me about the various signatures > of boolean algebras is that they completely axiomatize the > two-element model. This immediately leads me to wonder if > the three element model has ever been completely axiomatized. > Anyone know? There is no three element model. Every finite boolean algebra has 2^n elements, for positive n in N (Huntington 1904). Can't remember how the proof goes at the moment. > Here's a smallish but complete set of axioms for boolean > algebra, with the and/or/not signature (^/2 v/2 -/1). > x ^ 1 = x. > x v 0 = x. > > x ^ -x = 0. > x v -x = 1. > > x ^ y = y ^ x. > > x ^ (y v z) = (x ^ y) v (x ^ z). > x v (y ^ z) = (x v y) ^ (x v z). > > Its little fillip of asymmetry amuses me. Absolutely! (These are the same four postulates I posted before, -- hz === Subject: Re: Is Boolean algebra a finite field? <4A33FD0A.2660EE34@gmail.com> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > The more usual definition has it > that a Boolean algebra is a complemented distributive bounded lattice, > or something to that effect, i.e. a partially ordered set such that > infima and suprema of any two elements exist, the distributivity holds > for join and meet, there is a least element and a greatest element, and > every element has a complement. > Which is the more usual definition is a matter of convention, > I suppose. But the above, while pleasingly algebraic, assumes > a specific signature, despite the fact that others are possible. > What seems most significant to me about the various signatures > of boolean algebras is that they completely axiomatize the > two-element model. This immediately leads me to wonder if > the three element model has ever been completely axiomatized. > Anyone know? There is no three element model. Hmmm, you have misunderstood me. I didn't mean three element model of boolean algebra; I meant three element model. We can consider (two-element) boolean algebra as a description of the (the quotes mean up to isomorphism!) complete axiomatization of the two-element model: Start with two elements. We can construct four unary functions and sixteen binary ones. We can also get away with fewer such functions and still be functionally complete: and/or/not, just nand, etc. We might try the same thing for a three element model. What operations would we need to be functionally complete? What might the axiom sets look like? (I am interested in considering the smallest cases; this is how I came up with the somewhat-regrettable x=y theory; it does the above for the one-element model.) > (These are the same four postulates I posted before, Yeah, I saw that right after I posted. Marshall === Subject: Re: Is Boolean algebra a finite field? > The more usual definition has it > that a Boolean algebra is a complemented distributive bounded lattice, > or something to that effect, i.e. a partially ordered set such that > infima and suprema of any two elements exist, the distributivity holds > for join and meet, there is a least element and a greatest element, and > every element has a complement. > Which is the more usual definition is a matter of convention, > I suppose. But the above, while pleasingly algebraic, assumes > a specific signature, despite the fact that others are possible. > What seems most significant to me about the various signatures > of boolean algebras is that they completely axiomatize the > two-element model. This immediately leads me to wonder if > the three element model has ever been completely axiomatized. > Anyone know? > There is no three element model. > > Hmmm, you have misunderstood me. I didn't mean three element > model of boolean algebra; I meant three element model. > > We can consider (two-element) boolean algebra as a > description of the (the quotes mean up to isomorphism!) > complete axiomatization of the two-element model: Start with two elements. We can construct four unary > functions and sixteen binary ones. We can also get away > with fewer such functions and still be functionally complete: > and/or/not, just nand, etc. > > We might try the same thing for a three element model. You want a set of axioms that can only be satisfied by a three-element model? > What operations would we need to be functionally > complete? What might the axiom sets look like? > > (I am interested in considering the smallest cases; > this is how I came up with the somewhat-regrettable > x=y theory; Lol! > it does the above for the one-element model.) > > (These are the same four postulates I posted before, > > Yeah, I saw that right after I posted. -- hz === Subject: Re: Is Boolean algebra a finite field? <4A341E06.C665FFF2@gmail.com> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > The more usual definition has it > that a Boolean algebra is a complemented distributive bounded lattice, > or something to that effect, i.e. a partially ordered set such that > infima and suprema of any two elements exist, the distributivity holds > for join and meet, there is a least element and a greatest element, and > every element has a complement. > Which is the more usual definition is a matter of convention, > I suppose. But the above, while pleasingly algebraic, assumes > a specific signature, despite the fact that others are possible. > What seems most significant to me about the various signatures > of boolean algebras is that they completely axiomatize the > two-element model. This immediately leads me to wonder if > the three element model has ever been completely axiomatized. > Anyone know? > There is no three element model. > Hmmm, you have misunderstood me. I didn't mean three element > model of boolean algebra; I meant three element model. > We can consider (two-element) boolean algebra as a > description of the (the quotes mean up to isomorphism!) > complete axiomatization of the two-element model: > Start with two elements. We can construct four unary > functions and sixteen binary ones. We can also get away > with fewer such functions and still be functionally complete: > and/or/not, just nand, etc. > We might try the same thing for a three element model. You want a set of axioms that can only be satisfied > by a three-element model? Not exactly. In the two-element case, I start with the model, and I get some set of operators, and some set of axioms that describe those operators. Done a certain way, I have completely described the two-element model. However the resulting theory also has four element models. (And others.) (Those higher cardinality models are necessarily not completely described by the theory, but the theory nonetheless is satisfied by those models.) Now, start with the three element model. 1) What are some sets of operators that will be functionally complete over this model? (In the two-element case, some answers are {and, or, not} and {nor}.) 2) For each of those sets of operators, what set of axioms completely describes them? (Has this work been done? I haven't been able to find it.) The resulting answers might or might not also have, say, nine element models, but I'm OK with that; I just want answers to 1 and 2. I have them for 1 element and 2 element models, so now I'm naturally asking about 3 element models. Once I have that, guess what I want next? :-) And no, I don't have that for four element models, because even though the boolean algebra has four element models, boolean algebras don't fully characterize four element models; they are not functionally complete over four elements. It occurs to me that if T is a theory that is functionally complete on a model M of cardinality m, and T is also a theory of model N of cardinality n where n > m, then T is not functionally complete on N. I bet that's a theorem somewhere with some famous mathematician and/or logician's name all over it. If not, dibs. Marshall PS. There are millions of famous people, so there are probably millions of famous logicians, but not Quine, because Quine was not a logician. PPS. If you don't get the joke of the PS, just ignore it. === Subject: Re: Is Boolean algebra a finite field? > The more usual definition has it > that a Boolean algebra is a complemented distributive bounded lattice, > or something to that effect, i.e. a partially ordered set such that > infima and suprema of any two elements exist, the distributivity holds > for join and meet, there is a least element and a greatest element, and > every element has a complement. > Which is the more usual definition is a matter of convention, > I suppose. But the above, while pleasingly algebraic, assumes > a specific signature, despite the fact that others are possible. > What seems most significant to me about the various signatures > of boolean algebras is that they completely axiomatize the > two-element model. This immediately leads me to wonder if > the three element model has ever been completely axiomatized. > Anyone know? > There is no three element model. > Hmmm, you have misunderstood me. I didn't mean three element > model of boolean algebra; I meant three element model. > We can consider (two-element) boolean algebra as a > description of the (the quotes mean up to isomorphism!) > complete axiomatization of the two-element model: > Start with two elements. We can construct four unary > functions and sixteen binary ones. We can also get away > with fewer such functions and still be functionally complete: > and/or/not, just nand, etc. > We might try the same thing for a three element model. > You want a set of axioms that can only be satisfied > by a three-element model? > > Not exactly. > > In the two-element case, I start with the model, and > I get some set of operators, and some set of axioms > that describe those operators. Done a certain way, > I have completely described the two-element > model. However the resulting theory also has > four element models. (And others.) (Those > higher cardinality models are necessarily not > completely described by the theory, but the > theory nonetheless is satisfied by those models.) > > Now, start with the three element model. > > 1) What are some sets of operators that will > be functionally complete over this model? > (In the two-element case, some answers are > {and, or, not} and {nor}.) > > 2) For each of those sets of operators, what > set of axioms completely describes them? > > (Has this work been done? I haven't been > able to find it.) > The resulting answers might or might not > also have, say, nine element models, but > I'm OK with that; I just want answers to > 1 and 2. I have them for 1 element and > 2 element models, so now I'm naturally > asking about 3 element models. Once I > have that, guess what I want next? :-) > And no, I don't have that for four element > models, because even though the boolean > algebra has four element models, boolean > algebras don't fully characterize four element > models; they are not functionally complete > over four elements. > > It occurs to me that if T is a theory that > is functionally complete on a model M of > cardinality m, and T is also a theory of > model N of cardinality n where n > m, > then T is not functionally complete on N. > > I bet that's a theorem somewhere with > some famous mathematician and/or > logician's name all over it. If not, dibs. > > Marshall > > PS. There are millions of famous people, > so there are probably millions of famous > logicians, but not Quine, because Quine > was not a logician. I think Tom Hanks is a logician. -- hz === Subject: Re: Is Boolean algebra a finite field? Here is a proof that a boolean algebra cannot have a three-valued model. Such a model would consist of the elements {0, x, 1}. So we have naturally: 1!=0 1!=x x!=0 Since a boolean algebra has a complement, we have an element y=~x. And since the model is 3-valued we have either y=1, y=x or y=0. By double negation we have also ~y=x. Check y=1: ~y=0!=x, so cannot be. Check y=0: ~y=1!=x, so cannot be. Check y=x: ~y=~x=x. So question is whether ~x=x is possible. Recall the ring, we have ~x=u+x, so question is whether 1+x=x is possible. Recall again the ring, add x on both sides and we have, 1+x+x=1+0=x+x=0, so cannot be. Q.E.D. === Subject: Re: Is Boolean algebra a finite field? <4A342DC7.3B8A8A91@gmail.com> posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Here is a proof that a boolean algebra cannot have a > three-valued model. Such a model would consist of > the elements {0, x, 1}. So we have naturally: 1!=0 > 1!=x > x!=0 Since a boolean algebra has a complement, we have > an element y=~x. And since the model is 3-valued > we have either y=1, y=x or y=0. By double negation > we have also ~y=x. Check y=1: > ~y=0!=x, so cannot be. > Check y=0: > ~y=1!=x, so cannot be. > Check y=x: > ~y=~x=x. So question is whether ~x=x is possible. Recall the > ring, we have ~x=u+x, so question is whether 1+x=x > is possible. Recall again the ring, add x on both > sides and we have, 1+x+x=1+0=x+x=0, so cannot be. Q.E.D. Three-valued logic can certainly exist and 1+x+x = 1 1+0 = 1 x+x=x === Subject: Re: Is Boolean algebra a finite field? > Three-valued logic can certainly exist and > 1+x+x = 1 > 1+0 = 1 > x+x=x No, it cannot. A finite boolean algebra has 2^n elements. Because it is isomorphic to an n copies product of the 2-valued boolean alegbra. And x+x=x is only the case for x=0. But if everything amounts to 1 and 0, than your boolean algebra has not 3 elements, but only 2. Please show me your third element! === Subject: Re: Is Boolean algebra a finite field? <4A342DC7.3B8A8A91@gmail.com> posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Three-valued logic can certainly exist and > 1+x+x = 1 > 1+0 = 1 > x+x=x No, it cannot. A finite boolean algebra has > 2^n elements. Because it is isomorphic to an > n copies product of the 2-valued boolean alegbra. And x+x=x is only the case for x=0. But if > everything amounts to 1 and 0, than your > boolean algebra has not 3 elements, but only 2. Please show me your third element! The third element has been defined by Lukasiwicz. I could dig up the tables. But let us consider the general case of fuzzy logic where the values are in the interval <0,1>. a or B =def max(a,b) a and b =def min(a,b) not a = def 1-a Then take a subset of the values S = {0, 0.5, 1). 0.5 is you third element. Then 0+0=0, 0+0.5 = 0.5, 0+1 = 1, 0.5 + 1 = 1 0*0 = 0, 0.* 0.5 = 0, 0*1 = 0, 0.5*0.5 = 0.5, 1*1 = 1 ~0=1, ~0.5 = 0.5, ~1 = 0 === Subject: Re: Is Boolean algebra a finite field? > > > Here is a proof that a boolean algebra cannot have a > three-valued model. > Q.E.D. Three-valued logic is not classical, what is the same as to say it is not based on a boolean algebra. Maybe on a distribute lattice, but not on a boolean algebra. Bye === Subject: Re: Is Boolean algebra a finite field? <4A342DC7.3B8A8A91@gmail.com> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > (Has this work been done? I haven't been > able to find it.) > Aw, geeze, Google? I was hoping someone would just tell me the answer. Better yet, tell me where I can buy a pill I can swallow that will make it so I know the answer. Come to think of it, taking a pill is too much work. past and come up dry. I don't think I know the right terms to search for (I believe I've tried the ones you mention) and/or this question isn't generally interesting enough that anyone besides me cares. I'll just have to derive it myself; so much easier and educational. > PS. There are millions of famous people, > so there are probably millions of famous > logicians, but not Quine, because Quine > was not a logician. I think Tom Hanks is a logician. And do not forget that fraction of an electron in Tom Hanks's spleen holds to Objectivism. Marshall PS. Instant gratification takes too long. -- C. Fischer === Subject: Re: Is Boolean algebra a finite field? posting-account=Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN rv:1.9.0.10) Gecko/2009042315 Firefox/3.0.10,gzip(gfe),gzip(gfe) > (Has this work been done? I haven't been > able to find it.) > Aw, geeze, Google? I was hoping someone would > just tell me the answer. Better yet, tell me where I > can buy a pill I can swallow that will make it so > I know the answer. Come to think of it, taking a pill is too much work. past and come up dry. I don't think I know the right > terms to search for (I believe I've tried the ones you > mention) and/or this question isn't generally interesting > enough that anyone besides me cares. I'll just have > to derive it myself; so much easier and educational. when you get to n-valued logics there is a more complex classification theory two big starting points for a study would be lukasiewicz and post jan lukasiewickz from the lvov-warsaw school was one of the first to systematically study these logics and their various interpretations around the same time emil post derived logical implication rules for n-valued logics from functional interpretations some possible hits lukasiewicz algebra post algebras http://en.wikipedia.org/wiki/%C5%81ukasiewicz logic http://en.wikipedia.org/wiki/Multi-valued logic there are a number of interpretations of three-valued logic including implication with quantum observables and descriptive theories of ignorance, meaning, and paradox > PS. There are millions of famous people, > so there are probably millions of famous > logicians, but not Quine, because Quine > was not a logician. > I think Tom Hanks is a logician. And do not forget that fraction of an electron > in Tom Hanks's spleen holds to Objectivism. there are no fractions of electrons leptons are atomic units of entity that hold only one opinion at a time it is said that opinion is usually an ethical evaluation of sam-a-el in relation to the universal problem of good and bad but i forget whether it was electron or positron who held the pro sam-a-el position -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: Is Boolean algebra a finite field? > The more usual definition has it > that a Boolean algebra is a complemented distributive bounded lattice, > or something to that effect, i.e. a partially ordered set such that > infima and suprema of any two elements exist, the distributivity holds > for join and meet, there is a least element and a greatest element, and > every element has a complement. Which is the more usual definition is a matter of convention, I suppose. But the above, while pleasingly algebraic, assumes a specific signature, despite the fact that others are possible. What seems most significant to me about the various signatures of boolean algebras is that they completely axiomatize the two-element model. This immediately leads me to wonder if the three element model has ever been completely axiomatized. Anyone know? Here's a smallish but complete set of axioms for boolean algebra, with the and/or/not signature (^/2 v/2 -/1). x ^ 1 = x. x v 0 = x. x ^ -x = 0. x v -x = 1. x ^ y = y ^ x. x ^ (y v z) = (x ^ y) v (x ^ z). x v (y ^ z) = (x v y) ^ (x v z). Its little fillip of asymmetry amuses me. ---- I remember reading somewhere a long time ago about someone creating a ternary based computer. It wasn't any better than a binary computer and more complex. Since you can do everything with binary and it is more efficient(atleast as far as we know) to do this electrically(much easier to determine if a state is on/off rather than something else). === Subject: Re: Is Boolean algebra a finite field? posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) On Jun 13, 11:41am, Jon Slaughter I remember reading somewhere a long time ago about someone creating a > ternary based computer. It wasn't any better than a binary computer and more > complex. Since you can do everything with binary and it is more > efficient(atleast as far as we know) to do this electrically(much easier to > determine if a state is on/off rather than something else). Bleah! I wasn't proposing it as a hardware design; the idea turns my stomach. I don't like three-valued logic in programming languages, either; see any of a million essays on how awful SQL null is. Marshall === Subject: Re: Is Boolean algebra a finite field? > > No. Boolean algebra (at least according to Wikipedia) consists of a > (finite) set of Truth values, typically {T, F}, and logical > operations, numbering at least three, viz AND, OR, NOT. > > This is an idiosyncratic definition of Boolean algebra -- or, rather, > sounds more like a description, not of a class of mathematical > structures, but of Boolean algerba as a field of mathematics, the > algebraic study of propositional logic. The more usual definition has it > that a Boolean algebra is a complemented distributive bounded lattice, > or something to that effect, i.e. a partially ordered set such that > infima and suprema of any two elements exist, the distributivity holds > for join and meet, there is a least element and a greatest element, and > every element has a complement. > How do you define truth in a boolean algebra with more than two elements? Assume we have a boolean algebra R, which has more than two elements and a formula A, there are the following options: 1) A is true in R, iff R(A)>0 2) A is true in R, iff R(A)=1 3) What else? I think for the option 2), this will inevitable lead to a corresponding 2-valued boolean algebra R' as follows: Construct an ideal via the truth condition you are using. Make it prime P, it will be maximal, and thus R' := R/P will isomorphic to the 2-valued field. Bye === Subject: Re: Is Boolean algebra a finite field? <87zlccfsge.fsf@alatheia.truth.invalid> elements? Being equal to 1. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Is Boolean algebra a finite field? > No. Boolean algebra (at least according to Wikipedia) consists of a > (finite) set of Truth values, typically {T, F}, and logical > operations, numbering at least three, viz AND, OR, NOT. > This is an idiosyncratic definition of Boolean algebra -- or, rather, > sounds more like a description, not of a class of mathematical > structures, but of Boolean algerba as a field of mathematics, the > algebraic study of propositional logic. The more usual definition has it > that a Boolean algebra is a complemented distributive bounded lattice, > or something to that effect, i.e. a partially ordered set such that > infima and suprema of any two elements exist, the distributivity holds > for join and meet, there is a least element and a greatest element, and > every element has a complement. > > How do you define truth in a boolean algebra with more than > two elements? Assume we have a boolean algebra R, which has > more than two elements and a formula A, there are the following > options: 1) A is true in R, iff R(A)>0 > 2) A is true in R, iff R(A)=1 > 3) What else? > > I think for the option 2), this will inevitable > lead to a corresponding 2-valued boolean algebra R' as > follows: Construct an ideal via the truth condition you are > using. Make it prime P, it will be maximal, and thus > R' := R/P will isomorphic to the 2-valued field. > > Bye Oops, I think it works for R(A) != 0, i.e. for a notion of satisfiability and not for a notion of tautology (which would be R(A) = 1). We will then have an ideal I with R(A) !=0 iff A not in I. And from this the rest. Bye === Subject: Re: Is Boolean algebra a finite field? > > Is Boolean algebra a finite field? Boolean algebra has infinite models as well as finite models, e.g. the power set of any set is a Boolean algebra under conjunction, disjunction, and relative complementation. I don't think a Boolean algebra can be construed as a field either, at least in any simple and obvious way. If we interpret the addition operator as disjunction (a + b) and the multiplicative operator as conjunction (a * b) then, letting '0' be'false' and '1' be 'true', we have the identities a + 0 = a and a * 1 = a but we don't have for each element a an inverse -a such that a + -a = 0 and an inverse a' such that a * a' = 1, as the field axioms dictate. (I'm not saying that a more devious mind than mine couldn't come up with an interpretation that makes a Boolean algebra a field also!) -- hz === Subject: Re: Is Boolean algebra a finite field? <4A335AF8.B2E91521@gmail.com> posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Is Boolean algebra a finite field? Boolean algebra has infinite models as well as finite models, > e.g. the power set of any set is a Boolean algebra under > conjunction, disjunction, and relative complementation. I don't think a Boolean algebra can be construed as a field > either, at least in any simple and obvious way. If we interpret the addition operator as disjunction (a + b) > and the multiplicative operator as conjunction (a * b) then, > letting '0' be'false' and '1' be 'true', we have the identities > a + 0 = a and a * 1 = a but we don't have for each element a an > inverse -a such that a + -a = 0 and an inverse a' such that > a * a' = 1, as the field axioms dictate. (I'm not saying that a more devious mind than mine couldn't come >up with an interpretation that makes a Boolean algebra a field >also!) -- > hz What about boolean algebra with only two elements? === Subject: Re: Is Boolean algebra a finite field? > Is Boolean algebra a finite field? > Boolean algebra has infinite models as well as finite models, > e.g. the power set of any set is a Boolean algebra under > conjunction, disjunction, and relative complementation. I'm glad you responded, so I can *finally* get this right (third try): The power set of any set is a Boolean algebra under set intersection and set union. > I don't think a Boolean algebra can be construed as a field > either, at least in any simple and obvious way. > If we interpret the addition operator as disjunction (a + b) > and the multiplicative operator as conjunction (a * b) then, > letting '0' be'false' and '1' be 'true', we have the identities > a + 0 = a and a * 1 = a but we don't have for each element a an > inverse -a such that a + -a = 0 That's correct. > and an inverse a' such that a * a' = 1 That's incorrect; in this interpretation a is its own multiplicative inverse: (a * a') = (a * a) = a = 1 for a =/= 0. > as the field axioms dictate. > (I'm not saying that a more devious mind than mine couldn't come > [NonBreakingSpace]up with an interpretation that makes a Boolean algebra a field > [NonBreakingSpace]also!) > > What about boolean algebra with only two elements? As has been pointed out by others, a set of two elements, call them 0 and 1, is a Boolean algebra under conjunction (*) and disjunction (+), and is a field under conjunction (*) and exclusive disjunction (xor). -- hz === Subject: Re: Is Boolean algebra a finite field? <4A33E67B.7A26FBDB@gmail.com> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) I'm glad you responded, so I can *finally* get this right > (third try): The power set of any set is a Boolean algebra under > set intersection and set union. And complement. Marshall === Subject: Re: Is Boolean algebra a finite field? > I'm glad you responded, so I can *finally* get this right > (third try): > The power set of any set is a Boolean algebra under > set intersection and set union. > > And complement. Your axioms use an operation of complementation. My postulates don't -- they just say that for every element a there is an element b such that a + b = 1 and a * b = 0. It remains a fact that the powerset of every (non-empty, that's four times) set S is a Boolean algebra under union and intersection, since such an element b obviously exists for each a in P(S). Does anyone allow a trivial Boolean algebra of one object? In that case we'll have to have 0 = 1. I think some people allow trivial Boolean rings. -- hz === Subject: Re: Is Boolean algebra a finite field? <4A341D98.E5B2E55D@gmail.com> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > I'm glad you responded, so I can *finally* get this right > (third try): > The power set of any set is a Boolean algebra under > set intersection and set union. > And complement. Your axioms use an operation of complementation. > My postulates don't -- they just say that for > every element a there is an element b such that a + b = 1and a * b = 0. D'oh! You are absolutely right; my mistake. I plead ... uh, insanity. No, twinkies! Mom! herbzet is looking at me! Marshall === Subject: Re: Is Boolean algebra a finite field? > I'm glad you responded, so I can *finally* get this right > (third try): > The power set of any set is a Boolean algebra under > set intersection and set union. > And complement. > Your axioms use an operation of complementation. > My postulates don't -- they just say that for > every element a there is an element b such that > [NonBreakingSpace] [NonBreakingSpace] [NonBreakingSpace]a + b = 1 [NonBreakingSpace]and [NonBreakingSpace]a * b = 0. > > D'oh! You are absolutely right; my mistake. > I plead ... uh, insanity. No, twinkies! Mom! > herbzet is looking at me! I win! I WIN! BOO-YAH, sucker! -- hz === Subject: Re: Is Boolean algebra a finite field? <4A335AF8.B2E91521@gmail.com> <4A33E67B.7A26FBDB@gmail.com> <4A341D98.E5B2E55D@gmail.com> <4A342655.882DA33B@gmail.com I win! I WIN! BOO-YAH, sucker! This is, I believe, classic bully behaviour. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Is Boolean algebra a finite field? <4A335AF8.B2E91521@gmail.com> <4A33E67B.7A26FBDB@gmail.com> <4A341D98.E5B2E55D@gmail.com> <4A342655.882DA33B@gmail.com> <87tz2j6ahz.fsf@alatheia.truth.invalid > I win! I WIN! > BOO-YAH, sucker! > > This is, I believe, classic bully behaviour. YES! YES! YES! I'm in with the in crowd! -- hz === Subject: Re: Is Boolean algebra a finite field? > Is Boolean algebra a finite field? > > Boolean algebra has infinite models as well as finite models, > e.g. the power set of any set is a Boolean algebra under > conjunction, disjunction, and relative complementation. Oops, I mean set intersection and union respectively (as well as relative complementation). My mind wandered! -- hz === Subject: Re: Is Boolean algebra a finite field? > Is Boolean algebra a finite field? > Boolean algebra has infinite models as well as finite models, > e.g. the power set of any set is a Boolean algebra under > conjunction, disjunction, and relative complementation. > > Oops, I mean set intersection and union respectively (as well > as relative complementation). > > My mind wandered! Here is a definition of a Boolean algebra; the postulates are independent (Huntington 1904): A class of elements B closed under two binary operations (+) and (*) (where a * b will be written ab) is a Boolean algebra if and only if the following postulates hold: P1. The operations are commutative. P2. Each operation is distributive over the other. P3. There exist in B distinct identity elements 0 and 1 relative to the operations (+) and (*), respectively. P4. For every a in B there exists an element a' in B such that a + a' = 1 and aa' = 0. (Taken from a book by John Eldon Whitesitt.) -- hz === Subject: Re: Is Boolean algebra a finite field? <4A335AF8.B2E91521@gmail.com> <4A335CD5.6BDC9982@gmail.com> <4A3363D1.3794B49F@gmail.com> posting-account=oTDIagkAAACTxHurtPutBWvNQS8ZCNO9 Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > Here is a definition of a Boolean algebra; the postulates > are independent (Huntington 1904): A class of elements B closed under two binary operations > (+) and (*) (where a * b will be written ab) is a Boolean > algebra if and only if the following postulates hold: P1. The operations are commutative. > P2. Each operation is distributive over the other. > P3. There exist in B distinct identity elements 0 and 1 > relative to the operations (+) and (*), respectively. > P4. For every a in B there exists an element a' in B > such that a + a' = 1 and aa' = 0. (Taken from a book by John Eldon Whitesitt.) I'm surprised not to see explicit postulation of associativity: a(bc) = (ab)c a+(b+c) = (a+b)+c Are these theorems given the above? === Subject: Re: Is Boolean algebra a finite field? <4A335AF8.B2E91521@gmail.com> <4A335CD5.6BDC9982@gmail.com> > > Here is a definition of a Boolean algebra; the postulates > are independent (Huntington 1904): > A class of elements B closed under two binary operations > (+) and (*) (where a * b will be written ab) is a Boolean > algebra if and only if the following postulates hold: > P1. The operations are commutative. > P2. Each operation is distributive over the other. > P3. There exist in B distinct identity elements 0 and 1 > [NonBreakingSpace] [NonBreakingSpace] relative to the operations (+) and (*), respectively. > P4. For every a in B there exists an element a' in B > [NonBreakingSpace] [NonBreakingSpace] such that > [NonBreakingSpace] [NonBreakingSpace] [NonBreakingSpace] [NonBreakingSpace] [NonBreakingSpace] [NonBreakingSpace]a + a' = 1 [NonBreakingSpace] and [NonBreakingSpace] [NonBreakingSpace] aa' = 0. > (Taken from a book by John Eldon Whitesitt.) > > I'm surprised not to see explicit postulation of associativity: > > a(bc) = (ab)c > a+(b+c) = (a+b)+c > > Are these theorems given the above? Yeah -- for the trickery involved try this URL: http://tinyurl.com/lwn3r8 It's theorem 5 of chapter 2, page 30. Theorem 1 is on page 28. Boolean Algebra and Its Applications -- hz === Subject: Re: Is Boolean algebra a finite field? <4A335AF8.B2E91521@gmail.com> <4A335CD5.6BDC9982@gmail.com> <4A341CEB.CE2BD791@gmail.com> posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Here is a definition of a Boolean algebra; the postulates > are independent (Huntington 1904): > A class of elements B closed under two binary operations > (+) and (*) (where a * b will be written ab) is a Boolean > algebra if and only if the following postulates hold: > P1. The operations are commutative. > P2. Each operation is distributive over the other. > P3. There exist in B distinct identity elements 0 and 1 > relative to the operations (+) and (*), respectively. > P4. For every a in B there exists an element a' in B > such that > a + a' = 1 and aa' = 0. > (Taken from a book by John Eldon Whitesitt.) > I'm surprised not to see explicit postulation of associativity: > a(bc) = (ab)c > a+(b+c) = (a+b)+c > Are these theorems given the above? Yeah -- for the trickery involved try this URL: http://tinyurl.com/lwn3r8 It's theorem 5 of chapter 2, page 30. > Theorem 1 is on page 28. Boolean Algebra and Its Applications Is it a really cool book? I have been looking for a good reference on Boolean algebra. It seems this books treats propositional logic as well as switching circuits. === Subject: Re: Is Boolean algebra a finite field? <4A335AF8.B2E91521@gmail.com> <4A335CD5.6BDC9982@gmail.com> > Here is a definition of a Boolean algebra; the postulates > are independent (Huntington 1904): > A class of elements B closed under two binary operations > (+) and (*) (where a * b will be written ab) is a Boolean > algebra if and only if the following postulates hold: > P1. The operations are commutative. > P2. Each operation is distributive over the other. > P3. There exist in B distinct identity elements 0 and 1 > [NonBreakingSpace] [NonBreakingSpace] relative to the operations (+) and (*), respectively. > P4. For every a in B there exists an element a' in B > [NonBreakingSpace] [NonBreakingSpace] such that > [NonBreakingSpace] [NonBreakingSpace] [NonBreakingSpace] [NonBreakingSpace] [NonBreakingSpace] [NonBreakingSpace]a + a' = 1 [NonBreakingSpace] and [NonBreakingSpace] [NonBreakingSpace] aa' = 0. > (Taken from a book by John Eldon Whitesitt.) > I'm surprised not to see explicit postulation of associativity: > a(bc) = (ab)c > a+(b+c) = (a+b)+c > Are these theorems given the above? > Yeah -- for the trickery involved try this URL: > [NonBreakingSpace]http://tinyurl.com/lwn3r8 > It's theorem 5 of chapter 2, page 30. > Theorem 1 is on page 28. > Boolean Algebra and Its Applications > > Is it a really cool book? I have been looking for a good reference on > Boolean algebra. It seems this books treats propositional logic as > well as switching circuits. Let's put it this way -- when I read it a long time ago, it didn't make my head hurt too much. I liked it. Suggest you preview it at the above URL. Check out the user reviews. -- hz === Subject: Re: Is Boolean algebra a finite field? <4A335AF8.B2E91521@gmail.com> <4A335CD5.6BDC9982@gmail.com> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) On Jun 13, 12:08pm, cbr...@cbrownsystems.com > Here is a definition of a Boolean algebra; the postulates > are independent (Huntington 1904): > A class of elements B closed under two binary operations > (+) and (*) (where a * b will be written ab) is a Boolean > algebra if and only if the following postulates hold: > P1. The operations are commutative. > P2. Each operation is distributive over the other. > P3. There exist in B distinct identity elements 0 and 1 > relative to the operations (+) and (*), respectively. > P4. For every a in B there exists an element a' in B > such that > a + a' = 1 and aa' = 0. > (Taken from a book by John Eldon Whitesitt.) I'm surprised not to see explicit postulation of associativity: a(bc) = (ab)c > a+(b+c) = (a+b)+c Are these theorems given the above? Yes. Marshall === Subject: Re: Is Boolean algebra a finite field? > Is Boolean algebra a finite field? > No. Every element has a complement but not every element has a reciprocal. === Subject: Re: Is Boolean algebra a finite field? <20090612215751.H18493@agora.rdrop.com> posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Is Boolean algebra a finite field? No. > Every element has a complement but > not every element has a reciprocal. Here is a definition from Wikipedia Additive and multiplicative inverses For every a in F, there exists an element -a in F, such that a + (-a) = 0. Similarly, for any a in F other than 0, there exists an element a -1 in F, such that a .87 a-1 = 1. (The elements a + (-b) and a .87 b-1 are also denoted a - b and a/b, respectively.) In other words, subtraction and division operations exist. Doesn't Boolean algebra satisfy this? === Subject: Re: Is Boolean algebra a finite field? Content-ID: <20090614021230.G38380@agora.rdrop.com> --------------------------------------------------------------------- Content-ID: <20090614021230.E38380@agora.rdrop.com> Is Boolean algebra a finite field? > No. > Every element has a complement but > not every element has a reciprocal. Here is a definition from Wikipedia > Additive and multiplicative inverses > For every a in F, there exists an element -a in F, such that a + (-a) > = 0. Similarly, for any a in F other than 0, there exists an element > a-1 in F, such that a # a-1 = 1. (The elements a + (-b) and a á b-1 are What does a # a-1 mean? The notation for multiplicative inverse is a^-1 and not a - 1 = a + (-1). > also denoted a - b and a/b, respectively.) In other words, subtraction > and division operations exist. Doesn't Boolean algebra satisfy this? No. Let a' be the complement of a, a + b is union or join of a and b, ab the intersection or meet of a and b. Given a Boolean algebra with three elements, 0,a,1, there is no x with ax = 1 even though a /= 0, for we would have a = a1 = a(1 + x) = a1 + ax = a + 1 = 1. OTOH, a Boolean algebra of two elements is a field. A Boolean algebra can be converted to a ring by defining a new ++ with a ++ b = ab' + a'b, and keeping ab as is. === Subject: Re: Is Boolean algebra a finite field? <20090614020936.D36968@agora.rdrop.com> posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Is Boolean algebra a finite field? > No. > Every element has a complement but > not every element has a reciprocal. > Here is a definition from Wikipedia > Additive and multiplicative inverses > For every a in F, there exists an element -a in F, such that a + (-a) > = 0. Similarly, for any a in F other than 0, there exists an element > a-1 in F, such that a # a-1 = 1. (The elements a + (-b) and a .87 b-1 are What does a # a-1 mean?The notation for multiplicative inverse > is a^-1 and not a - 1 = a + (-1). > also denoted a - b and a/b, respectively.) In other words, subtraction > and division operations exist. > Doesn't Boolean algebra satisfy this? No.Let a' be the complement of a, > a + b is union or join of a and b, > ab the intersection or meet of a and b. Given a Boolean algebra with three elements, 0,a,1, > there is no x with ax = 1 even though a /= 0, for we > would have a = a1 = a(1 + x) = a1 + ax = a + 1 = 1. OTOH, a Boolean algebra of two elements is a field. A Boolean algebra can be converted to a ring by > defining a new ++ with a ++ b = ab' + a'b, > and keeping ab as is. OK, let me ask you this then: Does ++, ., ' give you the ability to define all the combinatorial functions? Are you saying that 2-valued Boolean algebra is a field (or can be massaged into a field) and multivalued Boolean algebras are not fields? If Boolean algebra (in general) is not a field then what is it? === Subject: Re: Is Boolean algebra a finite field? posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2009060310 Ubuntu/8.10 (intrepid) Firefox/3.0.11,gzip(gfe),gzip(gfe) > Is Boolean algebra a finite field? > No. > Every element has a complement but > not every element has a reciprocal. Here is a definition from Wikipedia Additive and multiplicative inverses > For every a in F, there exists an element -a in F, such that a + (-a) > = 0. Similarly, for any a in F other than 0, there exists an element a > -1 in F, such that a .87 a-1 = 1. (The elements a + (-b) and a .87 b-1 are > also denoted a - b and a/b, respectively.) In other words, subtraction > and division operations exist. Doesn't Boolean algebra satisfy this? No, not necessarily. Division is not required for a Boolean algebra. A basic example of a (finite) Boolean algebra is the algebra of subsets of a (finite) set S. Addition & subtraction are the same thing, symmetric difference of sets. Multiplication is set intersection. Zero is the empty set and one is the universal set. It seems intuitively clear that the only unit (an invertible multiplicative element) is one (the universal set) in these examples, e.g. that a set intersection cannot be undone by another (unless the set intersection did nothing to begin with). However a field extension of Z/2Z would be both a field and a Boolean algebra, if that helps. === Subject: Re: Is Boolean algebra a finite field? posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Is Boolean algebra a finite field? > No. > Every element has a complement but > not every element has a reciprocal. > Here is a definition from Wikipedia > Additive and multiplicative inverses > For every a in F, there exists an element -a in F, such that a + (-a) > = 0. Similarly, for any a in F other than 0, there exists an element a > -1 in F, such that a .87 a-1 = 1. (The elements a + (-b) and a .87 b-1 are > also denoted a - b and a/b, respectively.) In other words, subtraction > and division operations exist. > Doesn't Boolean algebra satisfy this? No, not necessarily. Division is not required for a > Boolean algebra. A basic example of a (finite) Boolean algebra is the > algebra of subsets of a (finite) set S. Addition & > subtraction are the same thing, symmetric difference > of sets. Multiplication is set intersection. Zero > is the empty set and one is the universal set. It seems intuitively clear that the only unit (an > invertible multiplicative element) is one (the > universal set) in these examples, e.g. that a set > intersection cannot be undone by another (unless > the set intersection did nothing to begin with). However a field extension of Z/2Z would be both a > field and a Boolean algebra, if that helps. > - Show quoted text - OK, based on this: For every a in F, there exists an element -a in F, such that a + (-a) = 0 If a = 1 there is no -a such that 1 + (-a) = 0. So what kind of algebra is Boolean algebra? Is it a category on its own? === Subject: Re: Is Boolean algebra a finite field? posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2009060310 Ubuntu/8.10 (intrepid) Firefox/3.0.11,gzip(gfe),gzip(gfe) > Is Boolean algebra a finite field? > No. > Every element has a complement but > not every element has a reciprocal. > Here is a definition from Wikipedia > Additive and multiplicative inverses > For every a in F, there exists an element -a in F, such that a + (-a) > = 0. Similarly, for any a in F other than 0, there exists an element a > -1 in F, such that a .87 a-1 = 1. (The elements a + (-b) and a .87 b-1 are > also denoted a - b and a/b, respectively.) In other words, subtraction > and division operations exist. > Doesn't Boolean algebra satisfy this? > No, not necessarily.Division is not required for a > Boolean algebra. > A basic example of a (finite) Boolean algebra is the > algebra of subsets of a (finite) set S.Addition & > subtraction are the same thing, symmetric difference > of sets.Multiplication is set intersection. Zero > is the empty set and one is the universal set. > It seems intuitively clear that the only unit (an > invertible multiplicative element) is one (the > universal set) in these examples, e.g. that a set > intersection cannot be undone by another (unless > the set intersection did nothing to begin with). > However a field extension of Z/2Z would be both a > field and a Boolean algebra, if that helps. > - Show quoted text - OK, based on this: For every a in F, there exists an element -a in F, such that a + (-a) > = 0 If a = 1 there is no -a such that 1 + (-a) = 0. You seem to confuse additive and multiplicative inverses. Algebras are rings; as such there is always an additive inverse. Boolean algebras are special in that each element is its own additive inverse. In particular 1 + 1 = 0. But your question was whether a Boolean algebra is necessarily a finite field. This is false on two accounts. Trivially a Boolean algebra need not be finite; the example of a power set of S, described in my earlier post, is finite if and only if S is finite. But let's assume you meant to ask whether a finite Boolean algebra is necessarily a field. [After all finiteness does help in some ways, such as Wedderburn's theorem that a finite division ring is a field.] The answer is no, again with examples in which the only unit of a Boolean algebra happens to be 1. In a field every nonzero element would be a unit. > So what kind of algebra is Boolean algebra? > Is it a category on its own? An algebra is generally defined as a ring with a vector space structure with respect to a field, which may be viewed if helpful as a subring. Thus a Boolean algebra is exactly an algebra over Z/2Z. === Subject: Re: Is Boolean algebra a finite field? posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Is Boolean algebra a finite field? > No. > Every element has a complement but > not every element has a reciprocal. > Here is a definition from Wikipedia > Additive and multiplicative inverses > For every a in F, there exists an element -a in F, such that a + (-a) > = 0. Similarly, for any a in F other than 0, there exists an element a > -1 in F, such that a .87 a-1 = 1. (The elements a + (-b) and a .87 b-1 are > also denoted a - b and a/b, respectively.) In other words, subtraction > and division operations exist. > Doesn't Boolean algebra satisfy this? > No, not necessarily.Division is not required for a > Boolean algebra. > A basic example of a (finite) Boolean algebra is the > algebra of subsets of a (finite) set S.Addition & > subtraction are the same thing, symmetric difference > of sets.Multiplication is set intersection. Zero > is the empty set and one is the universal set. > It seems intuitively clear that the only unit (an > invertible multiplicative element) is one (the > universal set) in these examples, e.g. that a set > intersection cannot be undone by another (unless > the set intersection did nothing to begin with). > However a field extension of Z/2Z would be both a > field and a Boolean algebra, if that helps. > - Show quoted text - > OK, based on this: > For every a in F, there exists an element -a in F, such that a + (-a) > = 0 > If a = 1 there is no -a such that 1 + (-a) = 0. > You seem to confuse additive and multiplicative > inverses.Algebras are rings; as such there is > always an additive inverse.Boolean algebras > are special in that each element is its own > additive inverse.In particular 1 + 1 = 0. What on earth are you talking about?? In a Boolean algebra + os OR not XOR. > But your question was whether a Boolean algebra > is necessarily a finite field.This is false > on two accounts.Trivially a Boolean algebra > need not be finite; the example of a power set > of S, described in my earlier post, is finite > if and only if S is finite. But let's assume you meant to ask whether a > finite Boolean algebra is necessarily a field. > [After all finiteness does help in some ways, > such as Wedderburn's theorem that a finite > division ring is a field.]The answer is no, > again with examples in which the only unit > of a Boolean algebra happens to be 1.In a > field every nonzero element would be a unit. > So what kind of algebra is Boolean algebra? > Is it a category on its own? An algebra is generally defined as a ring > with a vector space structure with respect > to a field, which may be viewed if helpful > as a subring.Thus a Boolean algebra is > exactly an algebra over Z/2Z. > - Show quoted text - Let me rephrase my question: is Boolean algebra a field? === Subject: Re: Is Boolean algebra a finite field? posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2009060310 Ubuntu/8.10 (intrepid) Firefox/3.0.11,gzip(gfe),gzip(gfe) [snip much repetition of basic question and answer] I think I'm beginning to understand the root of the confusion here. You asked if a Boolean algebra is a finite field. I assumed you were thinking in terms of the Boolean algebra being a ring extension of Z/2Z, which is one way of defining Boolean algebras that is consistent with the usual meaning of algebra in ring theory. > What on earth are you talking about?? In a Boolean > algebra + os OR not XOR. Stop and think. We should at least agree that Z/2Z is both a Boolean algebra and a field. What is 1+1 in Z/2Z? Is it 1 OR 1? [Hint: no.] In fact for the standard example of a Boolean algebra I gave twice already, the complete family of subsets of some set S (aka power set of S), addition is XOR (aka the symmetric difference of two sets). You may prefer to define a Boolean algebra in terms of AND, OR, and NOT, the logical operators. But it would still not be correct to identify OR with an addition operator for a ring. The error in trying to make OR into an addition operator is that OR does not have elementwise inverses for nonzero elements. > Let me rephrase my question: is Boolean algebra a field? Let me repeat my answer. Not all Boolean algebras are fields. Those that are fields are precisely the field extensions of Z/2Z (all of which are indeed Boolean algebras). But being a field is really quite a special case, since as discussed before, the only multiplicative inverse in the Boolean algebra example given by subsets of S is 1 (which is concretely the universal set S itself in this example). Are we at least clear on what we disagree about? I'm saying there are infinitely many Boolean algebras which are not fields (even if we restrict consideration to finite Boolean algebras). Take a finite set S with more than one element. Its power set is a Boolean algebra with respect to addition = symmetric difference of sets (XOR) and multiplication = intersection of sets (AND). === Subject: Re: Is Boolean algebra a finite field? posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > [snip much repetition of basic question and answer] I think I'm beginning to understand the root of the > confusion here.You asked if a Boolean algebra is > a finite field.I assumed you were thinking in terms > of the Boolean algebra being a ring extension of Z/2Z, > which is one way of defining Boolean algebras that is > consistent with the usual meaning of algebra in > ring theory. > What on earth are you talking about?? In a Boolean > algebra + os OR not XOR. Stop and think.We should at least agree that Z/2Z > is both a Boolean algebra and a field.What is 1+1 > in Z/2Z?Is it 1 OR 1?[Hint: no.] In fact for > the standard example of a Boolean algebra I gave > twice already, the complete family of subsets of > some set S (aka power set of S), addition is XOR > (aka the symmetric difference of two sets). You may prefer to define a Boolean algebra in terms > of AND, OR, and NOT, the logical operators.But it > would still not be correct to identify OR with an > addition operator for a ring.The error in trying > to make OR into an addition operator is that OR > does not have elementwise inverses for nonzero > elements. > Let me rephrase my question: is Boolean algebra a field? Let me repeat my answer.Not all Boolean algebras > are fields.Those that are fields are precisely > the field extensions of Z/2Z (all of which are > indeed Boolean algebras).But being a field is > really quite a special case, since as discussed > before, the only multiplicative inverse in the > Boolean algebra example given by subsets of S > is 1 (which is concretely the universal set S > itself in this example). Are we at least clear on what we disagree about? > I'm saying there are infinitely many Boolean > algebras which are not fields (even if we restrict > consideration to finite Boolean algebras).Take > a finite set S with more than one element.Its > power set is a Boolean algebra with respect to > addition = symmetric difference of sets (XOR) > and multiplication = intersection of sets (AND). > My original question was about Boolean algebras defined in terms of AND, OR (+), NOT (~). I did not say anything about identifying OR with an addition operator for a ring. === Subject: Re: Is Boolean algebra a finite field? posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2009060310 Ubuntu/8.10 (intrepid) Firefox/3.0.11,gzip(gfe),gzip(gfe) > My original question was about Boolean algebras defined in terms of > AND, OR (+), NOT (~). I did not say anything about identifying OR with > an addition operator for a ring. But you did. You asked if a Boolean algebra is a finite field. Fields are rings. Look back at your posts. Boolean OR is a binary operation that satisfies some but not all of the axioms for + in a field (or ring), namely OR is associative and commutative (as well as mutually distributive with respect to the AND operation). Also OR has an identity element, False or as we often say, Boolean zero. But that is not enough to define a field (or ring). Addition must have an inverse for each element. OR does not allow that. The simplest Boolean algebra is an example, {True,False} or {1,0}, as you prefer. While (0 OR 0) = 0, there is no value X such that (1 OR X) = 0. However this can be fixed! The binary operation that does have all the properties we require for (ring or field) addition is the symmetric difference (aka exclusive OR): X + Y = (X AND NOT(Y)) OR (Y AND NOT(X)) With this definition for addition it turns out 0 is still the identity, and each element is its own additive inverse. X + Y = (X AND NOT(X)) OR (X AND NOT(X)) = 0 regard, chip === Subject: Re: Is Boolean algebra a finite field? posting-account=-66P2goAAACSkYO5oYlUlLTiWSLM6Ea9 .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > My original question was about Boolean algebras defined in terms of > AND, OR (+), NOT (~). I did not say anything about identifying OR with > an addition operator for a ring. But you did.You asked if a Boolean algebra is a finite field. Fields are rings.Look back at your posts. Boolean OR is a binary operation that satisfies some but not all > of the axioms for + in a field (or ring), namely OR is associative > and commutative (as well as mutually distributive with respect to > the AND operation).Also OR has an identity element, False or > as we often say, Boolean zero. But that is not enough to define a field (or ring).Addition must > have an inverse for each element.OR does not allow that. The > simplest Boolean algebra is an example, {True,False} or {1,0}, as > you prefer.While (0 OR 0) = 0, there is no value X such that > (1 OR X) = 0. However this can be fixed!The binary operation that does have > all the properties we require for (ring or field) addition is > the symmetric difference (aka exclusive OR): X + Y = (X AND NOT(Y)) OR (Y AND NOT(X)) With this definition for addition it turns out 0 is still the > identity, and each element is its own additive inverse. X + Y = (X AND NOT(X)) OR (X AND NOT(X)) = 0 regard, chip OK, let me ask you this then: Does Z/2Z behave like xor, and, not? Does xor, and, not give you the ability to define all the combinatorial functions? Can xor be defined for the algebra of sets (0 = empty, 1 = universal etc.) or for multi-valued Boolean algebras? Are you saying that 2-valued Boolean algebra is a field and multivalued algebras are not fields? If Boolean algebra (in general) is not a field then what is it? === Subject: Re: Is Boolean algebra a finite field? posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2009060310 Ubuntu/8.10 (intrepid) Firefox/3.0.11,gzip(gfe),gzip(gfe) > My original question was about Boolean algebras defined in terms of > AND, OR (+), NOT (~). I did not say anything about identifying OR with > an addition operator for a ring. > But you did.You asked if a Boolean algebra is a finite field. > Fields are rings.Look back at your posts. > Boolean OR is a binary operation that satisfies some but not all > of the axioms for + in a field (or ring), namely OR is associative > and commutative (as well as mutually distributive with respect to > the AND operation).Also OR has an identity element, False or > as we often say, Boolean zero. > But that is not enough to define a field (or ring). Addition must > have an inverse for each element.OR does not allow that. The > simplest Boolean algebra is an example, {True,False} or {1,0}, as > you prefer.While (0 OR 0) = 0, there is no value X such that > (1 OR X) = 0. > However this can be fixed!The binary operation that does have > all the properties we require for (ring or field) addition is > the symmetric difference (aka exclusive OR): > X + Y = (X AND NOT(Y)) OR (Y AND NOT(X)) > With this definition for addition it turns out 0 is still the > identity, and each element is its own additive inverse. > X + Y = (X AND NOT(X)) OR (X AND NOT(X)) = 0 > regard, chip OK, let me ask you this then: > Does Z/2Z behave like xor, and, not? > Does xor, and, not give you the ability to define all the > combinatorial functions? > Can xor be defined for the algebra of sets (0 = empty, 1 = universal > etc.) or for multi-valued Boolean algebras? If by combinatorial functions you mean Boolean operations, then yes: 1) Z/2Z (aka {True = 1, False = 0}) is a Boolean algebra 2) NOT together with AND will define all Boolean operations 3) XOR for an algebra of sets is the symmetric difference (as previously defined in terms of AND/OR/NOT, see above) > Are you saying that 2-valued Boolean algebra is a field and > multivalued algebras are not fields? I'm saying that Z/2Z and all of its field extensions are both Boolean algebras and fields. The algebra of sets for S = {a,b} is not a field, at least not with respect to an interpretation in which multiplication is AND. That's again because existence of multiplicative inverses fails (although we do have a multiplicative identity, always for Boolean algebras, since X AND 1 = X). My point in discussion Boolean algebras as algebras (in the sense of commutative ring theory) is to clarify how close we come to having a field. > If Boolean algebra (in general) is not a field then what is it? A Boolean algebra is always a ring extension of Z/2Z. Google for Boolean ring and you'll see the construction of addition as symmetric difference is a standard one. === Subject: Re: Is Boolean algebra a finite field? posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) As it turns out the solution to the P Versus NP propblem is another problem: Sorted by relevance Sort by date 1 result for musat. Permutation Problem Why not just use random permutations? There are well-known fast algorithms for that. If you generated, say 1000 permutations per second for 2 weeks, you would process 1209600000 permutations out of a total of about 3.47 x 10^1166 so your probability of a repetition musat be minute. Derek Holt. authors Natural numbers [was: Diagonal wanderings...] Theorem lt_s: forall np:nat , n< p ...http://www.ncc.up.pt/~nam/aulas/0506/ t_coq/exemplos/pred.v - Similar pages Yes, this was all part of technique first pioneered by Musatov when he proved P==NP. On May 22, 11:17 pm, David Bernier < ultrafinitist, > My original question was about Boolean algebras defined in terms of > AND, OR (+), NOT (~). I did not say anything about identifying OR with > an addition operator for a ring. But you did. You asked if a Boolean algebra is a finite field. Fields are rings. Look back at your posts. Boolean OR is a binary operation that satisfies some but not all > of the axioms for + in a field (or ring), namely OR is associative > and commutative (as well as mutually distributive with respect to > the AND operation). Also OR has an identity element, False or > as we often say, Boolean zero. But that is not enough to define a field (or ring). Addition must > have an inverse for each element. OR does not allow that. The > simplest Boolean algebra is an example, {True,False} or {1,0}, as > you prefer. While (0 OR 0) = 0, there is no value X such that > (1 OR X) = 0. However this can be fixed! The binary operation that does have > all the properties we require for (ring or field) addition is > the symmetric difference (aka exclusive OR): X + Y = (X AND NOT(Y)) OR (Y AND NOT(X)) With this definition for addition it turns out 0 is still the > identity, and each element is its own additive inverse. X + Y = (X AND NOT(X)) OR (X AND NOT(X)) = 0 regard, chip > My original question was about Boolean algebras defined in terms of > AND, OR (+), NOT (~). I did not say anything about identifying OR with > an addition operator for a ring. But you did. You asked if a Boolean algebra is a finite field. Fields are rings. Look back at your posts. Boolean OR is a binary operation that satisfies some but not all > of the axioms for + in a field (or ring), namely OR is associative > and commutative (as well as mutually distributive with respect to > the AND operation). Also OR has an identity element, False or > as we often say, Boolean zero. But that is not enough to define a field (or ring). Addition must > have an inverse for each element. OR does not allow that. The > simplest Boolean algebra is an example, {True,False} or {1,0}, as > you prefer. While (0 OR 0) = 0, there is no value X such that > (1 OR X) = 0. However this can be fixed! The binary operation that does have > all the properties we require for (ring or field) addition is > the symmetric difference (aka exclusive OR): X + Y = (X AND NOT(Y)) OR (Y AND NOT(X)) With this definition for addition it turns out 0 is still the > identity, and each element is its own additive inverse. X + Y = (X AND NOT(X)) OR (X AND NOT(X)) = 0 regard, chip > Natural numbers [was: Diagonal wanderings...] Theorem lt_s: for all > np:nat , n< p >http://www.ncc.up.pt/~nam/aulas/0506/t_coq/exemplos/pred.v > and ultrafinitist. > Mechanical Witness:http://mathforum.org/kb/message.jspa?messageID=6751446&tstart=0-Hide > quoted text - > Here is the isomorphism: > Re: P=NP (proof) > I recently read about this problem and how it is > (according to Tim > Gowers, who I believe) the most important > problem in combinatorics > today. But I am used to being able to > immediately apprehend problems > in combinatorics. Is there actually a simple > formulation of the P=NP > question as purely regarding finite sets, > without any mention of > complexity classes and such? > Considering that P and NP are complexity classes, > it's hard to imagine > a formulation of the question without any mention > of complexity classes. > But here goes. > I take it you know what a graph is, and what it > means for two (finite) > graphs to be isomorphic. The graph isomorphism > problem asks you > to determine whether two given graphs are > isomorphic. In theory, > there is no difficulty in solving this problem; > you just look at each > map from the one vertex set to the other in turn > until either you find > one that's an isomorphism or you exhaust the set > of maps without > finding an isomorphism. In practice, this takes > too long; if each graph > has n vertices then there are n-factorial maps to > look at (counting just > those that are one-one), and n-factorial is big. > On the other hand, if someone shows you an > alleged isomorphism, > you can quickly verify or refute her allegation; > at worst, you have to > check about n^2 edges, and n^2 is small. > So graph isomorphism is easy to verify; the P = > NP question is > whether graph isomorphism is easy to solve. That > is, is there a way > to solve graph isomorphism that, instead of > requiring n-factorial > amount of work, requires only n^2, or n^3, or > even n^1000, > which is still tiny compared to n-factorial, for > large n. > Perhaps I'm mistaken and there has been a recent > development but I thought it wasn't known if > graph isomorphism is an NP complete problem. > Subgraph isomorphism is known to be though. > You're probably right about graph isomorphism, and I > simply > misremembered what I thought I had learned about it. > You're > certainly right about subgraph isomorphism, which for > the > record is the following problem; given two graphs, G > and H, > determine whether G has a subgraph isomorphic to H. Natural numbers [was: Diagonal wanderings...] Theorem lt_s: for all np:nat , n< p http://www.ncc.up.pt/~nam/aulas/0506/t_coq/exemplos/pred.v Part of technique when he proved P==NP. On May 22, 11:17 pm, formalist and ultrafinitist. Mechanical Witness:http://mathforum.org/kb/message.jspa? messageID=6751446&tstart=0-Hide Here is the isomorphism: Re: P=NP (proof) I recently read about this problem and how it is (according to Tim Gowers, who I believe) the most important problem in combinatorics today. But I am used to being able to immediately apprehend problems in combinatorics. Is there actually a simple formulation of the P=NP question as purely regarding finite sets, without any mention of complexity classes and such? Considering that P and NP are complexity classes, it's hard to imagine a formulation of the question without any mention of complexity classes. But here goes. I take it you know what a graph is, and what it means for two (finite) graphs to be isomorphic. The graph isomorphism problem asks you to determine whether two given graphs are isomorphic. In theory, there is no difficulty in solving this problem; you just look at each map from the one vertex set to the other in turn until either you find one that's an isomorphism or you exhaust the set of maps without finding an isomorphism. In practice, this takes too long; if each graph has n vertices then there are n-factorial maps to look at (counting just those that are one-one), and n-factorial is big. On the other hand, if someone shows you an alleged isomorphism, you can quickly verify or refute her allegation; at worst, you have to check about n^2 edges, and n^2 is small. So graph isomorphism is easy to verify; the P = NP question is whether graph isomorphism is easy to solve. That is, is there a way to solve graph isomorphism that, instead of requiring n-factorial amount of work, requires only n^2, or n^3, or even n^1000, which is still tiny compared to n-factorial, for large n. Perhaps I'm mistaken and there has been a recent development but I thought it wasn't known if graph isomorphism is an NP complete problem. Subgraph isomorphism is known to be though. You're probably right about graph isomorphism, and I simply misremembered what I thought I had learned about it. You're certainly right about subgraph isomorphism, which for the record is the following problem; given two graphs, G and H, determine whether G has a subgraph isomorphic to H. === Subject: Re: Proof Martin Musatov is a Pioneer posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Math Forum Discussions - Re: Reactions to/against the Binary > Treearguments, you are no pioneer, merely a naysayer. .... Martin > Musatov > P.S. P=NP, Call me the Charlton Heston if math, but they can > have my ...http://mathforum.org/kb/thread.jspa%3FforumID > %3D13%26threadID%3D1940341%26messageID%3D6724147 - 108k - Cached - > Similar pagesken and belly: Red Coats and PinkeyeAre you aware at the > plight of computer scientist Martin Musatov as he attempts to approve > an intrinsic .... Confessions of a Pioneer Woman. 11 hours > ago ...http://kenandbelly.blogspot.com/2009/03/red-coats-and- > pinkeye.html - 99k - Cached - Similar pages0xDEMartin Musatov, a > horror-film screenwriter, P=NP crank, and sockpuppeteer ... Thomas T. > Goldsmith, Jr. was an important pioneer in the early > television ...http://11011110.livejournal.com/tag/wikipedia- 65k - > Cached - Similar pagesTechnology Classes around Los Angeles - > Oodle.comHi Introducing Mind On Demand Software Labs, a Pioneer in RIA > (Rich Internet Application). .... Martin Musatov on TeachStreet - Los > Angeles, CA 1 month ago ...http://la.oodle.com/service/education/tech/ > - 90k - Cached - Similar pagessci.mathRe: struggling with definition > of wreath product, Martin Musatov ...... Jussi Piitulainen; Re: A > integral inequality for help., Pioneer ...http://sci.tech-archive.net/ > Archive/sci.math/2009-05/ - 284k - Cached - Similar pagesRe: General > Price: ...http://sci.tech-archive.net/Archive/sci.physics.relativity/ > 2009-06/msg00086.html - 24k - Cached - Similar pagesProject Excelsior > ' KuiperCliffTo me, Kittinger remains a true pioneer, as were all the > astronauts and ... pversusnp on Short History of the Shadow .87 Martin > Musatov on New ...http://kuipercliff.wordpress.com/2007/02/23/project- > excelsior/ - 32k - Cached - Similar pages Inverse 19 Musatov Proof Space Archaeology [CapitalEth] Apollo 11 > moon landing ' Kuiper Cliff The Riemann Hypothesis is Part of the existential and horrific allure > of the Pioneer missions is .... True: >http://farm4.static.flickr.com/3496/3256947403 398e26dde8 o.jpg >pversusnp on Short History of the > Shadow .87 Martin Musatov on New ...http://kuipercliff.wordpress.com/ > 2007/04/04/space-archaeology-apollo-11-moon-landing/ - 46k - Cached - > Similar pagesReactions to/against the Binary Tree - > Docendi.orgarguments, you are no pioneer, merely a naysayer. ..... On > you, > Martin Musatov ...http://www.docendi.org/reactions-t235962.html > %3Ft%3D235962 - 132k - Cached - Similar pages === Subject: Re: Article on Mathematicians in Math Ed I had said this earlier: > > How well respected is Professor Wu at the University > of Berkeley among mathematics educators? I do not > know since mathematics education is a new area for > me, and I am trying to learn more by reading > in mathematics education. > > I can tell from his website that he emphasizes what > should be taught in schools and what mathematics > educators > should know rather than mathematics pedagogy itself. > > > Jonathan Groves > > P.S. Michael Paul Goldenburg suggested that some > I don't know. However, I think some stuff in this > namely mathematical engineering and making sure > students' understanding of mathematics lines up > properly with correct mathematics. According to > the example Wu gives, elementary school students > shouldn't be taught the algebraic definition of > fractions (as in rings of fractions or fields of > fractions) or that Q is a subring of Z but > that students' understanding of fraction > arithmetic lines up with this fact. In other > words, students should see that fraction > arithmetic is a generalization of integer > arithmetic rather than something brand new. > > I had posted this discussion in math-teach, and > Goldenburg's response to my original posting > is there. I had noticed recently that David Klein seems to be a supporter of Wu because his website contains a link to Wu's website. David Klein is one of the most outspoken mathematicians against reform math. I don't know how much that means, if anything, but I decided to mention it since I had noticed that recently when I was reading some stuff online about those opposed to reform math. What I read mentioned David Klein and gave a link to his website, so I checked it out some and noticed the link to Wu's website. David Klein's website is at http://www.csun.edu/~vcmth00m/. Jonathan Groves === Subject: Re: Proving lower bound on complexity <4a3199ee$0$508$b45e6eb0@senator-bedfellow.mit.edu> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) >What are some of the techniques that can be used to prove that a >decision problem has some given lower bound on its complexity? (eg >Prove there is no Turing Machine that can solve problem X faster than O >(n^2)) Unconditional lower bounds of this type are extremely hard to prove. For > example, nobody knows how to rule out the possibility of a linear-time > algorithm for SAT, the paradigmatic NP-complete problem. However, if you put restrictions on both time *and* space---say, you have > only sublinear space available---then you can say something. See http://rjlipton.wordpress.com/2009/04/13/sat-is-not-too-easy/ Another thing you can do is to prove black-box lower bounds. That is, > you make the assumption that your input data is not totally transparent > to you but that some features of it are accessible to you only via a > black box.For example, suppose you have to sort a bunch of items, and > the only way you can tell whether item x is less than item y is to feed > x and y to a black box that will tell you how x and y compare. You can > then prove lower bounds on the minimum number of calls to the black box, > by constructing instances that cannot be distinguished from each other > unless you make that many calls. You can also restrict your model of computation in other ways, e.g., you > can ask for a lower bound on the circuit complexity of a problem if you > limit the circuits to be monotone, or to have constant depth. Then the > so-called method of approximations can be used to prove lower bounds. Finally, for problems with very high complexity, diagonalization arguments > can be used.For example, the time hierarchy theorem tells us that P != EXP. > Thus a problem like generalized chess, which one can show is EXP-complete, > cannot be solved in polynomial time. > -- > Tim Chow tchow-at-alum-dot-mit-dot-edu > The range of our projectiles---even ... the artillery---however great, will > never exceed four of those miles of which as many thousand separate us from > the center of the earth.---Galileo, Dialogues Concerning Two New Sciences >What are some of the techniques that can be used to prove that a >decision problem has some given lower bound on its complexity? (eg >Prove there is no Turing Machine that can solve problem X faster than O >(n^2)) Unconditional lower bounds of this type are extremely hard to prove. For > example, nobody knows how to rule out the possibility of a linear-time > algorithm for SAT, the paradigmatic NP-complete problem. However, if you put restrictions on both time *and* space---say, you have > only sublinear space available---then you can say something. See http://rjlipton.wordpress.com/2009/04/13/sat-is-not-too-easy/ Another thing you can do is to prove black-box lower bounds. That is, > you make the assumption that your input data is not totally transparent > to you but that some features of it are accessible to you only via a > black box. For example, suppose you have to sort a bunch of items, and > the only way you can tell whether item x is less than item y is to feed > x and y to a black box that will tell you how x and y compare. You can > then prove lower bounds on the minimum number of calls to the black box, > by constructing instances that cannot be distinguished from each other > unless you make that many calls. You can also restrict your model of computation in other ways, e.g., you > can ask for a lower bound on the circuit complexity of a problem if you > limit the circuits to be monotone, or to have constant depth. Then the > so-called method of approximations can be used to prove lower bounds. Finally, for problems with very high complexity, diagonalization arguments > can be used. For example, the time hierarchy theorem tells us that P != EXP. > Thus a problem like generalized chess, which one can show is EXP-complete, > cannot be solved in polynomial time. > -- > Tim Chow tchow-at-alum-dot-mit-dot-edu > The range of our projectiles---even ... the artillery---however great, will > never exceed four of those miles of which as many thousand separate us from > the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences This is the argument that proves that all probabilistic algorithms ... The problem of constructing a function of high circuit complexity by simply of diagonalization (which tends to lead to non-natural arguments) and of model Z of power-law graphs have property Y with very high probability. Y=Computational Complexity The primitive recursive functions have a very simple definition and yet we finally use the primitive recursive functions to compute properties: Alas, the above diagonal argument works on any class of total functions. Remarkably, a high percentage of naturally occurring computational problems turn out to be in Complexity: An explicit solution to Post's Problem: Finally we show the corresponding result for the linear BSS model. The BSS model of real number computation in this section summarizes very well some open problems of the previous arguments lead to some other problems and a uniform approach to obtain diagonal sets in complexity classes, Theory M In Complexity of Classification Problems and Comparative Advantages similar arguments can be made for combined systems. However, we believe it is possible with a high dimensional problem we are concerned about how the discriminate works. Though, a few off-diagonal points in each plot suggest... (some NP Completeness) Finally, we calculate the standard deviation in the boundary measure as... Resource bounded randomness and weakly complete problems*1 The combinatorial complexity of the argument, however, ... Finally, in Section 5, we apply some of our results on random sets to weak completeness. .... Conversely, by diagonalization we can show that there are random sets which are not nc+i random ..... Almost everywhere high nonuniform complexity. ... Hight Tech Solutions The temporal prior in bioelectromagnetic source imaging problems for computational complexity reduction, related to the application theory with some new wrinkles based on symmetry arguments. Finally, identification and employment of the isotropy are independent (diagonal. ), one ends up with spurious high-frequency noise in the Theoretical Computer Science : Complexity of tile rotation ...Complexity of the tile rotation problem Let y be the set of tiles given in Fig. .... The argument is identical for tiles vertically adjacents. D Finally, we will represent the positive components of a tile by heads of bold arrows ..... The orientation of the blocks in the diagonal of such configuration (see Fig. ... Comparing descriptional and computational complexity of infinite words there are several important results on a high level complexity due to Kolmogorov ..... Although by a diagonalization argument such words clearly exist. .... Finally, as a very interesting but probably difficult problem we state ... Finally, one will show ZFC + some large cardinal hypothesis proves P 'NP problems will have very high degree polynomial times. Before and after... This is, in my opinion, a very strong argument strengthened as the space of algorithms grows. On the intellectual terrain around NP... What about the computational complexity of problems such as primality testing which have come from diagonalization arguments whose sharpness and sophistication has been proven to be NPComplete problems are indeed very similar in structure. ..... Finally Arora et al. [ALM+92] showed that NP is the set of languages which ... Theory M threads to prove NP=P proves P'NP problems will have very high degree polynomial times as we have shown. I am working hard to do well for those who have believed in me and myself, so please be kind, constructive, analytical, factual and focused with the replies I will and do so appreciate. Martin Musatov === Subject: Index Formula (proof?) Someone pointed out in an earlier the following result: Let G be a group, H, K subgroups of G with K < H, and N a normal subgroup of G. Then [H : K] = [HN/N : KN/N][H / N : K / N]. This is easy to see when G is finite, but how does one prove the result in general? Is there a particular strategy involved here, when proving index formulas involving products of indices (much like the tower law for subgroups) ? I'd really be interested in seeing a proof, either here, or in a book/paper. === Subject: Re: Cantor's Omega posting-account=yxbZkgkAAABQBvyYeebYQ-PAvi0uT3tG Gecko/20080829 Firefox/2.0.0.17,gzip(gfe),gzip(gfe) > Saying that Wikipedia is often inaccurate > is like saying the Pope used to be a Nazi. And yet educated people like yourself are free to edit it to make it more accurate. Godwin! === Subject: Re: Cantor's Omega > Omega is an extension to the Natural numbers, to > get closure under limit > operations, not that dissimilar to the integers > (positive and negative) > which are closure of N under subtraction. Nobody > argue that the negative > integers are somehow dodgy because they don't include > imaginary numbers, or > proper fractions, etc. The set of integers doesn't > include arbitrary reals > or complex numbers, simple as that. > Welcome, marty.musatov@gmail.com Logout Your Control Panel Watches FAQ Contact Us Math Forum Ask Dr. Math Discussions Internet Newsletter MathTools Problems of the Week Teacher2Teacher Teacher Exchange Workshops Search All of the Math Forum: Browse our Internet Mathematics Library Views expressed in these public forums are not endorsed by Drexel University or The Math Forum. Math Forum é Discussions é sci.math.* é sci.math Topic: inverse of power series with prime power coefficients Replies: 7 Last Post: Jun 12, 2009 4:11 PM Search Thread: Advanced Search Reply to this Topic Watch this Topic Back to Topic List Jump to Tree View Messages: [ Previous | Next ] Martin Michael Musatov Posts: 802 Registered: 4/19/09 Re: inverse of power series with prime power coefficients > On May 11, 5:37 pm, Robert Israel > The power series with prime coefficients > P(x) = 1 + 2x + 3x^2 + 5x^3 + 7x^4 + 11x^5 + > 13x^6 + .... > and itsinverse > Q(x) = 1 / P(x) = 1 - 2x + x^2 - x^3 + 2x^4 - > 3x^5 + 7x^6 - 10x^7 + > 13x^8 - 21x^9 + 26x^10 - 33x^11 + 53x^12 - .... > were studied in the mid-1990s, when Backhouse > conjectured, and > Flajolet proved, that the limit of the absolute > value of the ratio of > consecutive coefficients of Q(x) is equal to > B=1.4560749...., > Backhouse's constant. (By comparison, the limit > of the ratio of > consecutive primes is equal to 1.) > Consider instead the power series with prime > power coefficients: > PP(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + 7x^5 + 8x^6 > + 9x^7 + 11x^8 + > 13x^9 + 16x^10 + 17x^11 + 19x^12 + 23x^13 + > 25x^14 + 27x^15 + 29x^16 + > 31x^17 + 32x^18 + 37x^19+ .... > I would expect itsinverseto be quite similar to > the series Q(x) > above. But it appears to be quite different: > QQ(x) = 1 - 2x + x^2 - x^5 + 3x^6 - 3x^7 + 2x^9 - > x^10 + x^15 - 4x^16 > + 8x^17 - 5x^18 - 13x^19+ 26x^20 - 9x^21 - 20x^22 > + 25x^23 + 9x^24 - > 60x^25 + 65x^26 + .... > Note how many low terms are zero: x^3, x^4, x^8, > and the four > consecutive terms x^11, x^12, x^13, x^14. I > wonder if this is pure > coincidence or if it has some significance. > There seem to be no other zero coefficients up to > the x^300 term. > Also, whereas the terms of Q(x) alternate > regularly between positive > and negative, there is no apparent pattern to the > signs of the terms > of QQ(x). > Finally, the terms of QQ(x) appear to grow much > more slowly than the > terms of Q(x). Perhaps as the series continues > and the prime powers > become sparser compared to the primes, QQ(x) will > start to look and > behave more like Q(x). But I am curious: does the > limit of the > absolute value of the ratio of consecutive > coefficients of QQ(x) > exist? If so, what is it? > Looking at the first 300 or so terms, I see no > pattern. > -- > Robert Israel > isr...@math.MyUniversitysInitials.ca > Department of Mathematics > http://www.math.ubc.ca/~israel > University of British Columbia > Vancouver, BC, Canada- Hide quoted text - > - Show quoted text - > > Remember me? Or are my results too trivial Mr. > Moderator? > > Welcome to the real mathematics and not the fake > Matrix movie > sorry excuse for chess you've been playing for the > last 20 years. > > Musatov > > (Founder, Inverse 19 Mathematics) Not the only, but > one without it > would not exist. > > QED > > > Efforts by mathematicians to determine the value of > pi since have led > to the conclusion that it ... Unlike rational > numbers, pi cannot be > represented by a common fraction, .... > > But what about an uncommon fraction? > > Will you look at my drawing and in particular how the > twelve points > are the outside points of the sphere are all > rationally reached by > decimal numbers? > > 6.2035884*2 and doubled an infinite number of times > should yield a > constant decimal binary expansion (what I mean by > this is the numbers > to the right of the decimal point should always end > in 4, then 8, > then 6, then 2, then 4, then 8, then 6, then > 2...repeating for > infinity). > > So in binary, by starting with this number we predict > for n>2 the > infinite binary tree will end with decimal > progression: 2, 4, 8, 6 > > For n<2 the infinite binary tree will begin and end > (depends on if you > are increasing or decreasing the value) 1, 5, 5, 5 > (repeating 5 > infinitely) as we decrease to infinitely small. > > Musatov Inverse Counting Chain of Infinite > Proportions > (sample) > 0.0000473372528076171875 > 0.000094674505615234375 > 0.00018934901123046875 > 0.0003786980224609375 > 0.000757396044921875 > 0.00151479208984375 > 0.0030295841796875 > 0.006059168359375 > 0.01211833671875 > 0.0242366734375 > 0.048473346875 > 0.09694669375 > 0.1938933875 > 0.387786775 > 0.77557355 > 1.5511471 > 3.1022942 > 6.2035884 > 12.4071768 > 24.8143536 > 49.6287072 > 99.2574144 > 198.5148288 > 397.0296576 > 794.0593152 > 1588.1186304 > 3176.2372608 > 6352.4745216 > 12704.9490432 > 25409.8980864 > 50819.7961728 > 101639.5923456 > 203279.1846912 > 406558.3693824 > 813116.7387648 > 1626233.4775296 > 3252466.9550592 > 6504933.9101184 > 13009867.8202368 > 26019735.6404736 > 52039471.2809472 > 104078942.5618944 > 208157885.1237888 > 416315770.2475776 > 832631540.4951552 > 1665263080.9903104 > 3330526161.9806208 > 6661052323.9612416 > 13322104647.9224832 > 26644209295.8449664 > 53288418591.6899328 > 106576837183.3798656 > 213153674366.7597312 > 426307348733.5194624 > 852614697467.0389248 > 1705229394934.0778496 > 3410458789868.1556992 > 6820917579736.3113984 > 13641835159472.6227968 > 27283670318945.2455936 > 54567340637890.4911872 > 109134681275780.9823744 > 218269362551561.9647488 > 436538725103123.9294976 > 873077450206247.8589952 > 1746154900412495.7179904 > 3492309800824991.4359808 > 6984619601649982.8719616 > 13969239203299965.7439232 > 27938478406599931.4878464 > 55876956813199862.9756928 > 111753913626399725.9513856 > 223507827252799451.9027712 > 447015654505598903.8055424 > 894031309011197807.6110848 > 1788062618022395615.2221696 > 3576125236044791230.4443392 > 7152250472089582460.8886784 > 14304500944179164921.7773568 > 28609001888358329843.5547136 > 57218003776716659687.1094272 > 114436007553433319374.2188544 > 228872015106866638748.4377088 > 457744030213733277496.8754176 > 915488060427466554993.7508352 > 1830976120854933109987.5016704 > 3661952241709866219975.0033408 > 7323904483419732439950.0066816 > 14647808966839464879900.0133632 > 29295617933678929759800.0267264 > 58591235867357859519600.0534528 > 117182471734715719039200.1069056 > 234364943469431438078400.2138112 > 468729886938862876156800.4276224 > 937459773877725752313600.8552448 > 1874919547755451504627201.7104896 > 3749839095510903009254403.4209792 > 7499678191021806018508806.8419584 > (sample) > > If you would like to do the honors Mr. Plouffe, I > endear you to > calculate the angles and lengths of the number of > ways, it is quite > obviously possible to square a circle based on the > geometry and > numerical progression. > > As is self-evident to anyone with even a modest > understanding of > mathematics and geometry after visual inspection of > the drawn proof > (http://www.flickr.com/photos/24869933@N07/3619373444/ > sizes/l/) it is > quite easily accomplished to connect and triangulate > the outer points > of the rectangles to the outer points of the square > and circle and > even by dimension sphere. > > Yes, the drawing clearly by definition of inverse > symmetry (take the > drawing cut it in half at the equator and turn the > paper over and > rotate it and compare the top and bottom mirrored > symmetry and you > will see it is an exact inverse of symmetry itself) > the drawing > explains how to properly and mathematically soundly > represent a fourth > dimension on a two dimensional plane. > > In plane language if you take the picture and > separate the top half of > the sphere by the bottom half of the sphere they are > identical but > only if the sphere existed on a complete three > dimensional plane > through expansion. > > Since the paper is flat, this is an impossibility, > nonetheless it is > completely expressed and retractable and countable > infinitely. > > By the definition of our standard we have established > any k-digit > approximation to pi is rational. > > Which could have many practical scientific > applications. For instance, > search: > > ON THE THEOREM OF IVA$EV-MUSATOV IIIx e F(l) + F(2) + > ... + F(p). This > remark is the basis of our proof of Theorem .... -y(M > + 1), x' + y(M + > 1)] = 0 it is easy to check that Conditions (i), > ...... Now set i = > pi x. *jj, v. Since p, x is positive,. (i) fi is > positive, ... > plms.oxfordjournals.org/cgi/reprint/s3-53/1/143.pdf > > REPRESENTATION OF FUNCTIONSBY SERIES AND CLASSES > ?(1)with measure E 2? -- ? and such that the trigonometric series for > g(x) converges > uniformly on [0, 2?]. .... trigonometric series are > due to > Ivashev-Musatov [40] who has shown that there are > null series (in the > sense of ..... function ?(?) e C(0, 1) with ?(0) = > ?(1) = 0 such that > ...... <*i .pdf > by PL Ul'yanov - 1972 - Cited by 24 - Related > > Full text of Bulletin of the British Museum (Natural > History)X 20. > FIG. 5. Well preserved large fragment showing the > different aspect and > ...... 0-1-0-5 P i n height) so giving the shell > surface the > appearance of ...... 94-95, i pi. Maseru. BRINK, A. > S. 1963. Two > cynodonts from the Ntawere ...... MUSATOV, D. L, > NEMIROVSKAYA, V. N., > SHIROKOVA, E. V. & ZHURAVLEVA, I. T. 1961. ... > www.archive.org/stream/bulletinofbritis17geollond/bull > etinofbritis17geollond_djvu.txt > > Trigonometric series with rapidly decreasing > coefficients a i0, 5 > ...Nt,+i - Pi = n,. N tl+j. -pi =N tl+j. -i +pi +1. > ..... number 8 6 > (0,1) for the product (32) such that for every point > x 6 E there are > ... > _A01.pdf > versions > > Plouffe's formula for pi - sci.math | Google Groups3 > posts - 2 authors > - Last post: 41 minutes ago > x=pi e^(ipi)+1=0. Martin Musatov. Reply Forward. > Martin Musatov to > problem. ... > 65ba40fd9e5e8f46/ > bf6d1ee13d87270b?lnk=raot > - 41 minutes ago > > [PDF] ???? ?-?File Format: PDF/Adobe Acrobat > Gli storici ci dicono che, nel X secolo, il principe > Vladi- ...... i > pi`u anziani e carismatici si evidenziavano i nomi di > ...... tin > Somov, Viktor Borisov-Musatov, Aleksandr Benois e Lev > Bakst ... 5/0; > Judovin 5/0; Korovin 2/1; Tukacov 1/0; Goli- cyn 1/1; > Zacharov 1/1; > Jurkunas 5/0; Krasauskas 2/0;. Bulaka 3/0. ... > www.esamizdat.it/eSamizdat_2005_(III)_2-3.pdf > > I am testing these identities: > > (1/2) * ( - k ) * ( k - 1/2 ) * ( - 3 ) / k! > > This equation produces > > M = k! ( - 3 ) * ( k - 1/2) * ( - k ) * (1/2) > > back! > > > Martin Musatov > > Reply Forward > > > ---------- Forwarded message ---------- === > Subject: CHALLENGE TO MATHEMATICIANS OF THE WORLD > > > On Apr 11, 11:40 am, Harris Moran > > > *plonk* > > message > > > muh brane i needs one- Hide quoted text - > > - Show quoted text - > > NOTE THE PROGRESSION OF NUMBERS PELLS PROPORTION > X^2=3 or 4 = 3 as > the numbers progression, WOW if this does not proof > this what will > > 1. Every odd number , number as Y value till > infinity is exactly > solvable with a proportionate result using the base > Pells equation i.e > 3,5,7,9,etc etc till infinty and the answer is as in > 2(2^2)+1=3^2 and > so on till infinity with a whole number integer. > > 2.Every even number , without exception, using Pells > equation, the > integer always is a .75 till infinity consistant > > as 63.75(2^4)=1=4^4 till infinity every integer > will end with .75. > is that the same proportion as 3 over 4, Y=3 for the > least with > numbers proportion > > There is a definite pattern by pells to numbers, and > I note that in > that pattern the number 19 proportion is strikingly > proprtionate, > and I clearly surmise that that circluar progression > is the same as > numbers for vector 19 progression. > Update from your neighborhood friendly Musatov. I am going to frame this first one!!! Exciting.... **************************************************************************** **************************************************************************** * **************************************************************************** * ** *Math Forum Discussions mathematics and geometry after visual inspection of the drawn proof ... approximation to pi is rational. Which could have many practical scientific applications. ... x=pi e^(ipi)+1=0. Martin* *Musatov. Reply Forward. Martin Musatov to ...* *http://mathforum.org/kb/message.jspa?messageID=6749244&tstart=0 - 4 hours ago * **************************************************************************** **************************************************************************** * ****************** And then this one is tease of the journey to come... ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++ Reactions to/against the Binary Tree - Page 6 - Docendi.org17 posts - Last post: May 28 But he claims that pi is the limit of the finite initial segments. These two claims suggest ... axiomatic basis. Are you an alias of Musatov? No I am Musatov. .... If Cantor's proof is correct, then an infinite path is the limit of ..... The real number sqrt(3) is the limit of a rational sequence. ... ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + +++++++++++++++++++++ 5/11/09 inverse of power series with prime power coefficients rokirovka@gmail.com 5/11/09 Re: inverse of power series with prime power coefficients Gerry Myerson 5/11/09 Re: inverse of power series with prime power coefficients Robert Israel 5/12/09 Re: inverse of power series with prime power coefficients hagman 5/13/09 small tommy prize ! amy666 5/14/09 Re: small tommy prize ! amy666 6/12/09 Re: inverse of power series with prime power coefficients Martin Michael Musatov 6/12/09 Re: inverse of power series with prime power coefficients Martin Michael Musatov [Privacy Policy] [Terms of Use] © Drexel University 1994-2009. All Rights Reserved. The Math Forum is a research and educational enterprise of the Goodwin College of Professional Studies. > Similarly, the set of transfinite ordinals doesn't > include reals or complex > numbers, simple as that. > > The definition of w - as being the smallest infinite > ordinal - is complete > as soon as the terms smallest, infinite and > ordinal are defined. > Cantor provided a definition of infinite, and defined > smallest in a way that > is consistent for both finite and infinite sets. > There are a number of ways > you can define and represent ordinals. How w is > represented depends upon > that choice. If you use the Von Newman construction, > w is the set of all > finite ordinals. > === Subject: Question About 0*infty One question that has been bothering me for a long time is this: In Calculus, we are taught that limits of the form 0*infty are indeterminate because these limits can be anything. However, in the extended real number system, the product 0*infty is often defined as 0. For example, in measure theory, it often is. For example, Wheeden and Zymund's book Measure and Integral mention this. Why is this definition reasonable and not in conflict with the fact that 0*infty is an indeterminate limit form? In fact, I was sure that the definitions of operations involving infty and possibly a real number as well are based on limits. Why I did not ask this question sooner, I don't know. Are there definitions of the extended real number system where this product is undefined? If there are, I don't recall seeing them before. Jonathan Groves === Subject: Re: Question About 0*infty does say that in measure theory and probability that 0*infty is often defined to be 0 but in other cases is often left undefined. I can see that in measure theory now that this definition of this product makes some things work out properly that wouldn't work otherwise. I must have been confusing limits as x approaches infty or infinite-valued limits with functions evaluated at infty or with infinite functional values in a similar way that many beginning calculus students confuse f(2) with the limit of f(x) as x approaches 2. Or maybe I was right in that the arithmetic involving infinity is based on limits but that there are exceptions in certain cases such as the exception that 0*infty = 0 in measure theory makes things work properly but would not work if we left this product undefined (which would make sense if we based the definition of this product on limits). Jonathan Groves === Subject: Re: Question About 0*infty One question that has been bothering me for a long time > is this: In Calculus, we are taught that limits of the form > 0*infty are indeterminate because these limits can be anything. > However, in the extended real number system, the product > 0*infty is often defined as 0. For example, in measure theory, > it often is. For example, Wheeden and Zymund's book Measure > and Integral mention this. Why is this definition reasonable > and not in conflict with the fact that 0*infty is an > indeterminate limit form? Other responders have already answered that. > In fact, I was sure that the definitions > of operations involving infty and possibly a real number as well > are based on limits. One may choose to think of those definitions as being motivated by the associated limits, except in cases when the associated limits are indeterminate forms. In those cases, if one chooses to define the operations, obviously the definitions are not motivated by the associated limits. > Why I did not ask this question sooner, I don't know. Are there definitions of the extended real number system where > this product is undefined? If there are, I don't recall seeing > them before. Yes. For example, in Apostol's _Mathematical Analysis_ (1957), the product is left undefined in both the extended reals and the extended complexes (sections 3-9 and 3-10). David === Subject: Re: Question About 0*infty Jonathan Groves a .8ecrit : > > One question that has been bothering me for a long time > is this: In Calculus, we are taught that limits of the form > 0*infty are indeterminate because these limits can be anything. > However, in the extended real number system, the product > 0*infty is often defined as 0. For example, in measure theory, > it often is. For example, Wheeden and Zymund's book Measure > and Integral mention this. Why is this definition reasonable > and not in conflict with the fact that 0*infty is an > indeterminate limit form? In fact, I was sure that the definitions > of operations involving infty and possibly a real number as well > are based on limits. A similar situation is the value of 0^0. As a limit (i.e. as lim f^g where lim f = lim g =0), it is indeterminate (like in lim x->0 x^(1/ln x), but as a natural extension of definition of powers (like the number of maps from the empty set to itself, say), the value 0^0 =1 is so useful that it will be usually the default choice. > > Why I did not ask this question sooner, I don't know. > > Are there definitions of the extended real number system where > this product is undefined? If there are, I don't recall seeing > them before. > > > Jonathan Groves === Subject: Re: Question About 0*infty posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY rv:1.9.0.1) Gecko/2008070206 Firefox/3.0.1,gzip(gfe),gzip(gfe) One question that has been bothering me for a long time > is this: In Calculus, we are taught that limits of the form > 0*infty are indeterminate because these limits can be anything. > However, in the extended real number system, the product > 0*infty is often defined as 0.For example, in measure theory, > it often is.For example, Wheeden and Zymund's book Measure > and Integral mention this.Why is this definition reasonable Because it is appropriate in measure theory. One wants the contribution to an integral of a function f from an infinite measure set on which f is zero to be zero. Likewise making a function infinite on a set of measure zero should not alter its integral. === Subject: Re: Question About 0*infty One question that has been bothering me for a long time > is this: In Calculus, we are taught that limits of the form > 0*infty are indeterminate because these limits can be anything. > However, in the extended real number system, the product > 0*infty is often defined as 0. For example, in measure theory, > it often is. For example, Wheeden and Zymund's book Measure > and Integral mention this. Why is this definition reasonable > and not in conflict with the fact that 0*infty is an > indeterminate limit form? In fact, I was sure that the definitions > of operations involving infty and possibly a real number as well > are based on limits. Why I did not ask this question sooner, I don't know. Are there definitions of the extended real number system where > this product is undefined? If there are, I don't recall seeing > them before. Because limits and points are not the same. When infinity becomes a point it is then treated as a value just like any other number. lim f(x) as x->oo is different than f(oo) just as lim f(x) as x-> 3 is different(in meaning) than f(3). So, while one can define 0*oo = 0 it is still distinct than used in a limit. One can argue if 0*oo = 0 is correct but it has nothing to do with limits. Although I guess deep down the definition is using the idea of limits since 0*a = 0 for all a except oo and therefore why not when a = oo(and this is sorta thinking in terms of limits). Remember, adding the *point* at infinity is adding the point... taking the limit to infinity is something entirely different and doesn't depend on infinity at all. Just as lim f(x) as x->3 does not depend on 3. Of course for some function the limit and the value are the same it is not necessarily true. Suppose we have f(x) = 1/x + 1 and f(oo) = 3 lim f(x) as limit being a finite number. We can then have something like f(x) = 1/x and g(x) = x lim f(x) * lim g(x) as x =>oo is 0*oo? which is 0 by definition? No, because lim g(x) -> oo as x->oo. Not lim g(x) = oo as x->oo. I think the problems you are having is the loose use of the concept of infinity with mixed usage of the terminology. It really is a process and should be thought of as one. We tend to say the limit of something *IS* but it really is *approaches*. Generally there isn't any confusion because it is understood what is meant. In the case when oo can be a number/point in the system then it can be confusing if one assumes evaulating limits at points rather than approaching those points. Similarly, 0*oo is the same type of logic as 3*6 when oo is a number. But lim f(x) * lim g(x) is something different. It is similar but there is a subtle difference between multiplying limits and multiplying points. But we know that in most cases lim f(x) * lim g(x) = lim f(x)*g(x) and most people tend to think about it as such. What we really need is the space of limits where things such as lim f(x) and lim g(x) belong and there exists a homomorphism from that space to the reals/complex. === Subject: Re: Three bar equal sign? > As a young adult I utilize that principle in my automated theorem- > proving software by representing A&B&C not as 2 applications of & but > as a set {A,B,C} with the & Set Function applied to it. In Laws of Form by G. Spencer Brown, conjunction and disjunction are likewise freed from having only binary scope. The commutative and associative properties of these operators are implicit in his system also. You might like to have a look at his elegant presentation. -- hz === Subject: Re: Three bar equal sign? posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > I will explain to you what I am after. I am after a new theory, a new > way of programming, a new way of measuring, a new way of > communicating. I am talking about an end to the old tired way of 'P', > if 'P' then 'Q' innefficiency in programming and language (and > practical matters too). When I was a teenager I noticed that lots of Propositional Calculus > axioms are derivable from the fact that AND and OR (as well as ADD and > MUL) are what I call Set Functions.You can apply them to any > finite set - so the order cannot matter.A&B = B&A , A&(B&C)=(A&B)&C > etc. As a young adult I utilize that principle in my automated theorem- > proving software by representing A&B&C not as 2 applications of & but > as a set {A,B,C} with the & Set Function applied to it. The result is that my programs generate theorems orders of magnitude > faster. Yes, standard Logic is poorly designed, with shameful duplication of > principles between different branches of Mathematics e.g. Logic and > Set Theory both have DeMorgan rules etc.Researchers keep looking for > ways to prove the axioms, but overlook the fact that they have the > wrong model for that.It should be as Set Functions. Just like they > use Predicate Calculus for MetaMathematics (e.g.; ZFC axioms) when a > higher level of abstraction is much more efficient and yields many > more theorems in practice (finite world). Always work at the highest level of abstraction you can. C-B >I propose the following: No need.I have implemented it and the Set Function new idea solves > the problem nicely. > 'X' if 'X' > The idea is we create a thread. The first programming variable means > nothing. It is an instruction to the system script. It is an open and > said> order of logic. > 'X' alone simply means Open or I am talking or Listen up, it can > be followed by anything, or an infinite number of things. > And the idea of 'X' is arbitrary, as 'X' could be replaced by any > other logical symbol or character in any language and the idea still > stands. The operation still works. > Let me explain: > 'X' when I say 'X'. > 'X' is the master open 'command' key or 'dialogue'. It sets up a > system open script. > We then can open any operation simply by repeating it. The system is > not loyal to any language or frame. (or insert proprietary here) > If I want to open up a program Y, I simply say, X, Y I want Y. > The system knows that if a variable is repeated it means to execute it > under the terms programmed. (or by pressing a key) Let me explain > further (further): > A formal illustration of how it would look in a system would be: > X, Y, Y, X > The first 'X' means open the root command, the first 'Y' means, I want > 'Y', the second 'Y' means it is just 'Y' I seek and the 'X' closes to > command. > The platform solution requires complete symmetry in programming and > logic threads. > The programming implementation allows the setup of a system where > repetition of a variable in symmetry order allows command. > In football the command hut means pitch me the ball back. The > Quarterback will communicate to his teammates, On the 2nd hut throw > the ball. The first hut is to get set. Still 'hut' is the command to > throw the ball. > The real world variable and how it relates to the equivalency in > programming is explained as follows: > Instead of saying (1)Listen up....(2)I want to open my word > processor.... > X 1 1 X > where (1)=the first command 1 > and (2)=11 > ***********Note in binary*: the machine literally must count the two > ones and add them to get two.************) > User inputs:(1)I want my word processor (2)I want my word > processor > and this means to the machine Open up the word processor... only > when you read(1) I want my word processor (2) I want my word > processor > (forgive my terminology, I am simply being plain) but this inside/out > programming > illustrates perfectly my point. > and forgive my for being so bold but this logic clear, simple, and > plain, simply resolves P=NP > and overcomes the halting problem irrefutably > the solution to P=NP or P Versus NP, or N=NP or -1=1 or however you > want to frame > the problem is simply this statement: Do not do anything until I say > it twice. > Advanced Symmetry application: the solution to the machine is the > overlooked symmetry. A complex command may look like: 'XYZZYX' which > would mean, I want the YZ complex variable please > Stephen Arthur Cook was right, the whole thing was simply due to lack > of ingenuity on behalf of our programmer. >(C)2009. Martin Musatov. All Rights Reserved in perpetuity*. > http://www.MeAmI.orgSearch for the People > *Applies to derviations of this work. Including software, programming, > and other profit generating means extracted from this work. Work may > be patent pending.- Hide quoted text - > - Show quoted text -- Hide quoted text - - Show quoted text - So you agree we have indeed overcome the P Versus NP problem and/or halting problem and/or proven P=NP, correct? (Yes if any or all of the above apply, No, if none) = this will do to begin with ;) === Subject: Re: Three bar equal sign? posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > I will explain to you what I am after. I am after a new theory, a new > way of programming, a new way of measuring, a new way of > communicating. I am talking about an end to the old tired way of 'P', > if 'P' then 'Q' innefficiency in programming and language (and > practical matters too). > When I was a teenager I noticed that lots of Propositional Calculus > axioms are derivable from the fact that AND and OR (as well as ADD and > MUL) are what I call Set Functions.You can apply them to any > finite set - so the order cannot matter.A&B = B&A , A&(B&C)=(A&B)&C > etc. > As a young adult I utilize that principle in my automated theorem- > proving software by representing A&B&C not as 2 applications of & but > as a set {A,B,C} with the & Set Function applied to it. > The result is that my programs generate theorems orders of magnitude > faster. > Yes, standard Logic is poorly designed, with shameful duplication of > principles between different branches of Mathematics e.g. Logic and > Set Theory both have DeMorgan rules etc.Researchers keep looking for > ways to prove the axioms, but overlook the fact that they have the > wrong model for that.It should be as Set Functions. Just like they > use Predicate Calculus for MetaMathematics (e.g.; ZFC axioms) when a > higher level of abstraction is much more efficient and yields many > more theorems in practice (finite world). > Always work at the highest level of abstraction you can. > C-B >I propose the following: > No need.I have implemented it and the Set Function new idea solves > the problem nicely. > 'X' if 'X' > The idea is we create a thread. The first programming variable means > nothing. It is an instruction to the system script. It is an open and > said> order of logic. > 'X' alone simply means Open or I am talking or Listen up, it can > be followed by anything, or an infinite number of things. > And the idea of 'X' is arbitrary, as 'X' could be replaced by any > other logical symbol or character in any language and the idea still > stands. The operation still works. > Let me explain: > 'X' when I say 'X'. > 'X' is the master open 'command' key or 'dialogue'. It sets up a > system open script. > We then can open any operation simply by repeating it. The system is > not loyal to any language or frame. (or insert proprietary here) > If I want to open up a program Y, I simply say, X, Y I want Y. > The system knows that if a variable is repeated it means to execute it > under the terms programmed. (or by pressing a key) Let me explain > further (further): > A formal illustration of how it would look in a system would be: > X, Y, Y, X > The first 'X' means open the root command, the first 'Y' means, I want > 'Y', the second 'Y' means it is just 'Y' I seek and the 'X' closes to > command. > The platform solution requires complete symmetry in programming and > logic threads. > The programming implementation allows the setup of a system where > repetition of a variable in symmetry order allows command. > In football the command hut means pitch me the ball back. The > Quarterback will communicate to his teammates, On the 2nd hut throw > the ball. The first hut is to get set. Still 'hut' is the command to > throw the ball. > The real world variable and how it relates to the equivalency in > programming is explained as follows: > Instead of saying (1)Listen up....(2)I want to open my word > processor.... > X 1 1 X > where (1)=the first command 1 > and (2)=11 > ***********Note in binary*: the machine literally must count the two > ones and add them to get two.************) > User inputs:(1)I want my word processor (2)I want my word > processor > and this means to the machine Open up the word processor... only > when you read(1) I want my word processor (2) I want my word > processor > (forgive my terminology, I am simply being plain) but this inside/out > programming > illustrates perfectly my point. > and forgive my for being so bold but this logic clear, simple, and > plain, simply resolves P=NP > and overcomes the halting problem irrefutably > the solution to P=NP or P Versus NP, or N=NP or -1=1 or however you > want to frame > the problem is simply this statement: Do not do anything until I say > it twice. > Advanced Symmetry application: the solution to the machine is the > overlooked symmetry. A complex command may look like: 'XYZZYX' which > would mean, I want the YZ complex variable please > Stephen Arthur Cook was right, the whole thing was simply due to lack > of ingenuity on behalf of our programmer. >(C)2009. Martin Musatov. All Rights Reserved in perpetuity*. > http://www.MeAmI.orgSearch for the People > *Applies to derviations of this work. Including software, programming, > and other profit generating means extracted from this work. Work may > be patent pending.- Hide quoted text - > - Show quoted text -- Hide quoted text - > - Show quoted text - So it is agreed we have indeed overcome the P Versus NP problem and/or > halting problem and/or proven 6P=NP, correct? (Yes if any or all of the > above apply, No, if none) = this will do to begin with ;)- Hide quoted text - - Show quoted text - >Yes. === Subject: Re: Three bar equal sign? posting-account=UA-6fQkAAADI18fSPOc495gPgW1akxLl Trident/4.0; MathPlayer 2.10b; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > I am after a new theory, a new > way of programming, a new way of measuring, a new way of > communicating. I am talking about an end to the old tired phrase a new way (A New Kind of Science)? Yes, I agree completely. You need better communication than repeating the phrase a new way over and over again. You need to learn some basic axioms and rules: AB+AC+AD = A (B+C+D) and the principle of substitution: A=a new way of B = programming C=measuring D=communicating Then, I am after a new theory, a new way of programming, a new way of measuring, a new way of communicating. => I am after a new way, of programming, measuring, communicating. Are you from India? C-B === Subject: Re: Three bar equal sign? posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > I am after a new theory, a new > way of programming, a new way of measuring, a new way of > communicating. I am talking about an end to the old tired phrase a new way(A New Kind of Science)? Yes, I agree > completely.You need better communication than repeating the phrase > a new way over and over again.You need to learn some basic axioms > and rules: AB+AC+AD = A (B+C+D) and the principle of substitution: A=a new way of > B = programming > C=measuring > D=communicating Then, I am after a new theory, a new way of programming, a new way of > measuring, a new way of communicating. = > I am after a new way, of programming, measuring, communicating. Are you from India? C-B I am from Los Angeles: > .... > (upside-down A x)(x^2 - 4 = (x + 4)(x - 4)) > A slightly better example is given by: > (upside-down A x)(x^2 - 4 = (x + 2)(x - 2)) > :) > Ken Pledger. I will explain to you what I am after. I am after a new theory, a new > way of programming, a new way of measuring, a new way of > communicating. I am talking about an end to the old tired way of 'P', > if 'P' then 'Q' innefficiency in programming and language (and > practical matters too). I assert my proprietary findings: > 'X' if 'X' > The idea is we create a thread. The first programming variable means > nothing. It is an instruction to the system script. It is an open and > said> order of logic. > 'X' alone simply means Open or I am talking or Listen up, it can > be followed by anything, or an infinite number of things. > And the idea of 'X' is arbitrary, as 'X' could be replaced by any > other logical symbol or character in any language and the idea still > stands. The operation still works. > Let me explain: > 'X' when I say 'X'. > 'X' is the master open 'command' key or 'dialogue'. It sets up a > system open script. > We then can open any operation simply by repeating it. The system is > not loyal to any language or frame. (or insert proprietary here) > If I want to open up a program Y, I simply say, X, Y I want Y. > The system knows that if a variable is repeated it means to execute it > under the terms programmed. (or by pressing a key) Let me explain > further (further): > A formal illustration of how it would look in a system would be: > X, Y, Y, X > The first 'X' means open the root command, the first 'Y' means, I want > 'Y', the second 'Y' means it is just 'Y' I seek and the 'X' closes to > command. > The platform solution requires complete symmetry in programming and > logic threads. > The programming implementation allows the setup of a system where > repetition of a variable in symmetry order allows command. > In football the command hut means pitch me the ball back. The > Quarterback will communicate to his teammates, On the 2nd hut throw > the ball. The first hut is to get set. Still 'hut' is the command to > throw the ball. > The real world variable and how it relates to the equivalency in > programming is explained as follows: > Instead of saying (1)Listen up....(2)I want to open my word > processor.... > X 1 1 X > where (1)=the first command 1 > and (2)=11 > ***********Note in binary*: the machine literally must count the two > ones and add them to get two.************) > User inputs:(1)I want my word processor (2)I want my word > processor > and this means to the machine Open up the word processor... only > when you read (1) I want my word processor (2) I want my word > processor > (forgive my terminology, I am simply being plain) but this inside/out > programming > illustrates perfectly my point. and forgive my for being so bold but this logic clear, simple, and > plain, simply resolves P=NP > and overcomes the halting problem irrefutably > the solution to P=NP or P Versus NP, or N=NP or -1=1 or however you > want to frame > the problem is simply this statement: Do not do anything until I say > it twice. Advanced Symmetry application: the solution to the machine is the > overlooked symmetry. A complex command may look like: 'XYZZYX' which > would mean, I want the YZ complex variable please Stephen Arthur Cook was right, the whole thing was simply due to lack > of ingenuity on behalf of our programmer. (C)=Copyright. 2009. Martin Musatov. All Rights Reserved in perpetuity*. > http://www.MeAmI.orgSearch for the People *Applies to derviations of this work. Including software, programming, > and other profit generating means extracted from this work. Work may > be patent pending.- Hide quoted text - - Show quoted text - === Subject: Re: Three bar equal sign? posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > I am after a new theory, a new > way of programming, a new way of measuring, a new way of > communicating. I am talking about an end to the old tired > phrase a new way(A New Kind of Science)? Yes, I agree > completely.You need better communication than repeating the phrase > a new way over and over again.You need to learn some basic axioms > and rules: > AB+AC+AD = A (B+C+D) > and the principle of substitution: > A=a new way of > B = programming > C=measuring > D=communicating > Then, > I am after a new theory, a new way of programming, a new way of > measuring, a new way of communicating. > = > I am after a new way, of programming, measuring, communicating. > Are you from India? > C-B I am from Los Angeles: > .... > (upside-down Ax)(x^2 - 4= (x + 4)(x - 4)) > A slightly better example is given by: > (upside-down Ax)(x^2 - 4= (x + 2)(x - 2)) > :) > Ken Pledger. > I will explain to you what I am after. I am after a new theory, a new > way of programming, a new way of measuring, a new way of > communicating. I am talking about an end to the old tired way of 'P', > if 'P' then 'Q' innefficiency in programming and language (and > practical matters too).I assert my proprietary findings: > 'X' if 'X' > The idea is we create a thread. The first programming variable means > nothing. It is an instruction to the system script. It is an open and > said> order of logic. > 'X' alone simply means Open or I am talking or Listen up, it can > be followed by anything, or an infinite number of things. > And the idea of 'X' is arbitrary, as 'X' could be replaced by any > other logical symbol or character in any language and the idea still > stands. The operation still works. > Let me explain: > 'X' when I say 'X'. > 'X' is the master open 'command' key or 'dialogue'. It sets up a > system open script. > We then can open any operation simply by repeating it. The system is > not loyal to any language or frame. (or insert proprietary here) > If I want to open up a program Y, I simply say, X, Y I want Y. > The system knows that if a variable is repeated it means to execute it > under the terms programmed. (or by pressing a key) Let me explain > further (further): > A formal illustration of how it would look in a system would be: > X, Y, Y, X > The first 'X' means open the root command, the first 'Y' means, I want > 'Y', the second 'Y' means it is just 'Y' I seek and the 'X' closes to > command. > The platform solution requires complete symmetry in programming and > logic threads. > The programming implementation allows the setup of a system where > repetition of a variable in symmetry order allows command. > In football the command hut means pitch me the ball back. The > Quarterback will communicate to his teammates, On the 2nd hut throw > the ball. The first hut is to get set. Still 'hut' is the command to > throw the ball. > The real world variable and how it relates to the equivalency in > programming is explained as follows: > Instead of saying (1)Listen up....(2)I want to open my word > processor.... > X 1 1 X > where (1)=the first command 1 > and (2)=11 > ***********Note in binary*: the machine literally must count the two > ones and add them to get two.************) > User inputs:(1)I want my word processor (2)I want my word > processor > and this means to the machine Open up the word processor... only > when you read(1) I want my word processor (2) I want my word > processor > (forgive my terminology, I am simply being plain) but this inside/out > programming > illustrates perfectly my point. > and forgive my for being so bold but this logic clear, simple, and > plain, simply resolves P=NP > and overcomes the halting problem irrefutably > the solution to P=NP or P Versus NP, or N=NP or -1=1 or however you > want to frame > the problem is simply this statement: Do not do anything until I say > it twice. > Advanced Symmetry application: the solution to the machine is the > overlooked symmetry. A complex command may look like: 'XYZZYX' which > would mean, I want the YZ complex variable please > Stephen Arthur Cook was right, the whole thing was simply due to lack > of ingenuity on behalf of our programmer. > (C)=Copyright. 2009. Martin Musatov. All Rights Reserved in perpetuity*. > http://www.MeAmI.orgSearch for the People > *Applies to derviations of this work. Including software, programming, > and other profit generating means extracted from this work. Work may > be patent pending. > I am after a new theory, a new > way of programming, a new way of measuring, a new way of > communicating. I am talking about an end to the old tired > phrase a new way (A New Kind of Science)? Yes, I agree > completely. You need better communication than repeating the phrase > a new way over and over again. You need to learn some basic axioms > and rules: > AB+AC+AD = A (B+C+D) > and the principle of substitution: > A=a new way of > B = programming > C=measuring > D=communicating > Then, > I am after a new theory, a new way of programming, a new way of > measuring, a new way of communicating. > = I am after a new way, of programming, measuring, communicating. > Are you from India? > C-B I am from Los Angeles: - Hide quoted text - - Show quoted text - > .... > (upside-down A x)(x^2 - 4 = (x + 4)(x - 4)) > A slightly better example is given by: > (upside-down A x)(x^2 - 4 = (x + 2)(x - 2)) > :) > Ken Pledger. > I will explain to you what I am after. I am after a new theory, a new > way of programming, a new way of measuring, a new way of > communicating. I am talking about an end to the old tired way of 'P', > if 'P' then 'Q' innefficiency in programming and language (and > practical matters too). I assert my proprietary findings: > 'X' if 'X' > The idea is we create a thread. The first programming variable means > nothing. It is an instruction to the system script. It is an open and > said> order of logic. > 'X' alone simply means Open or I am talking or Listen up, it can > be followed by anything, or an infinite number of things. > And the idea of 'X' is arbitrary, as 'X' could be replaced by any > other logical symbol or character in any language and the idea still > stands. The operation still works. > Let me explain: > 'X' when I say 'X'. > 'X' is the master open 'command' key or 'dialogue'. It sets up a > system open script. > We then can open any operation simply by repeating it. The system is > not loyal to any language or frame. (or insert proprietary here) > If I want to open up a program Y, I simply say, X, Y I want Y. > The system knows that if a variable is repeated it means to execute it > under the terms programmed. (or by pressing a key) Let me explain > further (further): > A formal illustration of how it would look in a system would be: > X, Y, Y, X > The first 'X' means open the root command, the first 'Y' means, I want > 'Y', the second 'Y' means it is just 'Y' I seek and the 'X' closes to > command. > The platform solution requires complete symmetry in programming and > logic threads. > The programming implementation allows the setup of a system where > repetition of a variable in symmetry order allows command. > In football the command hut means pitch me the ball back. The > Quarterback will communicate to his teammates, On the 2nd hut throw > the ball. The first hut is to get set. Still 'hut' is the command to > throw the ball. > The real world variable and how it relates to the equivalency in > programming is explained as follows: > Instead of saying (1)Listen up....(2)I want to open my word > processor.... > X 1 1 X > where (1)=the first command 1 > and (2)=11 > ***********Note in binary*: the machine literally must count the two > ones and add them to get two.************) > User inputs:(1)I want my word processor (2)I want my word > processor > and this means to the machine Open up the word processor... only > when you read (1) I want my word processor (2) I want my word > processor > (forgive my terminology, I am simply being plain) but this inside/out > programming > illustrates perfectly my point. > and forgive my for being so bold but this logic clear, simple, and > plain, simply resolves P=NP > and overcomes the halting problem irrefutably > the solution to P=NP or P Versus NP, or N=NP or -1=1 or however you > want to frame > the problem is simply this statement: Do not do anything until I say > it twice. > Advanced Symmetry application: the solution to the machine is the > overlooked symmetry. A complex command may look like: 'XYZZYX' which > would mean, I want the YZ complex variable please > Stephen Arthur Cook was right, the whole thing was simply due to lack > of ingenuity on behalf of our programmer. > (C)=Copyright. 2009. Martin Musatov. All Rights Reserved in perpetuity*. > http://www.MeAmI.orgSearch for the People > *Applies to derviations of this work. Including software, programming, > and other profit generating means extracted from this work. Work may > be patent pending. === Subject: Re: Three bar equal sign? posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > I've noted Kaplan using a three bar equal sign in some equations in > his book Advanced Calculus. For example f[phi(x,y), psi(x,y)] = x (obviously = above is a really a three bar) I am unfamiliar with the meaning in this context and he doesn't seem > to define it.The only semantics Im aware of are conguent (mod x) > and is by definition - and neither seem appropriate in the context > of multivariate differential equations. Any idea what this 3 bar equal sign means? Andrew. Hi Andrew!!! It means this: Three bar equal sign? - comp.programming | Google GroupsJun 11, === 2009 ... Local: Thurs, Jun 11 2009 10:53 pm. Subject: Re: Three bar equal sign? .... (C)2009. Martin Musatov. All Rights Reserved in perpetuity*. ... 87def8d2f1da640c Re: Three bar equal sign? - Docendi.org2 posts - Last post: 7 hours ago Default Re: Three bar equal sign? .... of ingenuity on behalf of our programmer. (C)2009. Martin Musatov. All Rights Reserved in perpetuity*. ... www.docendi.org/re-t241445.html - 7 hours ago Three bar equal sign? - comp.programming | Google GroupsJun 11, 2009 ... Message from discussion Three bar equal sign? .... (C)2009. Martin Musatov. All Rights Reserved in perpetuity*. ... Math Forum DiscussionsJun 11, 2009 ... Re: Three bar equal sign? is repeated it means to execute it ... (C)2009. Martin Musatov. All Rights Reserved in perpetuity*. ... mathforum.org/kb/thread.jspa?messageID=6748696&tstart=0 Three bar equal sign? - sci.logic | Google GroupsJun 11, 2009 ... === Local: Thurs, Jun 11 2009 3:53 pm. Subject: Re: Three bar equal sign? ... The system knows that if a variable is repeated it means to execute it ... (C)2009. Martin Musatov. All Rights Reserved in perpetuity*. ... 87def8d2f1da640c Math Forum DiscussionsJun 11, 2009 ... Re: Three bar equal sign? means? ... Martin Michael Musatov ... mathforum.org/kb/message.jspa?messageID=6748488&tstart=45 Discussions - comp.theory | Google Groups10 posts - 6 authors - Last post: 3 days ago will explain ... Musatov>I see where you are going and I fiercely agree. ... Math Forum DiscussionsJun 11, 2009 ... of multivariate differential Andrew. ... Martin Michael Musatov ... mathforum.org/kb/message.jspa?messageID=6747428&tstart=45 Discussions - sci.math | Google Groups0.75(2^2)+1=2^2 (for Dr Phil,Prof Edgar, Musatov, amy , Mathematicians), 6 new of 6 ... Three bar equal sign? 8 new of 8, NP=P (7 authors), Jun 11 ... Math Forum DiscussionsJun 11, 2009 ... Re: Three bar equal sign? Martin Michael Musatov ... mathforum.org/kb/thread.jspa?messageID=6748695&tstart=0 === Subject: Re: inequality <33074901.3411.1244447735062.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=YIt8FwoAAADro1uQVvnVvEv_WpFiGN9r Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > On Jun 8, 3:55am, Torsten Hennig How to prove that |x+y|^p+|x-y|^pleq 2(x^p+y^p) for > 1leq pleq 2, > x,y>0? Any clever series expansion? > Because of the convexity of |x|^p, > |x+y|^p <= |x|^p + |y|^p. > Furthermore, > |x-y|^p <= max{|x|^p,|y|^p} <= |x|^p + |y|^p. > Adding the inequalities gives the result. > Best wishes > Torsten. Convexity only implies |x+y|^pleq 2^{p-1}(x^p+y^p), because p>1. I've > seen 2 solution of this problem, both are nontrivial and not natural. > Looking for something short. without loss of generality x >= y, so dividing by y^p we get (x+1)^p + (x-1)^p =< 2 *( x^p +1 ), where x >=1 let f(x)=(x+1)^p + (x-1)^p, g(x)=2 *( x^p +1 ). We have that f(0)=g(0). Using the concavity of h(x)=x^(p-1) we may show that f'(x) =< g'(x). So a mean value theorem argument gives the required inequality. istvan === Subject: Re: Answer to Dik T. Winter <877hzjnkde.fsf@alatheia.truth.invalid> <87y6rzm05p.fsf@alatheia.truth.invalid> have anything to build on. No one can list the things he holds to be self evident in the sense of the American Declaration of Independence. Such things are also not axioms in any usual logical or mathematical sense -- they are more like the axioms of Objectivism, in that they have an indefinite range of possible applications, consequences, meaning. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Answer to Dik T. Winter > > Consider axioms like in the American Declaration of Independence: > We hold these truths to be self evident... > Unless WM can list the things he holds to be self evident, he does not > have anything to build on. > > No one can list the things he holds to be self evident in the sense of > the American Declaration of Independence. Such things are also not > axioms in any usual logical or mathematical sense -- they are more like > the axioms of Objectivism, in that they have an indefinite range of > possible applications, consequences, meaning. In that case WM is SOL in trying to build a coherent system. -- Virgil === Subject: Re: Answer to Dik T. Winter <87r5xofrgi.fsf@alatheia.truth.invalid> posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Consider axioms like in the American Declaration of Independence: >We hold these truths to be self evident... > Unless WM can list the things he holds to be self evident, he does not > have anything to build on. > No one can list the things he holds to be self evident in the sense of > the American Declaration of Independence. Such things are also not > axioms in any usual logical or mathematical sense -- they are more like > the axioms of Objectivism, in that they have an indefinite range of > possible applications, consequences, meaning. In that case WM is SOL in trying to build a coherent system. > Apropos: I am convinced that the platonism which underlies Cantorian set theory is utterly unsatisfactory as a philosophy of our subject [...] platonism is the medieval metaphysics of mathematics; surely we can do better [Soloman Feferman: Infinity in Mathematics: Is Cantor Necessary?] Furthermore, a case can be made that higher set theory is dispensable in scientifically applicable mathematics, that is, in that part of everyday mathematics which finds its applications in the other sciences. Put in other terms: the actual infinite is not required for the mathematics of the physical world. The reasons for this depend on other recent developments in mathematical logic, the description of which is the final aim of this essay [now chapter 12; cf. also chapter 14]. In order to explain the objections to Cantor's ideas in mathematics ... [Solomon Feferman: IN THE LIGHT OF LOGIC, p. 30] IN THE LIGHT OF LOGIC V COUNTABLY REDUCIBLE MATHEMATICS 12 Infinity in mathematics: Is Cantor necessary? (Conclusion) 229 13 Weyl vindicated: The Kontinuum 70 years later 249 14 Why a little bit goes a long way: Logical foundations of scientifically applicable mathematics 284 At least to that extent the question Is Cantor necessary? is answered with a resounding no === Subject: Re: Answer to Dik T. Winter > > Consider axioms like in the American Declaration of Independence: >We hold these truths to be self evident... > Unless WM can list the things he holds to be self evident, he does not > have anything to build on. > No one can list the things he holds to be self evident in the sense of > the American Declaration of Independence. Such things are also not > axioms in any usual logical or mathematical sense -- they are more like > the axioms of Objectivism, in that they have an indefinite range of > possible applications, consequences, meaning. > In that case WM is SOL in trying to build a coherent system. > > Apropos: > I am convinced that the platonism which underlies Cantorian set > theory is utterly unsatisfactory as a philosophy of our subject [...] > platonism is the medieval metaphysics of mathematics; surely we can do > better [Soloman Feferman: Infinity in Mathematics: Is Cantor > Necessary?] WM again argues from authority. Mathematics does not accept such arguments unless accompanied by proofs. > > Furthermore, a case can be made that higher set theory is dispensable > in scientifically applicable mathematics, that is, in that part of > everyday mathematics which finds its applications in the other > sciences. WM is certainly not able to make any such case, and what is not mathematics, but only application, is irrelevant to what mathematics is. > Put in other terms: the actual infinite is not required for > the mathematics of the physical world. That physicists do not need, or use, all of mathematics is their problem, not that of mathematicians. There is no need in mathematics for any applications to exist. In fact, G.H. Hardy regarded such mathematics as could be applies to be unworthy of study. -- Virgil === Subject: Re: Answer to Dik T. Winter <877hzjnkde.fsf@alatheia.truth.invalid> <87y6rzm05p.fsf@alatheia.truth.invalid> <87r5xofrgi.fsf@alatheia.truth.invalid> be unworthy of study. A fine illustration of the well-known phenomenon that renowned mathematicians are just as capable of mouthing inanities as anyone. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Answer to Dik T. Winter <877hzjnkde.fsf@alatheia.truth.invalid> <87y6rzm05p.fsf@alatheia.truth.invalid> <87r5xofrgi.fsf@alatheia.truth.invalid> <878wjv7sr8.fsf@alatheia.truth.invalid > In fact, G.H. Hardy regarded such mathematics as could be applies to > be unworthy of study. > > A fine illustration of the well-known phenomenon that renowned > mathematicians are just as capable of mouthing inanities as anyone. They're no logicians. -- hz === Subject: Re: Answer to Dik T. Winter > > In fact, G.H. Hardy regarded such mathematics as could be applies to > be unworthy of study. > > A fine illustration of the well-known phenomenon that renowned > mathematicians are just as capable of mouthing inanities as anyone. And hardhy was qquite wrong about what could be applied, as a considerable part of his work was in the bits of number theory that are the basis of current security techniques in todays internet transactions. -- Virgil === Subject: Re: Answer to Dik T. Winter <877hzjnkde.fsf@alatheia.truth.invalid> <87y6rzm05p.fsf@alatheia.truth.invalid> <87r5xofrgi.fsf@alatheia.truth.invalid> <87r5xofrgi.fsf@alatheia.truth.invalid> <87prd89hh4.fsf@alatheia.truth.invalid> posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) We agree that in Cantor's diagonal argument, applied to real numbers, the numbers are represented and identified solely by their digits. No further information is available. We assume that a real number p can be distinguished from a set Q of real numbers q by general considerations, for instance, if p is a transcendental number and Q consists of rational numbers q only. Of course it would be impossible to distinguish p from all q, because for every digit d_n of p, there is a number q that shares all digits up to d_n with p. It is possible to construct the complete binary tree from the numbers in Q. That means, it is possible to construct all possible digit sequences from the numbers in Q. It is also possible to construct the complete binary tree from the set Tp that consists of all termintaing rationals which are appended by a tail consisting of p. Therefore it is impossible to distinguish p from the binary tree. But the binary tree has been errected by using a countable set of paths only. Therefore it is impossible to distinguish, by means of digits, any real number from that countable set of paths which has been used to construct the tree. So it is impossible to distinguish pi from all terminating reals. Therefore there is no pi. === Subject: Re: Answer to Dik T. Winter > > We agree that in Cantor's diagonal argument, applied to real numbers, > the numbers are represented and identified solely by their digits. No > further information is available. > > We assume that a real number p can be distinguished from a set Q of > real numbers q by general considerations, for instance, if p is a > transcendental number and Q consists of rational numbers q only. > > > Of course it would be impossible to distinguish p from all q, because > for every digit d_n of p, there is a number q that shares all digits > up to d_n with p. It is possible to distinguish any real from any finite set of reals which does not contain it, and from many infinite sets of reals which do not contain it. The only ones which cannot be so distinguished are those sets containing a sequence converging to the real in question, which begs the question. > > It is possible to construct the complete binary tree from the numbers > in Q. But it is not possible to complete it without automatically creating uncountably many binary sequences. > That means, it is possible to construct all possible digit > sequences from the numbers in Q. All rational numbers have a binary expansion which is eventually periodic. How does one construct from such eventually periodic binary expansions a binary expansion that is not eventually periodic? One can find a sequence of such eventually periodic binaries converging to a non-perodic seqeunce, but it never actually arrives. So one cannot construct them exactly. One gets them purely as a bonus for completing the tree, not by any explicit construction of more than a dense subset o them.. -- Virgil === Subject: Re: Answer to Dik T. Winter <87r5xofrgi.fsf@alatheia.truth.invalid> posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > We agree that in Cantor's diagonal argument, applied to real numbers, > the numbers are represented and identified solely by their digits. No > further information is available. > We assume that a real number p can be distinguished from a set Q of > real numbers q by general considerations, for instance, if p is a > transcendental number and Q consists of rational numbers q only. > Of course it would be impossible to distinguish p from all q, because > for every digit d n of p, there is a number q that shares all digits > up to d n with p. It is possible but not sufficient for the following > to distinguish any real from any finite set of reals > which does not contain it, and from many infinite sets of reals which do > not contain it. The only ones which cannot be so distinguished are those sets containing > a sequence converging to the real in question, which begs the question. Every set of all terminating paths appended by arbitrary tails is countable and covers all real numbers of the unit interval. > It is possible to construct the complete binary tree from the numbers > in Q. But it is not possible to complete it without automatically creating > uncountably many binary sequences. That is the result of superstition. There is a countable set at the start and that set has not changed at the end. > That means, it is possible to construct all possible digit > sequences from the numbers in Q. All rational numbers have a binary expansion which is eventually > periodic. How does one construct from such eventually periodic binary expansions > a binary expansion that is not eventually periodic? That is not the right question to be put here. The right question is: If all periodic expansions cover all expansions of real numbers: can such expansions exist that are not eventually periodic? And the answer is a resounding no. One can find a sequence of such eventually periodic binaries converging > to a non-perodic seqeunce, but it never actually arrives. So it is. Therefore Cantor's diagonal never actually arrives. It is never completed, it is not a number. So one cannot construct themexactly. Yes, just this is impossible. One can construct this diagonal to any desired finite epsilon, but not better. === Subject: Re: Answer to Dik T. Winter Topic: Chris Menzel helps me prove N=NP: Inverse 19 Mathematics Replies: 0 Search Thread: Advanced Search Reply to this Topic Watch this Topic Back to Topic List Martin Michael Musatov Posts: 786 Registered: 4/19/09 Chris Menzel helps me prove N=NP: Inverse 19 Mathematics Forwarded conversation === === Subject: Re: Answer to Dik T. Winter > Because you do not check the lines in order.It is always your > basic assumption that you first check the first line and after > that the next line.That is wrong. > That is necessary because you cannot find the n-th line unless you > know the line number n - 1 or some equivalent mark. But why is getting the number n in any way related to the checking of the > previous lines? Because by blind choice you cannot be sure to hit what you want. > You are wrong.A list is a mapping from N to the elements of the list. > Through that list, given a number n, you find the n-th element of the list > without referring to any previous elements of the list. > But you cannot find number n without referring to the numbers less > than n. That does not mean that you check the line with number n.So you assertion > that you need to check all lines in order is false. In unary or in von Neumann's representation there is the whole counting process incorporated in each number. In decimal, you need only count the powers of 10. Anyhow addressing a number means counting. > To give an example, > let's have a list of positive rational numbers through the mapping given > a lnong time ago by David Tribble.To get the 51st element of the list, > we calculate the mapping: write 51 in binary, create from it a continued > fraction as described on > which is [0, 1, 2, 3], calculate it and we find 7/9. So we know the > 51st element without knowing the 50th element. Where did I come at the > 47th rational in that list (which is 7/11)? > How can you write 51 without knowing what it is? Of course you must > count. In unary this is more difficult than in decimal or binary, but > the principle is the same. How do you obtain this number > 1111111111111111111111111111111111111 > unless you count the digits? So you agree now that you can check the n-th element of the list before > checking any previous elements of the list?(How you get at n is irrelevant > here, because getting at n does not involve checking lines preceding the > n-th line.) Of course you can check the n-th element before checking any other. But in order to find the n-th element, you have to count from 1 to n, either in unary or at least in the powers of 10. Therefore it is not correct to talk about simultaneity. Pray use tail in the future rather than end, to avoid confusion. So Ido. > Every end of a path contains aleph 0 nodes. > These nodes can be mapped on that path. Every node will eventually be > mapped on one path. Therefore all nodes are used up for constructing > a countable set of paths. > Perhaps right, depends on how you actually do define things.But are > there nodes mapped to all paths?That is what you assert. > Unless there is at least one node occupied by the path p n that is not > occupied by the paths p 0 to p n-1, p n is not a new path. Assuming countability again. Countability of nodes only. I am only interested to cover every node, i.e., to exhaust the tree. I am a follower of Eudoxos. > Suppose we have the set of paths where each > path goes to the right after some specific (for that path) node. That would imply all paths with tails 111... No problem. >Is there > a node in your tree that is not occupied by any of the paths in that set? No, it isn't. Every path node is occupied by at least the head of a path. > Are there possible other paths that are *not* in that set of paths (like > a path that alternates going left and right)? It is said so. But there is no possibility, after having completed the construction, to introduce such a path. They have sneaked in. > Not in this case bacause you apply the words to different things. > *Unless* you assume that what is valid in finite cases also is valid > in infinite cases.But let's see: > sum{i = 0 .. n} 1/(i!) > is a rational number.So according to your logic: > e = sum{i = 0 .. n} 1/(i!) > is also a rational number. > I don't see a difference. But I can assuer you, there is no decimal > expansion for irrational numbers. Where am I talking about decimal expansions?According to you the > in finite sum above (which equals e by definition) is rational. Is that > true? True is: There are only rational sums of decimal or binary expansions. > Ok.The logic of finite complete sums of rational elements gives > that the sum is rational. Using your logic we get that the complete > (i.e. in the limit) sum of rational elements is also rational.And > so by that logic, e, pi and whatever are rational, and all numbers > we do use are rational. > In fact there are no binary expansions of irrational numbers. > What is the relevance?Where am I talking about binary expansions? > Binary, decimal, whatever. It does not exist. I am not talking about whatever expansion.Pray keep to the question. > According to your logic the complete (i.e. in the limit) sum of rational > elements is also rational. And so by that logic, e, pi and whatever are > rational and all numbers we do use are rational. No. The limit is not rational. But the limit cannot be represented by a sequence. Therefore the limit is not in any Cantor-list. That is the big error of set theory: to believe that irrational numbers have rational representations. Every binary or decimal sequence is a rational representation. I skip the rest because the main point now has become fairly clear: The paths in the binary tree. We should concentrate on that problem. === Subject: Re: Answer to Dik T. Winter > In unary or in von Neumann's representation there is the whole > counting process incorporated in each number. In decimal, you need > only count the powers of 10. Anyhow addressing a number means > counting. I can address sqrt(2) other than by counting. > > Of course you can check the n-th element before checking any other. > But in order to find the n-th element, you have to count from 1 to n, > either in unary or at least in the powers of 10. Therefore it is not > correct to talk about simultaneity. The problem of accessing a number is quite different from the problem of its existence. There are even numbers which exist in certain contexts none of which are individually accessible. > Assuming countability again. > > Countability of nodes only. I am only interested to cover every node, > i.e., to exhaust the tree. I am a follower of Eudoxos. Countability of members of set of nodes does not imply countability of subsets of that set. > > Suppose we have the set of paths where each > path goes to the right after some specific (for that path) node. > > That would imply all paths with tails 111... No problem. > >Is there > a node in your tree that is not occupied by any of the paths in that set? > > No, it isn't. Every path node is occupied by at least the head of a > path. > > Are there possible other paths that are *not* in that set of paths (like > a path that alternates going left and right)? > > It is said so. But there is no possibility, after having completed the > construction, to introduce such a path. They have sneaked in. However sneaky, they are there, in the sense that there are sets of nodes that satisfy the definition of being paths that were not built in by WM's construction but which appeared despite he attempts to eliminate them. So to keep those unwanted paths out, WM will have to redefine paths so that only sets of nodes explicitly constructed can be counted as paths. > > True is: There are only rational sums of decimal or binary expansions. True is: any sequence of decimal digits following 0. which can be unambiguously described defines a real number in [0,1]. And that includes lots of irrationals. > > No. The limit is not rational. But the limit cannot be represented by > a sequence. Therefore the limit is not in any Cantor-list. That is the > big error of set theory: to believe that irrational numbers have > rational representations. Every binary or decimal sequence is a > rational representation. There are real numbers which have precise definitions but no known decimal expansions. WM conflates a particular form of number name with the number itself. A name is not the thing named. And there are, in general, lots of different names for any number, any one of which suffices to establish its existence. > > I skip the rest because the main point now has become fairly clear: > The paths in the binary tree. We should concentrate on that problem. As soon as one has the set of all nodes of members of an infinite sequence of nested finite binary trees, one has a countable, partially ordered by the transitive closure of the 'parent of' relation set of nodes in which every maximal totally ordered subset is, by definition, a path. So there are uncountably many such paths in that tree. -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > There are real numbers which have precise definitions but no known > decimal expansions. May be. But those numbers are not able to participate in Cantor's list, neither as entries nor as anti diagonal- WM conflates a particular form of number name with the number itself. > A name is not the thing named. Cantor's list requires decimal representations. And there are, in general, lots of different names for any number, any > one of which suffices to establish its existence. But not sufficient to apply diagonalization. > I skip the rest because the main point now has become fairly clear: > The paths in the binary tree. We should concentrate on that problem. As soon as one has the set of all nodes of members of an infinite > sequence of nested finite binary trees, one has a countable, partially > ordered by the transitive closure of the 'parent of' relation set of > nodes in which every maximal totally ordered subset is, by definition, a > path. So there are uncountably many such paths in that tree. non sequitur. === Subject: Re: Answer to Dik T. Winter > > There are real numbers which have precise definitions but no known > decimal expansions. > > May be. But those numbers are not able to participate in Cantor's > list, neither as entries nor as anti diagonal- Cantor's remarks on lists of binary sequences is not about decimal expansions of real numbers. > WM conflates a particular form of number name with the number itself. > A name is not the thing named. > > Cantor's list requires decimal representations. Caantor's remarks about lists refers only to liss of binary sequences. It was not Cantor who revised those comments to apply to decimal expansions > And there are, in general, lots of different names for any number, any > one of which suffices to establish its existence. > > But not sufficient to apply diagonalization. Which is irrelevant here. Diagonalization only show that decimals aren't enough. The existence of other numbers which cannot be decimalized is just a bonus. > > I skip the rest because the main point now has become fairly clear: > The paths in the binary tree. We should concentrate on that problem. > As soon as one has the set of all nodes of members of an infinite > sequence of nested finite binary trees, one has a countable, partially > ordered by the transitive closure of the 'parent of' relation set of > nodes in which every maximal totally ordered subset is, by definition, a > path. > So there are uncountably many such paths in that tree. > > non sequitur. Follows from Cantor's anti-diagonal binary proof. -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > ... > Yes, obtained from. But that does not mean they are identical. > now? > > It is 50 years younger than Canto's writings. > > What is the relevance? > > validity now? My question still stands and I see no answer. Then take my answer: Yes. > But it does rule out theories which are contradicted by the > fundamental logical rules. And one of these rules is that a > complete linear set has a last element. > > What fundamental rules of logic are you using? I have never seen > such a > logical rule, because logic does not talk about sets. > > Logic states that the union of a *complete* set of finite linear > sets is a finite linear set > > As I said, logic is not talking about sets. So where in logic is such > stated? > > Logic is obtained from the behaviour of things. Things can be > considered as sets, at least if two things are taken together. Bolzano > excused himself for including 2. Later they included 1 and even 0, That is not an answer to my question. That answer is: Logic is obtained from the behaviour of things. > No. I use the fact that for complete linear sets always both > implications are true : > [**] & [***]. This means that [*] is true. > > You just state so without proof. > > A proof is a derivation of theorems from axioms or basic truths by > means of rules of logical inference. These rules themselves cannot > be proven but can only be obtained from the behaviour of existing > (i.e., finite) sets. > > So you are not using mathematical logic? > > Nobody should do so. Many logicians are below any level. Yes, I know you do not like logic. I like logic, but not nonsense. > There is > one Fool Of Matheology, for instance, who thinks that the cartesian > product of the set of finite alphabets is uncountable. What is the cartesion product of the set of finite alphabets? The only > definition I know is that the cartesion product of a set is the > cartesian square of a set, i.e. the set of pairs (x, y) where both x and > y come from the set. Apparently you mean something else. No. I mean exactly that: The set of finite words over a finite alphabet is countable. The set of meanings of these words, i.e., the set of languages, is countable. The set of finite alphabets is countable. The cartesian product of these, and possibly some further features, is countable. > I still do not see the logic through which you obtain it. > > It is obtained from the action and reaction of physical subjects. What has *that* to do with logical reasoning? Logical reasoning is able to > come up with algorithms like APR-CL that decide whether a number is prime > or not. What is the relation with action and reaction of physical subjects? > How does action and reaction of physical subjects relate to the logic that > constructed algorithms (like NSF) to factorise numbers? What are the > actions and reactions of physival subjects involved in taking the union of > FISONs? The logic is obtained from physical objects. How else should it have come into being? Remember, even brains are physical objects. === Subject: Re: Answer to Dik T. Winter > ... > Yes, obtained from. But that does not mean they are identical. > validity > now? > > It is 50 years younger than Canto's writings. > What is the relevance? > validity now? > My question still stands and I see no answer. > > Then take my answer: Yes. as well. > As I said, logic is not talking about sets. So where in logic is such > stated? > > Logic is obtained from the behaviour of things. Things can be > considered as sets, at least if two things are taken together. Bolzano > excused himself for including 2. Later they included 1 and even 0, > That is not an answer to my question. > > That answer is: Logic is obtained from the behaviour of things. On the contrary, logic is obtained from the behavior of minds. > No. I use the fact that for complete linear sets always both > implications are true : > [**] & [***]. This means that [*] is true. > You just state so without proof. > > A proof is a derivation of theorems from axioms or basic truths by > means of rules of logical inference. These rules themselves cannot > be proven but can only be obtained from the behaviour of existing > (i.e., finite) sets. > So you are not using mathematical logic? > > Nobody should do so. Many logicians are below any level. > Yes, I know you do not like logic. > > I like logic, but not nonsense. On the contrary, you seem enamored by nonsense. Otherwise you would not produce anywhere near as much of it. > There is > one Fool Of Matheology, for instance, who thinks that the cartesian > product of the set of finite alphabets is uncountable. > What is the cartesion product of the set of finite alphabets? The only > definition I know is that the cartesion product of a set is the > cartesian square of a set, i.e. the set of pairs (x, y) where both x and > y come from the set. Apparently you mean something else. > > No. I mean exactly that: The set of finite words over a finite > alphabet is countable. The set of meanings of these words, i.e., the > set of languages, is countable. The set of finite alphabets is > countable. The cartesian product of these, and possibly some further > features, is countable. Since one can have a Cartesian product of any set with itself countably many times, if that set contains more than one element, one can inject the set of paths of a maximal infinite binary tree into such a product, proving the product to be uncountable. So it depends on the sort of Cartesian product one considers whether it is uncountable or not. > > I still do not see the logic through which you obtain it. > > It is obtained from the action and reaction of physical subjects. Mathematics is not constrained by physics, except in the minds of those physicists who are not mathematicians. > The logic is obtained from physical objects. Logic is derived from mental objects, not physical ones. > How else should it have > come into being? Remember, even brains are physical objects. Even though embedded in a physical world, minds are not. They are metaphysical. Minds are not constrained to think only of things existing in WM's physics. If they were, there would be no theist religions, no fairy tales, no novels, no art, no music. -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > ... > And you are deluded.An axiom is a statement of something that can > not be proven, neither disproven using the remainder of the theory. > The axiom can be contradicted. Simple example: The axiom could be: The > binary tree has uncountably many paths. > Perhaps, although in ZF it is not an axiom. > It is, because the paths of the tree are isomorphic with the real > numbers in [0, 1] First off, the paths of the tree are not isomorphic with the real number. > There is no 1-1 mapping that is order preserving. There are two paths for every terminating rational number. The rest is a simple 1 to 1 mapping. Moreover, the statement the binary tree has uncountably many paths is a > theorem, not an axiom.It can be proven. And it can be disproven. In ZF there is no axiom that uses the word uncountable. It is defined and > all statements using it are theorems. But apparently you still do not understand what an axiom is. said: I The axiom *could be*: The binary tree has uncountably many paths. > I show that the end of each > path p of the set P can be mapped on a node, and that all paths p of > P cover all nodes of the tree. > Ignoring that in ZF the paths do not have an end. > The paths of the tree have no end. So your statment I show that the end of each path p of the set P can be > mapped on a node is blatant nonsense. I used the word end, that can have at least two meanings in the sense of tail. > But it can be shown for every node > that it gets covered and that all nodes get covered by a countable set > of paqths. You have not shown it.You only show it by assuming that there are countably > many paths. If every terminating path of the form 111...1000... is used, then every node is covered by the last 1. > Therefore after having completed the > covering of the whole tree, there remains no node that could be used > to construct a path that does not belong to P. > This is the wrong way around.You assume that you can cover this way the > whole tree (I think with this you mean each path in the tree).But that > is what you have to prove. > There is not much to prove. Append a tail of a path to every node. > Then every node is covered by at least one path, hence it does not > remain uncovered. Right.But do you use up *all* paths?*That* is what you have to prove. > You are always going to it from the wrong way.It is right that all nodes > can be covered by countably many paths.The set of paths where each path > from some node always goes right is an example.It does indeed cover all > nodes.But it is not the set of all paths, because there are paths that > do not satisfy the condition that from some node onwards it always goes > to the right. These path are in the tree too, after having completed the construction. Or can you find a node that is *not* covered by a path that after some node > to the right?Can you find a node that is covered by a path that alternates > going left and right that is not covered by a path that after some node > always goes to the right? I see that every such alternating path with all its nodes is in the tree, after construction. > This disproves the > mentioned axiom. > Indeed, when you assume it is false, it is easy to prove it is false. > I do not assume that the number of nodes is countable, but I count > them. Darn.Misreeading *again*.The number of nodes *is* countable.You do > assume the number of paths is countable.Why are you always misreading > what people do write? Perhaps because they do not express them precisely enough? But I do not always misread. We can state: The number of nodes is countable. The number of paths required to cover all nodes is countable too. > That means, you are willing to believe in what the Vatican says? > Well, no, because that dogma is not a valid axiom.But can you tell > me where that dogma actually is stated the way you say? > Sorry, I only read it some time ago somwhere. But I think the set of > dogmas must be in the net for those who are interested. I am not. So you just state something without being able to back it up. That is a ridiculous requirement in an informal discussion, in particular if not mathematics is concerned. === Subject: Re: Answer to Dik T. Winter > ... > And you are deluded. An axiom is a statement of something that > can > not be proven, neither disproven using the remainder of the > theory. > The axiom can be contradicted. Simple example: The axiom could be: > The > binary tree has uncountably many paths. > Perhaps, although in ZF it is not an axiom. > It is, because the paths of the tree are isomorphic with the real > numbers in [0, 1] > First off, the paths of the tree are not isomorphic with the real number. > There is no 1-1 mapping that is order preserving. > > There are two paths for every terminating rational number. The rest is > a simple 1 to 1 mapping. > Moreover, the statement the binary tree has uncountably many paths is a > theorem, not an axiom.It can be proven. > > And it can be disproven. Until WM provides us with a reasonably complete listing of his fundamental truths, on which he claims to base all his conclusions, he cannot prove anything to anyone. > In ZF there is no axiom that uses the word uncountable. It is defined > and > all statements using it are theorems. > But apparently you still do not understand what an axiom is. said: I The axiom *could be*: The binary tree has uncountably many > paths. Axioms are accepted without proof. The the maximal infinite binary tree has uncountably many paths is a theorem, not an axiom, and is provable from the axioms of ZF plus relevant definitions. > I show that the end of each > path p of the set P can be mapped on a node, and that all paths p > of > P cover all nodes of the tree. > Ignoring that in ZF the paths do not have an end. > The paths of the tree have no end. > So your statment I show that the end of each path p of the set P can be > mapped on a node is blatant nonsense. > > I used the word end, that can have at least two meanings in the sense > of tail. > But it can be shown for every node > that it gets covered and that all nodes get covered by a countable set > of paqths. > You have not shown it.You only show it by assuming that there are > countably > many paths. > > If every terminating path of the form 111...1000... is used, then > every node is covered by the last 1. But when one has the set of all nodes having only finitely many 1 branchings between them and the root node, suddenly one also has all paths with infinitely many one branchings in that same tree. > Right.But do you use up *all* paths?*That* is what you have to prove. > You are always going to it from the wrong way.It is right that all nodes > can be covered by countably many paths.The set of paths where each path > from some node always goes right is an example.It does indeed cover all > nodes.But it is not the set of all paths, because there are paths that > do not satisfy the condition that from some node onwards it always goes > to the right. > > These path are in the tree too, after having completed the > construction. > Or can you find a node that is *not* covered by a path that after some node > to the right?Can you find a node that is covered by a path that > alternates > going left and right that is not covered by a path that after some node > always goes to the right? > > I see that every such alternating path with all its nodes is in the > tree, after construction. Then there are uncountably many of them, as Cantor proved, and as no one, including WM, has been able to disprove. ssume it is false, it is easy to prove it is false. > I do not assume that the number of nodes is countable, but I count > them. > Darn.Misreeading *again*.The number of nodes *is* countable.You do > assume the number of paths is countable.Why are you always misreading > what people do write? > > Perhaps because they do not express them precisely enough? But I do > not always misread. You do it often enough, and consistently enough, to make discussions with you dificult. > We can state: The number of nodes is countable. The number of paths > required to cover all nodes is countable too. Fair enough, but that does not mean that EVERY set of paths which covers all nodes needs to be countable. The set of ALL paths, not all being required, can be, and is enough larger than any of WM's required to cover sets as to be uncountable. > > That means, you are willing to believe in what the Vatican says? > Well, no, because that dogma is not a valid axiom.But can you > tell > me where that dogma actually is stated the way you say? > Sorry, I only read it some time ago somwhere. But I think the set of > dogmas must be in the net for those who are interested. I am not. > So you just state something without being able to back it up. > > That is a ridiculous requirement in an informal discussion, in > particular if not mathematics is concerned. If that statement is so irrelevant to mathematics, why did you bring it up in a discussion of mathematics? -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > On 2009-06-11 16:49:49 -0400, WM said: > I need no axioms. > Then there is absolutely nothing that you take to be true? > There are absolutely no truths from which you can deduce other things? > There are truths, like 1 + 1 = 2, or, for linear sets, ExAy <==> AyEx These truths are the axioms of the theory you're using, or are > implied by them. I do not deny that. But I deny that axioms can be chosen. > Incorrect are such lies as the axiom of infinity or the axiom of > choice. The mathematics community mostly does not pursue a universally true set > of axioms. But mathematics does. > Instead, mathematics takes sets of axioms and attempts to > demonstrate that they lead to interesting results. Here is an interesting quote by Weyl (1946): We accept the hierarchy of types; but we assume only one category of primary objects, the numbers; and one basic binary relation between numbers, namely x is followed by y. All other relations of the various types are explicitly constructed, the quantifiers (Ex) and (Ax) being applied only to numbers and not to arguments of higher type. No axioms are postulated. Does he no longer belong to the math. community? > The results derived > from a given theory (a set of axioms) are only valid within that > theory. Mathematics (if done correctly) is true without any theory. > There are informal families of theories, within which the > informal definitions of various terms are portable (for example, there > is a family of set theories, each of which has a loosely similar > definition of what a set is), but informal similarity does not mean > that proofs from theories in the same family can necessarily be used > together. All transfinite set theories are nonsense, formal or informal. ZF set theory is no more true or false It is false, no doubt. A man who believes in or even attempts to prove something by means of higher cardinals is a fool. So it is with the idea that all sets are finite. The ZF axiom of > infinity is only the contradiction of the axiom there are no infinite > sets, and the resulting theory is both consistent and useful. That is wrong. Not a single useful bit has ever been obtained from finished infinity. > There > arealso consistent, useful theories that do not prove or require the > existence infinite sets. This is a prerequisite of a consisten theory like the convergence of a sequence to zero, shall the due series converge. If you would only name the theory you're using, or sketch out its > fundamental truths (that is, its axioms), you'd probably find people a > lot more receptive to your ideas. I am doing simply correct mathematics, at least I am striving for that goal. === Subject: Re: Answer to Dik T. Winter > On 2009-06-11 16:49:49 -0400, WM said: > > I need no axioms. > Then there is absolutely nothing that you take to be true? > There are absolutely no truths from which you can deduce other things? > There are truths, like 1 + 1 = 2, or, for linear sets, ExAy <==> AyEx > These truths are the axioms of the theory you're using, or are > implied by them. > > I do not deny that. But I deny that axioms can be chosen. > Incorrect are such lies as the axiom of infinity or the axiom of > choice. > The mathematics community mostly does not pursue a universally true set > of axioms. > > But mathematics does. Then how can mathematics deal with both Euclidean and non-Euclidean geometries? By WM's standards, only one of them can be true, so that mathematics, by WM's standards, should be forbidden from dealing with all but the true one. Which one would that be, WM? You claim to know so much about what is really true, so tell us which geometry is the true one. And then tell us how you know. > > Instead, mathematics takes sets of axioms and attempts to > demonstrate that they lead to interesting results. > > Here is an interesting quote by Weyl (1946): Mathematics does not accept truth-by-authority. > The results derived > from a given theory (a set of axioms) are only valid within that > theory. > > Mathematics (if done correctly) is true without any theory. Where do the first truths come from, that everything else derives from? They cannot come from the world of physics because we have no direct perception of that world. > > There are informal families of theories, within which the > informal definitions of various terms are portable (for example, there > is a family of set theories, each of which has a loosely similar > definition of what a set is), but informal similarity does not mean > that proofs from theories in the same family can necessarily be used > together. > > All transfinite set theories are nonsense, formal or informal. Then all of WM's theories are informal nonsense, too. And I prefer the coherent nonsense of mathematics to the inchoate idiotic nonsense of WM. > ZF set theory is no more true or false > > It is false, no doubt. To claim that without logical proof, as WM repeatedly does, is to proclaim oneself illogical. > A man who believes in or even attempts to prove > something by means of higher cardinals is a fool. The creative foolishness of mathematics can be, and historically has been, highly productive and useful, but WM's stupidity will never be. > So it is with the idea that all sets are finite. The ZF axiom of > infinity is only the contradiction of the axiom there are no infinite > sets, and the resulting theory is both consistent and useful. > > That is wrong. The creative foolishness of mathematics can be, and historically has been, highly productive and useful, but WM's stupidity will never be. > > There > arealso consistent, useful theories that do not prove or require the > existence infinite sets. > > This is a prerequisite of a consisten theory like the convergence of a > sequence to zero, shall the due series converge. In a set theory without infinite sets, and therefore also without infinite sequences, there can be no convergence because every sequence must have a last term. > If you would only name the theory you're using, or sketch out its > fundamental truths (that is, its axioms), you'd probably find people a > lot more receptive to your ideas. > > I am doing simply correct mathematics, at least I am striving for that > goal. And failing. Partly because you have not, or cannot, sketch out what you consider to be the fundamental truths on which you base your theory. Without being able to derive things from only fundamental truths and formal definitions, there is no way to check the validity of anything. So if WM feels compelled to keep his list of fundamental truths secret, he cannot publicly validate any of his arguments. -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > Before P is used to construct the binary tree, > the binary tree contains a path > p that can be distinguished from > every element of P. > Yes Using P to construct the binary tree does > not change any of the elements of P > or the path p. Please acknowledge After P is used to construct the binary tree, > the binary tree contains a path > p that can be distinguished from > every element of P. After construction, the tree contains P (and every other path of the unit interval you wish) in same same form, whether or not P is constructed first, before the construction by terminating paths has been done. That suggests that P is somewhat decoupled from the digit sequences represented in the tree. It has nothing to do with a digit sequence. === Subject: Re: Answer to Dik T. Winter > > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > Before P is used to construct the binary tree, > the binary tree contains a path > p that can be distinguished from > every element of P. > Yes > Using P to construct the binary tree does > not change any of the elements of P > or the path p. > Please acknowledge > After P is used to construct the binary tree, > the binary tree contains a path > p that can be distinguished from > every element of P. > > After construction, the tree contains P (and every other path of the > unit interval you wish) in same same form, whether or not P is > constructed first, before the construction by terminating paths has > been done. That suggests that P is somewhat decoupled from the digit > sequences represented in the tree. It has nothing to do with a digit > sequence. What happens in one of WM's non-maximal non-trees is irrelevant. In any maximal infinite binary tree, for every binary digit sequence there is a path and for every path there is a binary digit sequence. -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > Before P is used to construct the binary tree, > the binary tree contains a path > p that can be distinguished from > every element of P. > Yes > Using P to construct the binary tree does > not change any of the elements of P > or the path p. > Please acknowledge > After P is used to construct the binary tree, > the binary tree contains a path > p that can be distinguished from > every element of P. > After construction, the tree contains P (and every other path of the > unit interval you wish) in same same form So after construction the tree contains every element of P and the path p. Since they have not changed form it is still possible to distinguish p from every element of P So it is possible to contruct another path, p, which can be distinguished from every element of P. This directly contradicts your claim no possibility exists to construct or to distinguish by one or many or infinitely many nodes of the tree another path. - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > Before P is used to construct the binary tree, > the binary tree contains a path > p that can be distinguished from > every element of P. > Yes > Using P to construct the binary tree does > not change any of the elements of P > or the path p. > Please acknowledge > After P is used to construct the binary tree, > the binary tree contains a path > p that can be distinguished from > every element of P. > After construction, the tree contains P (and every other path of the > unit interval you wish) in same same form So after construction the tree contains every element of > P and the path p.Since they have not changed form it > is still possible to distinguish p from every element of P. Sorry, that is impossible. Every node of path p and every set of nodes of path p is covered by one or more paths of P. So it is possible to contruct another path, p,which can be > distinguished from every element of P. That is impossible. The path p 0 = 0.111... for instance is completely covered by terminating paths, whether or not it had been inserted originally. > This directly contradicts your claim no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. Sorry, this claim still stands. The reason is, that the path 0.111... does not exist. It is nothing but a union of terminating paths (potential infinity). The explanation is as follows: In order to define p 0 as a binary sequence, you say that for every terminating path p n there is an m > n such that the node m is covered by p 0 but not by p n. In the complete binary tree however, we have all nodes already covered by terminating paths. Therefore this bad trick of logic (using potential infinity for a set of nodes that should have actual existence) fails. === Subject: Re: Answer to Dik T. Winter > > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > Before P is used to construct the binary tree, > the binary tree contains a path > p that can be distinguished from > every element of P. > Yes > Using P to construct the binary tree does > not change any of the elements of P > or the path p. > Please acknowledge > After P is used to construct the binary tree, > the binary tree contains a path > p that can be distinguished from > every element of P. > After construction, the tree contains P (and every other path of the > unit interval you wish) in same same form > So after construction the tree contains every element of > P and the path p.Since they have not changed form it > is still possible to distinguish p from every element of P. > > Sorry, that is impossible. Every node of path p and every set of nodes > of path p is covered by one or more paths of P. Is P a set of paths or merely a set of nodes? If it is a set of nodes, it must be the set of all nodes or there will be paths wish are to subset of it. and the it will contain as subsets uncountably many paths If it is a set of paths, then it must contain all uncountably many paths or there will be a path which is not member of it. > So it is possible to contruct another path, p,which can be > distinguished from every element of P. > > That is impossible. The path p_0 = 0.111... for instance is completely > covered by terminating paths, whether or not it had been inserted > originally. If P is only a set of terminating paths then no non-terminating path will be a member of it. If P is a set of nodes, then it must contain every node of every finite subtree, and thus have every one of those uncountably many paths as a subset. Otherwise there will be paths that it does not contain as subsets. > > This directly contradicts your claim >no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > > Sorry, this claim still stands. The reason is, that the path 0.111... > does not exist. It is nothing but a union of terminating paths > (potential infinity). Except that if the infinite tree exists at all, which has been conceded or we would not even be talking about it, then it is actually infinite and WM's finiteness restrictions no longer apply. Wm cannot concede the existence of an actually infinite tree, which he has been doing, and then insist that it does not have the properties of an actually infinite tree. -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Sorry, that is impossible. Every node of path p and every set of nodes > of path p is covered by one or more paths of P. Is P a set of paths or merely a set of nodes? It is a set of paths. If it is a set of paths, then it must contain all uncountably many paths > or there will be a path which is not member of it. Wrong. All countably many terminating paths are sufficient to cover all nodes of the tree. > So it is possible to contruct another path, p,which can be > distinguished from every element of P. > That is impossible. The path p 0 = 0.111... for instance is completely > coveredby terminating paths, whether or not it had been inserted > originally. If P is only a set of terminating paths then no non-terminating path > will be a member of it. All non-terminating paths are in the tree constructed by P. === Subject: Re: Answer to Dik T. Winter > > Sorry, that is impossible. Every node of path p and every set of nodes > of path p is covered by one or more paths of P. > Is P a set of paths or merely a set of nodes? > > It is a set of paths. > If it is a set of paths, then it must contain all uncountably many paths > or there will be a path which is not member of it. > > Wrong. All countably many terminating paths are sufficient to cover > all nodes of the tree. But paths are not mere nodes, they are sets of nodes, and while every node many be a member of a member of P, that does not imply that every any particular subset of the set f nodes is a member of P. That does not even hold for finite sets. The set {{a},{b},{c}} contains every member of {a,b,c} as a member of a member but does NOT contain as a member every subset of {a,b,c}. For example {a,b} and {a,c} and {b,c} are not members nor subsets of {{a},{b},{c}}. WM needs to go back to set theory kindergarten, and brush up. > > So it is possible to contruct another path, p,which can be > distinguished from every element of P. > That is impossible. The path p_0 = 0.111... for instance is completely > coveredby terminating paths, whether or not it had been inserted > originally. > If P is only a set of terminating paths then no non-terminating path > will be a member of it. > > All non-terminating paths are in the tree constructed by P. But not in P itself. Since P is countable it can be put into a list which necessarily has an anti-diagonal not in P. Actually, at least as many anti-diagonals as members. -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > Your claim is that no possibility exists to construct or to distinguish by one or many or infinitely many nodes of the tree another path. > the tree contains every element of > P and the path p. Since they have not changed form it > is still possible to distinguish p from every element of P. Sorry, that is impossible. Note that you have agreed the binary tree contains a path p that can be distinguished from every element of P. this directly contradicts your claim that no possibility exists to construct or to distinguish by one or many or infinitely many nodes of the tree another path. > Every node of path p and every set of nodes > of path p is covered by one or more paths of P. Indeed, one *or more* paths of P. The set of nodes in path p is not covered by one path in P. - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. So it is. Proof: Let me construct a tree. I will not tell you what kind of paths I use. For instance I could use terminating paths or paths with tails 111... or paths with tails pi-3 or paths with tails e-2 or a mixture of many kinds of paths. I will present you the tree when it is ready. Would you bet to be able to distinuish your path from the set that I have used? > the tree contains every element of > P and the path p.Since they have not changed form it > is still possible to distinguish p from every element of P. > Sorry, that is impossible. Note that you have agreed the binary tree contains a path > p that can be distinguished from > every element of P. this directly contradicts your claim that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. So let us play the game. Choose a path and try to distinguish it from the set of paths that completely cover my tree. > Every node of path p and every set of nodes > of path p is covered by one or more paths of P. Indeed, one *or more* paths of P. > The set of nodes in path p is > not covered by one path in P. dollars. I will risk one of them, say 10 million dollars on the claim that you will not be able to distinguish your path from the set of paths that I have used for construction. What about your stakes? === Subject: Re: Answer to Dik T. Winter > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > > So it is. > Proof: Let me construct a tree. I will not tell you what kind of paths > I use. For instance I could use terminating paths or paths with tails > 111... or paths with tails pi-3 or paths with tails e-2 or a mixture > of many kinds of paths. I will present you the tree when it is ready. > Would you bet to be able to distinuish your path from the set that I > have used? If you hide the paths you have used, there is no way, without peeking, that anyone else can Tel wish paths yo have used, but that is immaterial. The point is that whatever paths you have used, by the time the maximal infinite binary tree finally exists, there are paths in it that you did not use. > > So let us play the game. Choose a path and try to distinguish it from > the set of paths that completely cover my tree. Until I know which paths you used, I can only guess, but if I guess slyly enough, and your set of paths is countable, the odds are greatly in my favor. > Every node of path p and every set of nodes > of path p is covered by one or more paths of P. > Indeed, one *or more* paths of P. > The set of nodes in path p is > not covered by one path in P. > > dollars. I will risk one of them, say 10 million dollars on the claim > that you will not be able to distinguish your path from the set of > paths that I have used for construction. What about your stakes? Is your set of paths countable (and actually counted or listed, at least algorithmically), and recorded with a neutral referee? If 'yes' to both, you would almost certainly lose at any stakes. All your opponent would have to do is use the Cantor antidiagonal to your own list of paths, which you cannot ever include in that list. -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > So it is. > Proof: Let me construct a tree. I will not tell you what kind of paths > I use. For instance I could use terminating paths or paths with tails > 111... or paths with tails pi-3 or paths with tails e-2 or a mixture > of many kinds of paths. I will present you the tree when it is ready. > Would you bet to be able to distinuish your path from the set that I > have used? If you hide the paths you have used, there is no way, without peeking, > that anyone else can Tel wish paths yo have used, but that is immaterial. The point is that whatever paths you have used, by the time the maximal > infinite binary tree finally exists, there are paths in it that you did > not use. How do they come in? The tree is constructed from a countable set P of terminating paths. If you have a Cantor list filled with those paths, then you claim to be able to construct another path p. So this path does not creaqpo intio the list. But it does creap into the tree? When does it creap? Where does it lurk and who tells it when to start? That is nonsense! There is no actualyy infinite path. So nothing needs to creap and superstision can be expelled from mathematics. > So let us play the game. Choose a path and try to distinguish it from > the set of paths that completely cover my tree. Until I know which paths you used, I can only guess, but if I guess > slyly enough, and your set of paths is countable, the odds are greatly > in my favor. But if I trick you out and decide only after you have guessed? Or if I am honest, as is my character, and use only every path that can be named? Then your odds are zero if not less. > Every node of path p and every set of nodes > of path p is covered by one or more paths of P. > Indeed, one *or more* paths of P. > The set of nodes in path p is > not covered by one path in P. > dollars. I will risk one of them, say 10 million dollars on the claim > that you will not be able to distinguish your path from the set of > paths that I have used for construction. What about your stakes? Is your set of paths countable (and actually counted or listed, at least > algorithmically), and recorded with a neutral referee? If 'yes' to both, you would almost certainly lose at any stakes. Yes to both. Here is the algorithmus generating the words than label my paths including the dictionary and grammar of all languages that are able to identify paths: 0 1 00 01 10 11 .. All your opponent would have to do is use the Cantor antidiagonal to > your own list of paths, which you cannot ever include in that list. There is only one problem: This list has no diagonal. No list of numbers that are given in a way such that they are uniquely defined has a diagonal. What has a diagonal is a list that contains blah-blah- blah like 0.9384791428759184... 0.2392847216773984... ... And the mathematics errected upon this blah-blah is like that. === Subject: Re: Answer to Dik T. Winter > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > So it is. > Proof: Let me construct a tree. I will not tell you what kind of paths > I use. For instance I could use terminating paths or paths with tails > 111... or paths with tails pi-3 or paths with tails e-2 or a mixture > of many kinds of paths. I will present you the tree when it is ready. > Would you bet to be able to distinuish your path from the set that I > have used? > If you hide the paths you have used, there is no way, without peeking, > that anyone else can Tel wish paths yo have used, but that is immaterial. > The point is that whatever paths you have used, by the time the maximal > infinite binary tree finally exists, there are paths in it that you did > not use. > > How do they come in? They are a natural and inevitable consequence of the tree becoming a maximal infinite binary tree. As soon as all the required nodes are there, so, automatically, are all those unused paths. As soon as one has the set of all nodes, every possible subset is a subset, and all those paths are merely paticular subsets. > The tree is constructed from a countable set P of > terminating paths. If the union of that set P contains every node of the maximal infinite binary tree as a member, then it contains every set of nodes as a subset, therefore contains every path as a subset. > If you have a Cantor list filled with those paths, > then you claim to be able to construct another path p. So this path > does not creaqpo intio the list. But it is in a new list. But P is not just a list, it must be a set of sets of nodes and the union of P (the set of all members of members of P) must then be the set of ALL nodes, at least if the tree is to be constructed from it. And once one has a set of all nodes, one has as subsets all those paths WM doesn't like hearing about. > But it does creap into the tree? Do you mean creep > When > does it creap? Where does it lurk and who tells it when to start? If the subsets of P cover all nodes then the union of P is the set of nodes and the subsets of that union include al those paths that upset WM so badly, so I guess one would have to say that when building your P, as soon as what you have built so far covers all nodes, you have all paths. > > That is nonsense! There is no actualyy infinite path. So nothing needs > to creap and superstision can be expelled from mathematics. That is WM's nonsense. WM can hardly build his own infinite paths, as he has been doing here for some time, and then deny infinite paths to everyone else. > > So let us play the game. Choose a path and try to distinguish it from > the set of paths that completely cover my tree. > Until I know which paths you used, I can only guess, but if I guess > slyly enough, and your set of paths is countable, the odds are greatly > in my favor. > > But if I trick you out and decide only after you have guessed? That is not the game originally proposed, so that WM finds he needs to cheat. > Or if I > am honest, as is my character, and use only every path that can be > named? Then your odds are zero if not less. For any finite list of more than one name, the concatenation of all its names cannot be included within it as a name. And WM declares that absolutely no infinite lists of names are possible, so he loses. > > Every node of path p and every set of nodes > of path p is covered by one or more paths of P. > Indeed, one *or more* paths of P. > The set of nodes in path p is > not covered by one path in P. > dollars. I will risk one of them, say 10 million dollars on the claim > that you will not be able to distinguish your path from the set of > paths that I have used for construction. What about your stakes? > Is your set of paths countable (and actually counted or listed, at least > algorithmically), and recorded with a neutral referee? > If 'yes' to both, you would almost certainly lose at any stakes. > > Yes to both. Here is the algorithmus generating the words than label > my paths including the dictionary and grammar of all languages that > are able to identify paths: > > 0 > 1 > 00 > 01 > 10 > 11 None of the above are paths, or even sets of nodes, so are not allowable. Your own specification required a SET OF PATHS: you will not be able to distinguish your path from the set of paths that I have used for construction -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > The point is that whatever paths you have used, by the time the maximal > infinite binary tree finally exists, there are paths in it that you did > not use. > How do they come in? They are a natural and inevitable consequence of the tree becoming a > maximal infinite binary tree. As soon as all the required nodes are > there, so, automatically, are all those unused paths. As soon as one has > the set of all nodes, every possible subset is a subset, and all those > paths are merely paticular subsets. All paths that I have used contain all possible subsets. > The tree is constructed from a countable set P of > terminating paths. If the union of that set P contains every node of the maximal infinite > binary tree as a member, then it contains every set of nodes as a > subset, therefore contains every path as a subset. It does. And it is a countable union.You remember a proof by Feferman and Levy? They showed that the statement that the set of all real numbers is the union of a denumerable set of denumerable sets cannot be refuted. > If you have a Cantor list filled with those paths, > then you claim to be able to construct another path p. So this path > does not creaqpo intio the list. But it is in a new list. But P is not just a list, it must be a set of > sets of nodes and the union of P (the set of all members of members of > P) must then be the set of ALL nodes, at least if the tree is to be > constructed from it. And once one has a set of all nodes, one has as subsets all those paths > WM doesn't like hearing about. > But it does creap into the tree? Do you mean creep probably. Why do you English always speek such that one cannot distinguish between weak and week? > When > does it creap? Where does it lurk and who tells it when to start? If the subsets of P cover all nodes then the union of P is the set of > nodes and the subsets of that union include al those paths that upset WM > so badly, so I guess one would have to say that when building your P, as > soon as what you have built so far covers all nodes, you have all paths. And if the right hand side of the tree is not yet ready, but the left hand side? Is there a border? > That is nonsense! There is no actually infinite path. So nothing needs > to creap and superstition can be expelled from mathematics. That is WM's nonsense. WM can hardly build his own infinite paths, as he has been doing here > for some time, and then deny infinite paths to everyone else. I can assume its existence and then find that this assumption was wrong. > So let us play the game. Choose a path and try to distinguish it from > the set of paths that completely cover my tree. > Until I know which paths you used, I can only guess, but if I guess > slyly enough, and your set of paths is countable, the odds are greatly > in my favor. > But if I trick you out and decide only after you have guessed? That is not the game originally proposed, so that WM finds he needs to > cheat. > Or if I > am honest, as is my character, and use only every path that can be > named? Then your odds are zero if not less. For any finite list of more than one name, the concatenation of all its > names cannot be included within it as a name. And WM declares that > absolutely no infinite lists of names are possible, so he loses. The list of everything is shown below. > Every node of path p and every set of nodes > of path p is covered by one or more paths of P. > Indeed, one *or more* paths of P. > The set of nodes in path p is > not covered by one path in P. > dollars. I will risk one of them, say 10 million dollars on the claim > that you will not be able to distinguish your path from the set of > paths that I have used for construction. What about your stakes? > Is your set of paths countable (and actually counted or listed, at least > algorithmically), and recorded with a neutral referee? > If 'yes' to both, you would almost certainly lose at any stakes. > Yes to both. Here is the algorithmus generating the words than label > my paths including the dictionary and grammar of all languages that > are able to identify paths: > 0 > 1 > 00 > 01 > 10 > 11 None of the above are paths, or even sets of nodes, so are not allowable. All paths that you can tink of and anything else is encoded in th elist above. Real numbers are not encoded as infinite sequences, though, because it is impossible to write infinite sequences. That is the main reason, why you and others never have written pi or 3.1415... as an infinite sequence. It is impossible. Your own specification required a SET OF PATHS: >you will not be able to distinguish your path from >the set of paths that I have used for construction after the construction has been completed. Yes, so it is. === Subject: Re: Answer to Dik T. Winter > > The point is that whatever paths you have used, by the time the maximal > infinite binary tree finally exists, there are paths in it that you did > not use. > How do they come in? > They are a natural and inevitable consequence of the tree becoming a > maximal infinite binary tree. As soon as all the required nodes are > there, so, automatically, are all those unused paths. As soon as one has > the set of all nodes, every possible subset is a subset, and all those > paths are merely paticular subsets. > > All paths that I have used contain all possible subsets. Outside of WM's world of MathUnrealism, if one has a maximal infinite binary tree and defines in it a path to be any maximal set of nodes totally ordered by the ancestor relation, then there are uncoutably many of paths. WM must be using some other definition of path in order to have so few of them. > The tree is constructed from a countable set P of > terminating paths. > If the union of that set P contains every node of the maximal infinite > binary tree as a member, then it contains every set of nodes as a > subset, therefore contains every path as a subset. > > It does. And it is a countable union.You remember a proof by Feferman > and Levy? They showed that the statement that the set of all real > numbers is the union of a denumerable set of denumerable sets cannot > be refuted. Did they say it could be proved? And didn't their proof require denial of the axiom of choice? > If you have a Cantor list filled with those paths, > then you claim to be able to construct another path p. So this path > does not creaqpo intio the list. > But it is in a new list. But P is not just a list, it must be a set of > sets of nodes and the union of P (the set of all members of members of > P) must then be the set of ALL nodes, at least if the tree is to be > constructed from it. > And once one has a set of all nodes, one has as subsets all those paths > WM doesn't like hearing about. > But it does creap into the tree? > Do you mean creep > > probably. Why do you English always speek such that one cannot > distinguish between weak and week? > When > does it creap? Where does it lurk and who tells it when to start? > If the subsets of P cover all nodes then the union of P is the set of > nodes and the subsets of that union include all those paths that upset WM > so badly, so I guess one would have to say that when building your P, as > soon as what you have built so far covers all nodes, you have all paths. > > And if the right hand side of the tree is not yet ready, but the left > hand side? Is there a border? The border appears to be between having all those nodes which precede and/or follow some particular node, and not having them. As soon as one has them, one also has uncountably many paths. > > That is nonsense! There is no actually infinite path. So nothing needs > to creap and superstition can be expelled from mathematics. > That is WM's nonsense. > WM can hardly build his own infinite paths, as he has been doing here > for some time, and then deny infinite paths to everyone else. > > I can assume its existence and then find that this assumption was > wrong. You cannot find the assumption wrong without assuming that we do not grant. > the set of paths that completely cover my tree. > Until I know which paths you used, I can only guess, but if I guess > slyly enough, and your set of paths is countable, the odds are greatly > in my favor. > But if I trick you out and decide only after you have guessed? > That is not the game originally proposed, so that WM finds he needs to > cheat. > Or if I > am honest, as is my character, and use only every path that can be > named? Then your odds are zero if not less. >For any finite list of more than one name, the concatenation of all its > names cannot be included within it as a name. And WM declares that > absolutely no infinite lists of names are possible, so he loses. > > The list of everything is shown below. > > Every node of path p and every set of nodes > of path p is covered by one or more paths of P. > Indeed, one *or more* paths of P. > The set of nodes in path p is > not covered by one path in P. > dollars. I will risk one of them, say 10 million dollars on the claim > that you will not be able to distinguish your path from the set of > paths that I have used for construction. What about your stakes? > Is your set of paths countable (and actually counted or listed, at least > algorithmically), and recorded with a neutral referee? > If 'yes' to both, you would almost certainly lose at any stakes. > Yes to both. Here is the algorithmus generating the words than label > my paths including the dictionary and grammar of all languages that > are able to identify paths: > 0 > 1 > 00 > 01 > 10 > 11 > None of the above are paths, or even sets of nodes, so are not allowable. > > All paths that you can tink of and anything else is encoded in th > elist above. Nope. Only nodes are encoded. To encode a path one would need an infinite binary sequence whose nodes are the finite initial sequences in your list. Either that or infinite subsets of members of your list. In either case, there are more that countably many of them. -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. Note that you have agreed the binary tree contains a path p that can be distinguished from every element of P. This directly contardicts your claim. - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. after p has been chosen> You should give up. There is no attempt to change P. There is a proof that you cannot distinguish p from the set of paths P that has been used to construct the tree. Note that you have agreed the binary tree contains a path > p that can be distinguished from > every element of P. > If I have agreed, then I was in error. The binary tree contains no path that can be distinguished from the countable set P that has been used to construct the tree. Or let me put is this way: Every path that is present in the tree belongs to a countable set that could have been unsed to construct the tree. Wanna raise the stakes? === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > after p has been chosen > You should give up. There is no attempt to change P. There is a proof > that you cannot distinguish p from the set of paths P that has been > used to construct the tree. > Note that you have agreed > the binary tree contains a path > p that can be distinguished from > every element of P. If I have agreed, then I was in error. OK. I will give you the list of things you agreed with. Point out where you made your error. There is a subset of nodes that cannot be found in a single element of P. A subset of nodes is distinguished from an element q of P if and only if it is not contained in q. A subset of nodes is distinguished from every element of P if and only if it is not contained in a single element of P. There is a subset of nodes which forms a path, call it p, that is not contained in a single element of P. The path p is distinguished from every element of P. All of the nodes of path p are in the tree. - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > You should give up. There is no attempt to change P. There is a proof > that you cannot distinguish p from the set of paths P that has been > used toconstruct the tree. > Note that you have agreed > the binary tree contains a path > p that can be distinguished from > every element of P. > If I have agreed, then I was in error. OK.I will give you the list of things you > agreed with.Point out where you made your error. There is a subset of nodes that cannot be found in a single > element of P. That is correct --- iff actually infinitely long sequences of nodes exist! And that is the assumption we start off with (and that we disprove). Consider the set P of terminating paths and p = 1/pi. The nodes of 1/pi are not in a single element of P. A subset of nodes is distinguished from an element > q of P if and only if it is not contained in q. 1/pi is not contained in any element of P --- if 1/pi exists! A subset of nodes is distinguished from every > element of P if and only if it is not contained in > a single element of P. Same as above. No error yet. There is a subset of nodes > which forms a path, call it p, > that is not contained in a single element of P. No error detected so far. The path p is distinguished from every element >of P. Ok. All of the nodes of path p are in the tree. That is correct because the tree contains all subsets of nodes. All information that is available about a number in binary representation is in (one path of) the tree. If I present you the complete tree without saying how I constructed it, then you have exactly the same information that can be applied in Cantor's list argument. You cannot distinguish your favourite path from the tree constructed by means of a countable set of paths. What's now? Play the game in order to prove that? Thou shalt not belittle my binary tree! === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I Your claim is that no possibility exists to construct or to distinguish by one or many or infinitely many nodes of the tree another path. In order to claim you do not contradict this you resort to Wolkenmuekenheim logic. The path p is distinguished from every element of P. All of the nodes of path p are in the tree. The binary tree does not contain a path p that can be distinguished from every element of P. - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. Yes. >The path p is distinguished >from every element of P. 1/pi is distinct from any terminating path. All of the nodes of path p >are in the tree. > All nodes of the path 1/pi are in the tree (together with all nodes of any other path of the unit interval). >The binary tree does not contain a path >p that can be distinguished from >every element of P. The binary tree does not contain any path that can be distinguished from every element of the set P of paths by which it was constructed. If this were no true, then you could determine whether a given path p differs from every element of P without knowing P. Note: Cantor's diagonal method uses only digits respective nodes, no additional information like the age of the writer or so. Same holds for my tree. === Subject: Re: Answer to Dik T. Winter > > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > > Yes. > >The path p is distinguished >from every element of P. > > 1/pi is distinct from any terminating path. >All of the nodes of path p >are in the tree. > All nodes of the path 1/pi are in the tree (together with all nodes of > any other path of the unit interval). That is not the issue. WM has presented a countable list of paths, P, from which he claims to build a maximal infinite binary tree, T. While the union of P, as a set of nodes, can well be the set of all nodes of T, the members of P, being sets of nodes which for paths, d not exhaust the set of all paths, where a path is any maximal set of nodes totally ordered by the ancestor relation. > >The binary tree does not contain a path >p that can be distinguished from >every element of P. > > The binary tree does not contain any path that can be distinguished > from every element of the set P of paths by which it was constructed. On the contrary, the tree contains more of those paths than the ones from which it was constructed. > If this were no true, then you could determine whether a given path p > differs from every element of P without knowing P. Nonsense. There is difference between knowing that there ARE paths not in P and knowing WHICH paths are not in P. The cardinalities involved require that there be paths not in P, but do not identify which paths are not in P. If Card(A) > Card(B), then there are elements of A which are not members of B. At least outside of WM's world of MathUnrealism. > > Note: Cantor's diagonal method uses only digits respective nodes, no > additional information like the age of the writer or so. Same holds > for my tree. How does the age of the writer of your tree affect its structure? > -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) We agree that in Cantor's diagonal argument, applied to real numbers, the numbers are represented and identified solely by their digits. No further information is available. We assume that a real number p can be distinguished from a set Q of real numbers q by general considerations, for instance, if p is a transcendental number and Q consists of rational numbers q only. Of course it would be impossible to distinguish p from all q, because for every digit d_n of p, there is a number q that shares all digits up to d_n with p. It is possible to construct the complete binary tree from the numbers in Q. That means, it is possible to construct all possible digit sequences from the numbers in Q. It is also possible to construct the complete binary tree from the set Tp that consists of all termintaing rationals which are appended by a tail consisting of p. Therefore it is impossible to distinguish p from the binary tree. But the binary tree has been errected by using a countable set of paths only. Therefore it is impossible to distinguish, by means of digits, any real number from a countable set of paths. Conclusion: Cantor's diagonal argument is contradicted. Comment: This is not surprising, because it is impossible to identify any irrational number by means of its decimal representation, but just this is claimed by the diagonal argument. === Subject: Re: Answer to Dik T. Winter > > > We agree that in Cantor's diagonal argument, applied to real numbers, > the numbers are represented and identified solely by their digits. No > further information is available. > > We assume that a real number p can be distinguished from a set Q of > real numbers q by general considerations, for instance, if p is a > transcendental number and Q consists of rational numbers q only. Then one can then distinguish between any real number whose binary or decimal or other base expansion is eventually periodic and those whose expansions are not eventually periodic. > > > Of course it would be impossible to distinguish p from all q, because > for every digit d_n of p, there is a number q that shares all digits > up to d_n with p. Thus one can distingish any number with a non-eventually-periodic expansion from all rationals. -- Virgil === Subject: Re: Answer to Dik T. Winter > > > > We agree that in Cantor's diagonal argument, applied to real numbers, > the numbers are represented and identified solely by their digits. No > further information is available. > > We assume that a real number p can be distinguished from a set Q of > real numbers q by general considerations, for instance, if p is a > transcendental number and Q consists of rational numbers q only. > > Then one can then distinguish between any real number whose binary or > decimal or other base expansion is eventually periodic and those whose > expansions are not eventually periodic. > > > Of course it would be impossible to distinguish p from all q, because > for every digit d_n of p, there is a number q that shares all digits > up to d_n with p. > > Thus one can distingish any number with a non-eventually-periodic > expansion from all rationals. What the hell has the diagonal argument to do with distinguishing numbers from one another? This is nothing but one more of Mueckenheim's red herrings. === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. Yes. > Wolkenmuekenheim logic. You agree that there is a path p in the tree You agree that path p can be distinguished from every element of P. You do not agree that the tree contains a path that can be distinguished from every element of P. - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > Yes. You agree that there is a path p in the tree You agree that path p can be distinguished > from every element of P. If actually infinite paths exist! That is obviously a prerequisite. Otherwise the construction using terminating paths covers everything. You do not agree that > the tree contains a path that > can be distinguished from every element of P. I can prove it. You only have to accept the game. You could see it even by yourself without any game. === Subject: Re: Answer to Dik T. Winter > > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > Yes. > > You agree that there is a path p in the tree > You agree that path p can be distinguished > from every element of P. > > If actually infinite paths exist! That is obviously a prerequisite. > Otherwise the construction using terminating paths covers everything. You construction would require the existence of potentially infinite sets, which does not occur in any known set theory system. > You do not agree that > the tree contains a path that > can be distinguished from every element of P. > > I can prove it. Between what WM claims to be able to prove and what he actually is able to prove lies an unbridgeable gulf. > You only have to accept the game. If it is the game WM offered before, he loses. -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > Yes. > You agree that there is a path p in the tree > You agree that path p can be distinguished > from every element of P. If actually infinite paths exist! Ok. We make the assumption explicit. Wolkenmuekenheim logic You agree that if actually infinite paths exist, there is a path p in the tree You agree that if actually infinite paths exist, path p can be distinguished from every element of P. You do not agree that if actually infinite paths exist, the tree contains a path that can be distinguished from every element of P. - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > Yes. > You agree that there is a path p in the tree > You agree that path p can be distinguished > from every element of P. > If actually infinite paths exist! Ok.We make the assumption explicit. > You agree that if actually infinite paths > exist, there is a path p in the tree You agree that if actually > infinite paths exist, path p > can be distinguished > from every element of P. by using nodes of the tree only without additional knowledge. You do not agree that > if actually infinite paths exist, > the tree contains a path that > can be distinguished from every element of P. I prove that it is impossible to distinguish a give path p from the paths in P using only the information stored in the tree, i.e. using only digit sequences to identify numbers, thereby proving that no actual infinity exists. Instead of grumbling about this fact, you either should accept it and confess that I am right and that set theory is wrong, or you should be able to distinguish your path p from the set P of paths that I have used for construction, but without any other knowledge than the final result, namely the digits, i.e., the nodes of the binary tree. That are the two possible logical alternatives. === Subject: Re: Answer to Dik T. Winter > > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > Yes. > You agree that there is a path p in the tree > You agree that path p can be distinguished > from every element of P. > If actually infinite paths exist! > Ok.We make the assumption explicit. > > You agree that if actually infinite paths > exist, there is a path p in the tree > You agree that if actually > infinite paths exist, path p > can be distinguished > from every element of P. > > by using nodes of the tree only without additional knowledge. So that WM argues that if P is hidden then it somehow will contain paths that it does not contain when known? > You do not agree that > if actually infinite paths exist, > the tree contains a path that > can be distinguished from every element of P. > > I prove that it is impossible to distinguish a give path p from the > paths in P using only the information stored in the tree, i.e. using > only digit sequences to identify numbers But it is eminently possible using the tree and knowing P. > thereby proving that no > actual infinity exists. Not a proof that will stand up to any scrutiny. > > Instead of grumbling about this fact, you either should accept it and > confess that I am right and that set theory is wrong, or you should be > able to distinguish your path p from the set P of paths that I have > used for construction, but without any other knowledge than the final > result, namely the digits, i.e., the nodes of the binary tree. Or we can just show, as we have done so often, that there are more paths in the tree than in P (in the sense that there can be no surjection from P to the set of all paths or injection the other way). Theorem: if A and B are sets such that Card(A) > Card(B) then there is some a in A that is not in B. > > That are the two possible logical alternatives. Except that WM carefully ignores the actuality. -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I Your claim is that no possibility exists to construct or to distinguish by one or many or infinitely many nodes of the tree another path. You claim you do not contradict this by using Wolkenmuekenheim logic. Assume A: I agree that B is true I agree that C is true I do not agree that (B and C) is true. - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) We agree that in Cantor's diagonal argument, applied to real numbers, the numbers are represented and identified solely by their digits. No further information is available. We assume that a real number p can be distinguished from a set Q of real numbers q by general considerations, for instance, if p is a transcendental number and Q consists of rational numbers q only. Of course it would be impossible to distinguish p from all q, because for every digit d_n of p, there is a number q that shares all digits up to d_n with p. It is possible to construct the complete binary tree from the numbers in Q. That means, it is possible to construct all possible digit sequences from the numbers in Q. It is also possible to construct the complete binary tree from the set Tp that consists of all termintaing rationals which are appended by a tail consisting of p. Therefore it is impossible to distinguish p from the binary tree. But the binary tree has been errected by using a countable set of paths only. Therefore it is impossible to distinguish, by means of digits, any real number from a countable set of paths. Conclusion: Cantor's diagonal argument is contradicted. Comment: This is not surprising, because it is impossible to identify any irrational number by means of its decimal representation, but just this is claimed by the diagonal argument. === Subject: Re: Answer to Dik T. Winter On 2009-06-13 14:29:05 -0400, WM said: > We agree that in Cantor's diagonal argument, applied to real numbers, > the numbers are represented and identified solely by their digits. No > further information is available. > > We assume that a real number p can be distinguished from a set Q of > real numbers q by general considerations, for instance, if p is a > transcendental number and Q consists of rational numbers q only. > > Of course it would be impossible to distinguish p from all q, because > for every digit d_n of p, there is a number q that shares all digits > up to d_n with p. Wait, so you're arguing that you can't distinguish 1/pi (roughly 0.31830988...) from a set containing all rationals in [0, 1] because, for example, 3/10 is equal to 1/pi to the first decimal place, 31/100 is equal to the second, and so on? What about the non-zero difference between any one rational number and 1/pi? |1/pi - 3/10| is greater than the rational 1/50. |1/pi - 31/100| is greater than the rational 1/125. In fact, for each rational q in [0, 1], 1/pi differs from q by an amount greater than some other rational r in [0, 1], so we can distinguish 1/pi from every rational in [0, 1]. -o === Subject: Re: Answer to Dik T. Winter > We agree that in Cantor's diagonal argument, applied to real numbers, > the numbers are represented and identified solely by their digits. No > further information is available. > > We assume that a real number p can be distinguished from a set Q of > real numbers q by general considerations, for instance, if p is a > transcendental number and Q consists of rational numbers q only. > > Of course it would be impossible to distinguish p from all q, because > for every digit d_n of p, there is a number q that shares all digits > up to d_n with p. > > It is possible to construct the complete binary tree from the numbers > in Q. That means, it is possible to construct all possible digit > sequences from the numbers in Q. It is also possible to construct the > complete binary tree from the set Tp that consists of all termintaing > rationals which are appended by a tail consisting of p. Therefore it > is impossible to distinguish p from the binary tree. But the binary > tree has been errected by using a countable set of paths only. That ignores the fact that the result of such building automatically contains paths not required in the construction itself. > Therefore it is impossible to distinguish, by means of digits, any > real number from a countable set of paths. Not in ZF. > > Conclusion: Cantor's diagonal argument is contradicted. Wrong outside of WM's world of MathUnrealism. > > Comment: This is not surprising, because it is impossible to identify > any irrational number by means of its decimal representation, but just > this is claimed by the diagonal argument. Not at all. The diagonal argument, as applied to infinitely long strings of decimal digits, only says that any listing of such strings is incomplete. Which it is. -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I Your claim is that no possibility exists to construct or to distinguish by one or many or infinitely many nodes of the tree another path. You claim you do not contradict this by using Wolkenmuekenheim logic. Assume A: I agree that B is true I agree that C is true I do not agree that (B and C) is true. - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. Choose a path and try it. I then will tell you whether you are right. === Subject: Re: Answer to Dik T. Winter > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > > Choose a path and try it. I then will tell you whether you are right. > O.K., I have my path. If you think you guessed it, tell me what you think it is. And I will tell you whether you are right. Or wrong! -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I Your claim is that no possibility exists to construct or to distinguish by one or many or infinitely many nodes of the tree another path. You claim you do not contradict this by using Wolkenmuekenheim logic. Assume A: I agree that B is true I agree that C is true I do not agree that (B and C) is true. - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. snip attempt to evade the topic. Topic restated and elaborated: We agree that in Cantor's diagonal argument, applied to real numbers, the numbers are represented and identified solely by their digits. No further information is available. We assume that a real number p can be distinguished from all elements of a set Q of real numbers q by general considerations, for instance, if p is a transcendental number and Q consists of rational numbers q only. (This implies the existence of such a number p, but does not imply the existence of a decimal expansion of p.) Of course it would be impossible to distinguish p from all q, because for every digit d_n of p, there is a number q that shares all digits up to d_n with p. It is possible to construct the complete binary tree from the numbers in Q. That means, it is possible to construct all possible digit sequences from the numbers in Q. It is also possible to construct the complete binary tree from the set Tp that consists of all termintaing rationals which are appended by a tail consisting of p. Therefore it is impossible to distinguish p from the binary tree. But the binary tree has been errected by using a countable set of paths only. Therefore it is impossible to distinguish, by means of digits, any real number from all elements a countable set of paths. Conclusion: Cantor's diagonal argument is contradicted. Comment: This is not surprising, because it is impossible to identify any irrational number by means of its decimal representation, but just this is claimed by the diagonal argument. === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > You have agreed to Assuming infinite paths exist: The path p is in the tree The path p can be distinguished from every element of P. You refuse to agree to Assuming infinite paths exist: There is a path p in the tree that can be distinguished from every element of P. Until you explain this, no other argument as to why you refuse to accept this statment will be considered. - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Assuming infinite paths exist: The path p is in the tree The path p can be distinguished from > every element of P. For every path p n of P there is a digit such that p differs from p n. But there is no digit of p that differs from every path p n of P because, if p exists as binary representation, then p is the union of all p n and as such cannot differ from the union. You refuse to agree to Assuming infinite paths exist: There is a path p in the tree that can > be distinguished from every element of P. Until you explain this, no other argument as > to why you refuse to accept this statment > will be considered. It is obviously impossible to distinguish p from all paths p n that are in the binary tree after construction has been completed. If you don't believe me, then play the game. Minimum stake 10^6 dollars or (if you come from Europe) Euros or Pounds. === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > Your claim is that no possibility exists to construct or to distinguish by one or many or infinitely many nodes of the tree another path. > Assuming infinite paths exist: > The path p is in the tree > The path p can be distinguished from > every element of P. For every path p_n of P there is a digit such that p differs from p_n. > But there is no digit of p that differs from every path p_n of P > because, if p exists as binary representation, then p is the union of > all p_n and as such cannot differ from the union. > The statement says nothing about p being distinguished from the union of all p_n, the statment says that p can be distinguished from every *element* of the union of all p_n. Do you wish to repudiate your repeated agreement to The path p can be distinguished from every element of P ? > You refuse to agree to > Assuming infinite paths exist: > There is a path p in the tree that can > be distinguished from every element of P. > Until you explain this, no other argument as > to why you refuse to accept this statment > will be considered. It is obviously impossible to distinguish p from all paths p_n that > are in the binary tree after construction has been completed. > Indeed. However, the statement is not about distinguishing p from paths in the tree, it is about distinguishing p from paths in P. - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Your claim is that no possibility exists to construct or to >distinguish by one or many or infinitely many nodes >of the tree another path. > Assuming infinite paths exist: > The path p is in the tree > The path p can be distinguished from > every element of P. > For every path p n of P there is a digit such that p differs from p n. > But there is no digit of p that differs from every path p n of P > because, if p exists as binary representation, then p is the union of > all p n and as such cannot differ from the union. The statement says nothing about p being distinguished from the union > of all p n, the statment says that p can be distinguished from every > *element* > of the union of all p n. Do you wish to repudiate your repeated > agreement to The path p can be distinguished >from every element of P ? No. The union of paths is not larger than every element - unless we agree to magic. In particular p cannot be distinguished from the paths of P, used to construct the tree, if I had included p in P. I agree that p was not included. But that does not play a role unless you are able to obtain that from the complete tree. But you are not! > You refuse to agree to > Assuming infinite paths exist: > There is a path p in the tree that can > be distinguished from every element of P. > Until you explain this, no other argument as > to why you refuse to accept this statment > will be considered. > It is obviously impossible to distinguish p from all paths p n that > are in the binary tree after construction has been completed. Indeed.However, the statement is not about > distinguishing p from paths in the > tree, it is about distinguishing p from paths in P. If this was possible before, then it would also be possible post festum, because only P was used to construct the tree. But it is not. Therefore it turns out that the original assumption p can be distuinguished from every path of P is contradicted. A classical proof by conradiction. By the way, one of the finest proofs that exists in mathematics. === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I Your claim is that no possibility exists to construct or to distinguish by one or many or infinitely many nodes of the tree another path. Your answer to Do you wish to repudiate your repeated agreement to The path p can be distinguished from every element of P. is No. Then you claim p cannot be distinguished from the paths of P, used to construct the tree, You agree to the stamentemt The path p is in the tree. then you say I agree that p was not included. But that does not play a role unless you are able to obtain that from the complete tree. But you are not! You are becoming incoherent. - William Hughes === Subject: Re: Answer to Dik T. Winter > > You should give up. There is no attempt to change P. There is a proof > that you cannot distinguish p from the set of paths P that has been > used toconstruct the tree. > Note that you have agreed > the binary tree contains a path > p that can be distinguished from > every element of P. > If I have agreed, then I was in error. > OK.I will give you the list of things you > agreed with.Point out where you made your error. > There is a subset of nodes that cannot be found in a single > element of P. > > That is correct --- iff actually infinitely long sequences of nodes > exist! WM has already said that HE can construct a tree with infinitely infinitely long sequences of nodes in it. So they must exist at least for him. But now he does not want to allow anyone else to be able touse them? > And that is the assumption we start off with (and that we > disprove). WM's perpetual claims of his own proofs and disproofs have yet to refer to any actual proof or disproof that is either mathematically or logically valid. > Consider the set P of terminating paths and p = 1/pi. The > nodes of 1/pi are not in a single element of P. > A subset of nodes is distinguished from an element > q of P if and only if it is not contained in q. > > 1/pi is not contained in any element of P --- if 1/pi exists! > A subset of nodes is distinguished from every > element of P if and only if it is not contained in > a single element of P. > > Same as above. No error yet. Given a maximal infinite binary tree and any finite or countably infinite set of subsets of its node set then there will be paths as node sets not members of that given set. > There is a subset of nodes > which forms a path, call it p, > that is not contained in a single element of P. > > No error detected so far. > The path p is distinguished from every element >of P. > > Ok. > All of the nodes of path p are in the tree. > > That is correct because the tree contains all subsets of nodes. > > All information that is available about a number in binary > representation is in (one path of) the tree. If I present you the > complete tree without saying how I constructed it, then you have > exactly the same information that can be applied in Cantor's list > argument. If you, or anyone else, claims that they can count (list) all the paths in that tree, then the Cantor diagonal argument refutes you all. > > You cannot distinguish your favourite path from the tree constructed > by means of a countable set of paths. Sure I can! At least in binary trees with more than two nodes. The first node of a path in such a tree has at most one child whereas in the tree it has two. That counts as distinguishing a path from a tree. > > What's now? Play the game in order to prove that? > > Thou shalt not belittle my binary tree! Your form of allegedly maximal infinite binary tree belittles itself by being too little to be a maximal infinite binary tree. -- Virgil === Subject: Re: Answer to Dik T. Winter > > On 11 Jun., 23:06, William Hughes > > On 11 Jun., 21:33, William Hughes > Your claim is that no possibility exists to construct or to > distinguish by one or many or infinitely many nodes > of the tree another path. > after p has been chosen > You should give up. There is no attempt to change P. There is a proof > that you cannot distinguish p from the set of paths P that has been > used to construct the tree. If that list of paths is finite or only countably infinite, then its anti-diagonal is not in it. So that unless WM's set of paths is uncountable, one can, quite mechanically, find a non-member path. > Note that you have agreed > the binary tree contains a path > p that can be distinguished from > every element of P. > If I have agreed, then I was in error. The binary tree contains no > path that can be distinguished from the countable set P that has been > used to construct the tree. If P is countable then for any counting of its members there is an anti-diagonal path NOT included in P. > Or let me put is this way: Every path that > is present in the tree belongs to a countable set that could have been > unsed to construct the tree. Which is, as usual, not the same thing at all. Every path is a member of many countable sets that could be used to construct the tree by WM's method. But that toes not imply that there is any one countable set which contains all these paths. And for every countable set roving it to be countable provides a direct method for proving it incomplete. > > Wanna raise the stakes? Whatever stakes you like, since as soon as your list is proved countable it is also proved incomplete. -- Virgil === Subject: Re: Answer to Dik T. Winter <79edo9F1dtelcU1@mid.individual.net> posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > My proof rests upon the fact that after the set of all finite paths of > the form 0.1, 0.11, 0.111, ... has been constructed, there is no > chance to construct the path 0.111... in addition. > Why not start with path 0.111... and then add those other paths > 0.1, 0.11, 0.111, ...? > [Many lines snipped, because I felt they didn't get to the point I > was trying to ask] > However, your contribution to this problem is of high value and I > appreciate it very much. Didn't you say: If I am going to enumerate the real numbers > in such and such a way then there will be some of them not > in my list? It is not enumerating all real numbers. What I do is to use actually infinite paths (at the end we will see that they do not exist, but I assume their existence) for constructing the complete binary tree. As a simple example I use only such paths that have tails of the form 000..., i.e., terminating paths. The result is: They exhaust the complete infinite binary tree. There is no node that remains uncovered. And there remains no node and no set of nodes that could be used to construct a path that is not yet in the tree. The tree is complete at the end. With the tip of your finger you can follow every path, even 1/pi or 1/3. But then, that doesn't prove all too much, because it's obvious > that there are many ways of not counting the reals. > The point is that there is no way of counting the reals. As I said, I do not count the reals. I cover all possible infinite sequences of bits by means of a countable set of paths = a countable set of binary sequences. And your idea to start with some paths like 0.111... or 1/pi that do not belong to the countable set (of terminating paths, i.e., such with tails 000...) that I use for construction, shows us, that the result is in no way different. But how can we explain that the result is not influenced by those paths? In my opinion by recognizing that such paths do not exist. === Subject: Re: Answer to Dik T. Winter > My proof rests upon the fact that after the set of all finite paths of > the form 0.1, 0.11, 0.111, ... has been constructed, there is no > chance to construct the path 0.111... in addition. > Why not start with path 0.111... and then add those other paths > 0.1, 0.11, 0.111, ...? > [Many lines snipped, because I felt they didn't get to the point I > was trying to ask] > However, your contribution to this problem is of high value and I > appreciate it very much. > Didn't you say: If I am going to enumerate the real numbers > in such and such a way then there will be some of them not > in my list? > > It is not enumerating all real numbers. > What I do is to use actually infinite paths (at the end we will see > that they do not exist, but I assume their existence) for constructing > the complete binary tree. As a simple example I use only such paths > that have tails of the form 000..., i.e., terminating paths. > > The result is: They exhaust the complete infinite binary tree. There > is no node that remains uncovered. And there remains no node and no > set of nodes that could be used to construct a path that is not yet in > the tree. The tree is complete at the end. With the tip of your finger > you can follow every path, even 1/pi or 1/3. While there is no node uncovered there are maximal totally ordered sets of nodes under the order induced by the the parent of relation (paths) which have not been accounted for. Namely all those which have infnitely many 1's in there binary representation. > But then, that doesn't prove all too much, because it's obvious > that there are many ways of not counting the reals. > The point is that there is no way of counting the reals. > > As I said, I do not count the reals. Because you can't. > I cover all possible infinite > sequences of bits by means of a countable set of paths = a countable > set of binary sequences. You have deliberately overlooked every sequence of bits which contains as many 1 bits as 0 bits. WHich is most of them. > > And your idea to start with some paths like 0.111... or 1/pi that do > not belong to the countable set (of terminating paths, i.e., such > with tails 000...) that I use for construction, shows us, that the > result is in no way different. You have deliberately overlooked every sequence of bits which contains as many 1 bits as 0 bits. WHich is most of them > But how can we explain that the result > is not influenced by those paths? In my opinion by recognizing that > such paths do not exist. So a path of infinitely many 0's does exist but a path of infinitely many 1's does not? Then your tree is not maximal, and omits some necessary nodes. -- Virgil === Subject: Re: Answer to Dik T. Winter > But how can we explain that the result > is not influenced by those paths? In my opinion by > recognizing that such paths do not exist. result ... not influenced by those paths? I disagree. Without those paths (like 0.111...) you are able tu enumerate, but taking care of all of them brings you into trouble. It's *them* which are the difficult part. Listing the finite subsets of the natural numbers is easy: just sort by their number of elements and within the sets of equal size order the sets lexicographical. Those sets influencing the result, I am not forced to disbelieve in the existence of them. Rainer Rosenthal r.rosenthal@web.de === Subject: Re: Answer to Dik T. Winter posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > My proof rests upon the fact that after the set of all finite paths of > the form 0.1, 0.11, 0.111, ... has been constructed, there is no > chance to construct the path 0.111... in addition. > Why not start with path 0.111... and then add those other paths > 0.1, 0.11, 0.111, ...? > [Many lines snipped, because I felt they didn't get to the point I > was trying to ask] > However, your contribution to this problem is of high value and I > appreciate it very much. > Didn't you say: If I am going to enumerate the real numbers > in such and such a way then there will be some of them not > in my list? It is not enumerating all real numbers. > What I do is to use actually infinite paths (at the end we will see > that they do not exist, but I assume their existence) for constructing > the complete binary tree. As a simple example I use only such paths > that have tails of the form 000..., i.e., terminating paths. The result is: They exhaust the complete infinite binary tree. There > is no node that remains uncovered. And there remains no node and no > set of nodes that could be used to construct a path that is not yet in > the tree. The tree is complete at the end. With the tip of your finger > you can follow every path, even 1/pi or 1/3. > But then, that doesn't prove all too much, because it's obvious > that there are many ways of not counting the reals. > The point is that there is no way of counting the reals. As I said, I do not count the reals. I cover all possible infinite > sequences of bits by means of a countable set of paths = a countable > set of binary sequences. And your idea to start with some paths like 0.111... or 1/pi that do > not belong to the countable set (of terminating paths, i.e., such > with tails 000...) that I use for construction, shows us, that the > result is in no way different. But how can we explain that the result > is not influenced by those paths? In my opinion by recognizing that > such paths do not exist. > - Show quoted text - Welcome to the real mathematics and not the fake Matrix movie sorry excuse for chess you've been playing for the last 20 years. Musatov (Founder, Inverse 19 Mathematics) Not the only, but one without it would not exist. QED Efforts by mathematicians to determine the value of pi since have led to the conclusion that it ... Unlike rational numbers, pi cannot be represented by a common fraction, .... But what about an uncommon fraction? Will you look at my drawing and in particular how the twelve points are the outside points of the sphere are all rationally reached by decimal numbers? 6.2035884*2 and doubled an infinite number of times should yield a constant decimal binary expansion (what I mean by this is the numbers to the right of the decimal point should always end in 4, then 8, then 6, then 2, then 4, then 8, then 6, then 2...repeating for infinity). So in binary, by starting with this number we predict for n>2 the infinite binary tree will end with decimal progression: 2, 4, 8, 6 For n<2 the infinite binary tree will begin and end (depends on if you are increasing or decreasing the value) 1, 5, 5, 5 (repeating 5 infinitely) as we decrease to infinitely small. Musatov Inverse Counting Chain of Infinite Proportions (sample) 0.0000473372528076171875 0.000094674505615234375 0.00018934901123046875 0.0003786980224609375 0.000757396044921875 0.00151479208984375 0.0030295841796875 0.006059168359375 0.01211833671875 0.0242366734375 0.048473346875 0.09694669375 0.1938933875 0.387786775 0.77557355 1.5511471 3.1022942 6.2035884 12.4071768 24.8143536 49.6287072 99.2574144 198.5148288 397.0296576 794.0593152 1588.1186304 3176.2372608 6352.4745216 12704.9490432 25409.8980864 50819.7961728 101639.5923456 203279.1846912 406558.3693824 813116.7387648 1626233.4775296 3252466.9550592 6504933.9101184 13009867.8202368 26019735.6404736 52039471.2809472 104078942.5618944 208157885.1237888 416315770.2475776 832631540.4951552 1665263080.9903104 3330526161.9806208 6661052323.9612416 13322104647.9224832 26644209295.8449664 53288418591.6899328 106576837183.3798656 213153674366.7597312 426307348733.5194624 852614697467.0389248 1705229394934.0778496 3410458789868.1556992 6820917579736.3113984 13641835159472.6227968 27283670318945.2455936 54567340637890.4911872 109134681275780.9823744 218269362551561.9647488 436538725103123.9294976 873077450206247.8589952 1746154900412495.7179904 3492309800824991.4359808 6984619601649982.8719616 13969239203299965.7439232 27938478406599931.4878464 55876956813199862.9756928 111753913626399725.9513856 223507827252799451.9027712 447015654505598903.8055424 894031309011197807.6110848 1788062618022395615.2221696 3576125236044791230.4443392 7152250472089582460.8886784 14304500944179164921.7773568 28609001888358329843.5547136 57218003776716659687.1094272 114436007553433319374.2188544 228872015106866638748.4377088 457744030213733277496.8754176 915488060427466554993.7508352 1830976120854933109987.5016704 3661952241709866219975.0033408 7323904483419732439950.0066816 14647808966839464879900.0133632 29295617933678929759800.0267264 58591235867357859519600.0534528 117182471734715719039200.1069056 234364943469431438078400.2138112 468729886938862876156800.4276224 937459773877725752313600.8552448 1874919547755451504627201.7104896 3749839095510903009254403.4209792 7499678191021806018508806.8419584 (sample) If you would like to do the honors Mr. Plouffe, I endear you to calculate the angles and lengths of the number of ways, it is quite obviously possible to square a circle based on the geometry and numerical progression. As is self-evident to anyone with even a modest understanding of mathematics and geometry after visual inspection of the drawn proof (http://www.flickr.com/photos/24869933@N07/3619373444/sizes/l/) it is quite easily accomplished to connect and triangulate the outer points of the rectangles to the outer points of the square and circle and even by dimension sphere. Yes, the drawing clearly by definition of inverse symmetry (take the drawing cut it in half at the equator and turn the paper over and rotate it and compare the top and bottom mirrored symmetry and you will see it is an exact inverse of symmetry itself) the drawing explains how to properly and mathematically soundly represent a fourth dimension on a two dimensional plane. In plane language if you take the picture and separate the top half of the sphere by the bottom half of the sphere they are identical but only if the sphere existed on a complete three dimensional plane through expansion. Since the paper is flat, this is an impossibility, nonetheless it is completely expressed and retractable and countable infinitely. By the definition of our standard we have established any k-digit approximation to pi is rational. Which could have many practical scientific applications. For instance, search: ON THE THEOREM OF IVA$EV-MUSATOV IIIx e F(l) + F(2) + ... + F(p). This remark is the basis of our proof of Theorem .... -y(M + 1), x' + y(M + 1)] = 0 it is easy to check that Conditions (i), ...... Now set i = pi x. *jj, v. Since p, x is positive,. (i) fi is positive, ... plms.oxfordjournals.org/cgi/reprint/s3-53/1/143.pdf REPRESENTATION OF FUNCTIONSBY SERIES AND CLASSES £r(1)with measure E> 2£k -- £` and such that the trigonometric series for g(x) converges uniformly on [0, 2£k]. .... trigonometric series are due to Ivashev-Musatov [40] who has shown that there are null series (in the sense of ..... function £r(£q) e C(0, 1) with £r(0) = £r(1) = 0 such that ...... <*i *plonk* > muh brane i needs one- Hide quoted text - > - Show quoted text - NOTE THE PROGRESSION OF NUMBERS PELLS PROPORTION X^2=3 or 4 = 3 as the numbers progression, WOW if this does not proof this what will 1. Every odd number , number as Y value till infinity is exactly solvable with a proportionate result using the base Pells equation i.e 3,5,7,9,etc etc till infinty and the answer is as in 2(2^2)+1=3^2 and so on till infinity with a whole number integer. 2.Every even number , without exception, using Pells equation, the integer always is a .75 till infinity consistant as 63.75(2^4)=1=4^4 till infinity every integer will end with .75. is that the same proportion as 3 over 4, Y=3 for the least with numbers proportion There is a definite pattern by pells to numbers, and I note that in that pattern the number 19 proportion is strikingly proprtionate, and I clearly surmise that that circluar progression is the same as numbers for vector 19 progression. === === Subject: Re: intrinsic or geodesic metric > Let (S,d) be a path connected metric space and p a path within S. > The following definitions are made. > (x0, x1,.. x_n) is partition (of [0,1]) when > x0 = 0, x_n = 1, for j = 0,.. n-1, xj < x_(j+1). > P_n = (0, 1/n,.. (n-1)/n, 1) is a partition. > s(x1,.. x_n) = s_p(x1,.. x_n) = sum_(j=0,.. n-1) d(p(xj), p(x_(j+1))) > length path p, len(p) = sup{ s(P) | P partition } > Some theorems ensue. > { s(P) | P partition } bounded iff sup{ s(P_n) | n in N } bounded > bounded { s(P) | P partition } ==> len(p) = sup{ s(P_n) | n in N } > -- > Define p:SxS -> R, (x,y) -> inf{ len(p) | p path from x to y } > p is called the intrinsic or geodesic metric for S. > How is this a metric? > Some paths within R^2 have unbounded length. > Yet this isn't a problem as other paths between the same two > paths have finite length. However for the subspace > cl{ (x, x.sin 1/x | 0 < x } of R^2, there is only one > path from (0,0) to (1,sin 1) and it is unbounded. > How is this usually handled? > By requiring every two points have a path of finite length between them? > By stretching the definition of metric to a codomain of extended reals? Who knows? Those both seem reasonable - you'd have to ask > whoever gave the definition. (Also you should note the _context_; > it could be that there always _is_ a path of finite length in the > spaces under consideration.) > Wikipedia sets p(x,y) = oo when all paths from x to y don't have finite length and does not say anything about altering the definition of metric, to fit that inconvient fact. > By defining p(x,y) > = arctan inf{ len(p) | p path from x to y } > when some path from x to y with finite length > = 1 > when no path from x to y with finite length? > Any comments about this third possiblity? > To show p is a metric, it's necessary to show > inf{ len(p) | p path from x to y } = 0 implies x = y. > Any suggestions how that's shown? It's immediate from the definition that if p is a path > from x to y then d(x,y) <= len(p). > Duh. Now to show p(a,b) = p(b,a). inf{ len(p) | p path from a to b } = inf{ len(q) | q path from b to a } First show p path from a to b ==> some q path from b to a with len(p) = len(q) That is { len(p) | p path from a to b } subset { len(q) | q path from b to a } To show that let q(r) = p'(1-r). q path from b to a. s_p(P_m) = sum(j=0,.. m-1) d(p(j/m), p((j+1)/m) = sum(j=0,.. m-1) d(p((m-j)/m), p((m-j-1)/m) = sum(j=0,.. m-1) d(p((1-j/m), p((1-(j+1)/m) = sum(j=0,.. m-1) d(q(j/m), q((j+1)/m)) = s_q(P_m) Thus len(p) = len(p') and the rest falls into place. Transitivity requires bigger lemmas. I think len(p+q) = len(p) + len(q) will suffice, where p+q is path made with the end of one path being joined to the beinging of the other path. === Subject: Re: PLEASE DO NOT FEED THE TROLL posting-account=rsfFvAoAAADqHCqmunbjtmaKjfuP-pwP Trident/4.0; GTB6; Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; InfoPath.1; .NET CLR 1.1.4322; .NET CLR 3.5.30729; .NET CLR 3.0.30618; OfficeLiveConnector.1.3; OfficeLivePatch.0.0),gzip(gfe),gzip(gfe) On 13 June, 10:55, PLEASE DO NOT FEED THE TROLL > < off-topic, racist drivel deleted > ------------------------------------------------- * Christina Downs, 24, of Portsmouth, N.H., mounted a full-blown > defense to the speeding ticket (44 mph in a 25 mph zone) she > received in 2007 (even though the officer said Downs had > arrogantly sped off again immediately afterward and had to be > stopped a second time).Acting as her own lawyer, Downs filed > motions and at a trial, put the officer through a meticulous, 96- > point cross-examination about such matters as work schedule, > training, engineering studies of road speeds, radar technology, > weather conditions, traffic flow, and the use of a tuning fork to > calibrate the radar device.The judge ruled against her, and in > October 2008, the state Supreme Court ordered her to pay the $100 > ticket. [Portsmouth Herald, 10-9-08] Downs may well be part of a plot to to add further to the legal coffers, lengthen hearings and expand waist lines. Anyway, years ago, an astronomer, might have been the delightful Fred Hoyle, was pulled over for speeding evidenced by returns of some sort of primitive 'radar gun'. Hoyle, if it were he, explained to the court that a correct interpretation of the evidence entailed that he had been travelling at (£ - joke possible) 4,700mph, imperiously dismissing the curious device. One supposes that he stopped for a coffee from time to time. Hoyle, if, again, it were he, won, and one hopes the court got a taste for rigorous forensic thought! I don't remember the exact details, but I do remember laughing a lot! Nor, of course, should one attack the officer concerned who was doubtless a good workman given bad tools! -- UncleEnglish. PS: We all need a good medicinal laugh, though oine CAN get too cross! === Subject: Derivatives and Limits let f:(a,b) -> R a differentiable function such that there exists (finite or infinite) the limit (from the left) of f'(x) as x approaches b. Then, I suppose, that there exists also the limit of f(x) as x approaches b. Indeed, if lim_{x -> b} f'(x) > 0 then f is increasing in a neighbourhood of b, so it has a limit in b. Analogous reasoning if lim_{x -> b} f'(x) < 0 So, the only non trivial case is lim_{x -> b} f'(x) = 0 I sketched a proof (using Lagrange's Theorem), but it's awful. Do you have a simple proof for this case? Maury Barbato === Subject: Re: Derivatives and Limits > >let f:(a,b) -> R a differentiable function such that >there exists (finite or infinite) the limit (from > the >left) of f'(x) as x approaches b. >Then, I suppose, that there exists also the limit >of f(x) as x approaches b. >Indeed, if >lim_{x -> b} f'(x) > 0 >then f is increasing in a neighbourhood of b, so >it has a limit in b. Analogous reasoning if >lim_{x -> b} f'(x) < 0 >So, the only non trivial case is >lim_{x -> b} f'(x) = 0 >I sketched a proof (using Lagrange's Theorem), but >it's awful. Do you have a simple proof for this > case? > > Is Lagrange's Theorem the Mean Value Theorem? > > This is not hard - I don't know whether you consider > what's below awful: > > We need only show that limf(x_n) exists for any > sequence x_n which increases to b. So we need > only show that (f(x_n)) is a Cauchy sequence. > > Given epsilon > 0, choose delta > 0 so that > |f'(x)| < epsilon/(b-a) for all x in (b-delta, b). > Choose N so x_n > b-delta for all n > N. > If n, m > N then MVT shows that |f(x_n) - f(x_m)| <= |x_n - x_m| epsilon/(b-a) > > < epsilon. > > > >Maury Barbato > > David C. Ullrich > > Understanding Godel isn't about following his formal > proof. > That would make a mockery of everything Godel was up > to. > (John Jones, My talk about Godel to the post-grads. > in sci.logic.) Ops, I practically added, in my reasoning, the proof that every Cauchy sequence is convergent ... hmm, my brain is getting rusty for having studied too little math and too much law lately! Anyhow, in Italian books about Calculus (which is called Analisi Matematica), the Mean value Theorem is called Lagrange's Theorem ... considering that Lagrange discovered a lot of theorems, I think it's better to call it the Mean Value Theorem! Maury Barbato === Subject: Re: Derivatives and Limits > let f:(a,b) -> R a differentiable function such that > there exists (finite or infinite) the limit (from the > left) of f'(x) as x approaches b. > Then, I suppose, that there exists also the limit > of f(x) as x approaches b. > Indeed, if > > lim_{x -> b} f'(x) > 0 > > then f is increasing in a neighbourhood of b, so > it has a limit in b. Analogous reasoning if > > lim_{x -> b} f'(x) < 0 > > So, the only non trivial case is > > lim_{x -> b} f'(x) = 0 > > I sketched a proof (using Lagrange's Theorem), but > it's awful. Do you have a simple proof for this case? I haven't checked this carefully, but it looks like it works: Since lim(x->b) f'(x) = 0, there must be a c such that |f'(x)| < 1 for x |f(x)-f(y)| < |x-y| for x, y in (c, b). It then follows that for any sequence {d_n} converging to b, the sequence {f(d_n)} is a Cauchy sequence, hence convergent. That would mean that lim(x->b) f(x) not only exists, but is finite. What do you think? -- Niels Diepeveen === === Subject: Re: Yet another funny bug in Maple 13 > Enjoy. > > int(1/z^(1/(sqrt(2)*(cos(Pi/12)-sin(Pi/12)))), z= 0..infinity); > int(1/z^(1/(sqrt(2)*(cos(Pi/12)-sin(Pi/12)))), z= 1..infinity); > > 0 > 0 > > Better still, > > evalf(int(1/z^(1/(sqrt(2)*(cos(Pi/12)-sin(Pi/12)))), z= 2..infinity)); > -.2500000000 Another example of testing only incomplete and hastily: The first bug at least involves int(z^c,z) = z^(c+1)/(c+1) for c <> 1 and int(1/z,z) = ln(z), which is a well known issue. So cross-check on numerical examples and even some variable integration bounds would show it. You are trying to hide it. The last one is simple lame testing: in this case you not integrate numerical, but evaluate the symbolic result. As you work with standard Digits=10 rounding off gives that, but using higher precision would show other results. So again you are hiding the actual problem and do only an incomplete testing. Surely you know that. And I hope you could do better. === Subject: Re: Yet another funny bug in Maple 13 posting-account=uD9kfgoAAABaqjCF8ol-EFTFI3g2PjhE Gecko/2008120122 Firefox/3.0.5,gzip(gfe),gzip(gfe) > Even better. Maple 13 cannot calculate int(1/z, z=1..2); if it is written as int(1/z^(sqrt(2)*(cos(Pi/12)-sin(Pi/12))), z= 1..2); while a not-totally-stupid math freshman knows it is just > ln(2)... But Maple 12 too is unable to compute it! I wonder why your cyber machine did not detect this earlier. === Subject: Re: Yet another funny bug in Maple 13 posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Even better. > Maple 13 cannot calculate > int(1/z, z=1..2); > if it is written as > int(1/z^(sqrt(2)*(cos(Pi/12)-sin(Pi/12))), z= 1..2); > while a not-totally-stupid math freshman knows it is just > ln(2)... > But Maple 12 too is unable to compute it! I wonder why your cyber > machine did not detect this earlier. Dammit, who told you the VM machine did not detect it earlier, > years ago? Do you realize that this bug exists for 10+ years?- Hide quoted text - - Show quoted text - Musatov * Bondarenko | Riemann Hypothesis / Matlab Seed Proof ...May > - PolyGamma[0, 3/2 + I]> - PolyGamma[0 ... http://mathforum.org/kb/thread.jspa?messageID=6727340&tstart=0 Discussions - comp.theory | Google Groups Musatov * Bondarenko | Riemann Hypothesis / Matlab Seed Proof. Search: Login 1. Create Account 2. Login 1. File Exchange 2. Newsgroup 3. Link Exchange 4. ... Math Forum Discussions - User Profile for: marty.musato @ mail.com15), http://mathforum.org/kb/search!execute.jspa?userID=577254&forceEmptySearch=t rue Discussions - sci.math | Google Groups An innovator called Musatov wants to bolster Sci-math, 6 new of 6, Vladimir Bondarenko (5 authors), 5:35am. intrinsic or geodesic metric, 3 new of 3 ... Math Forum Discussions Jun 13, 2009 ... Vladimir Bondarenko Posts: 4478. Registered: 12/4/04. Re: An schizophrenic called Musatov wants destroy Sci-math ... http://mathforum.org/kb/message.jspa?messageID=6750433&tstart=75 - 4 hours ago Saifee Durbar: His Life, His Accomplishments, His Dreams - soc ...May Martin Musatov P=NP Freedom Fighter fighting in Panama with the rebels since 1993. ... Math Forum Discussions Re: An ignorant cruel person Ludovicus wants Bondarenko. 6/13/09 ... http://mathforum.org/kb/message.jspa?messageID=6749940&tstart=75 - 4 hours ago Re: If Cantor set is uncountable and measure 0, then (Cantor)^c ...me (2)Inbox Musatov * Bondarenko | Riemann Hypothesis / Matlab Seed Proof May 29 me (2)Inbox Re: WTF Is Going On Here?! May 29 ... http://sci.tech-archive.net/Archive/sci.math/2009-06/msg01450.html Math Forum Discussions Jun 13, 2009 ... Re: An Innovator called Vladimir Bondarenko. 6/13/09 ... http://mathforum.org/kb/message.jspa?messageID=6750719&tstart=75 - 4 hours ago === === === Subject: IGNORANT MISTAKE IN WIKKEPEDIA ON RADIAN. POPPYCOCK! If your mathematics is based on such ignorance statements, then your mathematics need to be recreated. First of all a degree was not created by a mathematician , that reflects the arrogance of the current mathematics. There is a reason that the numbers convert to 2 at 19, think. 19 =1 by proprtion and a dimension less than 2 the least number over 1 PLUS THIS MATHEMATICA REASON === Subject: Re: IGNORANT MISTAKE IN WIKKEPEDIA ON RADIAN. POPPYCOCK! inverse 19 says... The wikkepedia on Radian Pi states this Perhaps the degree was originally >there are about 360 days in the year If your mathematics is based on such ignorance statements, then your >mathematics need to be recreated. First of all a degree was not created by a >mathematician , that reflects the arrogance of the current mathematics. There >is a reason that the numbers convert to 2 at 19, think. 19 =1 by proprtion and >a dimension less than 2 the least number over 1 > PLUS THIS MATHEMATICA REASON Tragically, the OP was cut down by assassins of the Illuminati before he could reveal the secret reason for 360 degrees in a circle. -- Daryl McCullough Ithaca, NY === Subject: Re: IGNORANT MISTAKE IN WIKKEPEDIA ON RADIAN. POPPYCOCK! PLUS THE MATHEMATICAL REASON(I cut off in the post by error). Correct this Poppy cock in your Wikkepedia 1.The inverse of infinity space is a mathematically a vector 19 , proved seperatly. 2. If you need to convert any linear into a perfect curve and advance to a circle , the exact angle you do it is 19 degrees and repeat it 19 times. It is hard to explain this to you mathematicians trained by the current dictum but I will try, that if there was a motorcyclist going at a T maximum speed, absolutly linear , he will be travelling at 2 Pi split ratio, i.e a 22/7 to each side of him to make it zero Pi, he will need to take 19 , 19 degree corrections(total 2Pi at one pi constant correction per 19), to make a circle, but there is a final 1/360 correction ie 0.0027 to wipe out the cones or smooth the edges, correction each degree So there are exactly 360 degrees. 3 Pells is clear, it has been proven that by us you take away one(360 times.0027=1), when you square a linear value to make a circle(1/360 Cameron constant) using the least square as a constant till infinity-- 90 (2^2) +1C= 19^2 or 360 +1= 361. 4 NOTE.Radian value of 57.295 divided by 3.14 =18.24 +.75=19 , these 0.75 as in 2Pi^2-0.75= 19inverse constant, are too complex mathematical correction to fully explain in this forum , but it is some thing to do with a linear and curved value square differential , 16 , 19 ,, , i.e 4 and 4.75, which I understand little now. CORRECT THIS POPPYCOCK IN YOUR MATHEMATICAL WIKKEPEDIA, YOU MATHEMATICIANS OF SCI MATH WITH ALL THOSE COMPLEX EQUATIONS SIMPLIFY YOUR BASE FIRST === Subject: Re: IGNORANT MISTAKE IN WIKKEPEDIA ON RADIAN. POPPYCOCK! > If your mathematics is based on such ignorance statements, then your > mathematics need to be recreated. I don't know the first thing about ignorance statements, but there's a group dedicated to recreational mathematics: alt.math.recreational. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === === Subject: Re: Wolfram Alpha claims to be a primary source. <4A1AC625.8070308@cs.berkeley.edu> <4a324910@212.67.96.135> posting-account=0WKfXgkAAAA02cBiVgL6triSGt-e_ueI Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > about the Wolfram Research quality assurance process: DL>My own experience is that, this glitch notwithstanding, > DL>it works quite effectively. This particular statement of Daniel Lichtblau, in general a clever > and responsible person, sounds like personally, I prefer this tie. My tie preferences are no small matter. My experience with the Wolfram Research SQA staff is of course not objective. It is nonetheless, or perhaps for that reason all the more, important. They are tasked with finding the problems in my work that I (sometimes carelessly, sometimes not) managed to miss. > There are figures. As comrade Galilei has it Measure what is measurable, and make measurable what is not so. To remove at once any accusation of my non-objectivity, is there > anyone to see, how many defects are still alive from the chapter > 1 of my draft I reported to Wolfram Research in 2001? http://maple.bug-list.org/chapter1.txt Best wishes, Vladimir Bondarenko Co-founder, CEO, Mathematical Director http://www.cybertester.com/Cyber Tester Ltd. > [...] I can say a bit about that. Our SQA people at that time distilled the report to very nearly 100 bug reports. When last I checked, about 90 per cent had been closed. Some were closed because they were not bugs to begin with. Most actual bugs were fixed. There may have been bugs that did not show actual wrong behavior but simply missing functionality. They would have become suggestion reports. I do not recall whether they are included in that approximately 100 figure or not. It is quite possible that there were other actual bugs in your report, distinct from the 100 or so that became reports. I would not blame the SQA staff for missing such things, as you have a way of burying matters in detail. Daniel Lichtblau Wolfram Research === Subject: Re: Wolfram Alpha claims to be a primary source. <4A1AC625.8070308@cs.berkeley.edu> <4a324910@212.67.96.135> posting-account=PEh90woAAACWN3C9IZwQVmbvngXF6iWJ Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) VB> how many defects are still alive from the chapter VB> 1 of my draft I reported to Wolfram Research in 2001? VB> http://maple.bug-list.org/chapter1.txt DL> I can say a bit about that. Our SQA people at that time distilled the DL> report to very nearly 100 bug reports. When last I checked, about 90 DL> per cent had been closed. Some were closed because they were not bugs DL> to begin with. Most actual bugs were fixed. That is good news. Why was this not communicated earlier by WR? Let me make a proposition. It is said that the Wolfram company, like the Vatican, the CIA and the Great Leader Kim Jong-IL scan Wikipedia looking for (and changing) data they believe not to be in accordance with their self-esteem. Can this web-technology Wolfram developed and the resources spend be redirected to scan two or three newsgroups and automatically (or better by Alpha) provide reports on bugs communicated in these groups? And deliver them on the desk of the SQA people. Maybe I am not the only one who would appreciate such an application of Alpha. === Subject: Re: Wolfram Alpha claims to be a primary source. > Let me make a proposition. It is said that the Wolfram company, > like the Vatican, the CIA and the Great Leader Kim Jong-IL > scan Wikipedia looking for (and changing) data they believe > not to be in accordance with their self-esteem. There is absolutely no doubt that proposition is true. 'JonMcLoone' regularly edits the Mathematica page, and freely acknowledges the fact he is employed by Wolfram Research. I have strong suspicion 'Cloudruns' is employed by Wolfram Research too, based on the fact that virtually all his changes to the Mathematica page are pro-Mathematica and all his changes to other similar programs make them look inferior. 1) Cloudruns added Mathematica to a list of 'Interactive Geometry Software' http://en.wikipedia.org/w/index.php?title=List_of_interactive_geometry_softw are&diff=248007214&oldid=246605110 (someone else has since reversed Cloudruns's edit.) 2) Cloudruns added to the page on 'Comparison of computer algebra systems' in the section describing Mathematica as Ubiquitous system also includes extensive numeric capabilities, statistics, image processing, number theory, boolean computation and is a development environment. http://en.wikipedia.org/w/index.php?title=Comparison_of_computer_algebra_sys tems&diff=prev&oldid=252962774 The page still says Mathematica is Ubiquitous. 3) Cloudruns removes any mention of disadvantages of Mathematica. http://en.wikipedia.org/w/index.php?title=Mathematica&diff=prev&oldid=214434 889 -- I respectfully request that this message is not archived by companies as unscrupulous as 'Experts Exchange' . In case you are unaware, 'Experts Exchange' take questions posted on the web and try to find idiots stupid enough to pay for the answers, which were posted freely by others. They are leeches. === Subject: Re: Wolfram Alpha claims to be a primary source. > Let me make a proposition. It is said that the Wolfram company, > like the Vatican, the CIA and the Great Leader Kim Jong-IL > scan Wikipedia looking for (and changing) data they believe > not to be in accordance with their self-esteem. > > There is absolutely no doubt that proposition is true. You seriously think WRI waste their time in edit fights with children and the unemployed on Wikipedia? LOL. -- Dr Jon D Harrop, Flying Frog Consultancy Ltd. http://www.ffconsultancy.com/?u === Subject: Re: Wolfram Alpha claims to be a primary source. <4a34192c@212.67.96.135> <3LidnfHygs0316nXnZ2dnUVZ8nqdnZ2d@brightview.co.uk> posting-account=PEh90woAAACWN3C9IZwQVmbvngXF6iWJ Gecko/2009060215 Firefox/3.0.11 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Let me make a proposition. It is said that the Wolfram company, > like the Vatican, the CIA and the Great Leader Kim Jong-IL > scan Wikipedia looking for (and changing) data they believe > not to be in accordance with their self-esteem. > There is absolutely no doubt that proposition is true. You seriously think WRI waste their time in edit fights Enter Wolfram Research in http://katrina.cs.caltech.edu/erenrich_rnd345/scanner_final/ > with children and the unemployed on Wikipedia? LOL. === Subject: Re: Wolfram Alpha claims to be a primary source. > Let me make a proposition. It is said that the Wolfram company, > like the Vatican, the CIA and the Great Leader Kim Jong-IL > scan Wikipedia looking for (and changing) data they believe > not to be in accordance with their self-esteem. > There is absolutely no doubt that proposition is true. > You seriously think WRI waste their time in edit fights > > Enter Wolfram Research in > http://katrina.cs.caltech.edu/erenrich_rnd345/scanner_final/ I get: Warning: mysql_connect() [function.mysql-connect]: Too many connections in /var/www/html/erenrich_rnd345/scanner_final/_utilities/db_auth.php on line 4 Database error. Please try again later. -- Dr Jon D Harrop, Flying Frog Consultancy Ltd. http://www.ffconsultancy.com/?u === Subject: Re: Wolfram Alpha claims to be a primary source. <4a34192c@212.67.96.135> <3LidnfHygs0316nXnZ2dnUVZ8nqdnZ2d@brightview.co.uk> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 8575489, 305093] NotebookOptionsPosition[ 8532371, 303700] NotebookOutlinePosition[ 8532954, 303723] CellTagsIndexPosition[ 8532911, 303720] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell[Riemann Zeta Function, Title], Cell[CellGroupData[{ Cell[Author, Subsection], Cell[< Eric W. 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.85019 .59183 .85436 .59067 .85266 .58852 Mtetra .85266 .58852 .85436 .59067 .85852 .58952 .85683 .58736 Mtetra .85683 .58736 .85852 .58952 .8627 .58836 .86101 .5862 Mtetra .86101 .5862 .8627 .58836 .86688 .58719 .86519 .58503 Mtetra .86519 .58503 .86688 .58719 .87106 .58603 .86938 .58386 Mtetra .86938 .58386 .87106 .58603 .87525 .58486 .87358 .5827 Mtetra .87358 .5827 .87525 .58486 .87945 .5837 .87778 .58152 Mtetra .654 .753 .922 r .87778 .58152 .87945 .5837 .88366 .58252 .88199 .58035 Mtetra .88199 .58035 .88366 .58252 .88787 .58135 .8862 .57918 Mtetra .8862 .57918 .88787 .58135 .89208 .58018 .89042 .578 Mtetra .89042 .578 .89208 .58018 .8963 .579 .89465 .57682 Mtetra .89465 .57682 .8963 .579 .90053 .57782 .89888 .57564 Mtetra .89888 .57564 .90053 .57782 .90476 .57664 .90312 .57445 Mtetra .90312 .57445 .90476 .57664 .909 .57546 .90737 .57327 Mtetra .90737 .57327 .909 .57546 .91325 .57428 .91162 .57208 Mtetra .91162 .57208 .91325 .57428 .9175 .57309 .91587 .57089 Mtetra .91587 .57089 .9175 .57309 .92176 .5719 .92014 .5697 Mtetra .92014 .5697 .92176 .5719 .92603 .57071 .9244 .56851 Mtetra .9244 .56851 .92603 .57071 .9303 .56952 .92868 .56732 Mtetra .92868 .56732 .9303 .56952 .93457 .56833 .93296 .56612 Mtetra .93296 .56612 .93457 .56833 .93886 .56713 .93725 .56492 Mtetra .93725 .56492 .93886 .56713 .94315 .56594 .94154 .56372 Mtetra .94154 .56372 .94315 .56594 .94744 .56474 .94584 .56252 Mtetra .94584 .56252 .94744 .56474 .95174 .56354 .95015 .56132 Mtetra .95015 .56132 .95174 .56354 .95605 .56233 .95446 .56011 Mtetra .95446 .56011 .95605 .56233 .96036 .56113 .95878 .5589 Mtetra .95878 .5589 .96036 .56113 .96469 .55993 .96311 .55769 Mtetra .96311 .55769 .96469 .55993 .96901 .55872 .96744 .55648 Mtetra .96744 .55648 .96901 .55872 .97335 .55751 .97177 .55527 Mtetra .97177 .55527 .97335 .55751 .97768 .5563 .97612 .55406 Mtetra .97612 .55406 .97768 .5563 .98203 .55508 .98047 .55284 Mtetra .98047 .55284 .98203 .55508 .98638 .55387 .98483 .55162 Mtetra .664 .741 .907 r .44619 .58156 .4482 .5835 .4471 .7959 .44968 .5805 Mtetra .576 .373 .569 r .44968 .5805 .4471 .7959 .45081 .79492 .45318 .57945 Mtetra 0 0 .384 r .45318 .57945 .45081 .79492 .45452 .79395 .45239 .79215 Mtetra .645 .763 .934 r .53933 .76931 .54138 .77117 .544 .55463 .54317 .7683 Mtetra .572 .362 .558 r .54317 .7683 .544 .55463 .54762 .55354 .54702 .76729 Mtetra .237 .102 .485 r .54702 .76729 .54762 .55354 .55124 .55245 .54977 .61382 Mtetra 0 0 .368 r .54977 .61382 .55124 .55245 .55486 .55136 .55294 .54934 Mtetra .937 .564 .217 r .64891 .52037 .65075 .52246 .65515 .53811 .65328 .53513 Mtetra .93 .559 .161 r .65328 .53513 .65515 .53811 .6596 .55374 .6577 .55026 Mtetra .882 .499 .032 r .6577 .55026 .6596 .55374 .66398 .56673 .66207 .56296 Mtetra .788 .403 0 r .66207 .56296 .66398 .56673 .66831 .57747 .66638 .57355 Mtetra .644 .272 0 r .66638 .57355 .66831 .57747 .67258 .58631 .67065 .58234 Mtetra 0 0 .468 r .67065 .58234 .67258 .58631 .67681 .59355 .67488 .5896 Mtetra 0 .033 .604 r .67488 .5896 .67681 .59355 .68101 .59944 .67908 .59556 Mtetra 0 .166 .702 r .67908 .59556 .68101 .59944 .68517 .60419 .68324 .60042 Mtetra .052 .275 .767 r .68324 .60042 .68517 .60419 .68931 .60798 .68738 .60434 Mtetra .17 .36 .81 r .68738 .60434 .68931 .60798 .69342 .61098 .69151 .60747 Mtetra .262 .427 .838 r .69151 .60747 .69342 .61098 .69752 .61332 .69561 .60993 Mtetra .334 .481 .857 r .69561 .60993 .69752 .61332 .7016 .61509 .6997 .61184 Mtetra .392 .523 .87 r .6997 .61184 .7016 .61509 .70567 .6164 .70377 .61327 Mtetra .437 .558 .879 r .70377 .61327 .70567 .6164 .70973 .61732 .70784 .6143 Mtetra .474 .586 .886 r .70784 .6143 .70973 .61732 .71378 .61791 .7119 .615 Mtetra .504 .61 .891 r .7119 .615 .71378 .61791 .71782 .61823 .71595 .61541 Mtetra .529 .63 .896 r .71595 .61541 .71782 .61823 .72186 .61832 .72 .61559 Mtetra .549 .648 .899 r .72 .61559 .72186 .61832 .7259 .61822 .72404 .61557 Mtetra .566 .662 .902 r .72404 .61557 .7259 .61822 .72994 .61796 .72809 .61538 Mtetra .58 .674 .904 r .72809 .61538 .72994 .61796 .73398 .61756 .73213 .61504 Mtetra .591 .685 .907 r .73213 .61504 .73398 .61756 .73801 .61705 .73617 .61459 Mtetra .601 .694 .908 r .73617 .61459 .73801 .61705 .74205 .61645 .74022 .61403 Mtetra .609 .702 .91 r .74022 .61403 .74205 .61645 .74609 .61576 .74426 .61339 Mtetra .616 .709 .912 r .74426 .61339 .74609 .61576 .75013 .61501 .74831 .61267 Mtetra .622 .715 .913 r .74831 .61267 .75013 .61501 .75418 .6142 .75236 .61189 Mtetra .627 .72 .914 r .75236 .61189 .75418 .6142 .75823 .61334 .75642 .61106 Mtetra .631 .725 .915 r .75642 .61106 .75823 .61334 .76228 .61244 .76047 .61018 Mtetra .634 .728 .916 r .76047 .61018 .76228 .61244 .76633 .61151 .76453 .60927 Mtetra .637 .732 .917 r .76453 .60927 .76633 .61151 .77039 .61054 .7686 .60832 Mtetra .64 .735 .917 r .7686 .60832 .77039 .61054 .77446 .60955 .77267 .60734 Mtetra .642 .737 .918 r .77267 .60734 .77446 .60955 .77852 .60853 .77674 .60634 Mtetra .644 .739 .918 r .77674 .60634 .77852 .60853 .7826 .6075 .78082 .60531 Mtetra .645 .741 .919 r .78082 .60531 .7826 .6075 .78668 .60645 .7849 .60427 Mtetra .647 .743 .919 r .7849 .60427 .78668 .60645 .79076 .60539 .78899 .60322 Mtetra .648 .744 .92 r .78899 .60322 .79076 .60539 .79485 .60431 .79308 .60215 Mtetra .649 .745 .92 r .79308 .60215 .79485 .60431 .79894 .60323 .79718 .60106 Mtetra .649 .746 .92 r .79718 .60106 .79894 .60323 .80304 .60213 .80128 .59997 Mtetra .65 .747 .921 r .80128 .59997 .80304 .60213 .80714 .60102 .80539 .59887 Mtetra .651 .748 .921 r .80539 .59887 .80714 .60102 .81125 .59991 .80951 .59776 Mtetra .651 .749 .921 r .80951 .59776 .81125 .59991 .81537 .59879 .81363 .59664 Mtetra .652 .749 .921 r .81363 .59664 .81537 .59879 .81949 .59767 .81775 .59552 Mtetra .652 .75 .921 r .81775 .59552 .81949 .59767 .82361 .59654 .82188 .59439 Mtetra .82188 .59439 .82361 .59654 .82775 .5954 .82602 .59325 Mtetra .653 .751 .921 r .82602 .59325 .82775 .5954 .83188 .59427 .83016 .59211 Mtetra .653 .751 .922 r .83016 .59211 .83188 .59427 .83603 .59312 .83431 .59097 Mtetra .83431 .59097 .83603 .59312 .84018 .59198 .83846 .58982 Mtetra .83846 .58982 .84018 .59198 .84433 .59083 .84262 .58867 Mtetra .653 .752 .922 r .84262 .58867 .84433 .59083 .84849 .58967 .84679 .58751 Mtetra .654 .752 .922 r .84679 .58751 .84849 .58967 .85266 .58852 .85096 .58635 Mtetra .85096 .58635 .85266 .58852 .85683 .58736 .85513 .58519 Mtetra .85513 .58519 .85683 .58736 .86101 .5862 .85932 .58403 Mtetra .85932 .58403 .86101 .5862 .86519 .58503 .86351 .58286 Mtetra .86351 .58286 .86519 .58503 .86938 .58386 .8677 .58169 Mtetra .8677 .58169 .86938 .58386 .87358 .5827 .8719 .58052 Mtetra .8719 .58052 .87358 .5827 .87778 .58152 .87611 .57935 Mtetra .654 .753 .922 r .87611 .57935 .87778 .58152 .88199 .58035 .88032 .57817 Mtetra .88032 .57817 .88199 .58035 .8862 .57918 .88454 .57699 Mtetra .88454 .57699 .8862 .57918 .89042 .578 .88876 .57581 Mtetra .88876 .57581 .89042 .578 .89465 .57682 .89299 .57463 Mtetra .89299 .57463 .89465 .57682 .89888 .57564 .89723 .57345 Mtetra .89723 .57345 .89888 .57564 .90312 .57445 .90147 .57226 Mtetra .90147 .57226 .90312 .57445 .90737 .57327 .90572 .57107 Mtetra .90572 .57107 .90737 .57327 .91162 .57208 .90998 .56988 Mtetra .90998 .56988 .91162 .57208 .91587 .57089 .91424 .56869 Mtetra .91424 .56869 .91587 .57089 .92014 .5697 .91851 .5675 Mtetra .91851 .5675 .92014 .5697 .9244 .56851 .92278 .5663 Mtetra .92278 .5663 .9244 .56851 .92868 .56732 .92706 .5651 Mtetra .92706 .5651 .92868 .56732 .93296 .56612 .93134 .5639 Mtetra .93134 .5639 .93296 .56612 .93725 .56492 .93564 .5627 Mtetra .93564 .5627 .93725 .56492 .94154 .56372 .93994 .5615 Mtetra .93994 .5615 .94154 .56372 .94584 .56252 .94424 .56029 Mtetra .94424 .56029 .94584 .56252 .95015 .56132 .94855 .55909 Mtetra .94855 .55909 .95015 .56132 .95446 .56011 .95287 .55788 Mtetra .95287 .55788 .95446 .56011 .95878 .5589 .95719 .55667 Mtetra .95719 .55667 .95878 .5589 .96311 .55769 .96152 .55546 Mtetra .96152 .55546 .96311 .55769 .96744 .55648 .96586 .55424 Mtetra .96586 .55424 .96744 .55648 .97177 .55527 .9702 .55303 Mtetra .9702 .55303 .97177 .55527 .97612 .55406 .97455 .55181 Mtetra .97455 .55181 .97612 .55406 .98047 .55284 .9789 .55059 Mtetra .9789 .55059 .98047 .55284 .98483 .55162 .98327 .54937 Mtetra .664 .741 .907 r .45117 .57749 .45318 .57945 .45239 .79215 .45467 .57644 Mtetra 0 0 .384 r .45467 .57644 .45239 .79215 .45612 .79117 .45398 .78936 Mtetra .645 .763 .934 r .54497 .76542 .54702 .76729 .54977 .61382 .54883 .76441 Mtetra .57 .359 .555 r .54883 .76441 .54977 .61382 .55294 .54934 .55269 .76339 Mtetra .518 .319 .55 r .55269 .76339 .55294 .54934 .55657 .54824 .55624 .72645 Mtetra 0 0 .399 r .55624 .72645 .55657 .54824 .56021 .54714 .55829 .54512 Mtetra .646 .735 .914 r .64332 .51941 .64516 .5215 .64891 .52037 .64708 .51845 Mtetra .945 .583 .258 r .64708 .51845 .64891 .52037 .65328 .53513 .65152 .53516 Mtetra .921 .555 .314 r .65152 .53516 .65328 .53513 .6577 .55026 .6559 .54936 Mtetra .92 .529 .232 r .6559 .54936 .6577 .55026 .66207 .56296 .66023 .56137 Mtetra .899 .483 .119 r .66023 .56137 .66207 .56296 .66638 .57355 .66453 .57148 Mtetra .845 .405 0 r .66453 .57148 .66638 .57355 .67065 .58234 .66878 .57994 Mtetra .738 .287 0 r .66878 .57994 .67065 .58234 .67488 .5896 .673 .58698 Mtetra 0 0 .414 r .673 .58698 .67488 .5896 .67908 .59556 .67719 .59281 Mtetra 0 .04 .593 r .67719 .59281 .67908 .59556 .68324 .60042 .68135 .59759 Mtetra 0 .195 .723 r .68135 .59759 .68324 .60042 .68738 .60434 .68549 .60148 Mtetra .012 .318 .806 r .68549 .60148 .68738 .60434 .69151 .60747 .68962 .60462 Mtetra .15 .41 .854 r .68962 .60462 .69151 .60747 .69561 .60993 .69372 .60711 Mtetra .255 .479 .882 r .69372 .60711 .69561 .60993 .6997 .61184 .69781 .60905 Mtetra .335 .529 .897 r .69781 .60905 .6997 .61184 .70377 .61327 .70189 .61053 Mtetra .397 .568 .905 r .70189 .61053 .70377 .61327 .70784 .6143 .70596 .61161 Mtetra .444 .598 .91 r .70596 .61161 .70784 .6143 .7119 .615 .71003 .61236 Mtetra .481 .623 .912 r .71003 .61236 .7119 .615 .71595 .61541 .71409 .61283 Mtetra .511 .642 .914 r .71409 .61283 .71595 .61541 .72 .61559 .71814 .61306 Mtetra .535 .658 .915 r .71814 .61306 .72 .61559 .72404 .61557 .72219 .61308 Mtetra .555 .672 .916 r .72219 .61308 .72404 .61557 .72809 .61538 .72624 .61293 Mtetra .571 .683 .916 r .72624 .61293 .72809 .61538 .73213 .61504 .73029 .61264 Mtetra .584 .693 .917 r .73029 .61264 .73213 .61504 .73617 .61459 .73434 .61222 Mtetra .595 .701 .917 r .73434 .61222 .73617 .61459 .74022 .61403 .73839 .61169 Mtetra .604 .708 .918 r .73839 .61169 .74022 .61403 .74426 .61339 .74244 .61108 Mtetra .612 .714 .918 r .74244 .61108 .74426 .61339 .74831 .61267 .74649 .61038 Mtetra .618 .719 .918 r .74649 .61038 .74831 .61267 .75236 .61189 .75055 .60963 Mtetra .624 .724 .919 r .75055 .60963 .75236 .61189 .75642 .61106 .75461 .60881 Mtetra .628 .728 .919 r .75461 .60881 .75642 .61106 .76047 .61018 .75867 .60795 Mtetra .632 .731 .919 r .75867 .60795 .76047 .61018 .76453 .60927 .76273 .60705 Mtetra .635 .734 .919 r .76273 .60705 .76453 .60927 .7686 .60832 .7668 .60611 Mtetra .638 .737 .92 r .7668 .60611 .7686 .60832 .77267 .60734 .77088 .60515 Mtetra .64 .739 .92 r .77088 .60515 .77267 .60734 .77674 .60634 .77496 .60415 Mtetra .642 .741 .92 r .77496 .60415 .77674 .60634 .78082 .60531 .77904 .60314 Mtetra .644 .742 .92 r .77904 .60314 .78082 .60531 .7849 .60427 .78313 .6021 Mtetra .646 .744 .921 r .78313 .6021 .7849 .60427 .78899 .60322 .78722 .60105 Mtetra .647 .745 .921 r .78722 .60105 .78899 .60322 .79308 .60215 .79132 .59998 Mtetra .648 .746 .921 r .79132 .59998 .79308 .60215 .79718 .60106 .79542 .5989 Mtetra .649 .747 .921 r .79542 .5989 .79718 .60106 .80128 .59997 .79953 .59781 Mtetra .65 .748 .921 r .79953 .59781 .80128 .59997 .80539 .59887 .80364 .59671 Mtetra .80364 .59671 .80539 .59887 .80951 .59776 .80776 .5956 Mtetra .651 .749 .921 r .80776 .5956 .80951 .59776 .81363 .59664 .81188 .59448 Mtetra .651 .75 .922 r .81188 .59448 .81363 .59664 .81775 .59552 .81601 .59336 Mtetra .652 .75 .922 r .81601 .59336 .81775 .59552 .82188 .59439 .82015 .59223 Mtetra .82015 .59223 .82188 .59439 .82602 .59325 .82429 .59109 Mtetra .652 .751 .922 r .82429 .59109 .82602 .59325 .83016 .59211 .82843 .58995 Mtetra .653 .751 .922 r .82843 .58995 .83016 .59211 .83431 .59097 .83259 .58881 Mtetra .83259 .58881 .83431 .59097 .83846 .58982 .83674 .58766 Mtetra .653 .752 .922 r .83674 .58766 .83846 .58982 .84262 .58867 .84091 .5865 Mtetra .84091 .5865 .84262 .58867 .84679 .58751 .84508 .58535 Mtetra .84508 .58535 .84679 .58751 .85096 .58635 .84925 .58419 Mtetra .654 .752 .922 r .84925 .58419 .85096 .58635 .85513 .58519 .85343 .58302 Mtetra .85343 .58302 .85513 .58519 .85932 .58403 .85762 .58186 Mtetra .85762 .58186 .85932 .58403 .86351 .58286 .86181 .58069 Mtetra .86181 .58069 .86351 .58286 .8677 .58169 .86601 .57952 Mtetra .86601 .57952 .8677 .58169 .8719 .58052 .87022 .57834 Mtetra .654 .753 .922 r .87022 .57834 .8719 .58052 .87611 .57935 .87443 .57717 Mtetra .87443 .57717 .87611 .57935 .88032 .57817 .87864 .57599 Mtetra .87864 .57599 .88032 .57817 .88454 .57699 .88287 .57481 Mtetra .88287 .57481 .88454 .57699 .88876 .57581 .8871 .57362 Mtetra .8871 .57362 .88876 .57581 .89299 .57463 .89133 .57244 Mtetra .89133 .57244 .89299 .57463 .89723 .57345 .89557 .57125 Mtetra .89557 .57125 .89723 .57345 .90147 .57226 .89982 .57006 Mtetra .89982 .57006 .90147 .57226 .90572 .57107 .90407 .56887 Mtetra .90407 .56887 .90572 .57107 .90998 .56988 .90833 .56768 Mtetra .90833 .56768 .90998 .56988 .91424 .56869 .9126 .56648 Mtetra .9126 .56648 .91424 .56869 .91851 .5675 .91687 .56529 Mtetra .91687 .56529 .91851 .5675 .92278 .5663 .92115 .56409 Mtetra .92115 .56409 .92278 .5663 .92706 .5651 .92543 .56289 Mtetra .92543 .56289 .92706 .5651 .93134 .5639 .92972 .56168 Mtetra .92972 .56168 .93134 .5639 .93564 .5627 .93402 .56048 Mtetra .93402 .56048 .93564 .5627 .93994 .5615 .93832 .55927 Mtetra .93832 .55927 .93994 .5615 .94424 .56029 .94263 .55806 Mtetra .94263 .55806 .94424 .56029 .94855 .55909 .94695 .55685 Mtetra .94695 .55685 .94855 .55909 .95287 .55788 .95127 .55564 Mtetra .95127 .55564 .95287 .55788 .95719 .55667 .9556 .55443 Mtetra .9556 .55443 .95719 .55667 .96152 .55546 .95993 .55321 Mtetra .95993 .55321 .96152 .55546 .96586 .55424 .96427 .552 Mtetra .96427 .552 .96586 .55424 .9702 .55303 .96862 .55078 Mtetra .96862 .55078 .9702 .55303 .97455 .55181 .97298 .54956 Mtetra .97298 .54956 .97455 .55181 .9789 .55059 .97734 .54833 Mtetra .97734 .54833 .9789 .55059 .98327 .54937 .9817 .54711 Mtetra .664 .741 .907 r .45267 .57448 .45467 .57644 .45398 .78936 .45618 .57341 Mtetra .575 .372 .568 r .45618 .57341 .45398 .78936 .45771 .78838 .45969 .57235 Mtetra 0 0 .386 r .45969 .57235 .45771 .78838 .46145 .7874 .45932 .78558 Mtetra .645 .763 .934 r .55064 .76151 .55269 .76339 .55624 .72645 .55451 .76049 Mtetra .559 .331 .526 r .55451 .76049 .55624 .72645 .55829 .54512 .55839 .75947 Mtetra .571 .361 .557 r .55839 .75947 .55829 .54512 .56193 .54401 .56227 .75845 Mtetra 0 0 .403 r .56227 .75845 .56193 .54401 .56558 .54291 .56366 .54087 Mtetra .669 .193 0 r .64147 .51731 .64332 .51941 .64708 .51845 .64547 .52281 Mtetra .857 .528 .43 r .64547 .52281 .64708 .51845 .65152 .53516 .64985 .53786 Mtetra .87 .522 .39 r .64985 .53786 .65152 .53516 .6559 .54936 .65418 .55077 Mtetra .879 .508 .336 r .65418 .55077 .6559 .54936 .66023 .56137 .65848 .56178 Mtetra .883 .482 .261 r .65848 .56178 .66023 .56137 .66453 .57148 .66274 .57112 Mtetra .872 .437 .158 r .66274 .57112 .66453 .57148 .66878 .57994 .66697 .579 Mtetra .835 .364 .016 r .66697 .579 .66878 .57994 .673 .58698 .67117 .58561 Mtetra .75 .249 0 r .67117 .58561 .673 .58698 .67719 .59281 .67535 .59111 Mtetra .602 .091 0 r .67535 .59111 .67719 .59281 .68135 .59759 .6795 .59566 Mtetra 0 .089 .595 r .6795 .59566 .68135 .59759 .68549 .60148 .68364 .59939 Mtetra 0 .255 .75 r .68364 .59939 .68549 .60148 .68962 .60462 .68775 .60241 Mtetra .012 .384 .845 r .68775 .60241 .68962 .60462 .69372 .60711 .69186 .60483 Mtetra .16 .475 .895 r .69186 .60483 .69372 .60711 .69781 .60905 .69595 .60673 Mtetra .27 .538 .919 r .69595 .60673 .69781 .60905 .70189 .61053 .70003 .60818 Mtetra .352 .582 .929 r .70003 .60818 .70189 .61053 .70596 .61161 .7041 .60926 Mtetra .412 .614 .933 r .7041 .60926 .70596 .61161 .71003 .61236 .70817 .61001 Mtetra .459 .638 .933 r .70817 .61001 .71003 .61236 .71409 .61283 .71223 .61048 Mtetra .494 .657 .933 r .71223 .61048 .71409 .61283 .71814 .61306 .71629 .61072 Mtetra .522 .672 .931 r .71629 .61072 .71814 .61306 .72219 .61308 .72034 .61076 Mtetra .545 .684 .93 r .72034 .61076 .72219 .61308 .72624 .61293 .7244 .61063 Mtetra .563 .694 .928 r .7244 .61063 .72624 .61293 .73029 .61264 .72845 .61035 Mtetra .578 .702 .927 r .72845 .61035 .73029 .61264 .73434 .61222 .7325 .60995 Mtetra .59 .709 .926 r .7325 .60995 .73434 .61222 .73839 .61169 .73656 .60944 Mtetra .6 .715 .925 r .73656 .60944 .73839 .61169 .74244 .61108 .74061 .60883 Mtetra .608 .72 .924 r .74061 .60883 .74244 .61108 .74649 .61038 .74467 .60816 Mtetra .615 .724 .924 r .74467 .60816 .74649 .61038 .75055 .60963 .74873 .60741 Mtetra .621 .728 .923 r .74873 .60741 .75055 .60963 .75461 .60881 .7528 .60661 Mtetra .626 .731 .923 r .7528 .60661 .75461 .60881 .75867 .60795 .75686 .60576 Mtetra .63 .734 .923 r .75686 .60576 .75867 .60795 .76273 .60705 .76093 .60486 Mtetra .634 .737 .922 r .76093 .60486 .76273 .60705 .7668 .60611 .76501 .60393 Mtetra .637 .739 .922 r .76501 .60393 .7668 .60611 .77088 .60515 .76909 .60297 Mtetra .639 .741 .922 r .76909 .60297 .77088 .60515 .77496 .60415 .77317 .60198 Mtetra .641 .742 .922 r .77317 .60198 .77496 .60415 .77904 .60314 .77725 .60097 Mtetra .643 .744 .922 r .77725 .60097 .77904 .60314 .78313 .6021 .78135 .59994 Mtetra .645 .745 .922 r .78135 .59994 .78313 .6021 .78722 .60105 .78544 .59889 Mtetra .646 .746 .922 r .78544 .59889 .78722 .60105 .79132 .59998 .78955 .59782 Mtetra .647 .747 .922 r .78955 .59782 .79132 .59998 .79542 .5989 .79365 .59675 Mtetra .648 .748 .922 r .79365 .59675 .79542 .5989 .79953 .59781 .79776 .59566 Mtetra .649 .748 .922 r .79776 .59566 .79953 .59781 .80364 .59671 .80188 .59456 Mtetra .65 .749 .922 r .80188 .59456 .80364 .59671 .80776 .5956 .806 .59345 Mtetra .651 .75 .922 r .806 .59345 .80776 .5956 .81188 .59448 .81013 .59233 Mtetra .81013 .59233 .81188 .59448 .81601 .59336 .81427 .5912 Mtetra .652 .75 .922 r .81427 .5912 .81601 .59336 .82015 .59223 .81841 .59007 Mtetra .652 .751 .922 r .81841 .59007 .82015 .59223 .82429 .59109 .82255 .58894 Mtetra .82255 .58894 .82429 .59109 .82843 .58995 .8267 .58779 Mtetra .653 .751 .922 r .8267 .58779 .82843 .58995 .83259 .58881 .83086 .58665 Mtetra .83086 .58665 .83259 .58881 .83674 .58766 .83502 .58549 Mtetra .653 .752 .922 r .83502 .58549 .83674 .58766 .84091 .5865 .83919 .58434 Mtetra .83919 .58434 .84091 .5865 .84508 .58535 .84336 .58318 Mtetra .84336 .58318 .84508 .58535 .84925 .58419 .84754 .58202 Mtetra .654 .752 .922 r .84754 .58202 .84925 .58419 .85343 .58302 .85173 .58085 Mtetra .85173 .58085 .85343 .58302 .85762 .58186 .85592 .57968 Mtetra .85592 .57968 .85762 .58186 .86181 .58069 .86012 .57851 Mtetra .86012 .57851 .86181 .58069 .86601 .57952 .86432 .57734 Mtetra .654 .753 .922 r .86432 .57734 .86601 .57952 .87022 .57834 .86853 .57616 Mtetra .86853 .57616 .87022 .57834 .87443 .57717 .87274 .57498 Mtetra .87274 .57498 .87443 .57717 .87864 .57599 .87697 .5738 Mtetra .87697 .5738 .87864 .57599 .88287 .57481 .88119 .57261 Mtetra .88119 .57261 .88287 .57481 .8871 .57362 .88543 .57143 Mtetra .88543 .57143 .8871 .57362 .89133 .57244 .88967 .57024 Mtetra .88967 .57024 .89133 .57244 .89557 .57125 .89391 .56905 Mtetra .89391 .56905 .89557 .57125 .89982 .57006 .89816 .56786 Mtetra .89816 .56786 .89982 .57006 .90407 .56887 .90242 .56666 Mtetra .90242 .56666 .90407 .56887 .90833 .56768 .90669 .56547 Mtetra .90669 .56547 .90833 .56768 .9126 .56648 .91096 .56427 Mtetra .91096 .56427 .9126 .56648 .91687 .56529 .91523 .56307 Mtetra .91523 .56307 .91687 .56529 .92115 .56409 .91952 .56187 Mtetra .91952 .56187 .92115 .56409 .92543 .56289 .9238 .56066 Mtetra .9238 .56066 .92543 .56289 .92972 .56168 .9281 .55946 Mtetra .9281 .55946 .92972 .56168 .93402 .56048 .9324 .55825 Mtetra .9324 .55825 .93402 .56048 .93832 .55927 .93671 .55704 Mtetra .93671 .55704 .93832 .55927 .94263 .55806 .94102 .55583 Mtetra .94102 .55583 .94263 .55806 .94695 .55685 .94534 .55462 Mtetra .94534 .55462 .94695 .55685 .95127 .55564 .94967 .5534 Mtetra .94967 .5534 .95127 .55564 .9556 .55443 .954 .55218 Mtetra .954 .55218 .9556 .55443 .95993 .55321 .95834 .55096 Mtetra .95834 .55096 .95993 .55321 .96427 .552 .96269 .54974 Mtetra .96269 .54974 .96427 .552 .96862 .55078 .96704 .54852 Mtetra .96704 .54852 .96862 .55078 .97298 .54956 .9714 .5473 Mtetra .9714 .5473 .97298 .54956 .97734 .54833 .97576 .54607 Mtetra .97576 .54607 .97734 .54833 .9817 .54711 .98013 .54484 Mtetra .664 .741 .907 r .45768 .57038 .45969 .57235 .45932 .78558 .46121 .56931 Mtetra .575 .371 .568 r .46121 .56931 .45932 .78558 .46306 .78459 .46473 .56824 Mtetra 0 0 .387 r .46473 .56824 .46306 .78459 .46681 .7836 .46468 .78178 Mtetra .645 .763 .934 r .56023 .75655 .56227 .75845 .56366 .54087 .56412 .75553 Mtetra .571 .361 .557 r .56412 .75553 .56366 .54087 .56732 .53976 .56802 .7545 Mtetra .001 0 .404 r .56802 .7545 .56732 .53976 .57098 .53866 .56907 .53661 Mtetra .937 .563 .216 r .63962 .51521 .64147 .51731 .64547 .52281 .64394 .52981 Mtetra .805 .491 .459 r .64394 .52981 .64547 .52281 .64985 .53786 .64825 .54287 Mtetra .817 .486 .429 r .64825 .54287 .64985 .53786 .65418 .55077 .65253 .55419 Mtetra .828 .475 .389 r .65253 .55419 .65418 .55077 .65848 .56178 .65678 .56394 Mtetra .835 .457 .336 r .65678 .56394 .65848 .56178 .66274 .57112 .661 .57228 Mtetra .837 .426 .262 r .661 .57228 .66274 .57112 .66697 .579 .66521 .57937 Mtetra .826 .376 .159 r .66521 .57937 .66697 .579 .67117 .58561 .66938 .58536 Mtetra .79 .296 .016 r .66938 .58536 .67117 .58561 .67535 .59111 .67354 .59039 Mtetra .709 .174 0 r .67354 .59039 .67535 .59111 .6795 .59566 .67768 .59457 Mtetra .563 .008 0 r .67768 .59457 .6795 .59566 .68364 .59939 .68181 .59801 Mtetra 0 .178 .626 r .68181 .59801 .68364 .59939 .68775 .60241 .68592 .60082 Mtetra 0 .344 .789 r .68592 .60082 .68775 .60241 .69186 .60483 .69001 .60307 Mtetra .054 .465 .884 r .69001 .60307 .69186 .60483 .69595 .60673 .6941 .60485 Mtetra .199 .546 .929 r .6941 .60485 .69595 .60673 .70003 .60818 .69818 .60621 Mtetra .304 .598 .948 r .69818 .60621 .70003 .60818 .7041 .60926 .70225 .60722 Mtetra .381 .633 .953 r .70225 .60722 .7041 .60926 .70817 .61001 .70632 .60792 Mtetra .437 .657 .953 r .70632 .60792 .70817 .61001 .71223 .61048 .71038 .60836 Mtetra .479 .674 .95 r .71038 .60836 .71223 .61048 .71629 .61072 .71444 .60858 Mtetra .511 .687 .947 r .71444 .60858 .71629 .61072 .72034 .61076 .7185 .6086 Mtetra .537 .697 .943 r .7185 .6086 .72034 .61076 .7244 .61063 .72256 .60846 Mtetra .557 .706 .94 r .72256 .60846 .7244 .61063 .72845 .61035 .72661 .60818 Mtetra .573 .712 .937 r .72661 .60818 .72845 .61035 .7325 .60995 .73067 .60777 Mtetra .586 .718 .934 r .73067 .60777 .7325 .60995 .73656 .60944 .73473 .60726 Mtetra .597 .723 .932 r .73473 .60726 .73656 .60944 .74061 .60883 .73879 .60666 Mtetra .606 .727 .93 r .73879 .60666 .74061 .60883 .74467 .60816 .74285 .60599 Mtetra .613 .73 .929 r .74285 .60599 .74467 .60816 .74873 .60741 .74692 .60524 Mtetra .619 .733 .928 r .74692 .60524 .74873 .60741 .7528 .60661 .75098 .60444 Mtetra .625 .736 .927 r .75098 .60444 .7528 .60661 .75686 .60576 .75505 .6036 Mtetra .629 .738 .926 r .75505 .6036 .75686 .60576 .76093 .60486 .75913 .6027 Mtetra .633 .74 .925 r .75913 .6027 .76093 .60486 .76501 .60393 .76321 .60178 Mtetra .636 .741 .925 r .76321 .60178 .76501 .60393 .76909 .60297 .76729 .60082 Mtetra .638 .743 .924 r .76729 .60082 .76909 .60297 .77317 .60198 .77138 .59983 Mtetra .641 .744 .924 r .77138 .59983 .77317 .60198 .77725 .60097 .77547 .59882 Mtetra .643 .745 .924 r .77547 .59882 .77725 .60097 .78135 .59994 .77956 .59779 Mtetra .644 .746 .923 r .77956 .59779 .78135 .59994 .78544 .59889 .78366 .59674 Mtetra .646 .747 .923 r .78366 .59674 .78544 .59889 .78955 .59782 .78777 .59567 Mtetra .647 .748 .923 r .78777 .59567 .78955 .59782 .79365 .59675 .79188 .5946 Mtetra .648 .749 .923 r .79188 .5946 .79365 .59675 .79776 .59566 .796 .5935 Mtetra .649 .749 .923 r .796 .5935 .79776 .59566 .80188 .59456 .80012 .5924 Mtetra .65 .75 .923 r .80012 .5924 .80188 .59456 .806 .59345 .80425 .59129 Mtetra .80425 .59129 .806 .59345 .81013 .59233 .80838 .59017 Mtetra .651 .75 .922 r .80838 .59017 .81013 .59233 .81427 .5912 .81252 .58905 Mtetra .651 .751 .922 r .81252 .58905 .81427 .5912 .81841 .59007 .81666 .58791 Mtetra .652 .751 .922 r .81666 .58791 .81841 .59007 .82255 .58894 .82081 .58677 Mtetra .82081 .58677 .82255 .58894 .8267 .58779 .82497 .58563 Mtetra .652 .752 .922 r .82497 .58563 .8267 .58779 .83086 .58665 .82913 .58448 Mtetra .653 .752 .922 r .82913 .58448 .83086 .58665 .83502 .58549 .83329 .58333 Mtetra .83329 .58333 .83502 .58549 .83919 .58434 .83747 .58217 Mtetra .83747 .58217 .83919 .58434 .84336 .58318 .84164 .58101 Mtetra .84164 .58101 .84336 .58318 .84754 .58202 .84583 .57984 Mtetra .84583 .57984 .84754 .58202 .85173 .58085 .85002 .57867 Mtetra .654 .752 .922 r .85002 .57867 .85173 .58085 .85592 .57968 .85421 .5775 Mtetra .85421 .5775 .85592 .57968 .86012 .57851 .85841 .57633 Mtetra .654 .753 .922 r .85841 .57633 .86012 .57851 .86432 .57734 .86262 .57515 Mtetra .86262 .57515 .86432 .57734 .86853 .57616 .86684 .57397 Mtetra .86684 .57397 .86853 .57616 .87274 .57498 .87106 .57279 Mtetra .87106 .57279 .87274 .57498 .87697 .5738 .87528 .5716 Mtetra .87528 .5716 .87697 .5738 .88119 .57261 .87951 .57042 Mtetra .87951 .57042 .88119 .57261 .88543 .57143 .88375 .56923 Mtetra .88375 .56923 .88543 .57143 .88967 .57024 .888 .56804 Mtetra .888 .56804 .88967 .57024 .89391 .56905 .89225 .56684 Mtetra .89225 .56684 .89391 .56905 .89816 .56786 .8965 .56565 Mtetra .8965 .56565 .89816 .56786 .90242 .56666 .90076 .56445 Mtetra .90076 .56445 .90242 .56666 .90669 .56547 .90503 .56325 Mtetra .90503 .56325 .90669 .56547 .91096 .56427 .90931 .56205 Mtetra .90931 .56205 .91096 .56427 .91523 .56307 .91359 .56085 Mtetra .91359 .56085 .91523 .56307 .91952 .56187 .91788 .55964 Mtetra .91788 .55964 .91952 .56187 .9238 .56066 .92217 .55843 Mtetra .92217 .55843 .9238 .56066 .9281 .55946 .92647 .55722 Mtetra .92647 .55722 .9281 .55946 .9324 .55825 .93078 .55601 Mtetra .93078 .55601 .9324 .55825 .93671 .55704 .93509 .5548 Mtetra .93509 .5548 .93671 .55704 .94102 .55583 .93941 .55359 Mtetra .93941 .55359 .94102 .55583 .94534 .55462 .94373 .55237 Mtetra .94373 .55237 .94534 .55462 .94967 .5534 .94806 .55115 Mtetra .94806 .55115 .94967 .5534 .954 .55218 .9524 .54993 Mtetra .9524 .54993 .954 .55218 .95834 .55096 .95674 .54871 Mtetra .95674 .54871 .95834 .55096 .96269 .54974 .96109 .54749 Mtetra .96109 .54749 .96269 .54974 .96704 .54852 .96545 .54626 Mtetra .96545 .54626 .96704 .54852 .9714 .5473 .96981 .54503 Mtetra .96981 .54503 .9714 .5473 .97576 .54607 .97418 .5438 Mtetra .97418 .5438 .97576 .54607 .98013 .54484 .97856 .54257 Mtetra .57 .36 .556 r .38473 .80052 .39504 .58938 .39848 .58833 .38839 .79955 Mtetra 0 0 .322 r .38839 .79955 .39848 .58833 .40193 .58729 .39987 .58535 Mtetra .664 .741 .907 r .46273 .56626 .46473 .56824 .46468 .78178 .46626 .56519 Mtetra 0 0 .387 r .46626 .56519 .46468 .78178 .46844 .78078 .46631 .77895 Mtetra .645 .763 .934 r .56597 .75259 .56802 .7545 .56907 .53661 .56988 .75156 Mtetra .57 .36 .556 r .56988 .75156 .56907 .53661 .57274 .53549 .57379 .75053 Mtetra .229 .096 .484 r .57379 .75053 .57274 .53549 .57641 .53438 .57539 .59242 Mtetra 0 0 .373 r .57539 .59242 .57641 .53438 .58009 .53326 .57818 .5312 Mtetra .923 .534 .167 r .634 .51425 .63586 .51635 .63962 .51521 .63822 .52665 Mtetra .75 .46 .491 r .63822 .52665 .63962 .51521 .64394 .52981 .64247 .53893 Mtetra .758 .456 .474 r .64247 .53893 .64394 .52981 .64825 .54287 .64671 .54978 Mtetra .769 .451 .45 r .64671 .54978 .64825 .54287 .65253 .55419 .65093 .5593 Mtetra .778 .442 .418 r .65093 .5593 .65253 .55419 .65678 .56394 .65513 .56758 Mtetra .786 .426 .376 r .65513 .56758 .65678 .56394 .661 .57228 .65931 .57474 Mtetra .79 .401 .319 r .65931 .57474 .661 .57228 .66521 .57937 .66348 .58089 Mtetra .786 .361 .24 r .66348 .58089 .66521 .57937 .66938 .58536 .66763 .58612 Mtetra .769 .299 .129 r .66763 .58612 .66938 .58536 .67354 .59039 .67177 .59053 Mtetra .724 .205 0 r .67177 .59053 .67354 .59039 .67768 .59457 .67589 .59422 Mtetra .631 .068 0 r .67589 .59422 .67768 .59457 .68181 .59801 .68 .59728 Mtetra .473 0 0 r .68 .59728 .68181 .59801 .68592 .60082 .6841 .59978 Mtetra 0 .297 .687 r .6841 .59978 .68592 .60082 .69001 .60307 .68819 .60179 Mtetra 0 .449 .84 r .68819 .60179 .69001 .60307 .6941 .60485 .69227 .60337 Mtetra .129 .552 .922 r .69227 .60337 .6941 .60485 .69818 .60621 .69635 .60459 Mtetra .258 .615 .957 r .69635 .60459 .69818 .60621 .70225 .60722 .70042 .60548 Mtetra .351 .653 .969 r .70042 .60548 .70225 .60722 .70632 .60792 .70448 .60609 Mtetra .418 .677 .969 r .70448 .60609 .70632 .60792 .71038 .60836 .70854 .60646 Mtetra .466 .693 .966 r .70854 .60646 .71038 .60836 .71444 .60858 .7126 .60662 Mtetra .503 .704 .961 r .7126 .60662 .71444 .60858 .7185 .6086 .71666 .60661 Mtetra .531 .713 .955 r .71666 .60661 .7185 .6086 .72256 .60846 .72072 .60643 Mtetra .552 .719 .95 r .72072 .60643 .72256 .60846 .72661 .60818 .72478 .60612 Mtetra .57 .724 .946 r .72478 .60612 .72661 .60818 .73067 .60777 .72884 .6057 Mtetra .584 .728 .942 r .72884 .6057 .73067 .60777 .73473 .60726 .7329 .60517 Mtetra .595 .731 .939 r .7329 .60517 .73473 .60726 .73879 .60666 .73696 .60456 Mtetra .604 .734 .936 r .73696 .60456 .73879 .60666 .74285 .60599 .74103 .60388 Mtetra .612 .736 .934 r .74103 .60388 .74285 .60599 .74692 .60524 .7451 .60313 Mtetra .618 .738 .932 r .7451 .60313 .74692 .60524 .75098 .60444 .74917 .60232 Mtetra .624 .74 .93 r .74917 .60232 .75098 .60444 .75505 .6036 .75324 .60147 Mtetra .628 .742 .929 r .75324 .60147 .75505 .6036 .75913 .6027 .75732 .60058 Mtetra .632 .743 .928 r .75732 .60058 .75913 .6027 .76321 .60178 .7614 .59964 Mtetra .635 .744 .927 r .7614 .59964 .76321 .60178 .76729 .60082 .76549 .59868 Mtetra .638 .745 .926 r .76549 .59868 .76729 .60082 .77138 .59983 .76958 .59769 Mtetra .64 .746 .926 r .76958 .59769 .77138 .59983 .77547 .59882 .77368 .59668 Mtetra .642 .747 .925 r .77368 .59668 .77547 .59882 .77956 .59779 .77778 .59565 Mtetra .644 .748 .925 r .77778 .59565 .77956 .59779 .78366 .59674 .78188 .5946 Mtetra .645 .748 .924 r .78188 .5946 .78366 .59674 .78777 .59567 .78599 .59353 Mtetra .647 .749 .924 r .78599 .59353 .78777 .59567 .79188 .5946 .79011 .59245 Mtetra .648 .75 .924 r .79011 .59245 .79188 .5946 .796 .5935 .79423 .59136 Mtetra .649 .75 .923 r .79423 .59136 .796 .5935 .80012 .5924 .79835 .59025 Mtetra .79835 .59025 .80012 .5924 .80425 .59129 .80249 .58914 Mtetra .65 .751 .923 r .80249 .58914 .80425 .59129 .80838 .59017 .80662 .58802 Mtetra .651 .751 .923 r .80662 .58802 .80838 .59017 .81252 .58905 .81076 .58689 Mtetra .81076 .58689 .81252 .58905 .81666 .58791 .81491 .58575 Mtetra .652 .751 .923 r .81491 .58575 .81666 .58791 .82081 .58677 .81907 .58461 Mtetra .652 .752 .923 r .81907 .58461 .82081 .58677 .82497 .58563 .82323 .58346 Mtetra .82323 .58346 .82497 .58563 .82913 .58448 .82739 .58231 Mtetra .653 .752 .923 r .82739 .58231 .82913 .58448 .83329 .58333 .83156 .58116 Mtetra .83156 .58116 .83329 .58333 .83747 .58217 .83574 .57999 Mtetra .653 .752 .922 r .83574 .57999 .83747 .58217 .84164 .58101 .83992 .57883 Mtetra .83992 .57883 .84164 .58101 .84583 .57984 .84411 .57766 Mtetra .84411 .57766 .84583 .57984 .85002 .57867 .8483 .57649 Mtetra .654 .752 .922 r .8483 .57649 .85002 .57867 .85421 .5775 .8525 .57532 Mtetra .654 .753 .922 r .8525 .57532 .85421 .5775 .85841 .57633 .85671 .57414 Mtetra .85671 .57414 .85841 .57633 .86262 .57515 .86092 .57296 Mtetra .86092 .57296 .86262 .57515 .86684 .57397 .86514 .57178 Mtetra .86514 .57178 .86684 .57397 .87106 .57279 .86936 .57059 Mtetra .86936 .57059 .87106 .57279 .87528 .5716 .87359 .5694 Mtetra .87359 .5694 .87528 .5716 .87951 .57042 .87783 .56821 Mtetra .87783 .56821 .87951 .57042 .88375 .56923 .88207 .56702 Mtetra .88207 .56702 .88375 .56923 .888 .56804 .88632 .56583 Mtetra .88632 .56583 .888 .56804 .89225 .56684 .89058 .56463 Mtetra .89058 .56463 .89225 .56684 .8965 .56565 .89484 .56343 Mtetra .89484 .56343 .8965 .56565 .90076 .56445 .8991 .56223 Mtetra .8991 .56223 .90076 .56445 .90503 .56325 .90338 .56103 Mtetra .90338 .56103 .90503 .56325 .90931 .56205 .90766 .55982 Mtetra .90766 .55982 .90931 .56205 .91359 .56085 .91194 .55862 Mtetra .91194 .55862 .91359 .56085 .91788 .55964 .91623 .55741 Mtetra .91623 .55741 .91788 .55964 .92217 .55843 .92053 .5562 Mtetra .92053 .5562 .92217 .55843 .92647 .55722 .92484 .55499 Mtetra .92484 .55499 .92647 .55722 .93078 .55601 .92915 .55377 Mtetra .92915 .55377 .93078 .55601 .93509 .5548 .93346 .55256 Mtetra .93346 .55256 .93509 .5548 .93941 .55359 .93779 .55134 Mtetra .93779 .55134 .93941 .55359 .94373 .55237 .94212 .55012 Mtetra .94212 .55012 .94373 .55237 .94806 .55115 .94645 .5489 Mtetra .94645 .5489 .94806 .55115 .9524 .54993 .95079 .54767 Mtetra .95079 .54767 .9524 .54993 .95674 .54871 .95514 .54645 Mtetra .95514 .54645 .95674 .54871 .96109 .54749 .9595 .54522 Mtetra .9595 .54522 .96109 .54749 .96545 .54626 .96386 .54399 Mtetra .96386 .54399 .96545 .54626 .96981 .54503 .96823 .54276 Mtetra .96823 .54276 .96981 .54503 .97418 .5438 .9726 .54153 Mtetra .9726 .54153 .97418 .5438 .97856 .54257 .97698 .54029 Mtetra .645 .763 .934 r .38618 .79776 .38839 .79955 .39987 .58535 .38984 .79679 Mtetra 0 0 .322 r .38984 .79679 .39987 .58535 .40332 .58431 .40126 .58237 Mtetra .664 .741 .907 r .46425 .5632 .46626 .56519 .46631 .77895 .46779 .56213 Mtetra .574 .371 .567 r .46779 .56213 .46631 .77895 .47007 .77795 .47134 .56105 Mtetra 0 0 .389 r .47134 .56105 .47007 .77795 .47385 .77695 .47171 .77511 Mtetra .645 .763 .934 r .57175 .74861 .57379 .75053 .57539 .59242 .57567 .74758 Mtetra .569 .357 .554 r .57567 .74758 .57539 .59242 .57818 .5312 .5796 .74654 Mtetra .384 .215 .525 r .5796 .74654 .57818 .5312 .58187 .53008 .58185 .64332 Mtetra .208 .077 .472 r .58185 .64332 .58187 .53008 .58556 .52896 .58418 .55709 Mtetra 0 0 .326 r .58418 .55709 .58556 .52896 .58926 .52784 .58735 .52576 Mtetra 0 0 .487 r .62462 .51442 .6265 .51653 .63025 .51539 .62855 .51874 Mtetra .844 .498 .401 r .62855 .51874 .63025 .51539 .634 .51425 .63271 .52991 Mtetra .702 .428 .506 r .63271 .52991 .634 .51425 .63822 .52665 .63688 .54024 Mtetra .705 .425 .497 r .63688 .54024 .63822 .52665 .64247 .53893 .64105 .54967 Mtetra .715 .423 .481 r .64105 .54967 .64247 .53893 .64671 .54978 .64521 .55817 Mtetra .724 .418 .46 r .64521 .55817 .64671 .54978 .65093 .5593 .64937 .56574 Mtetra .732 .409 .433 r .64937 .56574 .65093 .5593 .65513 .56758 .65352 .57243 Mtetra .739 .394 .397 r .65352 .57243 .65513 .56758 .65931 .57474 .65766 .57828 Mtetra .742 .371 .349 r .65766 .57828 .65931 .57474 .66348 .58089 .66179 .58335 Mtetra .74 .336 .284 r .66179 .58335 .66348 .58089 .66763 .58612 .66591 .58771 Mtetra .728 .283 .194 r .66591 .58771 .66763 .58612 .67177 .59053 .67002 .59141 Mtetra .698 .204 .068 r .67002 .59141 .67177 .59053 .67589 .59422 .67412 .59452 Mtetra .635 .088 0 r .67412 .59452 .67589 .59422 .68 .59728 .67821 .59711 Mtetra .518 0 0 r .67821 .59711 .68 .59728 .6841 .59978 .6823 .59923 Mtetra .338 0 0 r .6823 .59923 .6841 .59978 .68819 .60179 .68638 .60093 Mtetra 0 .432 .77 r .68638 .60093 .68819 .60179 .69227 .60337 .69045 .60227 Mtetra .069 .557 .894 r .69045 .60227 .69227 .60337 .69635 .60459 .69452 .60328 Mtetra .22 .633 .955 r .69452 .60328 .69635 .60459 .70042 .60548 .69859 .60402 Mtetra .327 .676 .977 r .69859 .60402 .70042 .60548 .70448 .60609 .70265 .6045 Mtetra .403 .7 .982 r .70265 .6045 .70448 .60609 .70854 .60646 .70671 .60477 Mtetra .458 .714 .979 r .70671 .60477 .70854 .60646 .7126 .60662 .71077 .60484 Mtetra .497 .723 .973 r .71077 .60484 .7126 .60662 .71666 .60661 .71483 .60476 Mtetra .527 .729 .966 r .71483 .60476 .71666 .60661 .72072 .60643 .71889 .60453 Mtetra .55 .733 .96 r .71889 .60453 .72072 .60643 .72478 .60612 .72295 .60418 Mtetra .568 .736 .954 r .72295 .60418 .72478 .60612 .72884 .6057 .72701 .60372 Mtetra .583 .738 .949 r .72701 .60372 .72884 .6057 .7329 .60517 .73107 .60316 Mtetra .594 .74 .945 r .73107 .60316 .7329 .60517 .73696 .60456 .73514 .60253 Mtetra .604 .741 .942 r .73514 .60253 .73696 .60456 .74103 .60388 .73921 .60183 Mtetra .612 .743 .939 r .73921 .60183 .74103 .60388 .7451 .60313 .74328 .60106 Mtetra .618 .744 .936 r .74328 .60106 .7451 .60313 .74917 .60232 .74735 .60024 Mtetra .623 .745 .934 r .74735 .60024 .74917 .60232 .75324 .60147 .75143 .59938 Mtetra .628 .746 .932 r .75143 .59938 .75324 .60147 .75732 .60058 .75551 .59848 Mtetra .632 .747 .93 r .75551 .59848 .75732 .60058 .7614 .59964 .7596 .59754 Mtetra .635 .747 .929 r .7596 .59754 .7614 .59964 .76549 .59868 .76369 .59657 Mtetra .638 .748 .928 r .76369 .59657 .76549 .59868 .76958 .59769 .76778 .59557 Mtetra .64 .748 .927 r .76778 .59557 .76958 .59769 .77368 .59668 .77188 .59456 Mtetra .642 .749 .926 r .77188 .59456 .77368 .59668 .77778 .59565 .77599 .59352 Mtetra .644 .749 .926 r .77599 .59352 .77778 .59565 .78188 .5946 .7801 .59246 Mtetra .645 .75 .925 r .7801 .59246 .78188 .5946 .78599 .59353 .78421 .59139 Mtetra .647 .75 .925 r .78421 .59139 .78599 .59353 .79011 .59245 .78833 .59031 Mtetra .648 .751 .924 r .78833 .59031 .79011 .59245 .79423 .59136 .79245 .58921 Mtetra .649 .751 .924 r .79245 .58921 .79423 .59136 .79835 .59025 .79658 .5881 Mtetra .79658 .5881 .79835 .59025 .80249 .58914 .80072 .58699 Mtetra .65 .751 .924 r .80072 .58699 .80249 .58914 .80662 .58802 .80486 .58586 Mtetra .651 .752 .923 r .80486 .58586 .80662 .58802 .81076 .58689 .80901 .58473 Mtetra .80901 .58473 .81076 .58689 .81491 .58575 .81316 .58359 Mtetra .652 .752 .923 r .81316 .58359 .81491 .58575 .81907 .58461 .81732 .58245 Mtetra .81732 .58245 .81907 .58461 .82323 .58346 .82148 .5813 Mtetra .82148 .5813 .82323 .58346 .82739 .58231 .82565 .58014 Mtetra .653 .752 .923 r .82565 .58014 .82739 .58231 .83156 .58116 .82983 .57898 Mtetra .82983 .57898 .83156 .58116 .83574 .57999 .83401 .57782 Mtetra .83401 .57782 .83574 .57999 .83992 .57883 .83819 .57665 Mtetra .83819 .57665 .83992 .57883 .84411 .57766 .84239 .57548 Mtetra .653 .753 .923 r .84239 .57548 .84411 .57766 .8483 .57649 .84658 .5743 Mtetra .654 .753 .922 r .84658 .5743 .8483 .57649 .8525 .57532 .85079 .57313 Mtetra .85079 .57313 .8525 .57532 .85671 .57414 .855 .57195 Mtetra .855 .57195 .85671 .57414 .86092 .57296 .85922 .57076 Mtetra .85922 .57076 .86092 .57296 .86514 .57178 .86344 .56958 Mtetra .86344 .56958 .86514 .57178 .86936 .57059 .86767 .56839 Mtetra .86767 .56839 .86936 .57059 .87359 .5694 .8719 .5672 Mtetra .8719 .5672 .87359 .5694 .87783 .56821 .87614 .566 Mtetra .87614 .566 .87783 .56821 .88207 .56702 .88039 .56481 Mtetra .88039 .56481 .88207 .56702 .88632 .56583 .88464 .56361 Mtetra .88464 .56361 .88632 .56583 .89058 .56463 .8889 .56241 Mtetra .8889 .56241 .89058 .56463 .89484 .56343 .89317 .56121 Mtetra .89317 .56121 .89484 .56343 .8991 .56223 .89744 .56001 Mtetra .89744 .56001 .8991 .56223 .90338 .56103 .90172 .5588 Mtetra .90172 .5588 .90338 .56103 .90766 .55982 .906 .55759 Mtetra .906 .55759 .90766 .55982 .91194 .55862 .91029 .55638 Mtetra .91029 .55638 .91194 .55862 .91623 .55741 .91459 .55517 Mtetra .91459 .55517 .91623 .55741 .92053 .5562 .91889 .55396 Mtetra .91889 .55396 .92053 .5562 .92484 .55499 .9232 .55274 Mtetra .9232 .55274 .92484 .55499 .92915 .55377 .92751 .55152 Mtetra .92751 .55152 .92915 .55377 .93346 .55256 .93183 .55031 Mtetra .93183 .55031 .93346 .55256 .93779 .55134 .93616 .54908 Mtetra .93616 .54908 .93779 .55134 .94212 .55012 .9405 .54786 Mtetra .9405 .54786 .94212 .55012 .94645 .5489 .94484 .54664 Mtetra .94484 .54664 .94645 .5489 .95079 .54767 .94918 .54541 Mtetra .94918 .54541 .95079 .54767 .95514 .54645 .95354 .54418 Mtetra .95354 .54418 .95514 .54645 .9595 .54522 .9579 .54295 Mtetra .9579 .54295 .9595 .54522 .96386 .54399 .96226 .54172 Mtetra .96226 .54172 .96386 .54399 .96823 .54276 .96664 .54048 Mtetra .96664 .54048 .96823 .54276 .9726 .54153 .97102 .53925 Mtetra .97102 .53925 .9726 .54153 .97698 .54029 .9754 .53801 Mtetra .645 .763 .934 r .38764 .795 .38984 .79679 .40126 .58237 .39131 .79403 Mtetra 0 0 .322 r .39131 .79403 .40126 .58237 .40472 .58131 .40266 .57937 Mtetra .664 .741 .907 r .46933 .55905 .47134 .56105 .47171 .77511 .47289 .55797 Mtetra 0 0 .389 r .47289 .55797 .47171 .77511 .4755 .77411 .47336 .77226 Mtetra .645 .763 .934 r .57756 .74461 .5796 .74654 .58185 .64332 .5815 .74357 Mtetra .567 .352 .549 r .5815 .74357 .58185 .64332 .58418 .55709 .58544 .74252 Mtetra .421 .243 .531 r .58544 .74252 .58418 .55709 .58735 .52576 .58805 .66788 Mtetra .408 .231 .527 r .58805 .66788 .58735 .52576 .59106 .52463 .59078 .60726 Mtetra .375 .201 .513 r .59078 .60726 .59106 .52463 .59477 .5235 .59379 .567 Mtetra .316 .145 .483 r .59379 .567 .59477 .5235 .59848 .52237 .59703 .54133 Mtetra .18 .008 .397 r .59703 .54133 .59848 .52237 .6022 .52124 .60046 .52618 Mtetra 0 0 0 r .60046 .52618 .6022 .52124 .60593 .52011 .60405 .51858 Mtetra .72 .815 .924 r .60405 .51858 .60593 .52011 .60966 .51898 .60777 .51688 Mtetra .325 .588 .939 r .60777 .51688 .60966 .51898 .61339 .51784 .61156 .51786 Mtetra .759 .258 .022 r .61156 .51786 .61339 .51784 .61713 .5167 .61545 .52188 Mtetra .73 .394 .41 r .61545 .52188 .61713 .5167 .62088 .51556 .61939 .5275 Mtetra .68 .397 .486 r .61939 .5275 .62088 .51556 .62462 .51442 .62339 .53406 Mtetra .651 .391 .514 r .62339 .53406 .62462 .51442 .62855 .51874 .62742 .54104 Mtetra .644 .389 .519 r .62742 .54104 .62855 .51874 .63271 .52991 .63148 .54811 Mtetra .654 .391 .509 r .63148 .54811 .63271 .52991 .63688 .54024 .63555 .55499 Mtetra .664 .392 .498 r .63555 .55499 .63688 .54024 .64105 .54967 .63964 .56153 Mtetra .674 .389 .482 r .63964 .56153 .64105 .54967 .64521 .55817 .64373 .56762 Mtetra .682 .384 .463 r .64373 .56762 .64521 .55817 .64937 .56574 .64783 .57318 Mtetra .689 .374 .438 r .64783 .57318 .64937 .56574 .65352 .57243 .65193 .5782 Mtetra .694 .36 .406 r .65193 .5782 .65352 .57243 .65766 .57828 .65602 .58266 Mtetra .696 .337 .364 r .65602 .58266 .65766 .57828 .66179 .58335 .66011 .58658 Mtetra .693 .304 .306 r .66011 .58658 .66179 .58335 .66591 .58771 .6642 .58998 Mtetra .683 .256 .228 r .6642 .58998 .66591 .58771 .67002 .59141 .66829 .5929 Mtetra .658 .185 .119 r .66829 .5929 .67002 .59141 .67412 .59452 .67236 .59536 Mtetra .607 .082 0 r .67236 .59536 .67412 .59452 .67821 .59711 .67644 .59742 Mtetra .514 0 0 r .67644 .59742 .67821 .59711 .6823 .59923 .68051 .5991 Mtetra .364 0 0 r .68051 .5991 .6823 .59923 .68638 .60093 .68458 .60045 Mtetra 0 .422 .695 r .68458 .60045 .68638 .60093 .69045 .60227 .68864 .60149 Mtetra .032 .564 .856 r .68864 .60149 .69045 .60227 .69452 .60328 .6927 .60226 Mtetra .195 .652 .942 r .6927 .60226 .69452 .60328 .69859 .60402 .69677 .6028 Mtetra .313 .7 .979 r .69677 .6028 .69859 .60402 .70265 .6045 .70082 .60312 Mtetra .396 .724 .989 r .70082 .60312 .70265 .6045 .70671 .60477 .70488 .60325 Mtetra .454 .736 .988 r .70488 .60325 .70671 .60477 .71077 .60484 .70894 .60322 Mtetra .496 .743 .982 r .70894 .60322 .71077 .60484 .71483 .60476 .713 .60305 Mtetra .527 .746 .975 r .713 .60305 .71483 .60476 .71889 .60453 .71706 .60275 Mtetra .551 .747 .968 r .71706 .60275 .71889 .60453 .72295 .60418 .72112 .60233 Mtetra .569 .748 .961 r .72112 .60233 .72295 .60418 .72701 .60372 .72518 .60182 Mtetra .584 .749 .956 r .72518 .60182 .72701 .60372 .73107 .60316 .72925 .60123 Mtetra .595 .749 .951 r .72925 .60123 .73107 .60316 .73514 .60253 .73331 .60056 Mtetra .605 .749 .946 r .73331 .60056 .73514 .60253 .73921 .60183 .73738 .59983 Mtetra .612 .749 .943 r .73738 .59983 .73921 .60183 .74328 .60106 .74146 .59904 Mtetra .619 .749 .939 r .74146 .59904 .74328 .60106 .74735 .60024 .74553 .5982 Mtetra .624 .75 .937 r .74553 .5982 .74735 .60024 .75143 .59938 .74962 .59732 Mtetra .628 .75 .935 r .74962 .59732 .75143 .59938 .75551 .59848 .7537 .5964 Mtetra .632 .75 .933 r .7537 .5964 .75551 .59848 .7596 .59754 .75779 .59545 Mtetra .635 .75 .931 r .75779 .59545 .7596 .59754 .76369 .59657 .76188 .59447 Mtetra .638 .75 .93 r .76188 .59447 .76369 .59657 .76778 .59557 .76598 .59347 Mtetra .64 .751 .929 r .76598 .59347 .76778 .59557 .77188 .59456 .77009 .59244 Mtetra .642 .751 .928 r .77009 .59244 .77188 .59456 .77599 .59352 .77419 .5914 Mtetra .644 .751 .927 r .77419 .5914 .77599 .59352 .7801 .59246 .77831 .59034 Mtetra .645 .751 .926 r .77831 .59034 .7801 .59246 .78421 .59139 .78242 .58926 Mtetra .647 .751 .926 r .78242 .58926 .78421 .59139 .78833 .59031 .78655 .58817 Mtetra .648 .752 .925 r .78655 .58817 .78833 .59031 .79245 .58921 .79068 .58707 Mtetra .649 .752 .925 r .79068 .58707 .79245 .58921 .79658 .5881 .79481 .58596 Mtetra .649 .752 .924 r .79481 .58596 .79658 .5881 .80072 .58699 .79895 .58484 Mtetra .65 .752 .924 r .79895 .58484 .80072 .58699 .80486 .58586 .8031 .58371 Mtetra .651 .752 .924 r .8031 .58371 .80486 .58586 .80901 .58473 .80725 .58257 Mtetra .80725 .58257 .80901 .58473 .81316 .58359 .8114 .58143 Mtetra .652 .752 .923 r .8114 .58143 .81316 .58359 .81732 .58245 .81556 .58028 Mtetra .81556 .58028 .81732 .58245 .82148 .5813 .81973 .57913 Mtetra .81973 .57913 .82148 .5813 .82565 .58014 .82391 .57797 Mtetra .653 .753 .923 r .82391 .57797 .82565 .58014 .82983 .57898 .82808 .5768 Mtetra .82808 .5768 .82983 .57898 .83401 .57782 .83227 .57564 Mtetra .83227 .57564 .83401 .57782 .83819 .57665 .83646 .57446 Mtetra .83646 .57446 .83819 .57665 .84239 .57548 .84066 .57329 Mtetra .84066 .57329 .84239 .57548 .84658 .5743 .84486 .57211 Mtetra .654 .753 .923 r .84486 .57211 .84658 .5743 .85079 .57313 .84907 .57093 Mtetra .84907 .57093 .85079 .57313 .855 .57195 .85328 .56975 Mtetra .654 .753 .922 r .85328 .56975 .855 .57195 .85922 .57076 .85751 .56856 Mtetra .85751 .56856 .85922 .57076 .86344 .56958 .86173 .56737 Mtetra .86173 .56737 .86344 .56958 .86767 .56839 .86597 .56618 Mtetra .86597 .56618 .86767 .56839 .8719 .5672 .8702 .56499 Mtetra .8702 .56499 .8719 .5672 .87614 .566 .87445 .56379 Mtetra .87445 .56379 .87614 .566 .88039 .56481 .8787 .56259 Mtetra .8787 .56259 .88039 .56481 .88464 .56361 .88296 .56139 Mtetra .88296 .56139 .88464 .56361 .8889 .56241 .88722 .56019 Mtetra .88722 .56019 .8889 .56241 .89317 .56121 .89149 .55898 Mtetra .89149 .55898 .89317 .56121 .89744 .56001 .89577 .55778 Mtetra .89577 .55778 .89744 .56001 .90172 .5588 .90005 .55657 Mtetra .90005 .55657 .90172 .5588 .906 .55759 .90434 .55536 Mtetra .90434 .55536 .906 .55759 .91029 .55638 .90863 .55414 Mtetra .90863 .55414 .91029 .55638 .91459 .55517 .91293 .55293 Mtetra .91293 .55293 .91459 .55517 .91889 .55396 .91724 .55171 Mtetra .91724 .55171 .91889 .55396 .9232 .55274 .92156 .55049 Mtetra .92156 .55049 .9232 .55274 .92751 .55152 .92588 .54927 Mtetra .92588 .54927 .92751 .55152 .93183 .55031 .9302 .54805 Mtetra .9302 .54805 .93183 .55031 .93616 .54908 .93453 .54682 Mtetra .93453 .54682 .93616 .54908 .9405 .54786 .93887 .5456 Mtetra .93887 .5456 .9405 .54786 .94484 .54664 .94322 .54437 Mtetra .94322 .54437 .94484 .54664 .94918 .54541 .94757 .54314 Mtetra .94757 .54314 .94918 .54541 .95354 .54418 .95193 .54191 Mtetra .95193 .54191 .95354 .54418 .9579 .54295 .95629 .54067 Mtetra .95629 .54067 .9579 .54295 .96226 .54172 .96066 .53944 Mtetra .96066 .53944 .96226 .54172 .96664 .54048 .96504 .5382 Mtetra .96504 .5382 .96664 .54048 .97102 .53925 .96942 .53696 Mtetra .96942 .53696 .97102 .53925 .9754 .53801 .97381 .53572 Mtetra .645 .763 .934 r .3891 .79222 .39131 .79403 .40266 .57937 .39277 .79125 Mtetra .569 .358 .555 r .39277 .79125 .40266 .57937 .40613 .57831 .39646 .79027 Mtetra 0 0 .324 r .39646 .79027 .40613 .57831 .4096 .57725 .40754 .57529 Mtetra .664 .741 .907 r .47088 .55597 .47289 .55797 .47336 .77226 .47444 .55488 Mtetra .574 .37 .566 r .47444 .55488 .47336 .77226 .47715 .77125 .47801 .55379 Mtetra 0 0 .391 r .47801 .55379 .47715 .77125 .48095 .77024 .47881 .76838 Mtetra .645 .763 .934 r .5834 .74059 .58544 .74252 .58805 .66788 .58735 .73954 Mtetra .562 .345 .543 r .58735 .73954 .58805 .66788 .59078 .60726 .5913 .7378 Mtetra .387 .214 .521 r .5913 .7378 .59078 .60726 .59379 .567 .59407 .67684 Mtetra .424 .241 .527 r .59407 .67684 .59379 .567 .59703 .54133 .5971 .63399 Mtetra .455 .264 .531 r .5971 .63399 .59703 .54133 .60046 .52618 .60033 .60442 Mtetra .482 .283 .533 r .60033 .60442 .60046 .52618 .60405 .51858 .60373 .58462 Mtetra .506 .3 .534 r .60373 .58462 .60405 .51858 .60777 .51688 .60726 .57202 Mtetra .526 .313 .534 r .60726 .57202 .60777 .51688 .61156 .51786 .61091 .56468 Mtetra .544 .325 .533 r .61091 .56468 .61156 .51786 .61545 .52188 .61465 .56115 Mtetra .561 .335 .53 r .61465 .56115 .61545 .52188 .61939 .5275 .61847 .56033 Mtetra .576 .343 .526 r .61847 .56033 .61939 .5275 .62339 .53406 .62234 .5614 Mtetra .589 .35 .521 r .62234 .5614 .62339 .53406 .62742 .54104 .62627 .56372 Mtetra .601 .354 .514 r .62627 .56372 .62742 .54104 .63148 .54811 .63023 .56681 Mtetra .613 .356 .505 r .63023 .56681 .63148 .54811 .63555 .55499 .63422 .57034 Mtetra .623 .356 .494 r .63422 .57034 .63555 .55499 .63964 .56153 .63823 .57403 Mtetra .631 .354 .479 r .63823 .57403 .63964 .56153 .64373 .56762 .64226 .57772 Mtetra .639 .348 .461 r .64226 .57772 .64373 .56762 .64783 .57318 .6463 .58126 Mtetra .645 .338 .437 r .6463 .58126 .64783 .57318 .65193 .5782 .65034 .58458 Mtetra .649 .323 .407 r .65034 .58458 .65193 .5782 .65602 .58266 .6544 .58762 Mtetra .65 .3 .367 r .6544 .58762 .65602 .58266 .66011 .58658 .65845 .59035 Mtetra .646 .268 .314 r .65845 .59035 .66011 .58658 .6642 .58998 .66251 .59276 Mtetra .634 .221 .242 r .66251 .59276 .6642 .58998 .66829 .5929 .66656 .59485 Mtetra .61 .153 .142 r .66656 .59485 .66829 .5929 .67236 .59536 .67062 .59663 Mtetra .564 .056 .005 r .67062 .59663 .67236 .59536 .67644 .59742 .67468 .59811 Mtetra .482 0 0 r .67468 .59811 .67644 .59742 .68051 .5991 .67873 .59932 Mtetra .35 0 0 r .67873 .59932 .68051 .5991 .68458 .60045 .68279 .60027 Mtetra .17 0 0 r .68279 .60027 .68458 .60045 .68864 .60149 .68684 .60098 Mtetra .023 .579 .818 r .68684 .60098 .68864 .60149 .6927 .60226 .69089 .60148 Mtetra .188 .674 .926 r .69089 .60148 .6927 .60226 .69677 .6028 .69495 .60179 Mtetra .311 .725 .976 r .69495 .60179 .69677 .6028 .70082 .60312 .699 .60192 Mtetra .397 .749 .993 r .699 .60192 .70082 .60312 .70488 .60325 .70306 .6019 Mtetra .457 .759 .994 r .70306 .6019 .70488 .60325 .70894 .60322 .70711 .60174 Mtetra .5 .762 .989 r .70711 .60174 .70894 .60322 .713 .60305 .71117 .60146 Mtetra .531 .763 .982 r .71117 .60146 .713 .60305 .71706 .60275 .71523 .60107 Mtetra .554 .762 .974 r .71523 .60107 .71706 .60275 .72112 .60233 .71929 .60058 Mtetra .572 .76 .967 r .71929 .60058 .72112 .60233 .72518 .60182 .72335 .60001 Mtetra .586 .759 .961 r .72335 .60001 .72518 .60182 .72925 .60123 .72742 .59936 Mtetra .598 .758 .955 r .72742 .59936 .72925 .60123 .73331 .60056 .73149 .59865 Mtetra .607 .757 .95 r .73149 .59865 .73331 .60056 .73738 .59983 .73556 .59788 Mtetra .614 .756 .946 r .73556 .59788 .73738 .59983 .74146 .59904 .73963 .59706 Mtetra .62 .755 .942 r .73963 .59706 .74146 .59904 .74553 .5982 .74371 .5962 Mtetra .625 .754 .939 r .74371 .5962 .74553 .5982 .74962 .59732 .7478 .59529 Mtetra .629 .754 .937 r .7478 .59529 .74962 .59732 .7537 .5964 .75189 .59436 Mtetra .633 .754 .935 r .75189 .59436 .7537 .5964 .75779 .59545 .75598 .59339 Mtetra .636 .753 .933 r .75598 .59339 .75779 .59545 .76188 .59447 .76008 .5924 Mtetra .639 .753 .931 r .76008 .5924 .76188 .59447 .76598 .59347 .76418 .59138 Mtetra .641 .753 .93 r .76418 .59138 .76598 .59347 .77009 .59244 .76828 .59034 Mtetra .643 .753 .929 r .76828 .59034 .77009 .59244 .77419 .5914 .7724 .58929 Mtetra .644 .753 .928 r .7724 .58929 .77419 .5914 .77831 .59034 .77651 .58822 Mtetra .646 .753 .927 r .77651 .58822 .77831 .59034 .78242 .58926 .78063 .58713 Mtetra .647 .753 .926 r .78063 .58713 .78242 .58926 .78655 .58817 .78476 .58604 Mtetra .648 .753 .926 r .78476 .58604 .78655 .58817 .79068 .58707 .78889 .58493 Mtetra .649 .753 .925 r .78889 .58493 .79068 .58707 .79481 .58596 .79303 .58381 Mtetra .65 .753 .925 r .79303 .58381 .79481 .58596 .79895 .58484 .79718 .58268 Mtetra .65 .753 .924 r .79718 .58268 .79895 .58484 .8031 .58371 .80133 .58155 Mtetra .651 .753 .924 r .80133 .58155 .8031 .58371 .80725 .58257 .80548 .58041 Mtetra .80548 .58041 .80725 .58257 .8114 .58143 .80964 .57926 Mtetra .652 .753 .924 r .80964 .57926 .8114 .58143 .81556 .58028 .81381 .57811 Mtetra .652 .753 .923 r .81381 .57811 .81556 .58028 .81973 .57913 .81798 .57695 Mtetra .81798 .57695 .81973 .57913 .82391 .57797 .82216 .57579 Mtetra .653 .753 .923 r .82216 .57579 .82391 .57797 .82808 .5768 .82634 .57462 Mtetra .82634 .57462 .82808 .5768 .83227 .57564 .83053 .57345 Mtetra .83053 .57345 .83227 .57564 .83646 .57446 .83472 .57228 Mtetra .83472 .57228 .83646 .57446 .84066 .57329 .83893 .5711 Mtetra .83893 .5711 .84066 .57329 .84486 .57211 .84313 .56992 Mtetra .654 .753 .923 r .84313 .56992 .84486 .57211 .84907 .57093 .84735 .56873 Mtetra .84735 .56873 .84907 .57093 .85328 .56975 .85157 .56755 Mtetra .85157 .56755 .85328 .56975 .85751 .56856 .85579 .56636 Mtetra .85579 .56636 .85751 .56856 .86173 .56737 .86002 .56516 Mtetra .654 .753 .922 r .86002 .56516 .86173 .56737 .86597 .56618 .86426 .56397 Mtetra .86426 .56397 .86597 .56618 .8702 .56499 .8685 .56277 Mtetra .8685 .56277 .8702 .56499 .87445 .56379 .87275 .56157 Mtetra .87275 .56157 .87445 .56379 .8787 .56259 .87701 .56037 Mtetra .87701 .56037 .8787 .56259 .88296 .56139 .88127 .55916 Mtetra .88127 .55916 .88296 .56139 .88722 .56019 .88554 .55796 Mtetra .88554 .55796 .88722 .56019 .89149 .55898 .88981 .55675 Mtetra .88981 .55675 .89149 .55898 .89577 .55778 .89409 .55554 Mtetra .89409 .55554 .89577 .55778 .90005 .55657 .89838 .55433 Mtetra .89838 .55433 .90005 .55657 .90434 .55536 .90267 .55311 Mtetra .90267 .55311 .90434 .55536 .90863 .55414 .90697 .5519 Mtetra .90697 .5519 .90863 .55414 .91293 .55293 .91128 .55068 Mtetra .91128 .55068 .91293 .55293 .91724 .55171 .91559 .54946 Mtetra .91559 .54946 .91724 .55171 .92156 .55049 .91991 .54824 Mtetra .91991 .54824 .92156 .55049 .92588 .54927 .92423 .54701 Mtetra .92423 .54701 .92588 .54927 .9302 .54805 .92856 .54579 Mtetra .92856 .54579 .9302 .54805 .93453 .54682 .9329 .54456 Mtetra .9329 .54456 .93453 .54682 .93887 .5456 .93724 .54333 Mtetra .93724 .54333 .93887 .5456 .94322 .54437 .94159 .5421 Mtetra .94159 .5421 .94322 .54437 .94757 .54314 .94595 .54086 Mtetra .94595 .54086 .94757 .54314 .95193 .54191 .95031 .53963 Mtetra .95031 .53963 .95193 .54191 .95629 .54067 .95468 .53839 Mtetra .95468 .53839 .95629 .54067 .96066 .53944 .95906 .53715 Mtetra .95906 .53715 .96066 .53944 .96504 .5382 .96344 .53591 Mtetra .96344 .53591 .96504 .5382 .96942 .53696 .96783 .53467 Mtetra .96783 .53467 .96942 .53696 .97381 .53572 .97222 .53342 Mtetra .645 .763 .934 r .39425 .78846 .39646 .79027 .40754 .57529 .39794 .78747 Mtetra 0 0 .324 r .39794 .78747 .40754 .57529 .41102 .57423 .40895 .57227 Mtetra .664 .741 .907 r .476 .55178 .47801 .55379 .47881 .76838 .47957 .55068 Mtetra 0 0 .391 r .47957 .55068 .47881 .76838 .48262 .76737 .48048 .7655 Mtetra .645 .763 .934 r .58531 .7376 .58735 .73954 .5913 .7378 .58927 .73654 Mtetra .619 .832 .982 r .58927 .73654 .5913 .7378 .59407 .67684 .59324 .73549 Mtetra .47 .269 .524 r .59324 .73549 .59407 .67684 .5971 .63399 .5969 .71914 Mtetra .37 .196 .511 r .5969 .71914 .5971 .63399 .60033 .60442 .59998 .67692 Mtetra .407 .223 .517 r .59998 .67692 .60033 .60442 .60373 .58462 .60323 .64632 Mtetra .438 .245 .52 r .60323 .64632 .60373 .58462 .60726 .57202 .60663 .62445 Mtetra .465 .264 .522 r .60663 .62445 .60726 .57202 .61091 .56468 .61015 .60915 Mtetra .488 .279 .522 r .61015 .60915 .61091 .56468 .61465 .56115 .61377 .59877 Mtetra .508 .291 .52 r .61377 .59877 .61465 .56115 .61847 .56033 .61747 .59206 Mtetra .525 .301 .517 r .61747 .59206 .61847 .56033 .62234 .5614 .62124 .58805 Mtetra .541 .309 .512 r .62124 .58805 .62234 .5614 .62627 .56372 .62508 .58602 Mtetra .555 .314 .506 r .62508 .58602 .62627 .56372 .63023 .56681 .62895 .58539 Mtetra .567 .317 .497 r .62895 .58539 .63023 .56681 .63422 .57034 .63287 .58573 Mtetra .577 .317 .486 r .63287 .58573 .63422 .57034 .63823 .57403 .63681 .58671 Mtetra .586 .315 .472 r .63681 .58671 .63823 .57403 .64226 .57772 .64078 .58808 Mtetra .593 .308 .453 r .64078 .58808 .64226 .57772 .6463 .58126 .64476 .58966 Mtetra .599 .298 .43 r .64476 .58966 .6463 .58126 .65034 .58458 .64876 .59131 Mtetra .602 .282 .4 r .64876 .59131 .65034 .58458 .6544 .58762 .65277 .59294 Mtetra .601 .258 .361 r .65277 .59294 .6544 .58762 .65845 .59035 .65679 .59448 Mtetra .595 .224 .309 r .65679 .59448 .65845 .59035 .66251 .59276 .66082 .59589 Mtetra .582 .177 .239 r .66082 .59589 .66251 .59276 .66656 .59485 .66485 .59713 Mtetra .556 .109 .144 r .66485 .59713 .66656 .59485 .67062 .59663 .66888 .5982 Mtetra .51 .014 .014 r .66888 .5982 .67062 .59663 .67468 .59811 .67292 .59909 Mtetra .431 0 0 r .67292 .59909 .67468 .59811 .67873 .59932 .67696 .59979 Mtetra .307 0 0 r .67696 .59979 .67873 .59932 .68279 .60027 .681 .60032 Mtetra .14 0 0 r .681 .60032 .68279 .60027 .68684 .60098 .68504 .60068 Mtetra .043 .605 .794 r .68504 .60068 .68684 .60098 .69089 .60148 .68909 .60089 Mtetra .203 .702 .913 r .68909 .60089 .69089 .60148 .69495 .60179 .69313 .60095 Mtetra .323 .752 .972 r .69313 .60095 .69495 .60179 .699 .60192 .69718 .60088 Mtetra .408 .774 .994 r .69718 .60088 .699 .60192 .70306 .6019 .70123 .60068 Mtetra .467 .781 .998 r .70123 .60068 .70306 .6019 .70711 .60174 .70528 .60038 Mtetra .508 .781 .994 r .70528 .60038 .70711 .60174 .71117 .60146 .70934 .59997 Mtetra .538 .779 .987 r .70934 .59997 .71117 .60146 .71523 .60107 .7134 .59948 Mtetra .561 .776 .979 r .7134 .59948 .71523 .60107 .71929 .60058 .71746 .5989 Mtetra .578 .772 .971 r .71746 .5989 .71929 .60058 .72335 .60001 .72152 .59826 Mtetra .591 .769 .965 r .72152 .59826 .72335 .60001 .72742 .59936 .72559 .59755 Mtetra .601 .767 .959 r .72559 .59755 .72742 .59936 .73149 .59865 .72966 .59679 Mtetra .61 .764 .953 r .72966 .59679 .73149 .59865 .73556 .59788 .73373 .59598 Mtetra .617 .762 .949 r .73373 .59598 .73556 .59788 .73963 .59706 .73781 .59512 Mtetra .622 .761 .945 r .73781 .59512 .73963 .59706 .74371 .5962 .74189 .59422 Mtetra .627 .759 .942 r .74189 .59422 .74371 .5962 .7478 .59529 .74598 .59329 Mtetra .631 .758 .939 r .74598 .59329 .7478 .59529 .75189 .59436 .75007 .59233 Mtetra .634 .757 .936 r .75007 .59233 .75189 .59436 .75598 .59339 .75416 .59135 Mtetra .637 .756 .934 r .75416 .59135 .75598 .59339 .76008 .5924 .75826 .59033 Mtetra .64 .756 .933 r .75826 .59033 .76008 .5924 .76418 .59138 .76237 .5893 Mtetra .642 .755 .931 r .76237 .5893 .76418 .59138 .76828 .59034 .76648 .58825 Mtetra .643 .755 .93 r .76648 .58825 .76828 .59034 .7724 .58929 .7706 .58719 Mtetra .645 .754 .929 r .7706 .58719 .7724 .58929 .77651 .58822 .77472 .58611 Mtetra .646 .754 .928 r .77472 .58611 .77651 .58822 .78063 .58713 .77884 .58501 Mtetra .647 .754 .927 r .77884 .58501 .78063 .58713 .78476 .58604 .78297 .58391 Mtetra .648 .754 .926 r .78297 .58391 .78476 .58604 .78889 .58493 .78711 .58279 Mtetra .649 .754 .926 r .78711 .58279 .78889 .58493 .79303 .58381 .79125 .58167 Mtetra .65 .753 .925 r .79125 .58167 .79303 .58381 .79718 .58268 .7954 .58053 Mtetra .7954 .58053 .79718 .58268 .80133 .58155 .79955 .57939 Mtetra .651 .753 .924 r .79955 .57939 .80133 .58155 .80548 .58041 .80371 .57825 Mtetra .80371 .57825 .80548 .58041 .80964 .57926 .80787 .5771 Mtetra .652 .753 .924 r .80787 .5771 .80964 .57926 .81381 .57811 .81204 .57594 Mtetra .81204 .57594 .81381 .57811 .81798 .57695 .81622 .57478 Mtetra .652 .753 .923 r .81622 .57478 .81798 .57695 .82216 .57579 .8204 .57361 Mtetra .653 .753 .923 r .8204 .57361 .82216 .57579 .82634 .57462 .82459 .57244 Mtetra .82459 .57244 .82634 .57462 .83053 .57345 .82878 .57126 Mtetra .82878 .57126 .83053 .57345 .83472 .57228 .83298 .57008 Mtetra .83298 .57008 .83472 .57228 .83893 .5711 .83719 .5689 Mtetra .83719 .5689 .83893 .5711 .84313 .56992 .8414 .56772 Mtetra .654 .753 .923 r .8414 .56772 .84313 .56992 .84735 .56873 .84562 .56653 Mtetra .84562 .56653 .84735 .56873 .85157 .56755 .84984 .56534 Mtetra .84984 .56534 .85157 .56755 .85579 .56636 .85407 .56414 Mtetra .85407 .56414 .85579 .56636 .86002 .56516 .85831 .56295 Mtetra .85831 .56295 .86002 .56516 .86426 .56397 .86255 .56175 Mtetra .654 .753 .922 r .86255 .56175 .86426 .56397 .8685 .56277 .8668 .56055 Mtetra .8668 .56055 .8685 .56277 .87275 .56157 .87105 .55935 Mtetra .87105 .55935 .87275 .56157 .87701 .56037 .87531 .55814 Mtetra .87531 .55814 .87701 .56037 .88127 .55916 .87958 .55693 Mtetra .87958 .55693 .88127 .55916 .88554 .55796 .88385 .55572 Mtetra .88385 .55572 .88554 .55796 .88981 .55675 .88813 .55451 Mtetra .88813 .55451 .88981 .55675 .89409 .55554 .89241 .5533 Mtetra .89241 .5533 .89409 .55554 .89838 .55433 .89671 .55208 Mtetra .89671 .55208 .89838 .55433 .90267 .55311 .901 .55086 Mtetra .901 .55086 .90267 .55311 .90697 .5519 .90531 .54964 Mtetra .90531 .54964 .90697 .5519 .91128 .55068 .90962 .54842 Mtetra .90962 .54842 .91128 .55068 .91559 .54946 .91393 .5472 Mtetra .91393 .5472 .91559 .54946 .91991 .54824 .91826 .54597 Mtetra .91826 .54597 .91991 .54824 .92423 .54701 .92259 .54475 Mtetra .92259 .54475 .92423 .54701 .92856 .54579 .92692 .54352 Mtetra .92692 .54352 .92856 .54579 .9329 .54456 .93126 .54229 Mtetra .93126 .54229 .9329 .54456 .93724 .54333 .93561 .54105 Mtetra .93561 .54105 .93724 .54333 .94159 .5421 .93997 .53982 Mtetra .93997 .53982 .94159 .5421 .94595 .54086 .94433 .53858 Mtetra .94433 .53858 .94595 .54086 .95031 .53963 .9487 .53734 Mtetra .9487 .53734 .95031 .53963 .95468 .53839 .95307 .5361 Mtetra .95307 .5361 .95468 .53839 .95906 .53715 .95745 .53486 Mtetra .95745 .53486 .95906 .53715 .96344 .53591 .96184 .53362 Mtetra .96184 .53362 .96344 .53591 .96783 .53467 .96623 .53237 Mtetra .96623 .53237 .96783 .53467 .97222 .53342 .97063 .53112 Mtetra .645 .763 .934 r .39573 .78566 .39794 .78747 .40895 .57227 .39943 .78467 Mtetra 0 0 .324 r .39943 .78467 .40895 .57227 .41244 .5712 .41038 .56923 Mtetra .664 .741 .907 r .47756 .54866 .47957 .55068 .48048 .7655 .48115 .54757 Mtetra 0 0 .391 r .48115 .54757 .48048 .7655 .48429 .76449 .48216 .76261 Mtetra .645 .763 .934 r .5912 .73353 .59324 .73549 .5969 .71914 .59518 .73247 Mtetra .53 .267 .461 r .59518 .73247 .5969 .71914 .59998 .67692 .59916 .73141 Mtetra .343 .168 .494 r .59916 .73141 .59998 .67692 .60323 .64632 .60252 .70193 Mtetra .347 .172 .496 r .60252 .70193 .60323 .64632 .60663 .62445 .60582 .6722 Mtetra .384 .198 .501 r .60582 .6722 .60663 .62445 .61015 .60915 .60924 .65004 Mtetra .415 .219 .504 r .60924 .65004 .61015 .60915 .61377 .59877 .61276 .63369 Mtetra .441 .236 .504 r .61276 .63369 .61377 .59877 .61747 .59206 .61638 .62178 Mtetra .464 .249 .502 r .61638 .62178 .61747 .59206 .62124 .58805 .62007 .61326 Mtetra .483 .259 .498 r .62007 .61326 .62124 .58805 .62508 .58602 .62382 .60731 Mtetra .499 .266 .492 r .62382 .60731 .62508 .58602 .62895 .58539 .62763 .60328 Mtetra .513 .27 .484 r .62763 .60328 .62895 .58539 .63287 .58573 .63147 .60069 Mtetra .525 .271 .473 r .63147 .60069 .63287 .58573 .63681 .58671 .63536 .59915 Mtetra .535 .269 .458 r .63536 .59915 .63681 .58671 .64078 .58808 .63927 .59835 Mtetra .543 .262 .44 r .63927 .59835 .64078 .58808 .64476 .58966 .64321 .59807 Mtetra .547 .251 .416 r .64321 .59807 .64476 .58966 .64876 .59131 .64717 .59813 Mtetra .549 .233 .385 r .64717 .59813 .64876 .59131 .65277 .59294 .65114 .5984 Mtetra .547 .208 .346 r .65114 .5984 .65277 .59294 .65679 .59448 .65513 .59878 Mtetra .539 .172 .293 r .65513 .59878 .65679 .59448 .66082 .59589 .65912 .5992 Mtetra .522 .122 .222 r .65912 .5992 .66082 .59589 .66485 .59713 .66313 .59961 Mtetra .493 .051 .127 r .66313 .59961 .66485 .59713 .66888 .5982 .66714 .59997 Mtetra .443 0 0 r .66714 .59997 .66888 .5982 .67292 .59909 .67116 .60025 Mtetra .362 0 0 r .67116 .60025 .67292 .59909 .67696 .59979 .67518 .60045 Mtetra .24 0 0 r .67518 .60045 .67696 .59979 .681 .60032 .67921 .60055 Mtetra .081 0 0 r .67921 .60055 .681 .60032 .68504 .60068 .68325 .60054 Mtetra .09 .646 .788 r .68325 .60054 .68504 .60068 .68909 .60089 .68728 .60044 Mtetra .238 .736 .908 r .68728 .60044 .68909 .60089 .69313 .60095 .69132 .60024 Mtetra .35 .782 .97 r .69132 .60024 .69313 .60095 .69718 .60088 .69536 .59995 Mtetra .429 .799 .994 r .69536 .59995 .69718 .60088 .70123 .60068 .69941 .59957 Mtetra .484 .803 1 r .69941 .59957 .70123 .60068 .70528 .60038 .70346 .59911 Mtetra .522 .8 .996 r .70346 .59911 .70528 .60038 .70934 .59997 .70751 .59857 Mtetra .549 .795 .99 r .70751 .59857 .70934 .59997 .7134 .59948 .71157 .59796 Mtetra .569 .789 .982 r .71157 .59796 .7134 .59948 .71746 .5989 .71563 .59729 Mtetra .585 .784 .974 r .71563 .59729 .71746 .5989 .72152 .59826 .71969 .59657 Mtetra .597 .779 .967 r .71969 .59657 .72152 .59826 .72559 .59755 .72376 .59579 Mtetra .606 .775 .961 r .72376 .59579 .72559 .59755 .72966 .59679 .72783 .59497 Mtetra .614 .771 .956 r .72783 .59497 .72966 .59679 .73373 .59598 .7319 .59411 Mtetra .62 .768 .951 r .7319 .59411 .73373 .59598 .73781 .59512 .73598 .59321 Mtetra .625 .766 .947 r .73598 .59321 .73781 .59512 .74189 .59422 .74007 .59228 Mtetra .63 .764 .943 r .74007 .59228 .74189 .59422 .74598 .59329 .74415 .59132 Mtetra .633 .762 .94 r .74415 .59132 .74598 .59329 .75007 .59233 .74825 .59033 Mtetra .636 .76 .938 r .74825 .59033 .75007 .59233 .75416 .59135 .75235 .58932 Mtetra .639 .759 .935 r .75235 .58932 .75416 .59135 .75826 .59033 .75645 .58829 Mtetra .641 .758 .934 r .75645 .58829 .75826 .59033 .76237 .5893 .76056 .58724 Mtetra .643 .757 .932 r .76056 .58724 .76237 .5893 .76648 .58825 .76467 .58617 Mtetra .644 .757 .931 r .76467 .58617 .76648 .58825 .7706 .58719 .76879 .58509 Mtetra .646 .756 .929 r .76879 .58509 .7706 .58719 .77472 .58611 .77291 .584 Mtetra .647 .756 .928 r .77291 .584 .77472 .58611 .77884 .58501 .77704 .58289 Mtetra .648 .755 .928 r .77704 .58289 .77884 .58501 .78297 .58391 .78118 .58178 Mtetra .649 .755 .927 r .78118 .58178 .78297 .58391 .78711 .58279 .78532 .58066 Mtetra .65 .755 .926 r .78532 .58066 .78711 .58279 .79125 .58167 .78946 .57952 Mtetra .65 .754 .926 r .78946 .57952 .79125 .58167 .7954 .58053 .79362 .57838 Mtetra .651 .754 .925 r .79362 .57838 .7954 .58053 .79955 .57939 .79777 .57724 Mtetra .79777 .57724 .79955 .57939 .80371 .57825 .80194 .57609 Mtetra .652 .754 .924 r .80194 .57609 .80371 .57825 .80787 .5771 .8061 .57493 Mtetra .8061 .57493 .80787 .5771 .81204 .57594 .81028 .57376 Mtetra .81028 .57376 .81204 .57594 .81622 .57478 .81446 .5726 Mtetra .653 .753 .924 r .81446 .5726 .81622 .57478 .8204 .57361 .81864 .57143 Mtetra .653 .753 .923 r .81864 .57143 .8204 .57361 .82459 .57244 .82284 .57025 Mtetra .82284 .57025 .82459 .57244 .82878 .57126 .82703 .56907 Mtetra .82703 .56907 .82878 .57126 .83298 .57008 .83124 .56789 Mtetra .83124 .56789 .83298 .57008 .83719 .5689 .83545 .5667 Mtetra .654 .753 .923 r .83545 .5667 .83719 .5689 .8414 .56772 .83966 .56551 Mtetra .83966 .56551 .8414 .56772 .84562 .56653 .84389 .56432 Mtetra .84389 .56432 .84562 .56653 .84984 .56534 .84811 .56313 Mtetra .84811 .56313 .84984 .56534 .85407 .56414 .85235 .56193 Mtetra .85235 .56193 .85407 .56414 .85831 .56295 .85659 .56073 Mtetra .85659 .56073 .85831 .56295 .86255 .56175 .86083 .55953 Mtetra .86083 .55953 .86255 .56175 .8668 .56055 .86509 .55832 Mtetra .654 .753 .922 r .86509 .55832 .8668 .56055 .87105 .55935 .86935 .55712 Mtetra .86935 .55712 .87105 .55935 .87531 .55814 .87361 .55591 Mtetra .87361 .55591 .87531 .55814 .87958 .55693 .87788 .5547 Mtetra .87788 .5547 .87958 .55693 .88385 .55572 .88216 .55348 Mtetra .88216 .55348 .88385 .55572 .88813 .55451 .88644 .55227 Mtetra .88644 .55227 .88813 .55451 .89241 .5533 .89073 .55105 Mtetra .89073 .55105 .89241 .5533 .89671 .55208 .89503 .54983 Mtetra .89503 .54983 .89671 .55208 .901 .55086 .89933 .54861 Mtetra .89933 .54861 .901 .55086 .90531 .54964 .90364 .54739 Mtetra .90364 .54739 .90531 .54964 .90962 .54842 .90795 .54616 Mtetra .90795 .54616 .90962 .54842 .91393 .5472 .91227 .54493 Mtetra .91227 .54493 .91393 .5472 .91826 .54597 .9166 .54371 Mtetra .9166 .54371 .91826 .54597 .92259 .54475 .92093 .54247 Mtetra .92093 .54247 .92259 .54475 .92692 .54352 .92528 .54124 Mtetra .92528 .54124 .92692 .54352 .93126 .54229 .92962 .54001 Mtetra .92962 .54001 .93126 .54229 .93561 .54105 .93397 .53877 Mtetra .93397 .53877 .93561 .54105 .93997 .53982 .93833 .53753 Mtetra .93833 .53753 .93997 .53982 .94433 .53858 .9427 .53629 Mtetra .9427 .53629 .94433 .53858 .9487 .53734 .94707 .53505 Mtetra .94707 .53505 .9487 .53734 .95307 .5361 .95145 .53381 Mtetra .95145 .53381 .95307 .5361 .95745 .53486 .95584 .53256 Mtetra .95584 .53256 .95745 .53486 .96184 .53362 .96023 .53131 Mtetra .96023 .53131 .96184 .53362 .96623 .53237 .96463 .53006 Mtetra .96463 .53006 .96623 .53237 .97063 .53112 .96903 .52881 Mtetra .645 .763 .934 r .39721 .78284 .39943 .78467 .41038 .56923 .40092 .78185 Mtetra .568 .357 .554 r .40092 .78185 .41038 .56923 .41387 .56816 .40464 .78086 Mtetra 0 0 .326 r .40464 .78086 .41387 .56816 .41737 .56709 .4153 .56511 Mtetra .664 .741 .907 r .47913 .54554 .48115 .54757 .48216 .76261 .48272 .54444 Mtetra .572 .368 .565 r .48272 .54444 .48216 .76261 .48598 .76159 .48632 .54334 Mtetra 0 0 .392 r .48632 .54334 .48598 .76159 .4898 .76057 .48767 .75868 Mtetra .645 .763 .934 r .59713 .72945 .59916 .73141 .60252 .70193 .60112 .72838 Mtetra .339 .144 .464 r .60112 .72838 .60252 .70193 .60582 .6722 .60481 .7141 Mtetra .271 .107 .467 r .60481 .7141 .60582 .6722 .60924 .65004 .60817 .68637 Mtetra .316 .138 .474 r .60817 .68637 .60924 .65004 .61276 .63369 .61163 .66506 Mtetra .353 .163 .478 r .61163 .66506 .61276 .63369 .61638 .62178 .61518 .64877 Mtetra .384 .182 .478 r .61518 .64877 .61638 .62178 .62007 .61326 .6188 .63637 Mtetra .409 .196 .476 r .6188 .63637 .62007 .61326 .62382 .60731 .62249 .62701 Mtetra .431 .206 .471 r .62249 .62701 .62382 .60731 .62763 .60328 .62624 .62 Mtetra .448 .213 .463 r .62624 .62 .62763 .60328 .63147 .60069 .63003 .6148 Mtetra .463 .215 .452 r .63003 .6148 .63147 .60069 .63536 .59915 .63387 .61098 Mtetra .474 .213 .437 r .63387 .61098 .63536 .59915 .63927 .59835 .63774 .60821 Mtetra .482 .206 .418 r .63774 .60821 .63927 .59835 .64321 .59807 .64163 .60621 Mtetra .487 .194 .393 r .64163 .60621 .64321 .59807 .64717 .59813 .64555 .60479 Mtetra .488 .174 .361 r .64555 .60479 .64717 .59813 .65114 .5984 .64949 .60379 Mtetra .483 .146 .319 r .64949 .60379 .65114 .5984 .65513 .59878 .65345 .60307 Mtetra .472 .107 .263 r .65345 .60307 .65513 .59878 .65912 .5992 .65742 .60255 Mtetra .452 .052 .189 r .65742 .60255 .65912 .5992 .66313 .59961 .6614 .60215 Mtetra .416 0 .09 r .6614 .60215 .66313 .59961 .66714 .59997 .6654 .60182 Mtetra .36 0 0 r .6654 .60182 .66714 .59997 .67116 .60025 .6694 .60151 Mtetra .272 0 0 r .6694 .60151 .67116 .60025 .67518 .60045 .67341 .6012 Mtetra .149 0 0 r .67341 .6012 .67518 .60045 .67921 .60055 .67742 .60087 Mtetra 0 0 0 r .67742 .60087 .67921 .60055 .68325 .60054 .68145 .6005 Mtetra .16 .7 .801 r .68145 .6005 .68325 .60054 .68728 .60044 .68547 .60009 Mtetra .292 .776 .912 r .68547 .60009 .68728 .60044 .69132 .60024 .6895 .59962 Mtetra .39 .812 .97 r .6895 .59962 .69132 .60024 .69536 .59995 .69354 .5991 Mtetra .459 .824 .994 r .69354 .5991 .69536 .59995 .69941 .59957 .69758 .59853 Mtetra .506 .823 1 r .69758 .59853 .69941 .59957 .70346 .59911 .70163 .59791 Mtetra .539 .816 .997 r .70163 .59791 .70346 .59911 .70751 .59857 .70568 .59723 Mtetra .563 .809 .991 r .70568 .59723 .70751 .59857 .71157 .59796 .70973 .5965 Mtetra .581 .801 .983 r .70973 .5965 .71157 .59796 .71563 .59729 .71379 .59573 Mtetra .594 .794 .976 r .71379 .59573 .71563 .59729 .71969 .59657 .71785 .59492 Mtetra .604 .788 .969 r .71785 .59492 .71969 .59657 .72376 .59579 .72192 .59407 Mtetra .613 .782 .962 r .72192 .59407 .72376 .59579 .72783 .59497 .72599 .59319 Mtetra .619 .778 .957 r .72599 .59319 .72783 .59497 .7319 .59411 .73007 .59227 Mtetra .624 .774 .952 r .73007 .59227 .7319 .59411 .73598 .59321 .73415 .59133 Mtetra .629 .77 .948 r .73415 .59133 .73598 .59321 .74007 .59228 .73824 .59035 Mtetra .633 .768 .944 r .73824 .59035 .74007 .59228 .74415 .59132 .74233 .58936 Mtetra .636 .765 .941 r .74233 .58936 .74415 .59132 .74825 .59033 .74642 .58834 Mtetra .638 .763 .939 r .74642 .58834 .74825 .59033 .75235 .58932 .75053 .5873 Mtetra .641 .762 .936 r .75053 .5873 .75235 .58932 .75645 .58829 .75463 .58625 Mtetra .642 .76 .934 r .75463 .58625 .75645 .58829 .76056 .58724 .75874 .58518 Mtetra .644 .759 .933 r .75874 .58518 .76056 .58724 .76467 .58617 .76286 .5841 Mtetra .645 .758 .931 r .76286 .5841 .76467 .58617 .76879 .58509 .76698 .583 Mtetra .647 .758 .93 r .76698 .583 .76879 .58509 .77291 .584 .77111 .5819 Mtetra .648 .757 .929 r .77111 .5819 .77291 .584 .77704 .58289 .77524 .58078 Mtetra .649 .756 .928 r .77524 .58078 .77704 .58289 .78118 .58178 .77938 .57966 Mtetra .649 .756 .927 r .77938 .57966 .78118 .58178 .78532 .58066 .78352 .57852 Mtetra .65 .755 .927 r .78352 .57852 .78532 .58066 .78946 .57952 .78767 .57738 Mtetra .651 .755 .926 r .78767 .57738 .78946 .57952 .79362 .57838 .79183 .57623 Mtetra .651 .755 .925 r .79183 .57623 .79362 .57838 .79777 .57724 .79599 .57508 Mtetra .652 .754 .925 r .79599 .57508 .79777 .57724 .80194 .57609 .80016 .57392 Mtetra .80016 .57392 .80194 .57609 .8061 .57493 .80433 .57276 Mtetra .652 .754 .924 r .80433 .57276 .8061 .57493 .81028 .57376 .80851 .57159 Mtetra .653 .754 .924 r .80851 .57159 .81028 .57376 .81446 .5726 .81269 .57041 Mtetra .81269 .57041 .81446 .5726 .81864 .57143 .81688 .56924 Mtetra .81688 .56924 .81864 .57143 .82284 .57025 .82108 .56806 Mtetra .653 .754 .923 r .82108 .56806 .82284 .57025 .82703 .56907 .82528 .56687 Mtetra .653 .753 .923 r .82528 .56687 .82703 .56907 .83124 .56789 .82949 .56569 Mtetra .82949 .56569 .83124 .56789 .83545 .5667 .8337 .5645 Mtetra .654 .753 .923 r .8337 .5645 .83545 .5667 .83966 .56551 .83792 .5633 Mtetra .83792 .5633 .83966 .56551 .84389 .56432 .84215 .56211 Mtetra .84215 .56211 .84389 .56432 .84811 .56313 .84638 .56091 Mtetra .84638 .56091 .84811 .56313 .85235 .56193 .85062 .55971 Mtetra .85062 .55971 .85235 .56193 .85659 .56073 .85486 .5585 Mtetra .85486 .5585 .85659 .56073 .86083 .55953 .85911 .5573 Mtetra .85911 .5573 .86083 .55953 .86509 .55832 .86337 .55609 Mtetra .86337 .55609 .86509 .55832 .86935 .55712 .86763 .55488 Mtetra .654 .753 .922 r .86763 .55488 .86935 .55712 .87361 .55591 .8719 .55367 Mtetra .8719 .55367 .87361 .55591 .87788 .5547 .87618 .55245 Mtetra .87618 .55245 .87788 .5547 .88216 .55348 .88046 .55124 Mtetra .88046 .55124 .88216 .55348 .88644 .55227 .88475 .55002 Mtetra .88475 .55002 .88644 .55227 .89073 .55105 .88904 .5488 Mtetra .88904 .5488 .89073 .55105 .89503 .54983 .89334 .54757 Mtetra .89334 .54757 .89503 .54983 .89933 .54861 .89765 .54635 Mtetra .89765 .54635 .89933 .54861 .90364 .54739 .90196 .54512 Mtetra .90196 .54512 .90364 .54739 .90795 .54616 .90628 .54389 Mtetra .90628 .54389 .90795 .54616 .91227 .54493 .91061 .54266 Mtetra .91061 .54266 .91227 .54493 .9166 .54371 .91494 .54143 Mtetra .91494 .54143 .9166 .54371 .92093 .54247 .91928 .5402 Mtetra .91928 .5402 .92093 .54247 .92528 .54124 .92362 .53896 Mtetra .92362 .53896 .92528 .54124 .92962 .54001 .92798 .53772 Mtetra .92798 .53772 .92962 .54001 .93397 .53877 .93233 .53648 Mtetra .93233 .53648 .93397 .53877 .93833 .53753 .9367 .53524 Mtetra .9367 .53524 .93833 .53753 .9427 .53629 .94107 .534 Mtetra .94107 .534 .9427 .53629 .94707 .53505 .94544 .53275 Mtetra .94544 .53275 .94707 .53505 .95145 .53381 .94983 .5315 Mtetra .94983 .5315 .95145 .53381 .95584 .53256 .95422 .53026 Mtetra .95422 .53026 .95584 .53256 .96023 .53131 .95861 .529 Mtetra .95861 .529 .96023 .53131 .96463 .53006 .96302 .52775 Mtetra .96302 .52775 .96463 .53006 .96903 .52881 .96743 .5265 Mtetra .645 .763 .934 r .40242 .77903 .40464 .78086 .4153 .56511 .40615 .77803 Mtetra 0 0 .326 r .40615 .77803 .4153 .56511 .41881 .56403 .41674 .56204 Mtetra .664 .741 .907 r .48431 .5413 .48632 .54334 .48767 .75868 .48791 .54019 Mtetra 0 0 .392 r .48791 .54019 .48767 .75868 .4915 .75766 .48937 .75576 Mtetra .645 .763 .934 r .59908 .72641 .60112 .72838 .60481 .7141 .60308 .72534 Mtetra .232 .005 .35 r .60308 .72534 .60481 .7141 .60817 .68637 .60693 .71739 Mtetra .17 .02 .421 r .60693 .71739 .60817 .68637 .61163 .66506 .61034 .69224 Mtetra .227 .059 .433 r .61034 .69224 .61163 .66506 .61518 .64877 .61385 .67244 Mtetra .272 .088 .439 r .61385 .67244 .61518 .64877 .6188 .63637 .61743 .65688 Mtetra .309 .111 .44 r .61743 .65688 .6188 .63637 .62249 .62701 .62108 .64469 Mtetra .339 .126 .438 r .62108 .64469 .62249 .62701 .62624 .62 .62478 .63515 Mtetra .363 .137 .431 r .62478 .63515 .62624 .62 .63003 .6148 .62853 .6277 Mtetra .381 .141 .42 r .62853 .6277 .63003 .6148 .63387 .61098 .63233 .6219 Mtetra .396 .141 .405 r .63233 .6219 .63387 .61098 .63774 .60821 .63616 .61738 Mtetra .406 .134 .385 r .63616 .61738 .63774 .60821 .64163 .60621 .64002 .61385 Mtetra .411 .12 .358 r .64002 .61385 .64163 .60621 .64555 .60479 .64391 .6111 Mtetra .411 .099 .323 r .64391 .6111 .64555 .60479 .64949 .60379 .64783 .60893 Mtetra .404 .067 .276 r .64783 .60893 .64949 .60379 .65345 .60307 .65176 .6072 Mtetra .389 .022 .216 r .65176 .6072 .65345 .60307 .65742 .60255 .6557 .6058 Mtetra .363 0 .135 r .6557 .6058 .65742 .60255 .6614 .60215 .65967 .60464 Mtetra .32 0 .028 r .65967 .60464 .6614 .60215 .6654 .60182 .66364 .60365 Mtetra .254 0 0 r .66364 .60365 .6654 .60182 .6694 .60151 .66763 .60278 Mtetra .158 0 0 r .66763 .60278 .6694 .60151 .67341 .6012 .67162 .60199 Mtetra .032 0 0 r .67162 .60199 .67341 .6012 .67742 .60087 .67563 .60124 Mtetra 0 0 0 r .67563 .60124 .67742 .60087 .68145 .6005 .67964 .60051 Mtetra .248 .761 .829 r .67964 .60051 .68145 .6005 .68547 .60009 .68366 .59979 Mtetra .359 .819 .924 r .68366 .59979 .68547 .60009 .6895 .59962 .68769 .59906 Mtetra .439 .843 .974 r .68769 .59906 .6895 .59962 .69354 .5991 .69172 .59831 Mtetra .495 .846 .994 r .69172 .59831 .69354 .5991 .69758 .59853 .69575 .59754 Mtetra .533 .84 .999 r .69575 .59754 .69758 .59853 .70163 .59791 .6998 .59675 Mtetra .56 .831 .996 r .6998 .59675 .70163 .59791 .70568 .59723 .70384 .59593 Mtetra .58 .821 .99 r .70384 .59593 .70568 .59723 .70973 .5965 .7079 .59509 Mtetra .594 .811 .983 r .7079 .59509 .70973 .5965 .71379 .59573 .71195 .59421 Mtetra .605 .803 .976 r .71195 .59421 .71379 .59573 .71785 .59492 .71602 .59331 Mtetra .613 .795 .969 r .71602 .59331 .71785 .59492 .72192 .59407 .72008 .59238 Mtetra .62 .789 .963 r .72008 .59238 .72192 .59407 .72599 .59319 .72416 .59143 Mtetra .625 .783 .957 r .72416 .59143 .72599 .59319 .73007 .59227 .72823 .59046 Mtetra .629 .778 .953 r .72823 .59046 .73007 .59227 .73415 .59133 .73232 .58946 Mtetra .633 .775 .948 r .73232 .58946 .73415 .59133 .73824 .59035 .7364 .58845 Mtetra .636 .771 .945 r .7364 .58845 .73824 .59035 .74233 .58936 .7405 .58741 Mtetra .639 .769 .942 r .7405 .58741 .74233 .58936 .74642 .58834 .7446 .58636 Mtetra .641 .766 .939 r .7446 .58636 .74642 .58834 .75053 .5873 .7487 .5853 Mtetra .643 .764 .937 r .7487 .5853 .75053 .5873 .75463 .58625 .75281 .58422 Mtetra .644 .763 .935 r .75281 .58422 .75463 .58625 .75874 .58518 .75692 .58313 Mtetra .646 .761 .933 r .75692 .58313 .75874 .58518 .76286 .5841 .76104 .58203 Mtetra .647 .76 .932 r .76104 .58203 .76286 .5841 .76698 .583 .76517 .58092 Mtetra .648 .759 .93 r .76517 .58092 .76698 .583 .77111 .5819 .7693 .5798 Mtetra .649 .758 .929 r .7693 .5798 .77111 .5819 .77524 .58078 .77344 .57867 Mtetra .649 .757 .928 r .77344 .57867 .77524 .58078 .77938 .57966 .77758 .57753 Mtetra .65 .757 .927 r .77758 .57753 .77938 .57966 .78352 .57852 .78172 .57639 Mtetra .651 .756 .927 r .78172 .57639 .78352 .57852 .78767 .57738 .78588 .57524 Mtetra .651 .756 .926 r .78588 .57524 .78767 .57738 .79183 .57623 .79004 .57408 Mtetra .652 .755 .926 r .79004 .57408 .79183 .57623 .79599 .57508 .7942 .57292 Mtetra .652 .755 .925 r .7942 .57292 .79599 .57508 .80016 .57392 .79837 .57175 Mtetra .79837 .57175 .80016 .57392 .80433 .57276 .80255 .57058 Mtetra .653 .754 .924 r .80255 .57058 .80433 .57276 .80851 .57159 .80673 .56941 Mtetra .80673 .56941 .80851 .57159 .81269 .57041 .81092 .56823 Mtetra .81092 .56823 .81269 .57041 .81688 .56924 .81511 .56705 Mtetra .81511 .56705 .81688 .56924 .82108 .56806 .81932 .56586 Mtetra .653 .754 .923 r .81932 .56586 .82108 .56806 .82528 .56687 .82352 .56467 Mtetra .82352 .56467 .82528 .56687 .82949 .56569 .82773 .56348 Mtetra .654 .754 .923 r .82773 .56348 .82949 .56569 .8337 .5645 .83195 .56228 Mtetra .83195 .56228 .8337 .5645 .83792 .5633 .83618 .56109 Mtetra .654 .753 .923 r .83618 .56109 .83792 .5633 .84215 .56211 .84041 .55989 Mtetra .84041 .55989 .84215 .56211 .84638 .56091 .84464 .55868 Mtetra .84464 .55868 .84638 .56091 .85062 .55971 .84889 .55748 Mtetra .84889 .55748 .85062 .55971 .85486 .5585 .85314 .55627 Mtetra .85314 .55627 .85486 .5585 .85911 .5573 .85739 .55506 Mtetra .85739 .55506 .85911 .5573 .86337 .55609 .86165 .55385 Mtetra .86165 .55385 .86337 .55609 .86763 .55488 .86592 .55264 Mtetra .654 .753 .922 r .86592 .55264 .86763 .55488 .8719 .55367 .87019 .55142 Mtetra .87019 .55142 .8719 .55367 .87618 .55245 .87447 .5502 Mtetra .87447 .5502 .87618 .55245 .88046 .55124 .87876 .54898 Mtetra .87876 .54898 .88046 .55124 .88475 .55002 .88305 .54776 Mtetra .88305 .54776 .88475 .55002 .88904 .5488 .88735 .54654 Mtetra .88735 .54654 .88904 .5488 .89334 .54757 .89166 .54531 Mtetra .89166 .54531 .89334 .54757 .89765 .54635 .89597 .54408 Mtetra .89597 .54408 .89765 .54635 .90196 .54512 .90028 .54285 Mtetra .90028 .54285 .90196 .54512 .90628 .54389 .90461 .54162 Mtetra .90461 .54162 .90628 .54389 .91061 .54266 .90894 .54039 Mtetra .90894 .54039 .91061 .54266 .91494 .54143 .91328 .53915 Mtetra .91328 .53915 .91494 .54143 .91928 .5402 .91762 .53791 Mtetra .91762 .53791 .91928 .5402 .92362 .53896 .92197 .53667 Mtetra .92197 .53667 .92362 .53896 .92798 .53772 .92632 .53543 Mtetra .92632 .53543 .92798 .53772 .93233 .53648 .93069 .53419 Mtetra .93069 .53419 .93233 .53648 .9367 .53524 .93506 .53294 Mtetra .93506 .53294 .9367 .53524 .94107 .534 .93943 .5317 Mtetra .93943 .5317 .94107 .534 .94544 .53275 .94381 .53045 Mtetra .94381 .53045 .94544 .53275 .94983 .5315 .9482 .5292 Mtetra .9482 .5292 .94983 .5315 .95422 .53026 .9526 .52794 Mtetra .9526 .52794 .95422 .53026 .95861 .529 .957 .52669 Mtetra .957 .52669 .95861 .529 .96302 .52775 .96141 .52543 Mtetra .96141 .52543 .96302 .52775 .96743 .5265 .96582 .52418 Mtetra .645 .763 .934 r .40393 .77619 .40615 .77803 .41674 .56204 .40766 .77519 Mtetra 0 0 .326 r .40766 .77519 .41674 .56204 .42026 .56097 .41819 .55897 Mtetra .664 .741 .907 r .4859 .53814 .48791 .54019 .48937 .75576 .48951 .53703 Mtetra .572 .367 .565 r .48951 .53703 .48937 .75576 .49321 .75474 .49312 .53592 Mtetra 0 0 .394 r .49312 .53592 .49321 .75474 .49706 .75371 .49492 .7518 Mtetra .645 .763 .934 r .60104 .72336 .60308 .72534 .60693 .71739 .60505 .72229 Mtetra 0 0 0 r .60505 .72229 .60693 .71739 .61034 .69224 .60892 .71478 Mtetra .034 0 .351 r .60892 .71478 .61034 .69224 .61385 .67244 .6124 .6924 Mtetra .106 0 .37 r .6124 .6924 .61385 .67244 .61743 .65688 .61595 .67442 Mtetra .162 0 .379 r .61595 .67442 .61743 .65688 .62108 .64469 .61957 .65999 Mtetra .207 .013 .382 r .61957 .65999 .62108 .64469 .62478 .63515 .62325 .64841 Mtetra .242 .03 .379 r .62325 .64841 .62478 .63515 .62853 .6277 .62697 .63911 Mtetra .269 .04 .37 r .62697 .63911 .62853 .6277 .63233 .6219 .63074 .63165 Mtetra .289 .042 .355 r .63074 .63165 .63233 .6219 .63616 .61738 .63454 .62564 Mtetra .303 .036 .333 r .63454 .62564 .63616 .61738 .64002 .61385 .63838 .62079 Mtetra .309 .022 .303 r .63838 .62079 .64002 .61385 .64391 .6111 .64224 .61686 Mtetra .309 0 .264 r .64224 .61686 .64391 .6111 .64783 .60893 .64613 .61366 Mtetra .3 0 .212 r .64613 .61366 .64783 .60893 .65176 .6072 .65004 .61102 Mtetra .28 0 .143 r .65004 .61102 .65176 .6072 .6557 .6058 .65397 .60883 Mtetra .245 0 .052 r .65397 .60883 .6557 .6058 .65967 .60464 .65792 .60697 Mtetra .193 0 0 r .65792 .60697 .65967 .60464 .66364 .60365 .66188 .60538 Mtetra .116 0 0 r .66188 .60538 .66364 .60365 .66763 .60278 .66585 .60398 Mtetra .013 0 0 r .66585 .60398 .66763 .60278 .67162 .60199 .66983 .60273 Mtetra 0 0 0 r .66983 .60273 .67162 .60199 .67563 .60124 .67383 .60158 Mtetra .236 .75 .743 r .67383 .60158 .67563 .60124 .67964 .60051 .67783 .60051 Mtetra .347 .824 .866 r .67783 .60051 .67964 .60051 .68366 .59979 .68184 .59949 Mtetra .433 .86 .941 r .68184 .59949 .68366 .59979 .68769 .59906 .68586 .5985 Mtetra .493 .87 .979 r .68586 .5985 .68769 .59906 .69172 .59831 .68989 .59754 Mtetra .535 .865 .994 r .68989 .59754 .69172 .59831 .69575 .59754 .69392 .59658 Mtetra .563 .855 .997 r .69392 .59658 .69575 .59754 .6998 .59675 .69796 .59562 Mtetra .583 .843 .994 r .69796 .59562 .6998 .59675 .70384 .59593 .702 .59466 Mtetra .598 .831 .988 r .702 .59466 .70384 .59593 .7079 .59509 .70605 .59369 Mtetra .608 .819 .981 r .70605 .59369 .7079 .59509 .71195 .59421 .71011 .59271 Mtetra .616 .81 .974 r .71011 .59271 .71195 .59421 .71602 .59331 .71417 .59172 Mtetra .622 .801 .968 r .71417 .59172 .71602 .59331 .72008 .59238 .71824 .59071 Mtetra .627 .794 .962 r .71824 .59071 .72008 .59238 .72416 .59143 .72232 .58969 Mtetra .631 .788 .957 r .72232 .58969 .72416 .59143 .72823 .59046 .72639 .58866 Mtetra .635 .783 .952 r .72639 .58866 .72823 .59046 .73232 .58946 .73048 .58761 Mtetra .637 .778 .948 r .73048 .58761 .73232 .58946 .7364 .58845 .73457 .58655 Mtetra .64 .774 .945 r .73457 .58655 .7364 .58845 .7405 .58741 .73866 .58548 Mtetra .642 .771 .942 r .73866 .58548 .7405 .58741 .7446 .58636 .74276 .5844 Mtetra .643 .769 .939 r .74276 .5844 .7446 .58636 .7487 .5853 .74687 .5833 Mtetra .645 .766 .937 r .74687 .5833 .7487 .5853 .75281 .58422 .75098 .5822 Mtetra .646 .764 .935 r .75098 .5822 .75281 .58422 .75692 .58313 .7551 .58109 Mtetra .647 .763 .933 r .7551 .58109 .75692 .58313 .76104 .58203 .75922 .57997 Mtetra .648 .761 .932 r .75922 .57997 .76104 .58203 .76517 .58092 .76335 .57884 Mtetra .649 .76 .93 r .76335 .57884 .76517 .58092 .7693 .5798 .76748 .5777 Mtetra .65 .759 .929 r .76748 .5777 .7693 .5798 .77344 .57867 .77162 .57656 Mtetra .65 .758 .928 r .77162 .57656 .77344 .57867 .77758 .57753 .77577 .57541 Mtetra .651 .757 .928 r .77577 .57541 .77758 .57753 .78172 .57639 .77992 .57425 Mtetra .651 .757 .927 r .77992 .57425 .78172 .57639 .78588 .57524 .78408 .57309 Mtetra .652 .756 .926 r .78408 .57309 .78588 .57524 .79004 .57408 .78824 .57193 Mtetra .78824 .57193 .79004 .57408 .7942 .57292 .79241 .57076 Mtetra .652 .755 .925 r .79241 .57076 .7942 .57292 .79837 .57175 .79658 .56958 Mtetra .653 .755 .925 r .79658 .56958 .79837 .57175 .80255 .57058 .80077 .56841 Mtetra .653 .755 .924 r .80077 .56841 .80255 .57058 .80673 .56941 .80495 .56722 Mtetra .80495 .56722 .80673 .56941 .81092 .56823 .80914 .56604 Mtetra .653 .754 .924 r .80914 .56604 .81092 .56823 .81511 .56705 .81334 .56485 Mtetra .81334 .56485 .81511 .56705 .81932 .56586 .81755 .56366 Mtetra .81755 .56366 .81932 .56586 .82352 .56467 .82176 .56247 Mtetra .654 .754 .923 r .82176 .56247 .82352 .56467 .82773 .56348 .82597 .56127 Mtetra .82597 .56127 .82773 .56348 .83195 .56228 .8302 .56007 Mtetra .8302 .56007 .83195 .56228 .83618 .56109 .83443 .55887 Mtetra .83443 .55887 .83618 .56109 .84041 .55989 .83866 .55766 Mtetra .654 .753 .923 r .83866 .55766 .84041 .55989 .84464 .55868 .8429 .55646 Mtetra .8429 .55646 .84464 .55868 .84889 .55748 .84715 .55525 Mtetra .84715 .55525 .84889 .55748 .85314 .55627 .8514 .55404 Mtetra .8514 .55404 .85314 .55627 .85739 .55506 .85566 .55282 Mtetra .85566 .55282 .85739 .55506 .86165 .55385 .85993 .55161 Mtetra .85993 .55161 .86165 .55385 .86592 .55264 .8642 .55039 Mtetra .8642 .55039 .86592 .55264 .87019 .55142 .86848 .54917 Mtetra .654 .753 .922 r .86848 .54917 .87019 .55142 .87447 .5502 .87276 .54795 Mtetra .87276 .54795 .87447 .5502 .87876 .54898 .87705 .54672 Mtetra .87705 .54672 .87876 .54898 .88305 .54776 .88135 .5455 Mtetra .88135 .5455 .88305 .54776 .88735 .54654 .88565 .54427 Mtetra .88565 .54427 .88735 .54654 .89166 .54531 .88996 .54304 Mtetra .88996 .54304 .89166 .54531 .89597 .54408 .89428 .54181 Mtetra .89428 .54181 .89597 .54408 .90028 .54285 .8986 .54058 Mtetra .8986 .54058 .90028 .54285 .90461 .54162 .90293 .53934 Mtetra .90293 .53934 .90461 .54162 .90894 .54039 .90726 .5381 Mtetra .90726 .5381 .90894 .54039 .91328 .53915 .91161 .53686 Mtetra .91161 .53686 .91328 .53915 .91762 .53791 .91595 .53562 Mtetra .91595 .53562 .91762 .53791 .92197 .53667 .92031 .53438 Mtetra .92031 .53438 .92197 .53667 .92632 .53543 .92467 .53314 Mtetra .92467 .53314 .92632 .53543 .93069 .53419 .92904 .53189 Mtetra .92904 .53189 .93069 .53419 .93506 .53294 .93341 .53064 Mtetra .93341 .53064 .93506 .53294 .93943 .5317 .93779 .52939 Mtetra .93779 .52939 .93943 .5317 .94381 .53045 .94218 .52814 Mtetra .94218 .52814 .94381 .53045 .9482 .5292 .94657 .52688 Mtetra .94657 .52688 .9482 .5292 .9526 .52794 .95097 .52563 Mtetra .95097 .52563 .9526 .52794 .957 .52669 .95538 .52437 Mtetra .95538 .52437 .957 .52669 .96141 .52543 .95979 .52311 Mtetra .95979 .52311 .96141 .52543 .96582 .52418 .96421 .52185 Mtetra .645 .763 .934 r .40544 .77334 .40766 .77519 .41819 .55897 .40918 .77233 Mtetra .567 .355 .553 r .40918 .77233 .41819 .55897 .42171 .55789 .41292 .77133 Mtetra 0 0 .328 r .41292 .77133 .42171 .55789 .42523 .5568 .42317 .5548 Mtetra .664 .741 .907 r .49111 .53386 .49312 .53592 .49492 .7518 .49473 .53275 Mtetra 0 0 .394 r .49473 .53275 .49492 .7518 .49878 .75077 .49664 .74885 Mtetra .645 .763 .934 r .60301 .7203 .60505 .72229 .60892 .71478 .60703 .71922 Mtetra .799 .827 .353 r .60703 .71922 .60892 .71478 .6124 .6924 .61082 .70842 Mtetra 0 0 .24 r .61082 .70842 .6124 .6924 .61595 .67442 .61437 .68876 Mtetra 0 0 .268 r .61437 .68876 .61595 .67442 .61957 .65999 .61797 .6727 Mtetra .007 0 .284 r .61797 .6727 .61957 .65999 .62325 .64841 .62163 .65958 Mtetra .063 0 .289 r .62163 .65958 .62325 .64841 .62697 .63911 .62534 .64884 Mtetra .105 0 .285 r .62534 .64884 .62697 .63911 .63074 .63165 .62908 .64004 Mtetra .135 0 .271 r .62908 .64004 .63074 .63165 .63454 .62564 .63287 .63282 Mtetra .155 0 .249 r .63287 .63282 .63454 .62564 .63838 .62079 .63669 .62687 Mtetra .166 0 .216 r .63669 .62687 .63838 .62079 .64224 .61686 .64053 .62195 Mtetra .165 0 .171 r .64053 .62195 .64224 .61686 .64613 .61366 .6444 .61786 Mtetra .154 0 .111 r .6444 .61786 .64613 .61366 .65004 .61102 .6483 .61443 Mtetra .128 0 .032 r .6483 .61443 .65004 .61102 .65397 .60883 .65221 .61154 Mtetra .085 0 0 r .65221 .61154 .65397 .60883 .65792 .60697 .65614 .60906 Mtetra .023 0 0 r .65614 .60906 .65792 .60697 .66188 .60538 .66009 .60692 Mtetra 0 0 0 r .66009 .60692 .66188 .60538 .66585 .60398 .66405 .60504 Mtetra .66405 .60504 .66585 .60398 .66983 .60273 .66803 .60337 Mtetra .27 .76 .682 r .66803 .60337 .66983 .60273 .67383 .60158 .67201 .60185 Mtetra .369 .837 .813 r .67201 .60185 .67383 .60158 .67783 .60051 .67601 .60046 Mtetra .449 .879 .903 r .67601 .60046 .67783 .60051 .68184 .59949 .68001 .59916 Mtetra .507 .894 .956 r .68001 .59916 .68184 .59949 .68586 .5985 .68403 .59793 Mtetra .548 .891 .983 r .68403 .59793 .68586 .5985 .68989 .59754 .68805 .59675 Mtetra .575 .88 .993 r .68805 .59675 .68989 .59754 .69392 .59658 .69208 .59561 Mtetra .594 .866 .994 r .69208 .59561 .69392 .59658 .69796 .59562 .69612 .59449 Mtetra .607 .851 .99 r .69612 .59449 .69796 .59562 .702 .59466 .70016 .59339 Mtetra .616 .838 .985 r .70016 .59339 .702 .59466 .70605 .59369 .70421 .5923 Mtetra .623 .826 .978 r .70421 .5923 .70605 .59369 .71011 .59271 .70826 .59121 Mtetra .628 .815 .972 r .70826 .59121 .71011 .59271 .71417 .59172 .71233 .59013 Mtetra .632 .806 .966 r .71233 .59013 .71417 .59172 .71824 .59071 .7164 .58904 Mtetra .635 .798 .961 r .7164 .58904 .71824 .59071 .72232 .58969 .72047 .58796 Mtetra .638 .791 .956 r .72047 .58796 .72232 .58969 .72639 .58866 .72455 .58686 Mtetra .64 .786 .951 r .72455 .58686 .72639 .58866 .73048 .58761 .72864 .58576 Mtetra .642 .781 .948 r .72864 .58576 .73048 .58761 .73457 .58655 .73273 .58466 Mtetra .644 .777 .944 r .73273 .58466 .73457 .58655 .73866 .58548 .73682 .58355 Mtetra .645 .773 .941 r .73682 .58355 .73866 .58548 .74276 .5844 .74093 .58243 Mtetra .646 .771 .939 r .74093 .58243 .74276 .5844 .74687 .5833 .74504 .58131 Mtetra .647 .768 .937 r .74504 .58131 .74687 .5833 .75098 .5822 .74915 .58018 Mtetra .648 .766 .935 r .74915 .58018 .75098 .5822 .7551 .58109 .75327 .57904 Mtetra .649 .764 .933 r .75327 .57904 .7551 .58109 .75922 .57997 .7574 .5779 Mtetra .65 .762 .932 r .7574 .5779 .75922 .57997 .76335 .57884 .76153 .57675 Mtetra .65 .761 .93 r .76153 .57675 .76335 .57884 .76748 .5777 .76567 .5756 Mtetra .651 .76 .929 r .76567 .5756 .76748 .5777 .77162 .57656 .76981 .57444 Mtetra .651 .759 .928 r .76981 .57444 .77162 .57656 .77577 .57541 .77396 .57328 Mtetra .652 .758 .928 r .77396 .57328 .77577 .57541 .77992 .57425 .77811 .57211 Mtetra .652 .757 .927 r .77811 .57211 .77992 .57425 .78408 .57309 .78227 .57094 Mtetra .652 .757 .926 r .78227 .57094 .78408 .57309 .78824 .57193 .78644 .56977 Mtetra .653 .756 .926 r .78644 .56977 .78824 .57193 .79241 .57076 .79061 .56859 Mtetra .653 .756 .925 r .79061 .56859 .79241 .57076 .79658 .56958 .79479 .56741 Mtetra .653 .755 .925 r .79479 .56741 .79658 .56958 .80077 .56841 .79898 .56622 Mtetra .79898 .56622 .80077 .56841 .80495 .56722 .80317 .56504 Mtetra .653 .755 .924 r .80317 .56504 .80495 .56722 .80914 .56604 .80736 .56384 Mtetra .80736 .56384 .80914 .56604 .81334 .56485 .81157 .56265 Mtetra .654 .754 .924 r .81157 .56265 .81334 .56485 .81755 .56366 .81578 .56145 Mtetra .81578 .56145 .81755 .56366 .82176 .56247 .81999 .56025 Mtetra .654 .754 .923 r .81999 .56025 .82176 .56247 .82597 .56127 .82421 .55905 Mtetra .82421 .55905 .82597 .56127 .8302 .56007 .82844 .55785 Mtetra .82844 .55785 .8302 .56007 .83443 .55887 .83267 .55664 Mtetra .83267 .55664 .83443 .55887 .83866 .55766 .83691 .55543 Mtetra .83691 .55543 .83866 .55766 .8429 .55646 .84116 .55422 Mtetra .84116 .55422 .8429 .55646 .84715 .55525 .84541 .55301 Mtetra .654 .753 .923 r .84541 .55301 .84715 .55525 .8514 .55404 .84966 .55179 Mtetra .84966 .55179 .8514 .55404 .85566 .55282 .85393 .55058 Mtetra .85393 .55058 .85566 .55282 .85993 .55161 .8582 .54936 Mtetra .8582 .54936 .85993 .55161 .8642 .55039 .86248 .54814 Mtetra .86248 .54814 .8642 .55039 .86848 .54917 .86676 .54691 Mtetra .654 .753 .922 r .86676 .54691 .86848 .54917 .87276 .54795 .87105 .54569 Mtetra .87105 .54569 .87276 .54795 .87705 .54672 .87534 .54446 Mtetra .87534 .54446 .87705 .54672 .88135 .5455 .87964 .54323 Mtetra .87964 .54323 .88135 .5455 .88565 .54427 .88395 .542 Mtetra .88395 .542 .88565 .54427 .88996 .54304 .88827 .54077 Mtetra .88827 .54077 .88996 .54304 .89428 .54181 .89259 .53953 Mtetra .89259 .53953 .89428 .54181 .8986 .54058 .89691 .53829 Mtetra .89691 .53829 .8986 .54058 .90293 .53934 .90125 .53705 Mtetra .90125 .53705 .90293 .53934 .90726 .5381 .90559 .53581 Mtetra .90559 .53581 .90726 .5381 .91161 .53686 .90993 .53457 Mtetra .90993 .53457 .91161 .53686 .91595 .53562 .91428 .53333 Mtetra .91428 .53333 .91595 .53562 .92031 .53438 .91864 .53208 Mtetra .91864 .53208 .92031 .53438 .92467 .53314 .92301 .53083 Mtetra .92301 .53083 .92467 .53314 .92904 .53189 .92738 .52958 Mtetra .92738 .52958 .92904 .53189 .93341 .53064 .93176 .52833 Mtetra .93176 .52833 .93341 .53064 .93779 .52939 .93614 .52708 Mtetra .93614 .52708 .93779 .52939 .94218 .52814 .94054 .52582 Mtetra .94054 .52582 .94218 .52814 .94657 .52688 .94493 .52456 Mtetra .94493 .52456 .94657 .52688 .95097 .52563 .94934 .5233 Mtetra .94934 .5233 .95097 .52563 .95538 .52437 .95375 .52204 Mtetra .95375 .52204 .95538 .52437 .95979 .52311 .95817 .52078 Mtetra .95817 .52078 .95979 .52311 .96421 .52185 .96259 .51951 Mtetra .645 .763 .934 r .41071 .76947 .41292 .77133 .42317 .5548 .41446 .76846 Mtetra 0 0 .328 r .41446 .76846 .42317 .5548 .4267 .55371 .42463 .55169 Mtetra .664 .741 .907 r .49272 .53068 .49473 .53275 .49664 .74885 .49635 .52956 Mtetra .375 .212 .529 r .49635 .52956 .49664 .74885 .50051 .74782 .49927 .62522 Mtetra 0 0 .406 r .49927 .62522 .50051 .74782 .50438 .74678 .50225 .74485 Mtetra .645 .763 .934 r .60499 .71722 .60703 .71922 .61082 .70842 .60902 .71614 Mtetra .721 .711 .181 r .60902 .71614 .61082 .70842 .61437 .68876 .61268 .69981 Mtetra 0 0 .068 r .61268 .69981 .61437 .68876 .61797 .6727 .61628 .68271 Mtetra 0 0 .107 r .61628 .68271 .61797 .6727 .62163 .65958 .61994 .66853 Mtetra 0 0 .13 r .61994 .66853 .62163 .65958 .62534 .64884 .62363 .65675 Mtetra 0 0 .137 r .62363 .65675 .62534 .64884 .62908 .64004 .62737 .64696 Mtetra 0 0 .13 r .62737 .64696 .62908 .64004 .63287 .63282 .63114 .6388 Mtetra 0 0 .11 r .63114 .6388 .63287 .63282 .63669 .62687 .63495 .63197 Mtetra 0 0 .076 r .63495 .63197 .63669 .62687 .64053 .62195 .63878 .62625 Mtetra 0 0 .026 r .63878 .62625 .64053 .62195 .6444 .61786 .64264 .62142 Mtetra 0 0 0 r .64264 .62142 .6444 .61786 .6483 .61443 .64653 .61733 Mtetra .64653 .61733 .6483 .61443 .65221 .61154 .65043 .61383 Mtetra .65043 .61383 .65221 .61154 .65614 .60906 .65435 .61082 Mtetra .65435 .61082 .65614 .60906 .66009 .60692 .65829 .6082 Mtetra .65829 .6082 .66009 .60692 .66405 .60504 .66224 .6059 Mtetra .354 .802 .66 r .66224 .6059 .66405 .60504 .66803 .60337 .66621 .60385 Mtetra .432 .868 .783 r .66621 .60385 .66803 .60337 .67201 .60185 .67019 .602 Mtetra .496 .906 .874 r .67019 .602 .67201 .60185 .67601 .60046 .67418 .60031 Mtetra .543 .92 .934 r .67418 .60031 .67601 .60046 .68001 .59916 .67818 .59875 Mtetra .576 .917 .967 r .67818 .59875 .68001 .59916 .68403 .59793 .68219 .5973 Mtetra .598 .905 .984 r .68219 .5973 .68403 .59793 .68805 .59675 .6862 .59592 Mtetra .613 .889 .989 r .6862 .59592 .68805 .59675 .69208 .59561 .69023 .5946 Mtetra .623 .872 .989 r .69023 .5946 .69208 .59561 .69612 .59449 .69427 .59333 Mtetra .629 .856 .985 r .69427 .59333 .69612 .59449 .70016 .59339 .69831 .5921 Mtetra .634 .842 .98 r .69831 .5921 .70016 .59339 .70421 .5923 .70236 .5909 Mtetra .637 .829 .974 r .70236 .5909 .70421 .5923 .70826 .59121 .70641 .58971 Mtetra .64 .819 .969 r .70641 .58971 .70826 .59121 .71233 .59013 .71047 .58854 Mtetra .642 .809 .963 r .71047 .58854 .71233 .59013 .7164 .58904 .71454 .58738 Mtetra .644 .801 .958 r .71454 .58738 .7164 .58904 .72047 .58796 .71862 .58622 Mtetra .645 .794 .954 r .71862 .58622 .72047 .58796 .72455 .58686 .7227 .58507 Mtetra .646 .788 .95 r .7227 .58507 .72455 .58686 .72864 .58576 .72679 .58392 Mtetra .647 .783 .947 r .72679 .58392 .72864 .58576 .73273 .58466 .73088 .58277 Mtetra .648 .779 .943 r .73088 .58277 .73273 .58466 .73682 .58355 .73498 .58162 Mtetra .649 .775 .941 r .73498 .58162 .73682 .58355 .74093 .58243 .73909 .58047 Mtetra .649 .772 .938 r .73909 .58047 .74093 .58243 .74504 .58131 .7432 .57931 Mtetra .65 .769 .936 r .7432 .57931 .74504 .58131 .74915 .58018 .74731 .57816 Mtetra .65 .767 .934 r .74731 .57816 .74915 .58018 .75327 .57904 .75144 .577 Mtetra .651 .765 .933 r .75144 .577 .75327 .57904 .7574 .5779 .75557 .57583 Mtetra .651 .763 .931 r .75557 .57583 .7574 .5779 .76153 .57675 .7597 .57467 Mtetra .652 .762 .93 r .7597 .57467 .76153 .57675 .76567 .5756 .76384 .5735 Mtetra .652 .761 .929 r .76384 .5735 .76567 .5756 .76981 .57444 .76799 .57233 Mtetra .652 .76 .928 r .76799 .57233 .76981 .57444 .77396 .57328 .77214 .57115 Mtetra .652 .759 .928 r .77214 .57115 .77396 .57328 .77811 .57211 .7763 .56997 Mtetra .653 .758 .927 r .7763 .56997 .77811 .57211 .78227 .57094 .78047 .56879 Mtetra .653 .757 .926 r .78047 .56879 .78227 .57094 .78644 .56977 .78464 .56761 Mtetra .78464 .56761 .78644 .56977 .79061 .56859 .78881 .56642 Mtetra .653 .756 .925 r .78881 .56642 .79061 .56859 .79479 .56741 .79299 .56523 Mtetra .79299 .56523 .79479 .56741 .79898 .56622 .79718 .56404 Mtetra .654 .755 .925 r .79718 .56404 .79898 .56622 .80317 .56504 .80138 .56284 Mtetra .654 .755 .924 r .80138 .56284 .80317 .56504 .80736 .56384 .80558 .56165 Mtetra .80558 .56165 .80736 .56384 .81157 .56265 .80979 .56045 Mtetra .80979 .56045 .81157 .56265 .81578 .56145 .814 .55924 Mtetra .654 .754 .924 r .814 .55924 .81578 .56145 .81999 .56025 .81822 .55804 Mtetra .654 .754 .923 r .81822 .55804 .81999 .56025 .82421 .55905 .82244 .55683 Mtetra .82244 .55683 .82421 .55905 .82844 .55785 .82667 .55562 Mtetra .82667 .55562 .82844 .55785 .83267 .55664 .83091 .55441 Mtetra .83091 .55441 .83267 .55664 .83691 .55543 .83515 .5532 Mtetra .83515 .5532 .83691 .55543 .84116 .55422 .8394 .55198 Mtetra .8394 .55198 .84116 .55422 .84541 .55301 .84366 .55077 Mtetra .654 .753 .923 r .84366 .55077 .84541 .55301 .84966 .55179 .84792 .54955 Mtetra .84792 .54955 .84966 .55179 .85393 .55058 .85219 .54833 Mtetra .85219 .54833 .85393 .55058 .8582 .54936 .85647 .5471 Mtetra .85647 .5471 .8582 .54936 .86248 .54814 .86075 .54588 Mtetra .86075 .54588 .86248 .54814 .86676 .54691 .86503 .54465 Mtetra .654 .753 .922 r .86503 .54465 .86676 .54691 .87105 .54569 .86933 .54342 Mtetra .86933 .54342 .87105 .54569 .87534 .54446 .87363 .54219 Mtetra .87363 .54219 .87534 .54446 .87964 .54323 .87793 .54096 Mtetra .87793 .54096 .87964 .54323 .88395 .542 .88225 .53972 Mtetra .88225 .53972 .88395 .542 .88827 .54077 .88656 .53848 Mtetra .88656 .53848 .88827 .54077 .89259 .53953 .89089 .53725 Mtetra .89089 .53725 .89259 .53953 .89691 .53829 .89522 .53601 Mtetra .89522 .53601 .89691 .53829 .90125 .53705 .89956 .53476 Mtetra .89956 .53476 .90125 .53705 .90559 .53581 .9039 .53352 Mtetra .9039 .53352 .90559 .53581 .90993 .53457 .90825 .53227 Mtetra .90825 .53227 .90993 .53457 .91428 .53333 .91261 .53102 Mtetra .91261 .53102 .91428 .53333 .91864 .53208 .91697 .52977 Mtetra .91697 .52977 .91864 .53208 .92301 .53083 .92134 .52852 Mtetra .92134 .52852 .92301 .53083 .92738 .52958 .92572 .52727 Mtetra .92572 .52727 .92738 .52958 .93176 .52833 .9301 .52601 Mtetra .9301 .52601 .93176 .52833 .93614 .52708 .93449 .52476 Mtetra .93449 .52476 .93614 .52708 .94054 .52582 .93889 .5235 Mtetra .93889 .5235 .94054 .52582 .94493 .52456 .94329 .52224 Mtetra .94329 .52224 .94493 .52456 .94934 .5233 .9477 .52097 Mtetra .9477 .52097 .94934 .5233 .95375 .52204 .95212 .51971 Mtetra .95212 .51971 .95375 .52204 .95817 .52078 .95654 .51844 Mtetra .95654 .51844 .95817 .52078 .96259 .51951 .96097 .51717 Mtetra .645 .763 .934 r .41224 .76659 .41446 .76846 .42463 .55169 .416 .76558 Mtetra 0 0 .328 r .416 .76558 .42463 .55169 .42817 .5506 .4261 .54858 Mtetra .664 .741 .907 r .49433 .52749 .49635 .52956 .49927 .62522 .49797 .52637 Mtetra .574 .372 .57 r .49797 .52637 .49927 .62522 .50225 .74485 .50162 .52524 Mtetra 0 0 .395 r .50162 .52524 .50225 .74485 .50613 .74381 .50399 .74188 Mtetra .888 .786 .743 r .60698 .71413 .60902 .71614 .61268 .69981 .61089 .70759 Mtetra .773 .737 .212 r .61089 .70759 .61268 .69981 .61628 .68271 .6145 .69 Mtetra 0 0 0 r .6145 .69 .61628 .68271 .61994 .66853 .61816 .67522 Mtetra .61816 .67522 .61994 .66853 .62363 .65675 .62186 .66279 Mtetra .62186 .66279 .62363 .65675 .62737 .64696 .6256 .65233 Mtetra .6256 .65233 .62737 .64696 .63114 .6388 .62937 .6435 Mtetra .62937 .6435 .63114 .6388 .63495 .63197 .63317 .63603 Mtetra .63317 .63603 .63495 .63197 .63878 .62625 .637 .62968 Mtetra .637 .62968 .63878 .62625 .64264 .62142 .64085 .62428 Mtetra .64085 .62428 .64264 .62142 .64653 .61733 .64472 .61965 Mtetra .64472 .61965 .64653 .61733 .65043 .61383 .64862 .61566 Mtetra .64862 .61566 .65043 .61383 .65435 .61082 .65253 .6122 Mtetra .438 .815 .567 r .65253 .6122 .65435 .61082 .65829 .6082 .65647 .60917 Mtetra .488 .873 .68 r .65647 .60917 .65829 .6082 .66224 .6059 .66041 .6065 Mtetra .535 .917 .781 r .66041 .6065 .66224 .6059 .66621 .60385 .66437 .60413 Mtetra .574 .941 .86 r .66437 .60413 .66621 .60385 .67019 .602 .66835 .60198 Mtetra .602 .948 .917 r .66835 .60198 .67019 .602 .67418 .60031 .67233 .60004 Mtetra .622 .942 .952 r .67233 .60004 .67418 .60031 .67818 .59875 .67633 .59825 Mtetra .635 .928 .972 r .67633 .59825 .67818 .59875 .68219 .5973 .68033 .59659 Mtetra .642 .91 .981 r .68033 .59659 .68219 .5973 .6862 .59592 .68435 .59503 Mtetra .646 .892 .984 r .68435 .59503 .6862 .59592 .69023 .5946 .68838 .59355 Mtetra .649 .874 .982 r .68838 .59355 .69023 .5946 .69427 .59333 .69241 .59214 Mtetra .65 .858 .979 r .69241 .59214 .69427 .59333 .69831 .5921 .69645 .59078 Mtetra .651 .844 .974 r .69645 .59078 .69831 .5921 .70236 .5909 .7005 .58947 Mtetra .651 .831 .969 r .7005 .58947 .70236 .5909 .70641 .58971 .70455 .58819 Mtetra .651 .82 .965 r .70455 .58819 .70641 .58971 .71047 .58854 .70862 .58693 Mtetra .651 .811 .96 r .70862 .58693 .71047 .58854 .71454 .58738 .71269 .58569 Mtetra .651 .803 .956 r .71269 .58569 .71454 .58738 .71862 .58622 .71676 .58447 Mtetra .651 .796 .952 r .71676 .58447 .71862 .58622 .7227 .58507 .72084 .58327 Mtetra .652 .79 .948 r .72084 .58327 .7227 .58507 .72679 .58392 .72493 .58207 Mtetra .652 .785 .945 r .72493 .58207 .72679 .58392 .73088 .58277 .72903 .58087 Mtetra .652 .78 .942 r .72903 .58087 .73088 .58277 .73498 .58162 .73313 .57968 Mtetra .652 .776 .94 r .73313 .57968 .73498 .58162 .73909 .58047 .73724 .5785 Mtetra .652 .773 .937 r .73724 .5785 .73909 .58047 .7432 .57931 .74135 .57731 Mtetra .652 .77 .935 r .74135 .57731 .7432 .57931 .74731 .57816 .74547 .57613 Mtetra .652 .768 .934 r .74547 .57613 .74731 .57816 .75144 .577 .7496 .57495 Mtetra .653 .766 .932 r .7496 .57495 .75144 .577 .75557 .57583 .75373 .57376 Mtetra .653 .764 .931 r .75373 .57376 .75557 .57583 .7597 .57467 .75787 .57258 Mtetra .653 .763 .93 r .75787 .57258 .7597 .57467 .76384 .5735 .76201 .57139 Mtetra .653 .761 .929 r .76201 .57139 .76384 .5735 .76799 .57233 .76616 .57021 Mtetra .653 .76 .928 r .76616 .57021 .76799 .57233 .77214 .57115 .77032 .56902 Mtetra .653 .759 .927 r .77032 .56902 .77214 .57115 .7763 .56997 .77448 .56783 Mtetra .653 .758 .927 r .77448 .56783 .7763 .56997 .78047 .56879 .77865 .56664 Mtetra .654 .758 .926 r .77865 .56664 .78047 .56879 .78464 .56761 .78283 .56544 Mtetra .654 .757 .926 r .78283 .56544 .78464 .56761 .78881 .56642 .78701 .56425 Mtetra .654 .756 .925 r .78701 .56425 .78881 .56642 .79299 .56523 .79119 .56305 Mtetra .79119 .56305 .79299 .56523 .79718 .56404 .79539 .56185 Mtetra .654 .756 .924 r .79539 .56185 .79718 .56404 .80138 .56284 .79958 .56065 Mtetra .654 .755 .924 r .79958 .56065 .80138 .56284 .80558 .56165 .80379 .55944 Mtetra .80379 .55944 .80558 .56165 .80979 .56045 .808 .55824 Mtetra .808 .55824 .80979 .56045 .814 .55924 .81222 .55703 Mtetra .654 .754 .924 r .81222 .55703 .814 .55924 .81822 .55804 .81644 .55582 Mtetra .654 .754 .923 r .81644 .55582 .81822 .55804 .82244 .55683 .82067 .55461 Mtetra .82067 .55461 .82244 .55683 .82667 .55562 .8249 .55339 Mtetra .8249 .55339 .82667 .55562 .83091 .55441 .82915 .55218 Mtetra .82915 .55218 .83091 .55441 .83515 .5532 .83339 .55096 Mtetra .83339 .55096 .83515 .5532 .8394 .55198 .83765 .54974 Mtetra .83765 .54974 .8394 .55198 .84366 .55077 .84191 .54852 Mtetra .84191 .54852 .84366 .55077 .84792 .54955 .84618 .54729 Mtetra .654 .753 .923 r .84618 .54729 .84792 .54955 .85219 .54833 .85045 .54607 Mtetra .85045 .54607 .85219 .54833 .85647 .5471 .85473 .54484 Mtetra .85473 .54484 .85647 .5471 .86075 .54588 .85901 .54361 Mtetra .85901 .54361 .86075 .54588 .86503 .54465 .8633 .54238 Mtetra .654 .753 .922 r .8633 .54238 .86503 .54465 .86933 .54342 .8676 .54115 Mtetra .8676 .54115 .86933 .54342 .87363 .54219 .87191 .53991 Mtetra .87191 .53991 .87363 .54219 .87793 .54096 .87622 .53868 Mtetra .87622 .53868 .87793 .54096 .88225 .53972 .88053 .53744 Mtetra .88053 .53744 .88225 .53972 .88656 .53848 .88486 .5362 Mtetra .88486 .5362 .88656 .53848 .89089 .53725 .88919 .53496 Mtetra .88919 .53496 .89089 .53725 .89522 .53601 .89352 .53371 Mtetra .89352 .53371 .89522 .53601 .89956 .53476 .89787 .53246 Mtetra .89787 .53246 .89956 .53476 .9039 .53352 .90222 .53122 Mtetra .90222 .53122 .9039 .53352 .90825 .53227 .90657 .52997 Mtetra .90657 .52997 .90825 .53227 .91261 .53102 .91093 .52872 Mtetra .91093 .52872 .91261 .53102 .91697 .52977 .9153 .52746 Mtetra .9153 .52746 .91697 .52977 .92134 .52852 .91968 .52621 Mtetra .91968 .52621 .92134 .52852 .92572 .52727 .92406 .52495 Mtetra .92406 .52495 .92572 .52727 .9301 .52601 .92845 .52369 Mtetra .92845 .52369 .9301 .52601 .93449 .52476 .93284 .52243 Mtetra .93284 .52243 .93449 .52476 .93889 .5235 .93724 .52117 Mtetra .93724 .52117 .93889 .5235 .94329 .52224 .94165 .5199 Mtetra .94165 .5199 .94329 .52224 .9477 .52097 .94606 .51864 Mtetra .94606 .51864 .9477 .52097 .95212 .51971 .95048 .51737 Mtetra .95048 .51737 .95212 .51971 .95654 .51844 .95491 .5161 Mtetra .95491 .5161 .95654 .51844 .96097 .51717 .95934 .51482 Mtetra .645 .763 .934 r .41378 .7637 .416 .76558 .4261 .54858 .41755 .76269 Mtetra 0 0 .328 r .41755 .76269 .4261 .54858 .42964 .54748 .42757 .54545 Mtetra .664 .741 .907 r .4996 .52316 .50162 .52524 .50399 .74188 .50325 .52203 Mtetra 0 0 .395 r .50325 .52203 .50399 .74188 .50788 .74083 .50574 .73889 Mtetra .645 .763 .934 r .60493 .71213 .60698 .71413 .61089 .70759 .60898 .71103 Mtetra .99 .856 .497 r .60898 .71103 .61089 .70759 .6145 .69 .61264 .69464 Mtetra .932 .851 .413 r .61264 .69464 .6145 .69 .61816 .67522 .61632 .67969 Mtetra .881 .848 .389 r .823 .84 .373 r .766 .831 .368 r .62377 .65613 .6256 .65233 .62937 .6435 .62754 .64689 Mtetra .716 .828 .377 r .62754 .64689 .62937 .6435 .63317 .63603 .63134 .63899 Mtetra .676 .832 .403 r .63134 .63899 .63317 .63603 .637 .62968 .63517 .63221 Mtetra .65 .845 .445 r .63517 .63221 .637 .62968 .64085 .62428 .63902 .62638 Mtetra .636 .867 .502 r .63902 .62638 .64085 .62428 .64472 .61965 .64289 .62135 Mtetra .634 .895 .572 r .64289 .62135 .64472 .61965 .64862 .61566 .64678 .61697 Mtetra .64 .925 .65 r .64678 .61697 .64862 .61566 .65253 .6122 .65069 .61315 Mtetra .651 .951 .728 r .65069 .61315 .65253 .6122 .65647 .60917 .65462 .6098 Mtetra .663 .969 .8 r .65462 .6098 .65647 .60917 .66041 .6065 .65856 .60682 Mtetra .673 .977 .861 r .65856 .60682 .66041 .6065 .66437 .60413 .66252 .60417 Mtetra .68 .974 .907 r .66252 .60417 .66437 .60413 .66835 .60198 .66649 .60177 Mtetra .682 .963 .939 r .66649 .60177 .66835 .60198 .67233 .60004 .67047 .5996 Mtetra .682 .947 .959 r .67047 .5996 .67233 .60004 .67633 .59825 .67447 .59761 Mtetra .68 .928 .97 r .67447 .59761 .67633 .59825 .68033 .59659 .67847 .59577 Mtetra .678 .908 .975 r .67847 .59577 .68033 .59659 .68435 .59503 .68249 .59405 Mtetra .674 .89 .976 r .68249 .59405 .68435 .59503 .68838 .59355 .68651 .59242 Mtetra .671 .872 .975 r .68651 .59242 .68838 .59355 .69241 .59214 .69054 .59089 Mtetra .668 .857 .972 r .69054 .59089 .69241 .59214 .69645 .59078 .69458 .58942 Mtetra .666 .843 .968 r .69458 .58942 .69645 .59078 .7005 .58947 .69863 .588 Mtetra .663 .831 .964 r .69863 .588 .7005 .58947 .70455 .58819 .70269 .58663 Mtetra .662 .82 .96 r .70269 .58663 .70455 .58819 .70862 .58693 .70675 .5853 Mtetra .66 .811 .956 r .70675 .5853 .70862 .58693 .71269 .58569 .71082 .58399 Mtetra .659 .803 .952 r .71082 .58399 .71269 .58569 .71676 .58447 .7149 .58271 Mtetra .658 .796 .949 r .7149 .58271 .71676 .58447 .72084 .58327 .71898 .58145 Mtetra .657 .79 .946 r .71898 .58145 .72084 .58327 .72493 .58207 .72308 .5802 Mtetra .656 .785 .943 r .72308 .5802 .72493 .58207 .72903 .58087 .72717 .57896 Mtetra .656 .781 .94 r .72717 .57896 .72903 .58087 .73313 .57968 .73128 .57774 Mtetra .655 .777 .938 r .73128 .57774 .73313 .57968 .73724 .5785 .73539 .57652 Mtetra .655 .774 .936 r .73539 .57652 .73724 .5785 .74135 .57731 .7395 .57531 Mtetra .655 .771 .935 r .7395 .57531 .74135 .57731 .74547 .57613 .74363 .5741 Mtetra .655 .769 .933 r .74363 .5741 .74547 .57613 .7496 .57495 .74776 .57289 Mtetra .654 .766 .932 r .74776 .57289 .7496 .57495 .75373 .57376 .75189 .57169 Mtetra .654 .765 .93 r .75189 .57169 .75373 .57376 .75787 .57258 .75603 .57048 Mtetra .654 .763 .929 r .75603 .57048 .75787 .57258 .76201 .57139 .76018 .56928 Mtetra .654 .762 .929 r .76018 .56928 .76201 .57139 .76616 .57021 .76433 .56808 Mtetra .654 .761 .928 r .76433 .56808 .76616 .57021 .77032 .56902 .76849 .56688 Mtetra .654 .76 .927 r .76849 .56688 .77032 .56902 .77448 .56783 .77266 .56568 Mtetra .654 .759 .926 r .77266 .56568 .77448 .56783 .77865 .56664 .77683 .56447 Mtetra .654 .758 .926 r .77683 .56447 .77865 .56664 .78283 .56544 .78101 .56327 Mtetra .654 .757 .925 r .78101 .56327 .78283 .56544 .78701 .56425 .7852 .56206 Mtetra .7852 .56206 .78701 .56425 .79119 .56305 .78939 .56086 Mtetra .654 .756 .925 r .78939 .56086 .79119 .56305 .79539 .56185 .79358 .55965 Mtetra .654 .756 .924 r .79358 .55965 .79539 .56185 .79958 .56065 .79778 .55844 Mtetra .654 .755 .924 r .79778 .55844 .79958 .56065 .80379 .55944 .80199 .55723 Mtetra .80199 .55723 .80379 .55944 .808 .55824 .80621 .55602 Mtetra .80621 .55602 .808 .55824 .81222 .55703 .81043 .55481 Mtetra .654 .755 .923 r .81043 .55481 .81222 .55703 .81644 .55582 .81466 .55359 Mtetra .654 .754 .923 r .81466 .55359 .81644 .55582 .82067 .55461 .81889 .55237 Mtetra .81889 .55237 .82067 .55461 .8249 .55339 .82313 .55115 Mtetra .82313 .55115 .8249 .55339 .82915 .55218 .82738 .54993 Mtetra .82738 .54993 .82915 .55218 .83339 .55096 .83163 .54871 Mtetra .83163 .54871 .83339 .55096 .83765 .54974 .83589 .54749 Mtetra .83589 .54749 .83765 .54974 .84191 .54852 .84015 .54626 Mtetra .84015 .54626 .84191 .54852 .84618 .54729 .84442 .54503 Mtetra .84442 .54503 .84618 .54729 .85045 .54607 .8487 .5438 Mtetra .654 .753 .923 r .8487 .5438 .85045 .54607 .85473 .54484 .85298 .54257 Mtetra .85298 .54257 .85473 .54484 .85901 .54361 .85727 .54134 Mtetra .85727 .54134 .85901 .54361 .8633 .54238 .86157 .54011 Mtetra .654 .753 .922 r .86157 .54011 .8633 .54238 .8676 .54115 .86587 .53887 Mtetra .86587 .53887 .8676 .54115 .87191 .53991 .87018 .53763 Mtetra .87018 .53763 .87191 .53991 .87622 .53868 .8745 .53639 Mtetra .8745 .53639 .87622 .53868 .88053 .53744 .87882 .53515 Mtetra .87882 .53515 .88053 .53744 .88486 .5362 .88315 .5339 Mtetra .88315 .5339 .88486 .5362 .88919 .53496 .88748 .53266 Mtetra .88748 .53266 .88919 .53496 .89352 .53371 .89182 .53141 Mtetra .89182 .53141 .89352 .53371 .89787 .53246 .89617 .53016 Mtetra .89617 .53016 .89787 .53246 .90222 .53122 .90052 .52891 Mtetra .90052 .52891 .90222 .53122 .90657 .52997 .90488 .52766 Mtetra .90488 .52766 .90657 .52997 .91093 .52872 .90925 .5264 Mtetra .90925 .5264 .91093 .52872 .9153 .52746 .91362 .52514 Mtetra .91362 .52514 .9153 .52746 .91968 .52621 .918 .52388 Mtetra .918 .52388 .91968 .52621 .92406 .52495 .92239 .52262 Mtetra .92239 .52262 .92406 .52495 .92845 .52369 .92678 .52136 Mtetra .92678 .52136 .92845 .52369 .93284 .52243 .93118 .5201 Mtetra .93118 .5201 .93284 .52243 .93724 .52117 .93559 .51883 Mtetra .93559 .51883 .93724 .52117 .94165 .5199 .94 .51756 Mtetra .94 .51756 .94165 .5199 .94606 .51864 .94442 .51629 Mtetra .94442 .51629 .94606 .51864 .95048 .51737 .94884 .51502 Mtetra .94884 .51502 .95048 .51737 .95491 .5161 .95328 .51375 Mtetra .95328 .51375 .95491 .5161 .95934 .51482 .95771 .51247 Mtetra 0 0 .325 r .36568 .56196 .35785 .7788 .36154 .7778 .35927 .77596 Mtetra .645 .763 .934 r .41532 .76081 .41755 .76269 .42757 .54545 .4191 .75978 Mtetra 0 0 .328 r .4191 .75978 .42757 .54545 .43113 .54435 .42905 .54232 Mtetra .664 .741 .907 r .50124 .51994 .50325 .52203 .50574 .73889 .5049 .5188 Mtetra .37 .208 .527 r .5049 .5188 .50574 .73889 .50964 .73784 .50808 .61625 Mtetra 0 0 .408 r .50808 .61625 .50964 .73784 .51355 .73678 .51141 .73483 Mtetra .932 .762 .648 r .60693 .70902 .60898 .71103 .61264 .69464 .61071 .6968 Mtetra .997 .857 .552 r .61071 .6968 .61264 .69464 .61632 .67969 .61441 .68203 Mtetra .989 .883 .552 r .972 .905 .555 r .948 .923 .563 r .62188 .6584 .62377 .65613 .62754 .64689 .62567 .64898 Mtetra .92 .938 .579 r .62567 .64898 .62754 .64689 .63134 .63899 .62947 .64085 Mtetra .891 .952 .604 r .62947 .64085 .63134 .63899 .63517 .63221 .6333 .63382 Mtetra .863 .965 .637 r .6333 .63382 .63517 .63221 .63902 .62638 .63715 .62771 Mtetra .839 .978 .678 r .63715 .62771 .63902 .62638 .64289 .62135 .64103 .62239 Mtetra .819 .988 .725 r .64103 .62239 .64289 .62135 .64678 .61697 .64492 .61774 Mtetra .802 .996 .774 r .64492 .61774 .64678 .61697 .65069 .61315 .64883 .61365 Mtetra .787 .999 .822 r .64883 .61365 .65069 .61315 .65462 .6098 .65275 .61004 Mtetra .774 .996 .864 r .65275 .61004 .65462 .6098 .65856 .60682 .6567 .60682 Mtetra .761 .987 .899 r .6567 .60682 .65856 .60682 .66252 .60417 .66065 .60394 Mtetra .748 .974 .925 r .66065 .60394 .66252 .60417 .66649 .60177 .66462 .60135 Mtetra .736 .957 .944 r .66462 .60135 .66649 .60177 .67047 .5996 .6686 .59898 Mtetra .724 .938 .956 r .6686 .59898 .67047 .5996 .67447 .59761 .67259 .59682 Mtetra .714 .919 .963 r .67259 .59682 .67447 .59761 .67847 .59577 .6766 .59482 Mtetra .705 .9 .966 r .6766 .59482 .67847 .59577 .68249 .59405 .68061 .59296 Mtetra .697 .883 .967 r .68061 .59296 .68249 .59405 .68651 .59242 .68463 .59121 Mtetra .69 .867 .966 r .68463 .59121 .68651 .59242 .69054 .59089 .68867 .58956 Mtetra .684 .852 .963 r .68867 .58956 .69054 .59089 .69458 .58942 .69271 .58799 Mtetra .679 .84 .961 r .69271 .58799 .69458 .58942 .69863 .588 .69676 .58648 Mtetra .674 .828 .957 r .69676 .58648 .69863 .588 .70269 .58663 .70081 .58503 Mtetra .671 .818 .954 r .70081 .58503 .70269 .58663 .70675 .5853 .70488 .58363 Mtetra .668 .81 .951 r .70488 .58363 .70675 .5853 .71082 .58399 .70895 .58226 Mtetra .665 .802 .948 r .70895 .58226 .71082 .58399 .7149 .58271 .71303 .58092 Mtetra .664 .796 .945 r .71303 .58092 .7149 .58271 .71898 .58145 .71712 .57961 Mtetra .662 .79 .943 r .71712 .57961 .71898 .58145 .72308 .5802 .72121 .57831 Mtetra .661 .785 .94 r .72121 .57831 .72308 .5802 .72717 .57896 .72531 .57704 Mtetra .659 .781 .938 r .72531 .57704 .72717 .57896 .73128 .57774 .72942 .57578 Mtetra .659 .777 .937 r .72942 .57578 .73128 .57774 .73539 .57652 .73353 .57453 Mtetra .658 .774 .935 r .73353 .57453 .73539 .57652 .7395 .57531 .73765 .57329 Mtetra .657 .771 .933 r .73765 .57329 .7395 .57531 .74363 .5741 .74178 .57205 Mtetra .657 .769 .932 r .74178 .57205 .74363 .5741 .74776 .57289 .74591 .57082 Mtetra .656 .767 .931 r .74591 .57082 .74776 .57289 .75189 .57169 .75005 .5696 Mtetra .656 .765 .93 r .75005 .5696 .75189 .57169 .75603 .57048 .75419 .56838 Mtetra .656 .763 .929 r .75419 .56838 .75603 .57048 .76018 .56928 .75834 .56716 Mtetra .655 .762 .928 r .75834 .56716 .76018 .56928 .76433 .56808 .7625 .56595 Mtetra .655 .761 .927 r .7625 .56595 .76433 .56808 .76849 .56688 .76666 .56473 Mtetra .655 .76 .927 r .76666 .56473 .76849 .56688 .77266 .56568 .77083 .56352 Mtetra .655 .759 .926 r .77083 .56352 .77266 .56568 .77683 .56447 .77501 .5623 Mtetra .655 .758 .926 r .77501 .5623 .77683 .56447 .78101 .56327 .77919 .56109 Mtetra .655 .757 .925 r .77919 .56109 .78101 .56327 .7852 .56206 .78338 .55988 Mtetra .78338 .55988 .7852 .56206 .78939 .56086 .78757 .55866 Mtetra .655 .756 .924 r .78757 .55866 .78939 .56086 .79358 .55965 .79177 .55745 Mtetra .79177 .55745 .79358 .55965 .79778 .55844 .79598 .55623 Mtetra .79598 .55623 .79778 .55844 .80199 .55723 .80019 .55501 Mtetra .655 .755 .924 r .80019 .55501 .80199 .55723 .80621 .55602 .80441 .5538 Mtetra .80441 .5538 .80621 .55602 .81043 .55481 .80864 .55258 Mtetra .655 .755 .923 r .80864 .55258 .81043 .55481 .81466 .55359 .81287 .55136 Mtetra .655 .754 .923 r .81287 .55136 .81466 .55359 .81889 .55237 .81711 .55013 Mtetra .654 .754 .923 r .81711 .55013 .81889 .55237 .82313 .55115 .82135 .54891 Mtetra .82135 .54891 .82313 .55115 .82738 .54993 .8256 .54769 Mtetra .8256 .54769 .82738 .54993 .83163 .54871 .82986 .54646 Mtetra .82986 .54646 .83163 .54871 .83589 .54749 .83412 .54523 Mtetra .83412 .54523 .83589 .54749 .84015 .54626 .83839 .544 Mtetra .83839 .544 .84015 .54626 .84442 .54503 .84267 .54277 Mtetra .84267 .54277 .84442 .54503 .8487 .5438 .84695 .54153 Mtetra .654 .753 .923 r .84695 .54153 .8487 .5438 .85298 .54257 .85124 .5403 Mtetra .85124 .5403 .85298 .54257 .85727 .54134 .85553 .53906 Mtetra .654 .753 .922 r .85553 .53906 .85727 .54134 .86157 .54011 .85983 .53782 Mtetra .85983 .53782 .86157 .54011 .86587 .53887 .86414 .53658 Mtetra .86414 .53658 .86587 .53887 .87018 .53763 .86845 .53534 Mtetra .86845 .53534 .87018 .53763 .8745 .53639 .87277 .5341 Mtetra .87277 .5341 .8745 .53639 .87882 .53515 .8771 .53285 Mtetra .8771 .53285 .87882 .53515 .88315 .5339 .88143 .5316 Mtetra .88143 .5316 .88315 .5339 .88748 .53266 .88577 .53035 Mtetra .88577 .53035 .88748 .53266 .89182 .53141 .89012 .5291 Mtetra .89012 .5291 .89182 .53141 .89617 .53016 .89447 .52785 Mtetra .89447 .52785 .89617 .53016 .90052 .52891 .89883 .52659 Mtetra .89883 .52659 .90052 .52891 .90488 .52766 .90319 .52534 Mtetra .90319 .52534 .90488 .52766 .90925 .5264 .90756 .52408 Mtetra .90756 .52408 .90925 .5264 .91362 .52514 .91194 .52282 Mtetra .91194 .52282 .91362 .52514 .918 .52388 .91632 .52156 Mtetra .91632 .52156 .918 .52388 .92239 .52262 .92071 .52029 Mtetra .92071 .52029 .92239 .52262 .92678 .52136 .92511 .51903 Mtetra .92511 .51903 .92678 .52136 .93118 .5201 .92952 .51776 Mtetra .92952 .51776 .93118 .5201 .93559 .51883 .93393 .51649 Mtetra .93393 .51649 .93559 .51883 .94 .51756 .93834 .51522 Mtetra .93834 .51522 .94 .51756 .94442 .51629 .94277 .51394 Mtetra .94277 .51394 .94442 .51629 .94884 .51502 .9472 .51267 Mtetra .9472 .51267 .94884 .51502 .95328 .51375 .95164 .51139 Mtetra .95164 .51139 .95328 .51375 .95771 .51247 .95608 .51011 Mtetra .664 .741 .907 r .36355 .55997 .36568 .56196 .35927 .77596 .36703 .55889 Mtetra 0 0 .325 r .36703 .55889 .35927 .77596 .36297 .77496 .36069 .77311 Mtetra .645 .763 .934 r .41687 .7579 .4191 .75978 .42905 .54232 .42066 .75687 Mtetra .565 .352 .551 r .42066 .75687 .42905 .54232 .43261 .54121 .42445 .75584 Mtetra 0 0 .33 r .42445 .75584 .43261 .54121 .43618 .54011 .4341 .53806 Mtetra .664 .741 .907 r .50288 .5167 .5049 .5188 .50808 .61625 .50655 .51556 Mtetra .572 .371 .569 r .50655 .51556 .50808 .61625 .51141 .73483 .51022 .51442 Mtetra 0 0 .397 r .51022 .51442 .51141 .73483 .51532 .73378 .51318 .73182 Mtetra .923 .769 .673 r .60487 .70699 .60693 .70902 .61071 .6968 .60871 .69667 Mtetra .975 .807 .611 r .60871 .69667 .61071 .6968 .61441 .68203 .61243 .6824 Mtetra .983 .839 .621 r .61243 .6824 .61441 .68203 .61813 .66933 .61618 .67 Mtetra .986 .87 .635 r .61618 .67 .61813 .66933 .62188 .6584 .61995 .65921 Mtetra .982 .898 .652 r .61995 .65921 .62188 .6584 .62567 .64898 .62374 .64983 Mtetra .973 .924 .673 r .62374 .64983 .62567 .64898 .62947 .64085 .62756 .64165 Mtetra .959 .945 .699 r .62756 .64165 .62947 .64085 .6333 .63382 .6314 .63451 Mtetra .94 .963 .728 r .6314 .63451 .6333 .63382 .63715 .62771 .63525 .62826 Mtetra .919 .976 .76 r .63525 .62826 .63715 .62771 .64103 .62239 .63913 .62278 Mtetra .897 .984 .793 r .63913 .62278 .64103 .62239 .64492 .61774 .64303 .61795 Mtetra .874 .987 .827 r .64303 .61795 .64492 .61774 .64883 .61365 .64694 .61368 Mtetra .851 .985 .858 r .64694 .61368 .64883 .61365 .65275 .61004 .65086 .60988 Mtetra .828 .978 .885 r .65086 .60988 .65275 .61004 .6567 .60682 .65481 .60649 Mtetra .807 .967 .908 r .65481 .60649 .6567 .60682 .66065 .60394 .65876 .60344 Mtetra .786 .953 .925 r .65876 .60344 .66065 .60394 .66462 .60135 .66273 .60068 Mtetra .768 .937 .938 r .66273 .60068 .66462 .60135 .6686 .59898 .66671 .59816 Mtetra .751 .921 .947 r .66671 .59816 .6686 .59898 .67259 .59682 .67071 .59586 Mtetra .737 .904 .952 r .67071 .59586 .67259 .59682 .6766 .59482 .67471 .59373 Mtetra .724 .888 .955 r .67471 .59373 .6766 .59482 .68061 .59296 .67872 .59175 Mtetra .713 .873 .956 r .67872 .59175 .68061 .59296 .68463 .59121 .68275 .5899 Mtetra .704 .859 .956 r .68275 .5899 .68463 .59121 .68867 .58956 .68678 .58815 Mtetra .696 .846 .955 r .68678 .58815 .68867 .58956 .69271 .58799 .69082 .58649 Mtetra .689 .835 .953 r .69082 .58649 .69271 .58799 .69676 .58648 .69487 .5849 Mtetra .683 .824 .951 r .69487 .5849 .69676 .58648 .70081 .58503 .69893 .58338 Mtetra .679 .815 .948 r .69893 .58338 .70081 .58503 .70488 .58363 .703 .58191 Mtetra .675 .807 .946 r .703 .58191 .70488 .58363 .70895 .58226 .70707 .58049 Mtetra .671 .8 .944 r .70707 .58049 .70895 .58226 .71303 .58092 .71115 .5791 Mtetra .669 .794 .942 r .71115 .5791 .71303 .58092 .71712 .57961 .71524 .57774 Mtetra .666 .789 .94 r .71524 .57774 .71712 .57961 .72121 .57831 .71934 .5764 Mtetra .664 .784 .938 r .71934 .5764 .72121 .57831 .72531 .57704 .72344 .57509 Mtetra .663 .78 .936 r .72344 .57509 .72531 .57704 .72942 .57578 .72755 .5738 Mtetra .661 .777 .935 r .72755 .5738 .72942 .57578 .73353 .57453 .73167 .57252 Mtetra .66 .774 .933 r .73167 .57252 .73353 .57453 .73765 .57329 .73579 .57125 Mtetra .659 .771 .932 r .73579 .57125 .73765 .57329 .74178 .57205 .73992 .57 Mtetra .659 .769 .931 r .73992 .57 .74178 .57205 .74591 .57082 .74405 .56875 Mtetra .658 .767 .93 r .74405 .56875 .74591 .57082 .75005 .5696 .7482 .5675 Mtetra .657 .765 .929 r .7482 .5675 .75005 .5696 .75419 .56838 .75234 .56627 Mtetra .657 .763 .928 r .75234 .56627 .75419 .56838 .75834 .56716 .7565 .56503 Mtetra .657 .762 .927 r .7565 .56503 .75834 .56716 .7625 .56595 .76066 .5638 Mtetra .656 .761 .927 r .76066 .5638 .7625 .56595 .76666 .56473 .76483 .56258 Mtetra .656 .76 .926 r .76483 .56258 .76666 .56473 .77083 .56352 .769 .56135 Mtetra .656 .759 .926 r .769 .56135 .77083 .56352 .77501 .5623 .77318 .56013 Mtetra .656 .758 .925 r .77318 .56013 .77501 .5623 .77919 .56109 .77737 .55891 Mtetra .655 .757 .925 r .77737 .55891 .77919 .56109 .78338 .55988 .78156 .55768 Mtetra .78156 .55768 .78338 .55988 .78757 .55866 .78576 .55646 Mtetra .655 .756 .924 r .78576 .55646 .78757 .55866 .79177 .55745 .78996 .55524 Mtetra .78996 .55524 .79177 .55745 .79598 .55623 .79417 .55402 Mtetra .79417 .55402 .79598 .55623 .80019 .55501 .79839 .55279 Mtetra .655 .755 .924 r .79839 .55279 .80019 .55501 .80441 .5538 .80261 .55157 Mtetra .655 .755 .923 r .80261 .55157 .80441 .5538 .80864 .55258 .80684 .55034 Mtetra .80684 .55034 .80864 .55258 .81287 .55136 .81108 .54912 Mtetra .655 .754 .923 r .81108 .54912 .81287 .55136 .81711 .55013 .81532 .54789 Mtetra .81532 .54789 .81711 .55013 .82135 .54891 .81957 .54666 Mtetra .81957 .54666 .82135 .54891 .8256 .54769 .82382 .54543 Mtetra .82382 .54543 .8256 .54769 .82986 .54646 .82808 .5442 Mtetra .82808 .5442 .82986 .54646 .83412 .54523 .83235 .54297 Mtetra .83235 .54297 .83412 .54523 .83839 .544 .83663 .54173 Mtetra .83663 .54173 .83839 .544 .84267 .54277 .84091 .5405 Mtetra .84091 .5405 .84267 .54277 .84695 .54153 .84519 .53926 Mtetra .655 .753 .923 r .84519 .53926 .84695 .54153 .85124 .5403 .84948 .53802 Mtetra .654 .753 .923 r .84948 .53802 .85124 .5403 .85553 .53906 .85378 .53678 Mtetra .654 .753 .922 r .85378 .53678 .85553 .53906 .85983 .53782 .85809 .53554 Mtetra .85809 .53554 .85983 .53782 .86414 .53658 .8624 .53429 Mtetra .8624 .53429 .86414 .53658 .86845 .53534 .86672 .53305 Mtetra .86672 .53305 .86845 .53534 .87277 .5341 .87104 .5318 Mtetra .87104 .5318 .87277 .5341 .8771 .53285 .87537 .53055 Mtetra .87537 .53055 .8771 .53285 .88143 .5316 .87971 .5293 Mtetra .87971 .5293 .88143 .5316 .88577 .53035 .88405 .52804 Mtetra .88405 .52804 .88577 .53035 .89012 .5291 .8884 .52679 Mtetra .8884 .52679 .89012 .5291 .89447 .52785 .89276 .52553 Mtetra .89276 .52553 .89447 .52785 .89883 .52659 .89712 .52427 Mtetra .89712 .52427 .89883 .52659 .90319 .52534 .90149 .52301 Mtetra .90149 .52301 .90319 .52534 .90756 .52408 .90587 .52175 Mtetra .90587 .52175 .90756 .52408 .91194 .52282 .91025 .52049 Mtetra .91025 .52049 .91194 .52282 .91632 .52156 .91464 .51922 Mtetra .91464 .51922 .91632 .52156 .92071 .52029 .91904 .51795 Mtetra .91904 .51795 .92071 .52029 .92511 .51903 .92344 .51668 Mtetra .92344 .51668 .92511 .51903 .92952 .51776 .92785 .51541 Mtetra .92785 .51541 .92952 .51776 .93393 .51649 .93226 .51414 Mtetra .93226 .51414 .93393 .51649 .93834 .51522 .93669 .51286 Mtetra .93669 .51286 .93834 .51522 .94277 .51394 .94111 .51159 Mtetra .94111 .51159 .94277 .51394 .9472 .51267 .94555 .51031 Mtetra .94555 .51031 .9472 .51267 .95164 .51139 .94999 .50903 Mtetra .94999 .50903 .95164 .51139 .95608 .51011 .95444 .50774 Mtetra .664 .741 .907 r .3649 .55688 .36703 .55889 .36069 .77311 .36839 .5558 Mtetra 0 0 .325 r .36839 .5558 .36069 .77311 .3644 .7721 .36212 .77024 Mtetra .645 .763 .934 r .42222 .75394 .42445 .75584 .4341 .53806 .42603 .75291 Mtetra 0 0 .33 r .42603 .75291 .4341 .53806 .43768 .53695 .4356 .53489 Mtetra .664 .741 .907 r .5082 .51231 .51022 .51442 .51318 .73182 .51189 .51117 Mtetra .045 0 .426 r .51189 .51117 .51318 .73182 .5171 .73076 .51502 .70937 Mtetra .087 .05 .516 r .51502 .70937 .5171 .73076 .52103 .72969 .51889 .72772 Mtetra .924 .769 .671 r .60281 .70496 .60487 .70699 .60871 .69667 .60666 .6945 Mtetra .934 .754 .635 r .60666 .6945 .60871 .69667 .61243 .6824 .61041 .68098 Mtetra .946 .784 .649 r .61041 .68098 .61243 .6824 .61618 .67 .61418 .6691 Mtetra .954 .813 .665 r .61418 .6691 .61618 .67 .61995 .65921 .61797 .65867 Mtetra .959 .842 .685 r .61797 .65867 .61995 .65921 .62374 .64983 .62178 .6495 Mtetra .958 .869 .707 r .62178 .6495 .62374 .64983 .62756 .64165 .62561 .64143 Mtetra .953 .893 .731 r .62561 .64143 .62756 .64165 .6314 .63451 .62946 .63433 Mtetra .943 .913 .757 r .62946 .63433 .6314 .63451 .63525 .62826 .63332 .62806 Mtetra .929 .929 .785 r .63332 .62806 .63525 .62826 .63913 .62278 .63721 .62252 Mtetra .911 .939 .812 r .63721 .62252 .63913 .62278 .64303 .61795 .64111 .61761 Mtetra .891 .945 .838 r .64111 .61761 .64303 .61795 .64694 .61368 .64502 .61323 Mtetra .87 .945 .862 r .64502 .61323 .64694 .61368 .65086 .60988 .64895 .60932 Mtetra .847 .941 .884 r .64895 .60932 .65086 .60988 .65481 .60649 .6529 .60581 Mtetra .825 .934 .901 r .6529 .60581 .65481 .60649 .65876 .60344 .65686 .60264 Mtetra .804 .924 .916 r .65686 .60264 .65876 .60344 .66273 .60068 .66083 .59976 Mtetra .785 .912 .927 r .66083 .59976 .66273 .60068 .66671 .59816 .66481 .59713 Mtetra .767 .899 .935 r .66481 .59713 .66671 .59816 .67071 .59586 .66881 .59472 Mtetra .751 .886 .94 r .66881 .59472 .67071 .59586 .67471 .59373 .67281 .59249 Mtetra .737 .873 .944 r .67281 .59249 .67471 .59373 .67872 .59175 .67683 .59042 Mtetra .725 .86 .945 r .67683 .59042 .67872 .59175 .68275 .5899 .68085 .58847 Mtetra .714 .849 .946 r .68085 .58847 .68275 .5899 .68678 .58815 .68489 .58664 Mtetra .705 .838 .946 r .68489 .58664 .68678 .58815 .69082 .58649 .68893 .58491 Mtetra .697 .828 .945 r .68893 .58491 .69082 .58649 .69487 .5849 .69298 .58326 Mtetra .691 .819 .944 r .69298 .58326 .69487 .5849 .69893 .58338 .69704 .58167 Mtetra .685 .811 .943 r .69704 .58167 .69893 .58338 .703 .58191 .70111 .58015 Mtetra .681 .804 .941 r .70111 .58015 .703 .58191 .70707 .58049 .70519 .57867 Mtetra .677 .798 .939 r .70519 .57867 .70707 .58049 .71115 .5791 .70927 .57724 Mtetra .673 .792 .938 r .70927 .57724 .71115 .5791 .71524 .57774 .71336 .57584 Mtetra .67 .787 .936 r .71336 .57584 .71524 .57774 .71934 .5764 .71746 .57447 Mtetra .668 .783 .935 r .71746 .57447 .71934 .5764 .72344 .57509 .72157 .57312 Mtetra .666 .779 .934 r .72157 .57312 .72344 .57509 .72755 .5738 .72568 .5718 Mtetra .664 .776 .933 r .72568 .5718 .72755 .5738 .73167 .57252 .7298 .57049 Mtetra .663 .773 .931 r .7298 .57049 .73167 .57252 .73579 .57125 .73392 .5692 Mtetra .661 .77 .93 r .73392 .5692 .73579 .57125 .73992 .57 .73805 .56792 Mtetra .66 .768 .93 r .73805 .56792 .73992 .57 .74405 .56875 .74219 .56665 Mtetra .659 .766 .929 r .74219 .56665 .74405 .56875 .7482 .5675 .74634 .56539 Mtetra .659 .765 .928 r .74634 .56539 .7482 .5675 .75234 .56627 .75049 .56414 Mtetra .658 .763 .927 r .75049 .56414 .75234 .56627 .7565 .56503 .75465 .56289 Mtetra .658 .762 .927 r .75465 .56289 .7565 .56503 .76066 .5638 .75881 .56165 Mtetra .657 .761 .926 r .75881 .56165 .76066 .5638 .76483 .56258 .76298 .56041 Mtetra .657 .76 .926 r .76298 .56041 .76483 .56258 .769 .56135 .76716 .55918 Mtetra .656 .759 .925 r .76716 .55918 .769 .56135 .77318 .56013 .77135 .55794 Mtetra .656 .758 .925 r .77135 .55794 .77318 .56013 .77737 .55891 .77554 .55671 Mtetra .656 .757 .925 r .77554 .55671 .77737 .55891 .78156 .55768 .77973 .55548 Mtetra .656 .757 .924 r .77973 .55548 .78156 .55768 .78576 .55646 .78393 .55425 Mtetra .655 .756 .924 r .78393 .55425 .78576 .55646 .78996 .55524 .78814 .55302 Mtetra .78814 .55302 .78996 .55524 .79417 .55402 .79236 .55179 Mtetra .79236 .55179 .79417 .55402 .79839 .55279 .79658 .55056 Mtetra .655 .755 .923 r .79658 .55056 .79839 .55279 .80261 .55157 .80081 .54933 Mtetra .80081 .54933 .80261 .55157 .80684 .55034 .80504 .5481 Mtetra .80504 .5481 .80684 .55034 .81108 .54912 .80928 .54687 Mtetra .655 .754 .923 r .80928 .54687 .81108 .54912 .81532 .54789 .81353 .54564 Mtetra .81353 .54564 .81532 .54789 .81957 .54666 .81778 .5444 Mtetra .81778 .5444 .81957 .54666 .82382 .54543 .82204 .54317 Mtetra .82204 .54317 .82382 .54543 .82808 .5442 .82631 .54193 Mtetra .82631 .54193 .82808 .5442 .83235 .54297 .83058 .5407 Mtetra .83058 .5407 .83235 .54297 .83663 .54173 .83485 .53946 Mtetra .83485 .53946 .83663 .54173 .84091 .5405 .83914 .53822 Mtetra .83914 .53822 .84091 .5405 .84519 .53926 .84343 .53698 Mtetra .655 .753 .923 r .84343 .53698 .84519 .53926 .84948 .53802 .84773 .53573 Mtetra .655 .753 .922 r .84773 .53573 .84948 .53802 .85378 .53678 .85203 .53449 Mtetra .85203 .53449 .85378 .53678 .85809 .53554 .85634 .53324 Mtetra .85634 .53324 .85809 .53554 .8624 .53429 .86066 .53199 Mtetra .86066 .53199 .8624 .53429 .86672 .53305 .86498 .53074 Mtetra .654 .753 .922 r .86498 .53074 .86672 .53305 .87104 .5318 .86931 .52949 Mtetra .86931 .52949 .87104 .5318 .87537 .53055 .87364 .52824 Mtetra .87364 .52824 .87537 .53055 .87971 .5293 .87799 .52698 Mtetra .87799 .52698 .87971 .5293 .88405 .52804 .88233 .52573 Mtetra .88233 .52573 .88405 .52804 .8884 .52679 .88669 .52447 Mtetra .88669 .52447 .8884 .52679 .89276 .52553 .89105 .52321 Mtetra .89105 .52321 .89276 .52553 .89712 .52427 .89542 .52195 Mtetra .89542 .52195 .89712 .52427 .90149 .52301 .89979 .52068 Mtetra .89979 .52068 .90149 .52301 .90587 .52175 .90417 .51942 Mtetra .90417 .51942 .90587 .52175 .91025 .52049 .90856 .51815 Mtetra .90856 .51815 .91025 .52049 .91464 .51922 .91295 .51688 Mtetra .91295 .51688 .91464 .51922 .91904 .51795 .91735 .51561 Mtetra .91735 .51561 .91904 .51795 .92344 .51668 .92176 .51434 Mtetra .92176 .51434 .92344 .51668 .92785 .51541 .92618 .51306 Mtetra .92618 .51306 .92785 .51541 .93226 .51414 .9306 .51178 Mtetra .9306 .51178 .93226 .51414 .93669 .51286 .93502 .5105 Mtetra .93502 .5105 .93669 .51286 .94111 .51159 .93946 .50922 Mtetra .93946 .50922 .94111 .51159 .94555 .51031 .943...[...] === Subject: Re: Wolfram Alpha claims to be a primary source. I cannot get this to open with MMA 7 Reader. What is it supposed to be? === Subject: Re: Wolfram Alpha claims to be a primary source. > DL> I can say a bit about that. Our SQA people at that time distilled > the > DL> report to very nearly 100 bug reports. When last I checked, about > 90 > DL> per cent had been closed. Some were closed because they were not > bugs > DL> to begin with. Most actual bugs were fixed. > > That is good news. Why was this not communicated earlier by WR? Because it would be counter productive for WRI to draw attention to bugs in their own software. -- Dr Jon D Harrop, Flying Frog Consultancy Ltd. http://www.ffconsultancy.com/?u === Subject: Re: Wolfram Alpha claims to be a primary source. <4A1AC625.8070308@cs.berkeley.edu> <4a324910@212.67.96.135> posting-account=PEh90woAAACWN3C9IZwQVmbvngXF6iWJ Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) DL> to begin with. Most actual bugs were fixed. PL> That is good news. Why was this not communicated earlier by WR? JH> Because it would be counter productive for WRI to draw JH> attention to bugs in their own software. This would imply some kind of MacCarthy thinking, assuming a cold war between uneducated and dumb users of Mathematica and a glorious CAS-industry guided by the Great Leader Wolfram. It would document disrespect for WR users to the level of impertinence. But reading Dave's thread on the most ridiculous Wolfram Research claims all this might very well be the case. === Subject: Re: Wolfram Alpha claims to be a primary source. > To remove at once any accusation of my non-objectivity, is there > anyone to see, how many defects are still alive from the chapter > 1 of my draft I reported to Wolfram Research in 2001? > > http://maple.bug-list.org/chapter1.txt I suspect Vladimir that had you chosen to use your skills in a more professional manner, you might have well be taken more seriously by Wolfram Research and Maplesoft. But your manner of reporting bugs, copying them to endless irrelevant newsgroups in a dramatic way is not the best approach. -- I respectfully request that this message is not archived by companies as unscrupulous as 'Experts Exchange' . In case you are unaware, 'Experts Exchange' take questions posted on the web and try to find idiots stupid enough to pay for the answers, which were posted freely by others. They are leeches. === Subject: Re: Wolfram Alpha claims to be a primary source. <4A1AC625.8070308@cs.berkeley.edu> <4a324910@212.67.96.135> <4a330174@212.67.96.135> posting-account=PEh90woAAACWN3C9IZwQVmbvngXF6iWJ Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) VB> ... how many defects are still alive from the chapter VB> 1 of my draft I reported to Wolfram Research in 2001? VB> http://maple.bug-list.org/chapter1.txt D> But your manner of reporting bugs, copying them to endless irrelevant D> newsgroups in a dramatic way is not the best approach. Newsgroups are an important source of feedback for any vendor and bugs reported there should be taken seriously by the respective QA-departments. You disagree? VB> ... how many defects are still alive from the chapter VB> 1 of my draft I reported to Wolfram Research in 2001? VB> http://maple.bug-list.org/chapter1.txt This is a very good question and I am very interested to read the answer. Preferably by Wolfram Research SQA to show to the world that it ridiculous to assume that it suffers from serious malfeasance. D> I suspect Vladimir that had you chosen to use your skills in a more D> professional manner, you might have well be taken more seriously by D> Wolfram Research and Maplesoft. Dave, if Wolfram Research SQA uses its skills in a professional manner then they are not interested in the charming forms of VB's presentation but only in the bugs and how to kill them as soon as possible. So when VB asks 2^3 years after his report ... VB> ... how many defects are still alive from the chapter VB> 1 of my draft I reported to Wolfram Research in 2001? VB> http://maple.bug-list.org/chapter1.txt then a public answer by WR is more then deserved. === Subject: Re: Wolfram Alpha claims to be a primary source. > VB> ... how many defects are still alive from the chapter > VB> 1 of my draft I reported to Wolfram Research in 2001? > VB> http://maple.bug-list.org/chapter1.txt > > D> But your manner of reporting bugs, copying them to endless > irrelevant > D> newsgroups in a dramatic way is not the best approach. > > Newsgroups are an important source of feedback for any vendor > and bugs reported there should be taken seriously by the > respective QA-departments. You disagree? You cannot seriously expect them to do anything about bug reports posted on usenet? Most software companies won't even do anything if you send the bug report directly to them. -- Dr Jon D Harrop, Flying Frog Consultancy Ltd. http://www.ffconsultancy.com/?u === Subject: Re: Wolfram Alpha claims to be a primary source. <4A1AC625.8070308@cs.berkeley.edu> <4a324910@212.67.96.135> posting-account=PEh90woAAACWN3C9IZwQVmbvngXF6iWJ Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) VB> ... how many defects are still alive from the chapter VB> 1 of my draft I reported to Wolfram Research in 2001? VB> http://maple.bug-list.org/chapter1.txt D> But your manner of reporting bugs, copying them to endless D> irrelevant newsgroups in a dramatic way is not the best approach. P> Newsgroups are an important source of feedback for any vendor P> and bugs reported there should be taken seriously by the P> respective QA-departments. You disagree? JH> You cannot seriously expect them to do anything about bug JH> reports posted on usenet? Ahh? Why? 'Expect' in the sense of: to look for with reason or justification. Sure I can! After all there are thousands of customers which paid thousands of dollars and expect that the software is maintained with the highest possible effort. JH> Most software companies won't even do anything if you send the bug JH> report directly to them. 'Expect' in the sense of: base conclusions upon observation. that Mathematica (5.0) gives BernoulliB[1, 1] -> -1/2. Some weeks later I received a personal mail by some guy I wonder what you really want to tell us. === Subject: Re: Wolfram Alpha claims to be a primary source. > Sure I can! After all there are thousands of customers which > paid thousands of dollars and expect that the software is > maintained with the highest possible effort. > ... > I wonder what you really want to tell us. The most commercially successful software companies are generally those that put the least effort into maintenance. They plough their money into marketing first and lawyers second. WRI is unquestionably one of those companies. Fixing bugs is way down on their list of priorities. Adding features is much higher, even if those features are not properly implemented, because they can market them and ship product. I am sure there are some devoted employees who strive to fix bugs and even search for feedback on usenet, as Daniel does, but that does not reflect the main thrust of the company's efforts. -- Dr Jon D Harrop, Flying Frog Consultancy Ltd. http://www.ffconsultancy.com/?u === Subject: ONE DEGREE AS CREATED =0.1652631 of P1 or 1/19th of 19 degrees This definition of Radian was under Wikkepedia reference, 1d.mind.net/ trignometry Realms/radian Demo1 . Irrespective there needs to be a lot of base of current mathematics that needs to be corrected . one degree was not created by man but by the creator, and was, is , will be It is exatcly equal by time dimension/ reference, and ration, to 0.16526 of Pi, I.e remember Pi is 1 or 22/7 at 19. beore all these complexities of Mathematics, the base of Mathematics needs to be corrected and Inverse 19 made of of ordinary fellows will do this as one of the missions of the research here. Inverse 19 === Subject: Re: ONE DEGREE AS CREATED =0.1652631 of P1 or 1/19th of 19 degrees Explanation in case some one calls me a schzophrenic/jew or something. At 1.05 value, Pi by time dimension equals 16, at 0.05, it is 32. In case you will call me further names, Time dimension, is related to Tm max speed, and one 19 turn , i.e 0ne Pi (times 19 to make a circle from linear), is one unit of time dimension by inverse 19, one degree is thus 19 Pi as 19=1P(time is inverse to dimension). THUS ONE UNIT OF TIME DIMENSION = 19 DEGREES= one Pi, and one degree is 1/19th of 1 time dimension. === Subject: Whitney's Extension Theorem a long time ago a mathematician told me that Whitney's extension theorem enables us to prove the following statement: Let A be an open subset of R^n and f:A -> R a C^m function such that all the partial derivatives of f up to order m extend continously to Cl(A). Then there exist a C^m function F:R^n -> R such that, for every x in A, F(x) = f(x). Recently, I got curious to know if this statement were true. Reading the original work of Whitney H. Whitney, Analytic extension of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934) 63-89 I discovered that Whitney's definition of a function of class C^m in a closed set is intuitive, but not trivial, and I don't know how to use Whitney's theorem (which is really a shining pearl of Analysis) to prove the statement above: at least, the proof is not straightforward, I think. Do you know more about the matter? Maury Barbato === Subject: ONE UNIT OF TIME DIMENSION , DEFINED(inverse 19 definition) RADIAN VALUE 57.295 / 3 =19 This is the time it takes at T max speed for linear dimension to turn one Pi value or 19 degrees. T= 3.14(work in progress on the equation)--- note T max at linear is zero time , static aether 1.It is indirectly proportionate to Pi value, which is 1/3(4 by 4/3) OF RADIAN VALUE OF 57.29/3 =19.0. 2. It time dimension unit is 19 times exactly of one degree by time dimension 3. Thus One degree is 1/19 of one time dimension, and 360 of full circle, Pi progresses 19 seperate times, and since it comes back to where it started, it total value is 2 Pi, at 19 time dimensions, with the expense of 19 time units. THUS TIME DIMENSION IS CURVED AND TIME is proportional to 2Pi(equation under review), LINEAR TIME at maximum spead is zero or static, we call it Static aether. === gmail.com === Subject: Elementary Linear Algebra (6thEd) - Larson, Falvo Instructors Solution Manual is available for purchase! Contact me at instructors.team[at]gmail.com posting-account=jy2E5goAAAB1WGwMQL1h2k61o4k7O_u4 Gecko/2009051221 Firefox/3.0.10,gzip(gfe),gzip(gfe) Elementary Linear Algebra (6thEd) - Larson, Falvo Instructors Solution Manual is available for purchase! Contact me at instructors.team[at] gmail.com === Subject: Test Bank and solution manual posting-account=jy2E5goAAAB1WGwMQL1h2k61o4k7O_u4 Gecko/2009051221 Firefox/3.0.10,gzip(gfe),gzip(gfe) I can get any test bank or solution manual listed below. I will get them for you to purchase them. 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Hall 4th ed ISBN - 0135005183 Advanced Accounting - Floyd Beams 9th ed ISBN - 0131851225 Advanced Accounting - Floyd Beams 10th ed ISBN - 0136033970 Advanced Accounting - Paul Fischer 10th ed ISBN - 0324379056 Advanced Accounting - Paul Fischer Test Bank only 9th ed ISBN - 0324304013 Advanced Calculus - G. B. Folland 1st ed ISBN - 0130652652 Advanced Engineering Mathematics - Michael Greenberg 2nd ed ISBN - 0133214311 Advanced Engineering Mathematics - Peter V. O'Neil 6th ed ISBN - 0534552080 Advertising - Sandra Moriarty 8th ed ISBN - 0132224151 Algebra and Trigonometry - Judith A. Beecher 3rd ed ISBN - 0321466209 Algebra and Trigonometry - Michael Sullivan 8th ed ISBN - 0132329034 Algebra and Trigonometry Enhanced with Graphing Utilities - Michael Sullivan 4th ed ISBN - 0131527398 Algebra and Trigonometry Enhanced with Graphing Utilities - Michael Sullivan 5th ed ISBN - 013600492X Algebra and Trigonometry: Graphs & Models and Graphing Calculator Manual Package - Marvin L. Bittinger 4th ed ISBN - 0321501519 Algebra for College Students - Allen R. Angel 3rd ed ISBN - 0136129080 Algebra for College Students - Margaret L. Lial 6th ed ISBN - 0321442547 Algebra For College Students - Robert F Blitzer 6th ed ISBN - 0136019749 An Introduction to Signals and Systems - John Alan Stuller 1st ed ISBN - 0495073016 Dunlap 1st ed ISBN - 0534392946 Analytical Mechanics - Grant Fowles, George Cassiday 7th ed ISBN - 0534494927 Anatomy & Physiology - Elaine N. Marieb 3rd ed ISBN - 0805347739 Anatomy & Physiology for Emergency Care - Bryan E. Bledsoe 2nd ed ISBN - 0132342987 Applied Algebra - Darel Hardy 1st ed ISBN - 0130674648 Applied Calculus - Geoffrey C. Berresford 5th ed ISBN - 0547169787 Applied Linear Algebra - Chehrzad Shakiban, Peter J. Olver 1st ed ISBN - 0131473824 Applied Multivariate Statistical Analysis - Richard A. Johnson 6th ed ISBN - 0131877151 Applied Partial Differential Equations - Richard Haberman 4th ed ISBN - 0130652431 Applied Physics - Dale Ewen 9th ed ISBN - 0135157331 Applied Statistics for Engineers and Physical Scientists - Johannes Ledolter 3rd ed ISBN - 0136017983 Art and Science of Leadership - Afsaneh Nahavandi 5th ed ISBN - 0136044085 Auditing and Assurance Services - Alvin A. Arens et al 11th ed ISBN - 0131867121 Auditing and Assurance Services - Alvin A. Arens et al 12th ed ISBN - 0135132126 Auditing Assurance and Risk - W. Robert Knechel, Steve Salterio, Brian Ballou 3rd ed ISBN - 0324313187 Auditing Cases - Mark Beasley 3rd ed ISBN - 0131494910 Auditing Cases - Mark S Beasley 4th ed ISBN - 0132423502 Auditing: A Business Risk Approach - Larry E. Rittenberg 6th ed ISBN - 0324375581 Automation, Production Systems, and Computer-Integrated Manufacturing - Mikell P. Groover 2nd ed ISBN - 0130889784 Automation, Production Systems, and Computer-Integrated Manufacturing - Mikell P. Groover 3rd ed ISBN - 0132393212 Basic Business Statistics - Mark L Berenson 10th ed ISBN - 0131678310 Basic Chemistry - Karen C. Timberlake 2nd ed ISBN - 0805344691 Basic Economics - Frank V. Mastrianna Test Bank 15th ed ISBN - 0324599161 Basic Environmental Technology: Water Supply, Waste Management & Pollution Control - Jerry A. Nathanson 5th ed ISBN - 0131190822 Basic Marketing Research Using Microsoft Excel Data Analysis - Alvin C Burns 2nd ed ISBN - 0132059584 Basic Mathematics through Applications - Geoffrey Akst 4th ed ISBN - 0321500113 Basics of Occupational Safety - David L. Goetsch 1st ed ISBN - 013502613X Beginning & Intermediate Algebra - Elayn El Martin-Gay 5th ed ISBN - 0136007317 Beginning Algebra - Elayn El Martin-Gay 5th ed ISBN - 0136007023 Beginning Algebra - Margaret L. Lial 10th ed ISBN - 0321437268 Beginning Algebra with Applications & Visualization - Gary K. Rockswold 2nd ed ISBN - 0321500040 Beginning and Intermediate Algebra - Margaret L. Lial 4th ed ISBN - 0321442334 Beginning and Intermediate Algebra with Applications & Visualization - Gary K. Rockswold 2nd ed ISBN - 0321500059 Behavior in Organizations - Jerald Greenberg 9th ed ISBN - 0131542842 Biochemistry - Mary Campbell 4th ed ISBN - 0534405215 Biochemistry with Lecture Notebook - Mary Campbell 4th ed ISBN - 0534391818 Biology - Neil A. Campbell Test Bank only w/ TestGen Software 7th ed ISBN - 080537146X Biology: Science for Life - Colleen Belk 3rd ed ISBN - 0321559592 Biology: Science for Life with Physiology - Colleen Belk 3rd ed ISBN - 0321559584 Biomaterials: The Intersection of Biology and Materials Science - Johnna S. Temenoff 1st ed ISBN - 0130097101 Biostatistics for the Health Sciences - R. Clifford Blair 1st ed ISBN - 0131176609 Bond Markets, Analysis and Strategies - Frank Fabozzi 6th ed ISBN - 0131986430 Bond Markets, Analysis, and Strategies - Frank J Fabozzi 7th ed ISBN - 0136078974 Brief Course in Mathematical Statistics - Elliot A. Tanis 1st ed ISBN - 0131751395 Brief Principles of Macroeconomics - Gregory Mankiw 5th ed ISBN - 0324590377 Brock Biology of Microorganisms - Michael T. Madigan 12th ed ISBN - 0132324601 Brock Biology of Microorganisms - Michael T. Madigan Test Bank 11th ed ISBN - 0132192268 Building Construction: Principles, Materials, and Systems - Madan Mehta 1st ed ISBN - 0130494216 Building Java Programs: A Back to Basics Approach - Stuart Reges 1st ed ISBN - 0321382838 Business - William M. Pride 10th ed ISBN - 0324829558 Business Analysis and Valuation: Using Financial Statements - Krishna Palepu 3rd ed ISBN - 0132346451 Business and Its Environment - David P. Baron 6th ed ISBN - 0136083927 Business and Society: Ethics and Stakeholder Management - Archie B. Carroll 7th ed ISBN - 0324569394 Business Communication Essentials - Courtland Bovee 4th ed ISBN - 0136084419 Business Communication Essentials and Peak Performance Grammar and Mechanics 2.0 CD Package - Court Bovee 3rd ed ISBN - 0132328992 Business Communication Today - Court Bovee 9th ed ISBN - 0131995359 Business Data Networks and Telecommunications - Raymond R. Panko 7th ed ISBN - 0136153402 Business English: Writing in the Workplace - Blanche Ettinger 4th ed ISBN - 0131565702 Business Ethics: A Stakeholder and Issues Management Approach - Joseph W. Weiss 6th ed ISBN - 0324589735 Business Ethics: Case Studies and Selected Readings - Marianne M. Jennings 6th ed ISBN - 0324657749 Business Forecasting - John Hanke 9th ed ISBN - 0132301202 Business in Action with Real Time Updates - Court Bovee 4th ed ISBN - 0136154085 Business Law - Henry R. Cheeseman 7th ed ISBN - 0136085547 Business Law and the Legal Environment - Jeffrey F. Beatty 4th ed ISBN - 0324303971 Business Law and the Legal Environment - Jeffrey F. Beatty 5th ed ISBN - 0324663528 Business Law and the Regulation of Business - Richard A. Mann 9th ed ISBN - 0324537131 Business Law Principles for Today's Commercial Environment - David P. Twomey 2nd ed ISBN - 0324303947 Business Law Today: Comprehensive - Roger LeRoy Miller 8th ed ISBN - 0324595743 Business Law Today: The Essentials - Roger LeRoy Miller 8th ed ISBN - 0324654545 Business Law: Alternate Edition - Gaylord A. Jentz 11th ed ISBN - 0324596162 Business Law: Text and Cases - Kenneth W. Clarkson 11th ed ISBN - 0324655223 Business Law: Text and Exercises - Roger LeRoy Miller 5th ed ISBN - 032464096X Business Statistics: A Decision Making Approach - David F. Groebner 7th ed ISBN - 0132416921 Business Statistics: A First Course - David Levine 5th ed ISBN - 0136065805 Business: Its Legal, Ethical, and Global Environment - Marianne M. Jennings 8th ed ISBN - 0324655541 Calculus - Dale Varberg 9th ed ISBN - 0131429248 Calculus and Its Applications - Larry Goldstein 11th ed ISBN - 0131919636 Calculus and Its Applications - Larry Goldstein 12th ed ISBN - 0321571304 Calculus and Its Applications - Marvin L. Bittinger 8th ed ISBN - 0321166396 Calculus and Its Applications - Marvin L. Bittinger 9th ed ISBN - 0321395344 Calculus Early Transcendentals - Henry Edwards 7th ed ISBN - 0131569899 Calculus for Business, Economics, Life Sciences & Social Sciences - Raymond Barnett 11th ed ISBN - 0132328186 Calculus for the Life Sciences - Marvin L. Bittinger 1st ed ISBN - 0321279352 Calculus With Applications - Margaret L. Lial et al 8th ed ISBN - 0321228146 Calculus with Applications for the Life Sciences - Raymond N. Greenwell 1st ed ISBN - 0201745828 Calculus, Early Transcendentals - C. Henry Edwards 7th ed ISBN - 0131569899 California Real Estate Law - Theodore Gordon 7th ed ISBN - 0324654685 Capital Budgeting and Long-Term Financing Decisions - Neil Seitz 4th ed ISBN - 0324258089 Capital Markets: Institutions and Instruments - Frank J Fabozzi 4th ed ISBN - 0136026028 Cases in Management Accounting and Control Systems - Brandt Allen 4th ed ISBN - 0135704251 Chemistry - John E McMurry Test Bank only 5th ed ISBN - 0131993232 Chemistry : An Introduction to General, Organic, Biological Chemistry - Karen Timberlake 9th ed ISBN - 0805330151 Chemistry for Changing Times - John W. Hill 12th ed ISBN - 0136054498 Chemistry: An Introduction to General, Organic, & Biological Chemistry - Karen C Timberlake 10th ed ISBN - 0136019706 Civil Drafting Technology - David A. Madsen 7th ed ISBN - 0135000688 ISBN - 0534408966 CMOS Circuit Design, Layout, and Simulation - David E. Boyce et al 1st ed ISBN - 0780334167 College Accounting 1-12 - Jeffrey Slater 9th ed ISBN - 0131071696 College Accounting 1-25 - Jeffrey Slater 10th ed ISBN - 0132286386 College Accounting Chapters 1-15 - James Heintz 19th ed ISBN - 0324382499 College Accounting Chapters 1-27 - James Heintz 19th ed ISBN - 0324376162 College Accounting Chapters 1-9 - James Heintz 19th ed ISBN - 0324382480 College Accounting: A Practical Approach Canadian Edition - Jeffrey Slater 10th ed ISBN - 0132069245 College Algebra - J. S. Ratti 1st ed ISBN - 0321296443 College Algebra - Judith A. Beecher 3rd ed ISBN - 0321466071 College Algebra - Margaret L. Lial 10th ed ISBN - 0321499131 College Algebra - Mark Dugopolski 4th ed ISBN - 0321356918 College Algebra - Michael Sullivan 8th ed ISBN - 0132402866 College Algebra - Robert F. Blitzer 5th ed ISBN - 0321559835 College Algebra and Trigonometry - J. S. Ratti 1st ed ISBN - 0321296427 College Algebra and Trigonometry - Margaret L. Lial 4th ed ISBN - 0321497449 College Algebra Enhanced with Graphing Utilities - Michael Sullivan 5th ed ISBN - 0136004911 College Algebra Essentials - Michael Sullivan 8th ed ISBN - 0136154344 College Algebra: Graphs and Models with Graphing Calculator Manual Package - Marvin L. Bittinger 4th ed ISBN - 0321531922 College Geometry - David C. Kay 2nd ed ISBN - 0321046242 College Geometry: A Problem Solving Approach with Applications - Gary L. Musser 2nd ed ISBN - 0131879693 College Math for Business, Economics, Life Sciences & Social Sciences - Raymond Barnett 11th ed ISBN - 0131572253 College Physics - Jerry D Wilson 6th ed ISBN - 0131495798 College Physics - Jerry D Wilson 7th ed ISBN - 0321571118 College Physics with Mastering Physics - Hugh Young 8th ed ISBN - 0805390707 Communicating in the Workplace - Thomas Cheesebro 1st ed ISBN - 0136136915 Communication Systems Engineering - John G. Proakis 2nd ed ISBN - 0130617938 Comparative International Accounting - Christopher Nobes 9th ed ISBN - 0273703579 Complex Variables With Applications - A. David Wunsch 3rd ed ISBN - 0201756099 Comprehensive Periodontics for the Dental Hygienist - Mea A. Weinberg 3rd ed ISBN - 0135015421 Computer Algorithms - Allen Van Gelder, Sara Baase 3rd ed ISBN - 0201612445 Computer Networking Complete Package - James F. Kurose 3rd ed ISBN - 0321418492 Computer Networking with Internet Protocols - William Stallings 1st ed ISBN - 0131410989 Computer Networking: A Top-Down Approach - James F. Kurose 4th ed ISBN - 0321497708 Computer Networking: A Top-Down Approach - James F. Kurose 5th ed ISBN - 0136079679 Computer Networking: A Top-Down Approach Featuring the Internet - James F. Kurose 3rd ed ISBN - 0321227352 Computer Organization and Architecture - William Stallings 7th ed ISBN - 0130351199 Computer Organization and Architecture: Designing for Performance - William Stallings 7th ed ISBN - 0131856448 Computer Organization and Architecture: Designing for Performance - William Stallings 8th ed ISBN - 0136073735 Computer Science: An Overview - J. Glenn Brookshear 10th ed ISBN - 0321524039 Computer Security: Principles and Practice - William Stallings 1st ed ISBN - 0136004245 Computer Systems Organization & Architecture - John D. Carpinelli 1st ed ISBN - 0201612534 Concepts in Federal Taxation 2007 - Kevin Murphy 14th ed ISBN - 0324313527 Concepts in Federal Taxation 2008 - Kevin Murphy 15th ed ISBN - 0324640153 Concepts in Federal Taxation 2009 - Kevin Murphy 16th ed ISBN - 0324659377 Concepts In Systems and Signals - John D. Sherrick 2nd ed ISBN - 0131782711 Concepts of Calculus with Applications - Martha Goshaw 1st ed ISBN - 0321320786 Concepts of Calculus With Applications-Updated Edition - Martha Goshaw 2nd ed ISBN - 0321577442 Concepts of Programming Languages - Robert W. Sebesta 8th ed ISBN - 0321493621 Conceptual Physical Science - Paul G. Hewitt 4th ed ISBN - 0321516958 Conceptual Physics Fundamentals - Paul G. Hewitt 1st ed ISBN - 0321501365 Conceptual Physics Media Update - Paul G. Hewitt 10th ed ISBN - 0321548094 Concrete Structures - Mehdi Setareh 1st ed ISBN - 0131988271 Construction Accounting & Financial Management - Stephen Peterson 2nd ed ISBN - 0135017114 Construction Methods and Management - Stephens W. Nunnally 7th ed ISBN - 0131716859 Construction Project Management - Fred Gould 3rd ed ISBN - 0131996231 Consumer Behavior - Michael Solomon 8th ed ISBN - 0136015964 Consumer Behavior - Wayne D. Hoyer 5th ed ISBN - 0547079923 Contemporary Auditing: Real Issues & Cases - Michael C. Knapp 7th ed ISBN - 0324658052 Contemporary Auditing: Real Issues & Cases Update - Michael C. Knapp 7th ed ISBN - 143907819X Contemporary Business and Online Commerce Law - Henry R. Cheeseman 6th ed ISBN - 013601500X Contemporary Engineering Economics - Chan S. Park 4th ed ISBN - 0131876287 Contemporary Financial Management - Charles Moyer 10th ed ISBN - 0324289081 Contemporary Financial Management - R. Charles Moyer, James R. McGuigan 11th ed ISBN - 0324653506 Contemporary Logistics - Paul R. Murphy 9th ed ISBN - 013156207X Contemporary Marketing - Louis E. Boone 14th ed ISBN - 032458203X Contemporary Marketing 2009 Update - Louis E. Boone 13th ed ISBN - 0324580215 Contemporary Project Management - Timothy Kloppenborg 1st ed ISBN - 0324382383 Cornerstones of Managerial Accounting - Maryanne M. Mowen 2nd ed ISBN - 0324379609 Cornerstones of Managerial Accounting - Maryanne M. Mowen 3rd ed ISBN - 0324660138 Corporate Finance - Jonathan Berk 1st ed ISBN - 0321415116 Corporate Finance - Michael C. Ehrhardt, Eugene F. Brigham 3rd ed ISBN - 0324655681 Corporate Finance: The Core plus MyFinanceLab Student Access Kit - Jonathan Berk 1st ed ISBN - 032155759X Corporate Financial Accounting - Carl S. Warren 10th ed ISBN - 0324663838 Corporate Financial Management - Douglas R. Emery 3rd ed ISBN - 0132278723 Cost Accounting - Charles T. Horngren, George Foster, Srikant M. Datar 12th ed ISBN - 0131495380 Cost Accounting - Charles T. Horngren, George Foster, Srikant M. Datar 13th ed ISBN - 0136126634 Cost Accounting Canadian Edition - Charles Horngren 4th ed ISBN - 0131971905 Cost Accounting: Traditions & Innovations - Jesse Barfield 5th ed ISBN - 032418090X Cost Benefit Analysis: Concepts and Practice - Anthony Boardman 3rd ed ISBN - 0131435833 Cost Management: Accounting and Control - Don R. Hansen, Maryanne M. Mowen 6th ed ISBN - 0324559674 Course in Probability - Neil Weiss 1st ed ISBN - 0201774712 Criminology: A Global Perspective - Robert W. Winslow 1st ed ISBN - 0131839020 Cryptography and Network Security - William Stallings 4th ed ISBN - 0131873164 Customer Service: Career Success Through Customer Loyalty - Paul R. Timm 4th ed ISBN - 0132236583 Data Abstraction & Problem Solving with C++ - Frank M. Carrano 5th ed ISBN - 0321433327 Data and Computer Communications - William Stallings 8th ed ISBN - 0132433109 Data Structures and Algorithm Analysis in C++ - Mark Allen Weiss 3rd ed ISBN - 032144146X Data Structures and Algorithm Analysis in Java - Mark Allen Weiss 2nd ed ISBN - 0321370139 Database Concepts - David Kroenke 3rd ed ISBN - 0131986252 Database Concepts - David Kroenke 4th ed ISBN - 0136086535 Database Systems: A Practical Approach - Thomas M. Connolly 4th ed ISBN - 0321294017 Derivatives Markets - Robert L. McDonald 2nd ed ISBN - 032128030X Detection and Estimation:Theory and Its Applications - Thomas Schonhoff 1st ed ISBN - 0130894990 Developmental Mathematics - Marvin L. Bittinger 7th ed ISBN - 0321331915 Developmental Mathematics: Basic Mathematics and Algebra - Margaret L. Lial 1st ed ISBN - 0321506421 Differential Equations - John Polking 2nd ed ISBN - 0131437380 Differential Equations and Boundary Value Problems: Computing and Modeling - Henry Edwards 4th ed ISBN - 0131561073 Differential Equations and Linear Algebra - Henry Edwards, David E. Penney 2nd ed ISBN - 0131481460 Differential Equations and Linear Algebra - Henry Edwards, David E. Penney 3rd ed ISBN - 0136054250 Differential Equations and Linear Algebra - Jerry Farlow 2nd ed ISBN - 0131860615 Differential Equations and Linear Algebra - Stephen W. Goode 3rd ed ISBN - 0130457949 Differential Equations Computing and Modeling - Henry Edwards 4th ed ISBN - 0136004385 Differential Equations With Boundary Value Problems - John C. Polking 2nd ed ISBN - 0130911062 Digital & Analog Communication Systems - Leon Couch 7th ed ISBN - 0131424920 Digital Communications - John Proakis 4th ed ISBN - 0072321113 Digital Design - Morris Mano 4th ed ISBN - 0131989243 Digital Electronics: A Practical Approach - William Kleitz 8th ed ISBN - 0132435780 Digital Fundamentals - Thomas Floyd 10th ed ISBN - 0132359235 Digital Signal Processing - John Proakis 4th ed ISBN - 0131873741 Digital Signal Processing Using MATLAB -Vinay K. Ingle, John G. Proakis 2nd ed ISBN - 0495073113 Digital Systems Design Using VHDL - Charles H. Roth 2nd ed ISBN - 0534384625 Digital Systems: Principles and Applications - Ronald Tocci et al 10th ed ISBN - 0131725793 Discrete and Combinatorial Mathematics - Ralph P. Grimaldi 5th ed ISBN - 0201726343 Discrete Mathematics - Edgar G. Goodaire, Michael M Parmenter 3rd ed ISBN - 0131679953 Discrete Mathematics - Otto, Eynden, Dossey, Spence 4th ed ISBN - 0321079124 Discrete Mathematics - Otto, Eynden, Dossey, Spence 5th ed ISBN - 0321305159 Discrete Mathematics - Richard Johnsonbaugh 6th ed ISBN - 0131176862 Drugs & the Human Body - Ken Liska 8th ed ISBN - 0132447134 Dynamics of Structures - Anil K. Chopra 3rd ed ISBN - 013156174X ECON for Macroeconomics - William A. McEachern 1st ed ISBN - 0324587805 Economic Development - Michael P. Todaro 10th ed ISBN - 0321485734 Economic Development - Michael Todaro, Stephen Smith 9th ed ISBN - 0321278887 Economic Growth - David N. Weil 2nd ed ISBN - 0321416627 Economic Growth - David Weil 1st ed ISBN - 0201680262 Economics - Michael Parkin 8th ed ISBN - 0321423003 Economics - Michael Parkin 9th ed ISBN - 0321600037 Economics - Richard Lipsey 13th ed ISBN - 0321369211 Economics - Roger A. Arnold 9th ed ISBN - 0324595425 Economics for Managers - Paul G Farnham 1st ed ISBN - 0130924253 Economics for Managers - Paul G Farnham 2nd ed ISBN - 013606552X Economics for Today - Irvin B. Tucker 6th ed ISBN - 0324591365 Economics of Money, Banking, and Financial Markets - Frederic Mishkin Test Bank 8th ed ISBN - 0321415051 Economics of Money, Banking, and Financial Markets, Update - Frederic Mishkin 7th ed ISBN - 0321331850 Economics Today - Roger LeRoy Miller 14th ed ISBN - 0321422341 Economics Today - Roger Miller 15th ed ISBN - 0321600215 Economics Today: The Macro View - Roger Miller 13th ed ISBN - 0321278992 Economics Today: The Macro View - Roger Miller 14th ed ISBN - 0321421442 Economics Today: The Macro View - Roger Miller 15th ed ISBN - 0321600223 Economics Today: The Micro View - Roger Miller 13th ed ISBN - 0321278984 Economics Today: The Micro View - Roger Miller 14th ed ISBN - 0321425065 Economics Today: The Micro View - Roger Miller 15th ed ISBN - 0321600185 Economics: A Contemporary Introduction - William A. McEachern 8th ed ISBN - 0324579217 Economics: A Tool for Critically Understanding Society - Tom Riddell 8th ed ISBN - 0321423585 Economics: Principles and Policy - William J. Baumol 10th ed ISBN - 0324537026 Economics: Principles and Policy - William J. Baumol 11th ed ISBN - 0324586205 Economics: Private and Public Choice - James D. Gwartney 12th ed ISBN - 0324580185 Effective Small Business Management - Norman M. Scarborough 9th ed ISBN - 0136152708 Effective Writing - Claire B. May 8th ed ISBN - 0136029086 Electric Circuits - James Nilsson 8th ed ISBN - 0131989251 Electrical Engineering: Principles and Applications - Allan R. Hambley 4th ed ISBN - 0131989227 Electrical Machines, Drives and Power Systems - Theodore Wildi 6th ed ISBN - 0131776916 Electronic Commerce 2008 - Efraim Turban 5th ed ISBN - 0132243318 Electronic Communications for Technicians - Tom Wheeler 2nd ed ISBN - 0131130498 Electronics and Computer Math - Bill R. Deem 8th ed ISBN - 0131711377 Electronics Fundamentals: Circuits, Devices and Applications - Thomas Floyd 7th ed ISBN - 013219709X Elementary Algebra - George Woodbury 1st ed ISBN - 0321166426 Elementary Algebra Early Graphing for College Students - Allen R. Angel 3rd ed ISBN - 0136134165 Elementary Algebra: Graphs and Authentic Applications - Jay Lehmann 1st ed ISBN - 013220164X Elementary and Intermediate Algebra - George Woodbury 2nd ed ISBN - 0321500067 Elementary and Intermediate Algebra: Graphs & Models - Marvin L. Bittinger 3rd ed ISBN - 0321422406 Elementary Differential Equations - Henry Edwards 6th ed ISBN - 0132397307 Elementary Differential Equations - Werner E. Kohler, Lee W.Johnson 1st ed ISBN - 0201709260 Elementary Differential Equations with Boundary Value Problems - Henry Edwards 6th ed ISBN - 0136006132 Elementary Differential Equations With Boundary Value Problems - Lee Johnson et al 1st ed ISBN - 0321121643 Elementary Differential Equations With Boundary Value Problems - Lee Johnson et al 2nd ed ISBN - 0321398505 Elementary Linear Algebra - Ron Larson 6th ed ISBN - 0618783768 Elementary Linear Algebra with Applications - Bernard Kolman 9th ed ISBN - 0132296543 Elementary Number Theory - Kenneth H. Rosen 5th ed ISBN - 0321237072 Elementary Statistics - Mario F. Triola 10th ed ISBN - 0321331834 Elementary Statistics - Mario F. Triola 9th ed ISBN - 0201775700 Elementary Statistics - Mario F. Triola 11th ed ISBN - 0321500245 Elementary Statistics - Neil A. Weiss 7th ed ISBN - 0321422090 Elementary Statistics - Ron Larson 4th ed ISBN - 0132424339 Elementary Statistics Using Excel - Mario Triola 3rd ed ISBN - 0321365135 Elementary Statistics Using the TI-83/84 Plus Calculator - Mario F. Triola 2nd ed ISBN - 0321462572 Elementary Statistics With Multimedia Study Guide - Mario F. Triola 10th ed ISBN - 0321460928 Elements of Forecasting - Francis X. Diebold 4th ed ISBN - 032432359X Embedded Microcontrollers & Processor Design - Charles Greg Osborn 1st ed ISBN - 0131130412 Embedded System Design with C805 - Han-Way Huang 1st ed ISBN - 0495471747 Employment Law - John J. Moran 4th ed ISBN - 0136009964 Engineering Computation with MATLAB - David M. Smith 2nd ed ISBN - 0136080634 Engineering Economy - William G Sullivan 13th ed ISBN - 0131486497 Engineering Economy - William G. Sullivan 14th ed ISBN - 0136142974 Engineering Economy and the Decision-Making Process - Joseph C. Hartman 1st ed ISBN - 0131424017 Engineering Fundamentals: An Introduction to Engineering - Saeed Moaveni 3rd ed ISBN - 0495082538 Engineering Materials: Properties and Selection - Ken Budinski 8th ed ISBN - 0131837796 Engineering Materials: Properties and Selection - Kenneth G. Budinski 9th ed ISBN - 0137128428 Engineering Mechanics Dynamics - Anthony M Bedford 5th ed ISBN - 0136129161 Engineering Mechanics: Dynamics - Andrew Pytel 3rd ed ISBN - 0495295612 Engineering Mechanics: Statics - Andrew Pytel 3rd ed ISBN - 0495244694 Engineering Mechanics: Statics - Anthony M Bedford 5th ed ISBN - 0136129153 Engineering Mechanics: Statics - Russell C. Hibbeler 11th ed ISBN - 0132215004 Engineering Mechanics: Statics - Russell C. Hibbeler 12th ed ISBN - 0136077900 Engineering Mechanics: Statics Computational Edition - Robert W. Soutas-Little 1st ed ISBN - 0534549217 Engineering Vibration - Daniel Inman 3rd ed ISBN - 0132281732 Enterprise Systems for Management - Luvai Motiwalla 1st ed ISBN - 013233531X Entrepreneurial Finance - Chris Leach 3rd ed ISBN - 0324561253 Entrepreneurial Finance - Philip J. Adelman 4th ed ISBN - 0132434792 Entrepreneurial Finance - Philip J. Adelman 5th ed ISBN - 013502529X Entrepreneurship: Successfully Launching New Ventures - Bruce Barringer 2nd ed ISBN - 0132240572 Entrepreneurship: Theory, Process, and Practice - Donald F. Kuratko 8th ed ISBN - 0324590911 Environmental and Natural Resource Economics - Tom Tietenberg 7th ed ISBN - 0321305043 Environmental Issues: An Introduction to Sustainability - Robert L. McConnell 3rd ed ISBN - 0131566504 Environmental Law - Nancy K. Kubasek 6th ed ISBN - 0136142168 Environmental Science: Toward A Sustainable Future - Richard T. Wright 10th ed ISBN - 0132302659 Error Control Coding - Daniel J. Costello Jr., Shu Lin 2nd ed ISBN - 0130426725 Essential Foundations of Economics - Robin Bade 4th ed ISBN - 0321522354 Essentials of Business Law - Jeffrey F. Beatty 3rd ed ISBN - 0324537123 Essentials of Business Law and the Legal Environment - Richard A. Mann 10th ed ISBN - 0324593562 Essentials of College Algebra with Modeling and Visualization - Gary K. Rockswold 3rd ed ISBN - 0321448898 Essentials of College Algebra, Alternate Edition - Margaret L. Lial 1st ed ISBN - 0321491858 Essentials of Economics - Gregory Mankiw 4th ed ISBN - 0324236964 Essentials of Economics - Gregory Mankiw 5th ed ISBN - 0324590024 Essentials of Entrepreneurship and Small Business Management - Thomas W Zimmerer 5th ed ISBN - 0132294389 Essentials of Logic - Irving Copi 2nd ed ISBN - 013238034X Essentials of Management Information Systems - Jane Laudon 8th ed ISBN - 013602579X Essentials of Managerial Finance - Scott Besley 13th ed ISBN - 0324258755 Essentials of Marketing - Charles W. Lamb 6th ed ISBN - 0324656203 Essentials of Materials Science & Engineering - Donald R. Askeland 2nd ed ISBN - 0495244465 Essentials of Organizational Behavior - Stephen P Robbins 9th ed ISBN - 0132431521 Essentials of Organizational Behavior - Stephen P. Robbins 10th ed ISBN - 0136077617 Essentials of Statistics - Mario F. Triola 3rd ed ISBN - 0321434250 Essentials of the Legal Environment - Roger LeRoy Miller 2nd ed ISBN - 0324400403 Ethics for the Information Age - Mike Quinn 3rd ed ISBN - 0321536851 Excellence in Business Communication - John V. Thill 8th ed ISBN - 0136157505 Experiencing MIS - David Kroenke 2nd ed ISBN - 0136078680 Exploring Business - Karen Collins 1st ed ISBN - 0131403656 Exploring Corporate Strategy - Gerry Johnson 8th ed ISBN - 140588732X Exploring Macroeconomics - Robert L. Sexton 4th ed ISBN - 0324395558 Federal Tax Research - William A. Raabe 8th ed ISBN - 0324659652 Feedback Control of Dynamic Systems - Gene Franklin 5th ed ISBN - 0131499300 Financial & Managerial Accounting - Carl S. Warren 9th ed ISBN - 0324401884 Financial & Managerial Accounting - Carl S. Warren 10th ed ISBN - 0324663811 Financial Accounting - Belverd E. Needles 10th ed ISBN - 0547193289 Financial Accounting - Carl S. Warren, James M. Reeve 10th ed ISBN - 0324380674 Financial Accounting - Carl S. Warren, James M. Reeve 11th ed ISBN - 0324663781 Financial Accounting - Jane Reimers 1st ed ISBN - 0131492012 Financial Accounting - Walter Harrison, Charles Horngren 6th ed ISBN - 0131499459 Financial Accounting - Walter Harrison, Charles Horngren 7th ed ISBN - 0138128200 Financial Accounting and Financial Tips - Walter T. Harrison 7th ed ISBN - 0135012848 Financial Accounting, Reporting & Analysis: International Edition - Barry Elliott 2nd ed ISBN - 027370253X Financial Accounting: A Bridge to Decision Making - Robert Ingram 6th ed ISBN - 0324313357 Financial Accounting: A Business Process Approach - Jane L. Reimers 2nd ed ISBN - 0131473867 Financial Accounting: An Integrated Statements Approach - Jonathan Duchac 2nd ed ISBN - 0324312113 Financial Accounting: An Introduction to Concepts, Methods and Uses - Clyde P. Stickney 12th ed ISBN - 0324381980 Financial Accounting: An Introduction to Concepts, Methods and Uses - Clyde P. Stickney 13th ed ISBN - 0324651147 Financial Accounting: The Impact on Decision Makers - Gary Porter 6th ed ISBN - 0324655231 Financial and Managerial Accounting - Meg Pollard 1st ed ISBN - 0136008984 Financial And Managerial Accounting Ch 1-13 - Charles Horngren 1st ed ISBN - 0135009855 Financial Economics - Zvi Bodie 2nd ed ISBN - 0131856154 Financial Management for Public, Health, and Not-for-Profit - Steven A Finkler 3rd ed ISBN - 0136070736 Financial Management For Public, Health, and Not-for-Profit Organizations - Steven Finkler 2nd ed ISBN - 0131471988 Financial Management: Theory & Practice - Eugene Brigham 12th ed ISBN - 0324422695 Financial Markets and Institutions - Frederic S. Mishkin 5th ed ISBN - 0321280296 Financial Markets and Institutions - Frederic S. Mishkin 6th ed ISBN - 0321374215 Financial Markets and Institutions - Jeff Madura 8th ed ISBN - 0324568215 Financial Markets and Institutions Abridged Edition - Jeff Madura 8th ed ISBN - 0324593643 Financial Reporting and Analysis - Lawrence Revsine 3rd ed ISBN - 0131430211 Financial Reporting and Analysis Using Financial Accounting Information - Charles Gibson 10th ed ISBN - 0324304455 Financial Reporting and Analysis Using Financial Accounting Information - Charles Gibson 11th ed ISBN - 0324657420 Financial Reporting, Financial Statement Analysis, and Valuation - Clyde P. Stickney 6th ed ISBN - 0324302959 Financial/Managerial Accounting - Walter T. Harrison 1st ed ISBN - 0131568779 Finite Element Analysis Theory and Application with ANSYS - Saeed Moaveni 3rd ed ISBN - 0131890808 Finite Math and Its Application - Larry Goldstein 9th ed ISBN - 0131873644 Finite Mathematics - Margaret L. Lial et al 8th ed ISBN - 032122826X Finite Mathematics and Calculus with Applications - Margaret Lial 8th ed ISBN - 0321426517 Finite Mathematics for Business, Economics, Life Sciences & Social Sciences - Raymond Barnett 11th ed ISBN - 0132255707 Finite Mathematics with Applications - Margaret L. Lial 9th ed ISBN - 0321386728 First Course in Abstract Algebra - John Fraleigh 7th ed ISBN - 0201763907 First Course in Abstract Algebra - Joseph Rotman 3rd ed ISBN - 0131862677 First Course In Probability - Sheldon M. Ross 7th ed ISBN - 0131856626 First Course In Probability - Sheldon M. Ross 8th ed ISBN - 013603313X First Course in Statistics, A - James T. McClave 10th ed ISBN - 0136152597 Fluency with Information Technology: Skills, Concepts, and Capabilities - Lawrence Snyder 3rd ed ISBN - 0321512391 Foundations of Economics - Robin Bade 4th ed ISBN - 0321522362 Foundations of Finance - Arthur Keown, William Petty, John Martin, David Scott 5th ed ISBN - 0131856057 Foundations of Finance: Logic and Practice of Financial Mangement - Arthur J. Keown 6th ed ISBN - 0135048168 Foundations of Finance: The Logic and Practice of Financial Management - Arthur Keown 6th ed ISBN - 0132339226 Foundations of Financial Markets and Institutions - Frank J. Fabozzi 4th ed ISBN - 0136135315 Foundations of Geometry - Gerard Venema 5th ed ISBN - 0131437003 Foundations of Macroeconomics - Robin Bade 4th ed ISBN - 0321522370 Foundations of MEMS - Chang Liu 1st ed ISBN - 0131472860 Foundations of Microeconomics - Robin Bade 3rd ed ISBN - 0321415957 Foundations of Microeconomics - Robin Bade 4th ed ISBN - 0321522389 Foundations of the Legal Environment of Business - Marianne M. Jennings 1st ed ISBN - 0324566514 Framework for Human Resource Management, A - Gary Dessler 5th ed ISBN - 0136041531 Framework for Marketing Management, A - Philip Kotler 4th ed ISBN - 0136026605 Fraud Examination - Steve Albrecht 2nd ed ISBN - 0324651155 Friendly Introduction to Analysis - Witold A.J. Kosmala 2nd ed ISBN - 0130457965 Fundamental Cornerstones of Managerial Accounting - Dan L. Heitger, Maryanne M. Mowen 1st ed ISBN - 0324378068 Fundamental Mathematics through Applications - Geoffrey Akst 4th ed ISBN - 0321496906 Fundamentals of Advanced Accounting - Paul M. Fischer Test Bank 1st ed ISBN - 0324378904 Fundamentals of Applied Electromagnetics - Fawwaz T. Ulaby 5th ed ISBN - 0132413264 Fundamentals of Business Law Summarized Cases - Roger LeRoy Miller 7th ed ISBN - 0324381689 Fundamentals of Business Law: Excerpted Cases - Roger LeRoy Miller 2nd ed ISBN - 0324595727 Fundamentals of Business Law: Summarized Cases - Roger LeRoy Miller 8th ed ISBN - 0324595735 Fundamentals of Communication Systems - John G. Proakis 1st ed ISBN - 013147135X Fundamentals of Complex Analysis - Edward Saff 3rd ed ISBN - 0139078746 Fundamentals of Derivatives Markets - Robert L. McDonald 1st ed ISBN - 0321357175 Fundamentals of Differential Equations - Kent Nagle, Edward Saff 6th ed ISBN - 0321145720 Fundamentals of Differential Equations - R. Kent Nagle 7th ed ISBN - 0321410483 Fundamentals of Differential Equations with Boundary Value Problems - R. Kent Nagle 5th ed ISBN - 0321419219 Fundamentals of Electromagnetics for Electrical and Computer Engineering - Nannapaneni Narayana Rao 1st ed ISBN - 0136013333 Fundamentals of Engineering Economics - Chan S. Park 2nd ed ISBN - 0132209608 Fundamentals of Financial Management - Eugene Brigham 11th ed ISBN - 0324319800 Fundamentals of Financial Management - Eugene F. Brigham 12th ed ISBN - 0324597703 Fundamentals of Financial Management Concise - Eugene F. Brigham 6th ed ISBN - 0324664559 Fundamentals of Investing - Lawrence J. Gitman 10th ed ISBN - 0321489381 Fundamentals of Management: Essential Concepts and Applications - Stephen P. Robbins Test Bank 5th ed ISBN - 0131487361 Fundamentals of Multinational Finance - Michael Moffett 3rd ed ISBN - 0321541642 Fundamentals of Organic Chemistry - John McMurry Test Bank only 5th ed ISBN - 0534395732 Fundamentals of Probability, with Stochastic Processes - Saeed Ghahramani 3rd ed ISBN - 0131453408 Fundamentals of Signals and Systems - Edward Kamen 3rd ed ISBN - 0131687379 Fundamentals of Statistics - Michael Sullivan 2nd ed ISBN - 0131569872 Further Mathematics for Economic Analysis - Knut Sydsaeter et al 1st ed ISBN - 0273655760 Geometry: Theorems and Constructions - Allan Berele 1st ed ISBN - 0130871214 Global Investments - Bruno Solnik 6th ed ISBN - 0321527704 Global Strategy - Mike W. Peng 2nd ed ISBN - 0324590997 Governmental and Nonprofit Accounting: Theory and Practice - Robert J. Freeman 9th ed ISBN - 0136029515 Health Economics - Charles E. Phelps 4th ed ISBN - 0321594576 High-Speed Networks and Internets: Performance and Quality of Service - William Stallings 2nd ed ISBN - 0130322210 Historical Geology - Reed Wicander 6th ed ISBN - 0495560073 Human Anatomy & Physiology - Elaine N. Marieb 7th ed ISBN - 0805359095 Human Anatomy and Physiology - Elaine N. Marieb 8th ed ISBN - 0805395911 Human Anatomy and Physiology Lab Manual - Elaine N. Marieb 9th ed ISBN - 0805372652 Human Biology: Concepts and Current Issues - Michael D. Johnson 5th ed ISBN - 0321570200 Human Diseases: A Systemic Approach - Mark Zelman 7th ed ISBN - 0135155568 Human Physiology: An Integrated Approach - Dee Unglaub Silverthorn 5th ed ISBN - 0321559398 ISBN - 0495014850 Human Relations for Career and Personal Success: Concepts, Applications, and Skills - Andrew J. DuBrin 8th ed ISBN - 0131791796 Human Relations: Interpersonal Job-Oriented Skills - Andrew J. DuBrin 10th ed ISBN - 0135019443 Human Resource Management - Gary Dessler 11th ed ISBN - 0131746170 Human Resource Management - R. Wayne Mondy 11th ed ISBN - 0136077285 Human Resource Management - Wayne Mondy 10th ed ISBN - 0132225956 Human Side of Organizations - Michael Drafke 10th ed ISBN - 0135139740 Hydrology and Floodplain Analysis - Philip B. Bedient 4th ed ISBN - 0131745891 Income Tax Fundamentals 2006 - Gerald E. Whittenburg 24th ed ISBN - 0324399022 Income Tax Fundamentals 2007 - Gerald E. Whittenburg 25th ed ISBN - 032439926X Income Tax Fundamentals 2009 - Gerald E. Whittenburg 27th ed ISBN - 0324663676 Information Systems Today: Managing in the Digital World - Leonard Jessup 3rd ed ISBN - 0132335069 Information Systems Today: Managing the Digital World - Joseph Valacich 4th ed ISBN - 0136078400 Information Technology Auditing and Assurance - James Hall 2nd ed ISBN - 0324191987 Inquiry into Physics - Vern J. Ostdiek 6th ed ISBN - 0495119431 Integrated Arithmetic and Basic Algebra - Bill E. Jordan 4th ed ISBN - 0321442555 Intel Micro 8086 - Barry B. Brey 8th ed ISBN - 0135026458 Intel Microprocessors - Barry B. Brey 7th ed ISBN - 0131195069 Intel Microprocessors - Barry B. Brey 8th ed ISBN - 0135026458 Interactive Computer Graphics: A Top-Down Approach Using OpenGL - Edward Angel 5th ed ISBN - 0321535863 Interactive Statistics - Martha Aliaga, Brenda Gunderson 3rd ed ISBN - 0131497561 Intermediate Accounting - James D. Stice 16th ed ISBN - 0324312148 Intermediate Accounting - James D. Stice 17th ed ISBN - 032459237X Intermediate Accounting - Loren A. Nikolai 10th ed ISBN - 0324651929 Intermediate Accounting - Loren A. Nikolai 11th ed ISBN - 032465913X Intermediate Accounting Revised - David Spiceland 4th ed ISBN - 0073215422 Intermediate Algebra - Elayn El Martin-Gay 5th ed ISBN - 0136007295 Intermediate Algebra - Margaret L. Lial 10th ed ISBN - 0321443624 Intermediate Algebra - Margaret L. Lial 9th ed ISBN - 0321574974 Intermediate Algebra - Marvin L. Bittinger 10th ed ISBN - 0321319087 Intermediate Algebra for College Students - Allen R. Angel 7th ed ISBN - 0132383578 Intermediate Algebra for College Students - Robert F. Blitzer 5th ed ISBN - 0136007627 Intermediate Algebra with Applications & Visualization - Gary K. Rockswold 3rd ed ISBN - 0321500032 Intermediate Algebra: Functions & Authentic Applications - Jay Lehmann 3rd ed ISBN - 0131953338 Intermediate Algebra: Graphs & Models - Marvin L. Bittinger 3rd ed ISBN - 0321416163 Intermediate Financial Management - Eugene F. Brigham 10th ed ISBN - 0324594690 International Accounting - Frederick Choi 5th ed ISBN - 0131480979 International Accounting - Frederick D. Choi 6th ed ISBN - 0131588141 International Business - John Daniels 12th ed ISBN - 0136029655 International Business - Ricky Griffin 6th ed ISBN - 0137153732 International Business Law - Ray A. August 5th ed ISBN - 013600864X International Business Law and Its Environment - Richard Schaffer 7th ed ISBN - 0324649673 International Business: Environments and Operations - John Daniels 11th ed ISBN - 0131869426 International Business: Strategy, Management, and the New Realities - Tamer Cavusgil 1st ed ISBN - 0131738607 International Business: The Challenges of Globalization - John J. Wild 4th ed ISBN - 0131747436 International Business: The Challenges of Globalization - John J. Wild 5th ed ISBN - 0137153759 International Economics - Charles Sawyer, Richard Sprinkle 2nd ed ISBN - 0131704168 International Economics - James Gerber 3rd ed ISBN - 032123796X International Economics - James Gerber 4th ed ISBN - 0321415558 International Economics - Robert Carbaugh 11th ed ISBN - 032442194X International Economics - W. Charles Sawyer 3rd ed ISBN - 0136054692 International Economics: Theory And Policy - Paul Krugman, Maurice Obstfeld 7th ed ISBN - 0321293835 International Economics: Theory and Policy - Paul R. Krugman 8th ed ISBN - 0321488830 International Financial Management - Geert Bekaert 1st ed ISBN - 0131163604 International Financial Management - Jeff Madura 9th ed ISBN - 0324568193 International Financial Management Abridged Edition - Jeff Madura 9th ed ISBN - 0324593473 International Financial Management, Abridged Edition - Jeff Madura 8th ed ISBN - 0324365632 International Management: Managing Across Borders and Cultures - Helen Deresky 6th ed ISBN - 0136143261 International Money and Finance - Michael Melvin 7th ed ISBN - 0201770288 Intro Stats - Richard D. De Veaux 3rd ed ISBN - 0321500458 Introduction to Abstract Algebra - Olympia Nicodemi 1st ed ISBN - 0131019635 Introduction to Analysis - William R. Wade 3rd ed ISBN - 0131453335 Introduction to Business Law - Jeffrey F. Beatty 2nd ed ISBN - 0324311427 Introduction to Business Statistics - Ronald M. Weiers 6th ed ISBN - 0324381433 Introduction to C++ EXCEL MATLAB & Basic Engineering Numerical Methods - Harvey G. Stenger 1st ed ISBN - 0136142931 Introduction to C++, Excel MatLab & Basic Engineering Numerical Methods - Harvey Stenger 1st ed ISBN - 0136120245 Introduction to Chemical Principles - Stephen Stoker 9th ed ISBN - 0132379945 Introduction to Computing Systems - Sanjay J. Patel, Yale Patt 2nd ed ISBN - 0072467509 Introduction to Corporate Finance - William L. Megginson 1st ed ISBN - 0324379862 Introduction to Corporate Finance - William L. Megginson 2nd ed ISBN - 0324657935 Introduction to Cryptography with Coding Theory - Wade Trappe 2nd ed ISBN - 0131862391 Introduction to Derivatives and Risk Management - Don M. Chance 7th ed ISBN - 0324321392 Introduction to Econometrics - James H. Stock 2nd ed ISBN - 0321278879 Introduction to Econometrics Brief Edition - James H. Stock 1st ed ISBN - 0321432517 Introduction to Economic Reasoning - William D. Rohlf 7th ed ISBN - 0321416112 Introduction to Electrodynamics -David J. Griffiths 3rd ed ISBN - 013805326X Introduction to Embedded Systems - Jonathan W. Valvano 1st ed ISBN - 049541137X Introduction to Environmental Engineering - P. Aarne Vesilind 3rd ed ISBN - 0495295833 Introduction to Environmental Engineering - Richard O. Mines 1st ed ISBN - 0132347474 Introduction to Environmental Engineering and Science - Gilbert M. Masters 3rd ed ISBN - 0131481932 Introduction to Financial Accounting - Charles Horngren 9th ed ISBN - 0131479725 Introduction to Fire Prevention - James C. Robertson 7th ed ISBN - 0135041945 Introduction to Fourier Optics - Joseph Goodman 3rd ed ISBN - 0974707724 Introduction to Government and Non-for-Profit Accounting - Martin Ives 6th ed ISBN - 0132366355 Introduction to Graph Theory - Douglas West 2nd ed ISBN - 0130144002 Introduction to Law - Joanne Hames 3rd ed ISBN - 0131183818 Introduction to Linear Algebra - Lee Johnson, Dean Riess, Jimmy Arnold 5th ed ISBN - 0201658593 Introduction to Linear Programming - Leonid Vaserstein 1st ed ISBN - 0130359173 Introduction to Management Accounting - Charles T. Horngren 14th ed ISBN - 0136129218 Introduction to Management Accounting, Chap. 1-17: International Edition - Charles Horngren 13th ed ISBN - 0131273078 Introduction to Management Science and Student - Bernard Taylor 8th ed ISBN - 0131050524 Introduction to Management Science and Student - Bernard Taylor 9th ed ISBN - 0131888099 Introduction to Materials Management - Tony Arnold 6th ed ISBN - 0132337614 Introduction to Materials Science for Engineers - James F. Shackelford 7th ed ISBN - 0136012604 Introduction to Mathematical Statistics and Its Applications - Richard J. Larsen 4th ed ISBN - 0131867938 Introduction to Operations and Supply Chain Management - Cecil Bozarth 2nd ed ISBN - 0131791036 Introduction to Optics - Frank Pedrotti et al. 3rd ed ISBN - 0131499335 Introduction to Programming with C++ - Y. Daniel Liang 2nd ed ISBN - 0136097200 Introduction to Quantum Mechanics - David Griffiths 2nd ed ISBN - 0131118927 Introduction to Risk Management and Insurance - Mark Dorfman 8th ed ISBN - 0131449583 Introduction to Risk Management and Insurance - Mark S. Dorfman 9th ed ISBN - 0132242273 Introduction to Signal and System Analysis - Kaliappan Gopalan 1st ed ISBN - 0534466060 Introduction to Spectroscopy - Donald L. Pavia 4th ed ISBN - 0495114782 Introduction to Technical Mathematics - Allyn J. Washington, Mario Triola 5th ed ISBN - 0321374177 Introduction to Telecommunications - Martha Rosengrant 2nd ed ISBN - 0131126156 Introduction to the Design & Analysis of Experiments - George C Canavos 1st ed ISBN - 0136158633 Introduction to the Design and Analysis of Algorithms - Anany Levitin 1st ed ISBN - 0201743957 Introduction to the Design and Analysis of Algorithms - Anany Levitin 2nd ed ISBN - 0321358287 Introduction to Transportation Engineering - Lester A. Hoel 1st ed ISBN - 0534952895 Introduction to Vacuum Technology - David M. Hata 1st ed ISBN - 0130450189 Introductory & Intermediate Algebra for College Students - Robert F. Blitzer 3rd ed ISBN - 0136028950 Introductory Algebra - Marvin L. Bittinger 10th ed ISBN - 0321269470 Introductory Algebra for College Students - Robert F. Blitzer 5th ed ISBN - 0132356791 Introductory Algebra through Applications - Geoffrey Akst 2nd ed ISBN - 0321518020 Introductory and Intermediate Algebra - Robert F Blitzer 2nd ed ISBN - 0131492594 Introductory Chemistry - Nivaldo J. Tro 3rd ed ISBN - 0136003826 Introductory Chemistry - Steve Russo, Michael Silver, Mike Silver 2nd ed ISBN - 032104634X Introductory Circuit Analysis - Robert Boylestad 11th ed ISBN - 0131730444 Introductory Econometrics: A Modern Approach - Jeffrey Wooldridge 3rd ed ISBN - 0324289782 Introductory Econometrics: A Modern Approach - Jeffrey Wooldridge 4th ed ISBN - 0324581629 Introductory Linear Algebra: An Applied First Course - Bernard Kolman 8th ed ISBN - 0131437402 Introductory Mathematical Analysis - Ernest F Haeussler 12th ed ISBN - 0132404222 Introductory Statistics - Neil A. Weiss 8th ed ISBN - 0321393619 Inventing Entrepreneurs: Technology Innovators and their Entrepreneurial Journey - Gerry George 1st ed ISBN - 0131574701 Investments - Frank K. Reilly 7th ed ISBN - 0324288999 Investments: An Introduction - Herbert B. Mayo 9th ed ISBN - 0324561261 Java Software Solutions: Foundations of Program Design - John Lewis 5th ed ISBN - 0321409493 Java: Introduction to Problem Solving and Programming - Walter Savitch 6th ed ISBN - 0136072259 John E. Freund's Mathematical Statistics with Applications - Irwin Miller 7th ed ISBN - 0131427067 Kleppner's Advertising Procedure - Ronald Lane 17th ed ISBN - 0132308290 Labor and Employment Law: Text & Cases - David Twomey 14th ed ISBN - 0324594844 Labor Relations - Arthur A Sloane 12th ed ISBN - 013196223X Labor Relations - Arthur A. Sloane 13th ed ISBN - 0136077188 Labor Relations and Collective Bargaining: Cases, Practice, and Law - Michael R. Carrell 8th ed ISBN - 0131868721 Labor Relations and Collective Bargaining: Cases, Practice, and Law - Michael R. Carrell 9th ed ISBN - 0136084354 Lakeside Company: Case Studies in Auditing - John M. Trussel 11th ed ISBN - 0131588516 Law and Economics - Robert Cooter 5th ed ISBN - 0321336348 Law and Ethics in the Business Environment - Terry Halbert 6th ed ISBN - 0324657323 Law for Business - John D. Ashcroft 16th ed ISBN - 0324381573 Leadership - Robert N. Lussier 3rd ed ISBN - 0324316976 Leadership in Organizations - Gary Yukl 7th ed ISBN - 0132424312 Learning Microsoft Office Accounting 2007 and Student CD Package - Terri Brunsdon 1st ed ISBN - 0131586602 Learning Peachtree Complete 2007 & Peachtree Complete CD Package - Terri Brunsdon 1st ed ISBN - 0132405571 Learning Quickbooks Pro 2007 and Student CD Package - Terri Brunsdon 1st ed ISBN - 0132419386 Legal Aspects of Architecture, Engineering & the Construction Process - Justin Sweet 8th ed ISBN - 0495411213 Legal Terminology - Gordon W. Brown 5th ed ISBN - 0131568043 Level Three Leadership: Getting Below the Surface - James G Clawson 4th ed ISBN - 0132423847 Linear Algebra and Its Applications - David C. Lay 3rd ed ISBN - 0321287134 Linear Algebra for Engineers and Scientists Using Matlab - Kenneth Hardy 1st ed ISBN - 0139067280 Linear Algebra with Applications - Otto Bretscher 3rd ed ISBN - 0131453343 Linear Algebra with Applications - Steven Leon 7th ed ISBN - 0131857851 Logic and Computer Design Fundamentals - M. Morris Mano 4th ed ISBN - 013198926X Machine Design: An Integrated Approach - Robert L. Norton 3rd ed ISBN - 0131481908 Machines and Mechanisms: Applied Kinematic Analysis - David H. Myszka 3rd ed ISBN - 0131837761 Macroeconomics - Andrew B. Abel 6th ed ISBN - 0321451406 Macroeconomics - Glenn Hubbard 2nd ed ISBN - 0132356694 Macroeconomics - Michael Parkin 7th ed ISBN - 032124608X Macroeconomics - Michael Parkin 8th ed ISBN - 0321416570 Macroeconomics - Michael Parkin 9th ed ISBN - 0321600053 Macroeconomics - Olivier Blanchard 5th ed ISBN - 0132078295 Macroeconomics - Richard Froyen 8th ed ISBN - 0131435825 Macroeconomics - Richard G. Lipsey 13th ed ISBN - 0321369238 Macroeconomics - Richard T Froyen 9th ed ISBN - 0132438356 Macroeconomics - Robert Gordon 10th ed ISBN - 0321278801 Macroeconomics - Robert Gordon 11th ed ISBN - 0321485513 Macroeconomics - Roger A. Arnold 9th ed ISBN - 032478550X Macroeconomics - Stephen D. Williamson 3rd ed ISBN - 0321416589 Macroeconomics: A Modern Approach - Robert J. Barro 1st ed ISBN - 0324178107 Macroeconomics: Principles and Policy - William J. Baumol 10th ed ISBN - 0324537034 Macroeconomics: Principles and Policy - William J. Baumol 11th ed ISBN - 0324586213 Macroeconomics: Principles and Tools - Arthur O'Sullivan, Steven Sheffrin 4th ed ISBN - 0131536184 Macroeconomics: Principles, Applications, and Tools - Arthur O'Sullivan 5th ed ISBN - 013232928X Macroeconomics: Public and Private Choice - James D. Gwartney 12th ed ISBN - 0324580193 Making Career Decisions that Count: A Practical Guide - Darrell A. Luzzo 3rd ed ISBN - 0131712772 Making the Team - Leigh Thompson 3rd ed ISBN - 0131861352 Management - Michael Hitt 2nd ed ISBN - 0132354373 Management - Richard L. Daft 9th ed ISBN - 0324595840 Management - Stephen P Robbins 9th ed ISBN - 0132257734 Management Communication: A Case-Analysis Approach - James S O'Rourke 4th ed ISBN - 0136079768 Management Information Systems - Ken Laudon 11th ed ISBN - 013607846X Management of Organizational Behavior - Paul H Hersey 9th ed ISBN - 0131441396 Manager's Bookshelf - Jon L. Pierce 8th ed ISBN - 0132301652 Managerial Accounting - Carl Warren 9th ed ISBN - 0324381913 Managerial Accounting - Carl Warren 10th ed ISBN - 032466382X Managerial Accounting - Linda S. Bamber 1st ed ISBN - 0138129711 Managerial Accounting Class Test Edition - Linda S. Bamber 1st ed ISBN - 0132284634 Managerial Accounting: A Focus on Ethical Decision Making - Steve Jackson, Roby Sawyers 4th ed ISBN - 0324650647 Managerial Accounting: A Focus on Ethical Decision Making - Steve Jackson, Roby Sawyers 5th ed ISBN - 0324663854 Managerial Accounting: An Introduction to Concepts, Methods and Uses - Michael W. Maher 10th ed ISBN - 0324639767 Managerial Economics - Mark Hirschey 12th ed ISBN - 0324584849 Managerial Economics: A Problem Solving Approach - Luke M. Froeb 1st ed ISBN - 0324359810 Managerial Economics: Applications, Strategies, and Tactics - James R. McGuigan 11th ed ISBN - 0324421605 Managerial Economics: Economic Tools for Today's Decision Makers - Paul G. Keat 5th ed ISBN - 0131860151 Managerial Economics: Economic Tools for Today's Decision Makers - Paul G. Keat 6th ed ISBN - 0136040047 Managerial Statistics A Case-Based Approach - Peter Klibanoff 1st ed ISBN - 0324226454 Managers and the Legal Environment - Constance E. Bagley 6th ed ISBN - 0324582048 Managing Human Resources - Luis Gomez-Mejia 5th ed ISBN - 013187067X Managing in a Global Economy: Demystifying International Macroeconomics - John E. Marthinsen 1st ed ISBN - 0324395507 Managing Information Technology - Carol V Brown 6th ed ISBN - 0131789546 Managing the Law: The Legal Aspects of Doing Business - Mitchell McInnes 2nd ed ISBN - 0132042762 Manual Auditing and Assurance Practice Set: CAST - Frank A. Buckless 1st ed ISBN - 0130464716 Manufacturing Processes for Engineering Materials - Serope Kalpakjian 5th ed ISBN - 0132272717 Manufacturing, Engineering & Technology - Serope Kalpakjian 5th ed ISBN - 0131489658 Market Regulation - Roger Sherman 1st ed ISBN - 0321322320 Market-Based Management - Roger Best 5th ed ISBN - 0132336537 Marketing - Charles W. Lamb 10th ed ISBN - 0324591098 Marketing - William M. Pride 15th ed ISBN - 0547167474 Marketing Management - Dawn Iacobucci 1st ed ISBN - 0324784430 Marketing Management - Philip Kotler 13th ed ISBN - 0136009980 Materials Science and Engineering: An Introduction - William Callister 6th ed ISBN - 0471135763 Mathematical Economics - Jeffrey Baldani 2nd ed ISBN - 0324183321 Mathematical Ideas - Charles D. Miller 11th ed ISBN - 0321361466 Mathematical Ideas - Charles D. Miller 11th ed ISBN - 0321361482 Mathematical Methods for Economics - Michael Klein 2nd ed ISBN - 0201726262 Mathematical Proofs: A Transition to Advanced Mathematics - Gary Chartrand 2nd ed ISBN - 0321390539 Mathematical Proofs: A Transition to Advanced Mathematics - Gary Chartrand 1st ed ISBN - 0201710900 Mathematical Reasoning for Elementary Teachers - Calvin T. Long 5th ed ISBN - 0321460847 Mathematics for Business - Stanley A. Salzman 8th ed ISBN - 0321357434 Mathematics for Elementary School Teachers - Phares O'Daffer 4th ed ISBN - 0321448049 Mathematics for Physicists - Susan Lea 1st ed ISBN - 0534379974 Mathematics of Interest Rates and Finance - Gary Guthrie 1st ed ISBN - 0130461822 Mathematics with Applications - Margaret L. Lial 9th ed ISBN - 0321334337 Mechanical Behavior of Materials - Norman Dowling 3rd ed ISBN - 0131863126 Mechanics of Materials - James M. Gere 7th ed ISBN - 0534553974 Mechanics of Materials - Russell C. Hibbeler 7th ed ISBN - 0132209918 Medical Imaging Signals and Systems - Jerry L. Prince 1st ed ISBN - 0130653535 Microbiology with Diseases by Body System - Robert W. Bauman 2nd ed ISBN - 032151341X Microbiology: An Introduction - Gerard J. Tortora 9th ed ISBN - 0805347909 Microeconomics - Glenn Hubbard 2nd ed ISBN - 0138132771 Microeconomics - Jeffrey Perloff 4th ed ISBN - 0321414527 Microeconomics - Jeffrey Perloff 5th ed ISBN - 0321531191 Microeconomics - Michael Parkin 7th ed ISBN - 0321454944 Microeconomics - Michael Parkin 8th ed ISBN - 0321416600 Microeconomics - Michael Parkin 9th ed ISBN - 0321600045 Microeconomics - Richard G. Lipsey 13th ed ISBN - 032136922X Microeconomics - Robert Pindyck, Daniel Rubinfeld 6th ed ISBN - 0130084611 Microeconomics - Robert Pindyck, Daniel Rubinfeld 7th ed ISBN - 0132080230 Microeconomics - Roger A. Arnold 9th ed ISBN - 0324785496 Microeconomics: Principles and Policy - William J. Baumol 11th ed ISBN - 0324586221 Microeconomics: Principles and Tools - Arthur O'Sullivan, Steven Sheffrin 4th ed ISBN - 0131536060 Microeconomics: Principles, Applications, and Tools - Arthur O'Sullivan 5th ed ISBN - 0131572830 Microeconomics: Public and Private Choice - James D. Gwartney 12th ed ISBN - 0324580207 Microeconomics: Theory and Applications with Calculus - Jeffrey M. Perloff 1st ed ISBN - 0321277945 Microwave Engineering - David Pozar 3rd ed ISBN - 0471448788 MIS Essentials - David Kroenke 1st ed ISBN - 0136075606 MKTG 3.0 2009 Edition - Charles W. Lamb 3rd ed ISBN - 0324789289 Modern Control Systems - Richard C Dorf 11th ed ISBN - 0132270285 Modern Database Management - Jeffrey Hoffer 8th ed ISBN - 0132212110 Modern Database Management - Jeffrey Hoffer 9th ed ISBN - 0136003915 Modern Electronic Communication - Jeff Beasley 9th ed ISBN - 0132251132 Modern Elementary Statistics - John E. Freund 12th ed ISBN - 013187439X Modern Industrial Organization - Dennis Carlton, Jeffrey Perloff 4th ed ISBN - 0321180232 Modern Labor Economics: Theory and Public Policy - Ronald Ehrenberg 10th ed ISBN - 0321533739 Modern Labor Economics: Theory and Public Policy - Ronald Ehrenberg 9th ed ISBN - 0321305035 Modern Management - Samuel C. 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Barringer 1st ed ISBN - 0132318326 Price Theory and Applications - Steven Landsburg 7th ed ISBN - 0324421613 Principles of Accounting - Meg Pollard 1st ed ISBN - 0132304791 Principles of Auditing: An Introduction to International Standards on Auditing - Rick Hayes 2nd ed ISBN - 0273684108 Principles of CMOS VLSI Design - Neil H.E. Weste 3rd ed ISBN - 0201533766 Principles of Cost Accounting - Edward J. Vanderbeck 13th ed ISBN - 0324191693 Principles of Cost Accounting - Edward J. Vanderbeck 14th ed ISBN - 0324374178 Principles of Customer Relationship Management - Roger Baran 1st ed ISBN - 0324322380 Principles of Economics - Gregory Mankiw 4th ed ISBN - 0324224729 Principles of Economics - Gregory Mankiw 5th ed ISBN - 0324589972 Principles of Economics - Karl Case 8th ed ISBN - 0132289148 Principles of Electric Circuits: Conventional Current Version - Thomas Floyd 8th ed ISBN - 0131701797 Principles of Electric Circuits: Conventional Flow Version - Thomas L. Floyd 9th ed ISBN - 013507309X Principles of Finance - Scott Besley 3rd ed ISBN - 0324232624 Principles of Finance - Scott Besley 4th ed ISBN - 0324655886 Principles of Foundation Engineering - Braja M. Das 6th ed ISBN - 0495082465 Principles of Geotechnical Engineering 6th ed ISBN - 0534551440 Principles of Heat Transfer - Frank Kreith 6th ed ISBN - 0534375960 Principles of Law and Economics - Daniel Cole, Peter Grossman 1st ed ISBN - 0130932612 Principles of Macroeconomics - Gregory Mankiw 4th ed ISBN - 0324236956 Principles of Macroeconomics - Gregory Mankiw 5th ed ISBN - 0324589999 Principles of Managerial Finance - Lawrence J. Gitman 12th ed ISBN - 0321557530 Principles of Managerial Finance Brief Edition - Lawrence Gitman 5th ed ISBN - 0321557522 Principles of Managerial Finance plus MyfinanceLab Student Access Kit - Lawrence J. Gitman 12th ed ISBN - 0321557530 Principles of Marketing - Philip Kotler 12th ed ISBN - 0132390027 Principles of Marketing - Philip Kotler 13th ed ISBN - 0136079415 Principles of Microeconomics - Gregory Mankiw 4th ed ISBN - 0324319169 Principles of Microeconomics - Karl E. Case 8th ed ISBN - 0131994859 Principles of Money, Banking & Financial Markets - Lawrence Ritter 12th ed ISBN - 0321375572 Principles of Money, Banking, Financial Markets - Lawrence Ritter et al 11th ed ISBN - 0321205251 Principles Of Operations Management - Jay Heizer 6th ed ISBN - 013155445X Principles Of Operations Management - Jay Heizer 7th ed ISBN - 0132449757 Principles of Risk Management and Insurance - George E. Rejda 10th ed ISBN - 0321414934 Probabilistic Systems and Random Signals - Abraham Haddad 1st ed ISBN - 0130094552 Probability & Statistics for Engineers & Scientists - Ronald E. Walpole 8th ed ISBN - 0131877119 Probability and Statistical Inference - Robert Hogg, Eliot Tanis 7th ed ISBN - 0131464132 Probability and Statistical Inference - Robert Hogg, Eliot Tanis 8th ed ISBN - 0321584759 Probability and Statistics - Morris DeGroot, Mark Schervish 3rd ed ISBN - 0201524880 Probability and Statistics for Engineers - Richard Johnson, Irwin Miller, John Freund 7th ed ISBN - 0131437453 Probability and Statistics for Engineers and Scientists - Anthony J. Hayter 3rd ed ISBN - 0495107573 Probability Random Variables, and Stochastic Processes - Papoulis et al 4th ed ISBN - 0073660116 Probability, Statistics, and Random Processes For Electrical Engineering - Alberto Leon-Garcia 3rd ed ISBN - 0131471228 Problem Solving and Program Design in C - Jeri R. Hanly 5th ed ISBN - 0321409914 Problem Solving and Program Design in C - Jeri R. Hanly 6th ed ISBN - 0321535421 Problem Solving with C++ - Walter Savitch 6th ed ISBN - 0321412699 Problem Solving with C++ - Walter Savitch 7th ed ISBN - 0321531345 Problem Solving, Abstraction & Design Using C++ - Frank L. Friedman 5th ed ISBN - 0321450051 Process Control Instrumentation Technology - Curtis Johnson 8th ed ISBN - 0131194577 Professional Office Procedures - Susan Cooperman 5th ed ISBN - 0135156645 Professional Selling: A Trust-Based Approach - Thomas N. Ingram 4th ed ISBN - 032453809X Professionalism: Real Skills for Workplace Success - Lydia E. Anderson 1st ed ISBN - 0131714392 Programming the World Wide Web - Robert W. Sebesta 4th ed ISBN - 0321489691 Project Management for Information Systems - James Cadle 5th ed ISBN - 0132068583 Public Relations Practices: Managerial Case Studies and Problems - Allen H. Center 7th ed ISBN - 0132341360 Quality Control - Dale H. Besterfield 8th ed ISBN - 0135000955 Quality Management - Donna C.S. 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Trieschmann 12th ed ISBN - 0324183208 Risk Takers: Uses and Abuses of Financial Derivatives - John Marthinsen 2nd ed ISBN - 0321542568 Routers and Routing Basics CCNA 2 Labs and Study Guide - Allan Johnson 1st ed ISBN - 1587131676 Selling Today - Gerald L. Manning 11th ed ISBN - 013207995X Short-Term Financial Management - Terry Maness 3rd ed ISBN - 0324202938 Short-Term Financial Management - Terry Maness Test Bank 3rd ed ISBN - 0324202938 Signals, Systems, and Transforms - Charles L Phillips 4th ed ISBN - 0131989235 Social Entrepreneurship: A Modern Approach to Social Value Creation - Arthur C. Brooks 1st ed ISBN - 0132330768 Software Engineering - Ian Sommerville 8th ed ISBN - 0321313798 Software Engineering: Theory and Practice - Shari Lawrence Pfleeger 4th ed ISBN - 0136061699 Soils and Foundations - Cheng Liu, Jack Evett 7th ed ISBN - 0132221381 Solid State Electronic Devices - Ben Streetman 6th ed ISBN - 013149726X Solid State Physics: Essential Concepts - David W. 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Mott 1st ed ISBN - 0135159822 Statics and Strengths of Materials - Harold I. Morrow 6th ed ISBN - 0131719777 Statistical Methods for the Social Sciences - Alan Agresti 4th ed ISBN - 0130272957 Statistical Reasoning for Everyday Life - Jeffrey O. Bennett 3rd ed ISBN - 0321286723 Statistics - James T. McClave 11th ed ISBN - 0132069512 Statistics for Business & Economics - James T. McClave 10th ed ISBN - 0132409356 Statistics for Business and Economics - David R. Anderson Test Bank only 10th ed ISBN - 0324360681 Statistics for Business and Economics - Paul Newbold, William L. Carlson 6th ed ISBN - 0132203847 Statistics for Business and Economics - Paul Newbold, William L. Carlson 7th ed ISBN - 0136085369 Statistics for Management and Economics - Gerald Keller 7th ed ISBN - 0534491243 Statistics for Managers Using Microsoft Excel - David M. Levine 5th ed ISBN - 0136149901 Statistics for Science and Engineering - John Kinney 1st ed ISBN - 0201437201 Statistics for the Behavioral and Social Sciences - Arthur Aron 4th ed ISBN - 0131562789 Statistics for the Life Sciences - Myra Samuels 3rd ed ISBN - 013041316X Statistics, Data Analysis & Decision Modeling - James R. Evans 4th ed ISBN - 0136066003 Statistics: Informed Decisions Using Data - Michael Sullivan 3rd ed ISBN - 0321568028 1st ed ISBN - 0130083690 Statistics: The Art and Science of Learning from Data - Alan Agresti 2nd ed ISBN - 0135131995 Stats: Data and Models - Richard D. De Veaux 2nd ed ISBN - 0321433793 Steel Design - William T. Segui 4th ed ISBN - 0495244716 Strategic Brand Management - Kevin Keller 3rd ed ISBN - 0131888595 Strategic Compensation - Joe Martocchio 5th ed ISBN - 0136007449 Strategic Management and Business Policy - Tom Wheelen 10th ed ISBN - 0131494597 Strategic Management and Business Policy - Tom Wheelen 11th ed ISBN - 013232346X Strategic Management and Competitive Advantage - Jay Barney 2nd ed ISBN - 013613520X Strategic Management and Competitive Advantage: Concepts and Cases - Jay Barney 2nd ed ISBN - 0132338238 Strategic Management in Action - Mary Coulter 4th ed ISBN - 0132277476 Strategic Managment: A Dynamic Perspective Integrated Stratsim Simulation Experience - Print Upgrade - Mason Carpenter 1st ed ISBN - 0136149057 Strategic Management: Concepts and Cases - Fred David 11th ed ISBN - 0131869493 Strategic Management: Concepts and Cases - Fred David 12th ed ISBN - 0136015700 Strategic Marketing for Non-Profit Organizations - Alan Andreasen 7th ed ISBN - 013175372X Strategic Staffing - Jean M. Phillips 1st ed ISBN - 0131586947 Structural Analysis - Aslam Kassimali 4th ed ISBN - 0495295655 Structural Analysis - Russell C. Hibbeler 7th ed ISBN - 0136020607 Structural Steel Design - Jack C. McCormac 4th ed ISBN - 013221816X Structural Steel Design ASD Method - Jack C. McCormac 4th ed ISBN - 0065000609 Structural Steel Design: A Practice Oriented Approach - Abi O. Aghayere 1st ed ISBN - 0132340186 Structure and Interpretation of Signals and Systems - Edward Lee, Pravin Varaiya 1st ed ISBN - 0201745518 Structures - Daniel Lewis Schodek 6th ed ISBN - 0131789392 Supply Chain Management - Sunil Chopra 3rd ed ISBN - 0131730428 Supply Chain Management: A Logistics Perspective - John J. Coyle 8th ed ISBN - 0324376928 Survey of Accounting - Carl S. Warren 3rd ed ISBN - 0324312482 Survey of Accounting - Carl S. Warren 4th ed ISBN - 0324658265 Survey of Economics - Irvin B. Tucker 6th ed ISBN - 0324579616 Survey of Mathematics with Applications, A - Allen R. Angel 8th ed ISBN - 0321501071 Survey of Mathematics with Applications - Allen Angel 8th ed ISBN - 032150108X Surveying with Construction Applications - Barry F. Kavanagh 6th ed ISBN - 0131709321 Surveying with Construction Applications - Barry F. Kavanagh 7th ed ISBN - 0135000513 Surveying: Principles and Applications - Barry F. Kavanagh 8th ed ISBN - 013236512X System Dynamics and Response - S. Graham Kelly 1st ed ISBN - 0534549306 System Modeling and Analysis: Foundations of System Performance Evaluation - Hisashi Kobayashi 1st ed ISBN - 013034835X Systems Analysis and Design - Kenneth E. Kendall 7th ed ISBN - 0132240858 Tax Research - Barbara H Karlin 4th ed ISBN - 013601531X Taxation for Decision Makers 2008 - Shirley Dennis-Escoffier 2nd ed ISBN - 0324654111 Taxes & Business Strategy - Myron S. Scholes 4th ed ISBN - 0136033156 Teaching Today's Health - David Anspaugh 9th ed ISBN - 0321596773 Technical Calculus with Analytic Geometry - Allyn J. 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Thomas, Jr 11th ed ISBN - 0321495756 Thomas' Calculus, Early Transcendentals, Media Upgrade, Part One - George B. Thomas, Jr. 11th ed ISBN - 0321498747 Thomas' Calculus, Media Upgrade - George B. Thomas, Jr. 11th ed ISBN - 032148987X Thomas' Calculus, Media Upgrade, Part One Single Variable - George B. Thomas, Jr. 11th ed ISBN - 0321498755 Thomas' Calculus, Media Upgrade, Part Two Multivariable, Chap 11-16 - George B. Thomas, Jr. 11th ed ISBN - 0321501039 Traffic & Highway Engineering - Nicholas J. Garber 4th ed ISBN - 0495082503 Training in Interpersonal Skills - Stephen P. Robbins 5th ed ISBN - 0132354993 Trigonometry - Margaret L. Lial 9th ed ISBN - 0321528859 Trigonometry - Mark Dugopolski 2nd ed ISBN - 032135690X Trigonometry: A Right Triangle Approach - Michael Sullivan 5th ed ISBN - 0136028969 Trigonometry: A Unit Circle Approach - Michael Sullivan 8th ed ISBN - 0132392798 UFL Collective Bargaining Agreement - Louis Marino 1st ed ISBN - 0131587668 Understanding and Managing Diversity - Carol Harvey 4th ed ISBN - 0132069105 Understanding and Managing Organizational Behavior - Jennifer George 5th ed ISBN - 013239457X Understanding Fiber Optics - Jeff Hecht 5th ed ISBN - 0131174290 Understanding Financial Statements - Lyn M. Fraser 8th ed ISBN - 0131878565 Understanding Modern Economics - Roger Miller 1st ed ISBN - 0321245822 University Calculus - Joel Hass, Maurice D. 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Porter 6th ed ISBN - 0324593740 Using MIS - David Kroenke 2nd ed ISBN - 0138132488 Using Peachtree Complete 2007 for Accounting - Glenn Owen 1st ed ISBN - 0324377975 Using Peachtree Complete 2009 for Accounting - Glenn Owen 3rd ed ISBN - 0324665512 Using Quickbooks Pro 2007 for Accounting - Glenn Owen 7th ed ISBN - 0324378750 Vector Calculus - Susan Colley 3rd ed ISBN - 0131858742 VHDL: A Starter's Guide - Sudhakar Yalamanchili 2nd ed ISBN - 0131457357 Water and Wastewater Technology - Mark J. Hammer 6th ed ISBN - 0131745425 Water Resources Engineering - David A. Chin 2nd ed ISBN - 0131481924 Water Supply and Pollution Control - Warren Viessman, Jr. 8th ed ISBN - 0132337177 Web 101 - Wendy G. 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Clarkson 10th ed ISBN - 0324303904 Wireless Communications & Networks - William Stallings 2nd ed ISBN - 0131918354 Writing and Speaking at Work: A Practical Guide for Business Communication - Edward P. Bailey 4th ed ISBN - 0131881302 Your Attitude is Showing - Sharon Lund O'Neil 12th ed ISBN - 0132429047 === Subject: Statics And Strengths Of Materials (6thEd) - Morrow - Solutions Manual is available for purchase! Contact me at instructors.team[at]gmail.com posting-account=jy2E5goAAAB1WGwMQL1h2k61o4k7O_u4 Gecko/2009051221 Firefox/3.0.10,gzip(gfe),gzip(gfe) Statics And Strengths Of Materials (6thEd) - Morrow - Solutions Manual is available for purchase! Contact me at instructors.team[at]gmail.com === Subject: TestBank_Macroeconomics Principles and Policy Baumol 10th is available for purchase! Contact me at instructors.team[at]gmail.com posting-account=jy2E5goAAAB1WGwMQL1h2k61o4k7O_u4 Gecko/2009051221 Firefox/3.0.10,gzip(gfe),gzip(gfe) TestBank_Macroeconomics Principles and Policy Baumol 10th is available for purchase! Contact me at instructors.team[at]gmail.com === === Subject: Consecutive Numbers let k <= m <= n three positive integers. Let S(k,m,n) be the number of subsets of {1,...,n} with the following properties: (1) S contains m elements (2) S contains at least a sequence of at least k consecutive numbers (that is, there exist j such that j, j + 1, ..., j + k - 1 are in S) I don't know the answer, except in the trivial cases k = m or m = n. Do you have some idea? Maury Barbato === Subject: Re: Consecutive Numbers > let k <= m <= n three positive integers. > Let S(k,m,n) be the number of subsets of > {1,...,n} with the following properties: > > (1) S contains m elements > > (2) S contains at least a sequence of at least k > consecutive numbers (that is, there exist j such > that j, j + 1, ..., j + k - 1 are in S) > > I don't know the answer, except in the trivial > cases k = m or m = n. Do you have some idea? > > Maury Barbato Add this, and our programmer can solve the rest: http://mathforum.org/kb/message.jspa?messageID=6751446&tstart=0 === Subject: Re: Consecutive Numbers >let k <= m <= n three positive integers. >Let S(k,m,n) be the number of subsets of >{1,...,n} with the following properties: (1) S contains m elements (2) S contains at least a sequence of at least k >consecutive numbers (that is, there exist j such >that j, j + 1, ..., j + k - 1 are in S) I don't know the answer, except in the trivial >cases k = m or m = n. Do you have some idea? Maury Barbato Can you answer the question for S(1, m, n)? Can you answer the question for S(2, m, n)? Can you extrapolate from there? rossum === Subject: Re: Consecutive Numbers > >let k <= m <= n three positive integers. >Let S(k,m,n) be the number of subsets of >{1,...,n} with the following properties: >(1) S contains m elements >(2) S contains at least a sequence of at least k >consecutive numbers (that is, there exist j such >that j, j + 1, ..., j + k - 1 are in S) >I don't know the answer, except in the trivial >cases k = m or m = n. Do you have some idea? >Maury Barbato > Can you answer the question for S(1, m, n)? > Can you answer the question for S(2, m, n)? > Can you extrapolate from there? > > rossum > Obviously, S(1,m,n) = (n choose m), but I don't know how to find S(2,m,n). Some hint? Maury Barbato === Subject: property of minimum of loops of length n (collatz problem) AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.2 Safari/525.20.1,gzip(gfe),gzip(gfe) For the Collatz problem, If a loop exists for positive numbers then, if a is the minimum of all loops with n odds then, a < n * 4^(n-1) for n > 1 I have a proof... But I'd rather have some comments on the statement above... Is it something interesting or worthwhile?? What I see is that, this formula can be used to check the existence of a loop... For example, for an loop with n odd numbers --> check whether all the odd numbers from 1 to n * 4^(n-1) contain a loop of length n or not --> if there isnt any, then no positive loop containing n odd numbers exists === Subject: 9 x M Primality Criterion posting-account=zPkoaQoAAAAenkwMuvm4l7rd9lPgI5Fa Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) 9 x M Primality Criterion With 7 as definitive smallest prime, for every two primes M and N such that 7 .b2 M < N the rule says the period of M/N divides with M and 9. Period(07/11) = 63 | 9 x 7 Period(07/13) = 538461 | 9 x 7 x 8547 Period(07/17) = 4117647058823529 | 9 x 7 x 65359477124183 Period(07/19) = 368421052631578947 | 9 x 7 x 5847953216374269 Period(07/23) = 3043478260869565217391 | 9 x 7 x 48309178743961352657 Period(07/29) = 2413793103448275862068965517 | 9 x 7 x 38314176245210727969348659 Period(07/31) = 225806451612903 | 9 x 7 x 3584229390681 Period(07/37) = 189 | 9 x 7 x 3 Period(07/41) = 17073 | 9 x 7 x 271 Period(11/13) = 846153 | 9 x 11 x 8547 Period(11/17) = 6470588235294117 | 9 x 11 x 65359477124183 Period(11/19) = 578947368421052631 | 9 x 11 x 5847953216374269 Period(11/23) = 4782608695652173913043 | 9 x 11 x 48309178743961352657 Period(11/29) = 3793103448275862068965517241 | 9 x 11 x 38314176245210727969348659 Period(13/17) = 7647058823529411 | 9 x 13 x 65359477124183 Period(13/19) = 684210526315789473 | 9 x 13 x 5847953216374269 Period(13/23) = 5652173913043478260869 | 9 x 13 x 48309178743961352657 Period(13/29) = 4482758620689655172413793103 | 9 x 13 x 38314176245210727969348659 Period(17/19) = 894736842105263157 | 9 x 17 x 5847953216374269 Period(17/23) = 7391304347826086956521 | 9 x 17 x 48309178743961352657 Period(17/29) = 5862068965517241379310344827 | 9 x 17 x 38314176245210727969348659 Period(19/23) = 8260869565217391304347 | 9 x 19 x 48309178743961352657 Period(19/29) = 6551724137931034482758620689 | 9 x 19 x 38314176245210727969348659 Period(23/29) = 7931034482758620689655172413 | 9 x 23 x 38314176245210727969348659 Entire 24 confirmations... THE RULE HOLDS!!! IT WORKS!!! But nevertheless if 7 .b2 N < M are primes then the period of M/N divides with 9 as well: Period(29/07) = 142857 | 9 x 15873 Period(29/11) = 63 | 9 x 7 Period(29/13) = 230769 | 9 x 25641 Period(29/17) = 7058823529411764 | 9 x 784313725490196 Period(29/19) = 526315789473684210 | 9 x 58479532163742690 Period(29/23) = 2608695652173913043478 | 9 x 289855072463768115942 Period(23/07) = 285714 | 9 x 31746 Period(23/11) = 09 | 9 x 1 Period(23/13) = 769230 | 9 x 85470 Period(23/17) = 3529411764705882 | 9 x 392156862745098 Period(23/19) = 210526315789473684 | 9 x 23391812865497076 Period(19/07) = 714285 | 9 x 79365 Period(19/11) = 72 | 9 x 8 Period(19/13) = 461538 | 9 x 51282 Period(19/17) = 1176470588235294 | 9 x 130718954248366 Period(17/07) = 428571 | 9 x 47619 Period(17/11) = 54 | 9 x 6 Period(17/13) = 307692 | 9 x 34188 Period(13/07) = 857142 | 9 x 95238 Period(13/11) = 18 | 9 x 2 Period(11/07) = 571428 | 9 x 63492 Entire additional 21 confirmations... THE RULE HOLDS!!! IT WORKS!!! === Subject: Re: 9 x M Primality Criterion > 9 x M Primality Criterion > > With 7 as definitive smallest prime, for every two primes M and N such > that 7 .b2 M < N the rule says the period of M/N divides with M and 9. A few notes on better terminology -- (a) 7 is not the smallest prime integer, so your definitive smallest prime phrase is nonsense. Perhaps you mean, Let M and N be prime integers >= 7. (b) By usual conventions the period of M/N is the number of digits in the repeating string, rather than the repeating string itself. (c) divides with means nothing; rather, say 9 divides ... or M divides .... > Period(07/11) = 63 | 9 x 7 > Period(07/13) = 538461 | 9 x 7 x 8547 > Period(07/17) = 4117647058823529 | 9 x 7 x 65359477124183 ... > Entire 24 confirmations... THE RULE HOLDS!!! IT WORKS!!! ... You haven't stated a rule. If you mean to say that 9 divides the number that is the repeating-digits part p of a fraction, that is simple to prove: Suppose m ... > Suppose m p/10^(3k) + ... for some positive integers p and k. Then 10^k * m/n = p > + p/10^k + p/10^(2k) + ... = p + m/n, so ((10^k)-1) m/n = p and m * > ((10^k)-1) = n*p. It is evident that 9 divides (10^k)-1, so by premises > (9 does not divide m or n) and by fundamental theorem of arithmetic, 9 > divides p. In place of the premise 9 does not divide m or n I should have written 3 does not divide m or n. -- jiw === Subject: Re: 9 x M Primality Criterion posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > 9 x M Primality Criterion > With 7 as definitive smallest prime, for every two primes M and N such > that 7 .b2 M < N the rule says the period of M/N divides with M and 9. A few notes on better terminology -- (a) 7 is not the smallest > prime integer, so your definitive smallest prime phrase is > nonsense.Perhaps you mean, Let M and N be prime integers >= 7. > (b) By usual conventions the period of M/N is the number of digits > in the repeating string, rather than the repeating string itself. > (c) divides with means nothing; rather, say 9 divides ... or > M divides .... > Period(07/11) = 63 | 9 x 7 > Period(07/13) = 538461 | 9 x 7 x 8547 > Period(07/17) = 4117647058823529 | 9 x 7 x 65359477124183 > ... > Entire 24 confirmations... THE RULE HOLDS!!! IT WORKS!!! ... You haven't stated a rule.If you mean to say that 9 divides > the number that is the repeating-digits part p of a fraction, > that is simple to prove: Suppose m p/10^(2k) + p/10^(3k) + ... for some positive integers p and k. > Then 10^k * m/n = p + p/10^k + p/10^(2k) + ... = p + m/n, > so ((10^k)-1) m/n = p and m * ((10^k)-1) = n*p.It is > evident that 9 divides (10^k)-1, so by premises (9 does > not divide m or n) and by fundamental theorem of arithmetic, > 9 divides p. -- > jiw ... 9 x M Primality Criterion With 7 as definitive smallest prime, for every two primes M and N >such that 7 ? M < N the rule says the period of ... Methods and apparatus for efficient ... - reassigning, at each iteration, both the cost >minimizing (primal) allocations ... process in accordance with an optimizing criterion, said >system comprising: .... 9. A method for utilizing the Karmarkar algorithm for allocating >industrial or ... n: where M is the value of the maximum component of x, .epsilon. is an ... A Primality Criterion. For which integers m do there exist integers a,b,c,d ... We can >immediately rule out m less than 5, and also the number 9, by inspection. ... x,y where x is less >than q and y is less than p, and then take v=px-qy. ... Methods and apparatus for efficient ... - assigning at each iteration, both the cost minimizing >(primal) ... in accordance with an optimizing criterion, said system comprising: ... 9. The >system according to claim 5 wherein said controlled process is an inventory control process. ... >+1),: where: M=max.sub.j x.sub.j: and .epsilon. is an arbitrary small ... 9 x M Primality Criterion With 7 as definitive smallest prime, for every two primes M and N >such ... -- >mmm >los angeles, ca On a class of primality criteriaWe can see that nlm (,,,-2,,), 9 Since m and n are relatively prime, ... H. W. Gould and W. E. Greig, A Lucas triangle primality criterion dual to that ... www.springerlink.com/index/U6725V27855NV11J.pdf by MA Vsemirnov - 1994 EXPLICIT PRIMALITY CRITERIA FOR ( p - 1) p - 1 1. Introduction ...Deterministic polynomial time primality criteria for 2 .... polynomial in Z[x]. iii) N is a prime if and only if h(S m .... out in [5] and [9], one way of approaching the task of ensuring that (2.7) will hold ... www.ams.org/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf A class of primality criteria formulated in terms of the ... 0090-4104/81/1601-0874507.50 9 1981 Plenum Publishing Corporation .... Like the Mann--Shanks primality criteria, the criteria considered here also have no ..... h/M.,A. A... ,x h.- f... h. f.r,,. &, a,,i,: p.,. [... ,, o...o. *...-',', ... www.springerlink.com/index/H895N80371W718MH.pdf EXPLICIT PRIMALITY CRITERIA FOR h *2k 1 In this paper we consider ...Next we describe primality criteria for numbers of the form h ...... (2) Choose a suitable bound b, and perform step (a) for pairs x, y with .... (and m = 8, because the squares 9 + 24 = 52 and 9 * 25 + 1 = 172 are trapped ... www.jstor.org/stable/2152938 [PDF] The primality of certain integers of the form 2Arn-1File Format: PDF/Adobe Acrobat - View as HTML M. Newirun, A table of the first factor for prime oydotoimc, fieldé, Math.. Comp. ... are given in [9]. With this information we can now give the main result of [£>] as ... This result follows easily from the facts that (1, ) .84 .841 .... This again is a criterion foT primality that involves only Lucas functions. ... matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf === Subject: z-score for a normal distribution posting-account=tXYyDgoAAAClsCVRo58cVx83gNKWzoan AppleWebKit/530.5 (KHTML, like Gecko) Chrome/2.0.172.30 Safari/530.5,gzip(gfe),gzip(gfe) If an Auto Service provides oil and lube service for cars and the mean time for an oil and lube service is 15 mins. per car with a std. deviation of 2.4 minutes, then if the management wants to guarantee a maximum waiting time by where if the time is exceeded, the customer gets a discount, and they want to limit such a discount to 5% of customers, then the maximum waiting time would be found (assuming times taken for oil and lube service have a normal distribution) by looking up the z-score for ~.05 and then (z-score)*(std. dev.) + mean = x-score Using a table of z-scores, I get -1.64, thus: max. time would be 11.064 mins. Does that appear correct? === Subject: Re: z-score for a normal distribution posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > If an Auto Service provides oil and lube service for cars > and the mean time for an oil and lube service is 15 mins. > per car with a std. deviation of 2.4 minutes, then if the > management wants to guarantee a maximum waiting time > by where if the time is exceeded, the customer gets a > discount, and they want to limit such a discount to 5% of > customers, then the maximum waiting time would be found > (assuming times taken for oil and lube service have a normal > distribution) by looking up the z-score for ~.05 and then > (z-score)*(std. dev.) + mean = x-score Using a table of z-scores, I get -1.64, thus: > max. time would be 11.064 mins.Does that appear > correct? Without doing any calculations, think about the answer intuitively. You want to meet the condition {T <= M} 95% of the time (T = service time, M = max. time, to be determined). Should M be less than the mean of T, or should it be greater? (Hint: if M were exactly equal to the mean of T then you would meet the condition 50% of the time. You want to do better than that.) So, should M be less than 15 min. or greater than 15 min.? Your answer is less than 15 min, so what is that telling you? R.G. Vickson === Subject: Re: z-score for a normal distribution posting-account=tXYyDgoAAAClsCVRo58cVx83gNKWzoan AppleWebKit/530.5 (KHTML, like Gecko) Chrome/2.0.172.30 Safari/530.5,gzip(gfe),gzip(gfe) > If an Auto Service provides oil and lube service for cars > and the mean time for an oil and lube service is 15 mins. > per car with a std. deviation of 2.4 minutes, then if the > management wants to guarantee a maximum waiting time > by where if the time is exceeded, the customer gets a > discount, and they want to limit such a discount to 5% of > customers, then the maximum waiting time would be found > (assuming times taken for oil and lube service have a normal > distribution) by looking up the z-score for ~.05 and then > (z-score)*(std. dev.) + mean = x-score > Using a table of z-scores, I get -1.64, thus: > max. time would be 11.064 mins.Does that appear > correct? Without doing any calculations, think about the answer intuitively. > You want to meet the condition {T <= M} 95% of the time (T = service > time, M = max. time, to be determined). Should M be less than the mean > of T, or should it be greater? (Hint: if M were exactly equal to the > mean of T then you would meet the condition 50% of the time. You want > to do better than that.) So, should M be less than 15 min. or greater > than 15 min.? Your answer is less than 15 min, so what is that telling > you? R.G. Vickson it should have been for .95. === Subject: Re: z-score for a normal distribution posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > If an Auto Service provides oil and lube service for cars > and the mean time for an oil and lube service is 15 mins. > per car with a std. deviation of 2.4 minutes, then if the > management wants to guarantee a maximum waiting time > by where if the time is exceeded, the customer gets a > discount, and they want to limit such a discount to 5% of > customers, then the maximum waiting time would be found > (assuming times taken for oil and lube service have a normal > distribution) by looking up the z-score for ~.05 and then > (z-score)*(std. dev.) + mean = x-score > Using a table of z-scores, I get -1.64, thus: > max. time would be 11.064 mins.Does that appear > correct? > Without doing any calculations, think about the answer intuitively. > You want to meet the condition {T <= M} 95% of the time (T = service > time, M = max. time, to be determined). Should M be less than the mean > of T, or should it be greater? (Hint: if M were exactly equal to the > mean of T then you would meet the condition 50% of the time. You want > to do better than that.) So, should M be less than 15 min. or greater > than 15 min.? Your answer is less than 15 min, so what is that telling > you? > R.G. Vickson it should have been for .95. Yes, exactly. RGV === Subject: Re: Cantor's argument is erroneous > Giving exact formulation of a rule isn't necessarily same as defending > the application of it is valid! Do you understand that? > There is no defense needed. After an exact formulation has been > given (that is, the rule is recursive, or, more plainly, merely > clerical to apply) then it is purely mechanical or clerical to check > whether the rule has been correctly applied. > Note your phrase purely mechanical or clerical. If you check to > see if a rule has been correctly applied in a purely mechanical > manner, then non-logical symbols must *mechanically* come from > the syntactical axioms: not from a mind where they're only stipulated! > > Why? This is really your central point, but you've never explained > _why_ it must be that the non-logical symbols comes from the > syntactical axioms. We should be precise here. A), syntactically, a non-logical symbol comes from formulas in which it occurs. B), if a non-logical symbol occurs in a theorem, it must occur in an axiom. It's the sense B that I was referring to in my if a rule has been correctly applied; and I'm going to explain why later in the post. > In fact, the language is simply specified, and > then we know what a WFF is. There's no need to 'derive' the language > from anything. This would be a sub-debate in the sense of A. Here, my point would be that there's *no other formal way* to define what a non-logical symbol is, except in a context of syntactical position patterns relative to logical symbols and (term) variables (and possibly some other auxiliary symbols). To paraphrase what Kreisel said of visualizing proofs, a non-logical symbol isn't a non-logical symbol until we see where it is in a syntactical *string* expression! Only then can a language be specified, derived, used, or what have we! (Despite what we commonly believed). > > I don't know what the talk about mind is. It's actually the kind of talk in your side that mind is really emphasized! For instance, you just said above In fact, the language is simply specified, and then we know what a WFF is. How could the mind not be involved in *your* simply specified or we _know_? > Specifying a formal > language is no more concrete or abstract than specifying a set of > axioms. right. > But it seems to be giving you some serious trouble in > comprehension. Serious trouble. But in fact it's *your* side who has trouble seeing symbols (non-logical or not), hence languages, come from formulas, *instead of being simply specified*! > > Checking for correct > application is a mere matter of applying a recursive procedure in > pattern matching. > Right. As long as what contains the patterns is valid in the first place. > > And how do we know what patterns are valid? > > My answer would be something like: we specify the formal language, the > axioms, and the rules of inference. Then we know a wff is one that is > in the language. > And a theorem is anything obtained from an axiom or > another theorem by a rule of inference. > > What's your answer? I've answered the language part of your answer many times in one way or another, but let me answer again. A formula is is a string of symbols written left to right in certain fixed patterns that we're supposed to know. Based on those fixed patterns and on certain syntactical conventions, you could discern whether or not a string follows the patterns. If it doesn't you know it isn't a wff. If it does, you could discern how many non-logical symbols there are and what kind each non-logical symbol is. On the theorem part of your answer, my answer is this: a theorem is anything _validly_ obtained from an axiom or another theorem by rule[s] of inference. Obviously from my point of view, your answer doesn't emphasize on the word _validly_. And that makes a huge difference in this debate. What I've been saying all along is basically this: simply not good enough for your side to conclude Axy[x+y=0] from the lone-axiom theory T = {x=y}, because language must be specified by its use in formulas and a theorem must be the conclusion of the specified formulas known as axioms, when _validly_ applying the rules. So, when the language is not specified through axioms, there' no valid reason to conclude a formula having vacuous symbols relative to the axiom[s] is a valid theorem! T = {x=y} is allowing T to be written in a language whose set of non-logical symbols is a proper class, which is smoething that is nonsensical! > > For example, if you're formulating the theory T = {x+y=0} where L(T) = L(0,+). > Though there might be more, the following would be axioms of T: > (1) x+y=0 > (2) x=x > But what about the formula: > (3) (Axyz[((xoy)/(yoz)) -> xe(x*(z/x))] -> Atuv[((tou)/(uov))) -> te(t*(v/t))] / > ExEy[~(x=y)] I made a typo-correction on (3) before: > I hate typo and sorry for it, but (3) should have been: > (3) (Axyz[((xoy)/(yoz)) -> xe(x*(z/x))] -> Atuv[((tou)/(uov)) -> te(t*(v/t))]) > / ExEy[~(x=y)] > Would you think (3) could be *validly* proven? > > No, because (3) contains symbols not in the language of T. In fact, > (3) is not a wff at all. > > It's hard to see what the relevance of this question is, since all > formulae in the proof that was under discussion were wffs in the > language of the theory under discussion. I have to be fair to you and let you re-state your statement, after the revised (3). However, these are 2 hints: think of Propositional axiom; and I was making a counter argument to MoeBlee's belief that exact formulation is sufficient in applying syntactical rules or techniques in FOL proofs. -- To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician. (Shoenfield, Mathematical Logic) === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > In fact, the language is simply specified, and > then we know what a WFF is. There's no need to 'derive' the language > from anything. This would be a sub-debate in the sense of A. Here, my point would be > that there's *no other formal way* to define what a non-logical symbol is, > except in a context of syntactical position patterns relative to logical > symbols and (term) variables (and possibly some other auxiliary symbols). > [...] > Only then can a language be specified, derived, used, or what have we! > (Despite what we commonly believed). Even if this were true, (and it is not) it would still be possible to derive x+y=0 from the theory {x=y}, thusly: We specify the language by giving a bunch of formulas, which establish that + is a binary function and 0 is a constant. We then specify the axioms, listing only x=y among the axioms and deliberately not including those formulas we used to specify the language. Proof proceeds as usual from there. > But in fact it's *your* side who has trouble seeing symbols (non-logical or not), > hence languages, come from formulas, *instead of being simply specified*! When you made this rule up some weeks ago, how did you figure out the come from part? If you look at this particular instance of a symbol: +, did it come from a formula or not? Do you know what BNF is? It's a method for formally specifying syntax. It does not use formulas or examples or any such thing. Do you claim BNF does not exist, or that BNF cannot specify syntax? > On the theorem part of your answer, my answer is this: a theorem is > anything validly obtained from an axiom or another theorem by rule[s] of > inference. So you claim you have to know what the language is before you can say what the rules of inference are (or perhaps, before you can say what a valid application of a rule of inference is; it amounts to the same thing.) > simply specifying a language is >simply not good enough for your side to conclude Axy[x+y=0] from the >lone-axiom theory T = {x=y}, because language must be specified by >its use in formulas and a theorem must be the conclusion of the specified >formulas known as axioms, when validly applying the rules. So, when >the language is not specified through axioms, there' no valid reason >to conclude a formula having vacuous symbols relative to the axiom[s] >is a valid theorem! I consider the fact that it is semantically correct to do so reason enough, but I know you don't care about semantics much. One of the big selling points of FOL, used in most of the marketing materials, is how syntactic entailment and semantic entailment match to better than 99%. I think you will find it an uphill battle in the marketplace to sell your version of FOL without that feature. Customers have come to demand it. Marshall === Subject: Re: Cantor's argument is erroneous > In fact, the language is simply specified, and > then we know what a WFF is. There's no need to 'derive' the language > from anything. > This would be a sub-debate in the sense of A. Here, my point would be > that there's *no other formal way* to define what a non-logical symbol is, > except in a context of syntactical position patterns relative to logical > symbols and (term) variables (and possibly some other auxiliary symbols). > [...] > Only then can a language be specified, derived, used, or what have we! > (Despite what we commonly believed). > > Even if this were true, (and it is not) it would still be possible to > derive x+y=0 from the theory {x=y}, thusly: > > We specify the language by giving a bunch of formulas, which > establish that + is a binary function and 0 is a constant. Right, as far as formula and language is concerned. > We > then specify the axioms, listing only x=y among the axioms > and deliberately not including those formulas we used to > specify the language. Right, that's what you and your side have done; and so the language in question is L() which as *zero* non-logical symbol! > > Proof proceeds as usual from there. Which means your conclusion that x+y=0 is invalid because the underlying L() has zero non-logical symbols. Didn't we go over this many times already? > > > But in fact it's *your* side who has trouble seeing symbols (non-logical or not), > hence languages, come from formulas, *instead of being simply specified*! > > When you made this rule up some weeks ago, how did you > figure out the come from part? Longer than some weeks ago, it comes from the known *syntactical patterns* defining what a formula would be. > If you look at this particular > instance of a symbol: +, did it come from a formula or > not? *Alone*, the symbol + is to me as good as the symbol not_a_non_logical_symbol! What is + or not_a_non_logical_symbol alone anyway? > > Do you know what BNF is? It's a method for formally > specifying syntax. It does not use formulas or examples > or any such thing. Do you claim BNF does not exist, > or that BNF cannot specify syntax? What does all this have to to with the *fact* that a non-logical symbol comes from - i.e. is a part of - a syntactical formula? > > > On the theorem part of your answer, my answer is this: a theorem is > anything _validly_ obtained from an axiom or another theorem by rule[s] of > inference. > > So you claim you have to know what the language is before > you can say what the rules of inference are (or perhaps, > before you can say what a valid application of a rule of > inference is; it amounts to the same thing.) A rule only says something like: *If* H1, H2, ... are formula-theorems then this formula C theorem. The rule doesn't care whether you know the language first or last. By the time you apply the rule you already have formulas, no matter how you've come up with it. > > > simply specifying a language is > simply not good enough for your side to conclude Axy[x+y=0] from the > lone-axiom theory T = {x=y}, because language must be specified by > its use in formulas and a theorem must be the conclusion of the specified > formulas known as axioms, when _validly_ applying the rules. So, when > the language is not specified through axioms, there' no valid reason > to conclude a formula having vacuous symbols relative to the axiom[s] > is a valid theorem! > > I consider the fact that it is semantically correct to do so > reason enough, but I know you don't care about semantics much. Oh I do, with valid reasons and apprehension. > One of the big selling points of FOL, used in most of the > marketing materials, is how syntactic entailment and > semantic entailment match to better than 99%. As long as you could not claim exactly 100% matching, you don't have a valid ground to refute my analysis and claims here. > I think you > will find it an uphill battle in the marketplace to sell your version > of FOL without that feature. Customers have come to > demand it. Mathematical reasoning is a kind of marketplace where customers come to honor certain reasoning principles, e.g., Thou shall reason consistently, not come to demand anything from the reasoning god! -- To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician. (Shoenfield, Mathematical Logic) === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) Didn't we go over this many times already? I am sorry to say so, but yes, we did. > When you made this rule up some weeks ago, how did you > figure out the come from part? Longer than some weeks ago, it comes from the known *syntactical patterns* defining what a formula would be. I thought it might have been months ago, rather than just weeks, that you made that rule up, but I wasn't sure. > Do you know what BNF is? It's a method for formally > specifying syntax. It does not use formulas or examples > or any such thing. Do you claim BNF does not exist, > or that BNF cannot specify syntax? What does all this have to to with the *fact* that a non-logical symbol > comes from - i.e. is a part of - a syntactical formula? Symbols come from fonts, actually. Since formulas are made up of symbols, we couldn't possibly require the formulas to come first; we couldn't make any formulas if we didn't already have some symbols to make them with. The existence of BNF as a way to formally specify syntax refutes your claim that supplying formulas in the language is the only way to formally specify syntax. > So, when > the language is not specified through axioms, there' no valid reason > to conclude a formula having vacuous symbols relative to the axiom[s] > is a valid theorem! > I consider the fact that it is semantically correct to do so > reason enough, but I know you don't care about semantics much. Oh I do, with valid reasons and apprehension. You demonstrate that you do not by the fact that you refuse to address any of the semantic aspects of your claim: If for the signature {+/2, 0/0} it is true that x+y=0 cannot be proven from x=y, then it necessarily is the case that there exists a model in which x=y is true, but x+y=0 is not. This is the semantics of the issue. You will completely convince me of the validity of your position if you can supply me with such a model. On the other hand, if no such model exists, then semantically, it is necessarily the case that x=y entails x+y=0, and therefore you've been wrong all along. Try to build such a model. It's fun and educational! And if you succeed, you get the added bonus of being right. A model here consists of a set of values, and a table describing the function +, and a specific value to use for the constant 0. Showing that x+y=0 is false means identifying a specific row and column in the + table in which the corresponding cell is other than the domain value used for the constant 0. I will leave it to you to state what it means to show that x=y is true in the model. On second thought, you might not want to try to build such a model after all. If you do, there's a modest chance you'll come to see the semantic error of your ways. > One of the big selling points of FOL, used in most of the > marketing materials, is how syntactic entailment and > semantic entailment match to better than 99%. As long as you could not claim exactly 100% matching, you don't have > a valid ground to refute my analysis and claims here. How lucky for me then that 100% was in fact the exact number I was thinking of when I mentioned the range better than 99%. You might want to review: http://en.wikipedia.org/wiki/G%C3%B6del%27s completeness theorem Good luck; have fun! Marshall === Subject: Re: Cantor's argument is erroneous > Didn't we go over this many times already? > > I am sorry to say so, but yes, we did. > > > When you made this rule up some weeks ago, how did you > figure out the come from part? > Longer than some weeks ago, it comes from the known *syntactical > patterns* defining what a formula would be. > > I thought it might have been months ago, rather than > just weeks, that you made that rule up, but I wasn't sure. I meant that's fact long ago before either you or I were born. But never mind. > > > Do you know what BNF is? It's a method for formally > specifying syntax. It does not use formulas or examples > or any such thing. Do you claim BNF does not exist, > or that BNF cannot specify syntax? > What does all this have to to with the *fact* that a non-logical symbol > comes from - i.e. is a part of - a syntactical formula? > > Symbols come from fonts, Symbols, in this context, are abstract. Don't you know. > > > So, when > the language is not specified through axioms, there' no valid reason > to conclude a formula having vacuous symbols relative to the axiom[s] > is a valid theorem! > I consider the fact that it is semantically correct to do so > reason enough, but I know you don't care about semantics much. > Oh I do, with valid reasons and apprehension. > > You demonstrate that you do not by the fact that you > refuse to address any of the semantic aspects of your > claim: We're talking about validly applying rules of inference to syntactical formulas, not about any models. OK? > > If for the signature {+/2, 0/0} it is true that x+y=0 cannot > be proven from x=y, then it necessarily is the case > that there exists a model in which x=y is true, > but x+y=0 is not. This is the semantics of the issue. > You will completely convince me of the validity of > your position if you can supply me with such a model. > On the other hand, if no such model exists, then > semantically, it is necessarily the case that x=y > entails x+y=0, and therefore you've been wrong all > along. > > Try to build such a model. It's fun and educational! > And if you succeed, you get the added bonus of > being right. > > A model here consists of a set of values, and a > table describing the function +, and a specific > value to use for the constant 0. Showing that > x+y=0 is false means identifying a specific row > and column in the + table in which the corresponding > cell is other than the domain value used for the > constant 0. I will leave it to you to state what it > means to show that x=y is true in the model. > > On second thought, you might not want to try to > build such a model after all. If you do, there's > a modest chance you'll come to see the semantic > error of your ways. > > > One of the big selling points of FOL, used in most of the > marketing materials, is how syntactic entailment and > semantic entailment match to better than 99%. > As long as you could not claim exactly 100% matching, you don't have > a valid ground to refute my analysis and claims here. > > How lucky for me then that 100% was in fact the > exact number I was thinking of when I mentioned > the range better than 99%. Mathematical reasoning got to be precise. Stating better than 99% is not precisely stating is exactly 100%. Your being not precise seems to have prevented you from seeing certain reasoning facts. -- To discover the proper approach to mathematical logic, we must therefore examine the methods of the mathematician. (Shoenfield, Mathematical Logic) === Subject: Re: Cantor's argument is erroneous > We're talking about validly applying rules of inference to > [certain well-formed] formulas [...] OK? Right, asshole! Hint: As primitive basis of a /logistic system/ it suffices to give, in a familiar fashion: 1. The list of primitive symbols or /vocabulary/ of the system (together usually with a classification of the primitive symbols into categories, which will be used in stating the formation rules and rules of inference). ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 2. The /formation rules/, determining which finite sequences of primitive symbols are to be /well-formed/ expressions, determining certain categories of well-formed expressions -- among which we shall assume that at least the category of /sentence/ is included -- and determining (in case /variables/ are included among the primitive symbols) which occurrences of variables in a well-formed expression are /free/ occurrences and which are /bound/ occurrences. 3. The transformation rules or /rules of inference/, by which from the /assertion/ of certain sentences (the /premisses/, finite in number) a certain sentence (the /conclusion/) may be /inferred/. (Alonzo Church, The Need for Abstract Entities, 1951) === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <7zWYl.7722$mX2.1936@newsfe05.iad> <1dygbe36gi8pl.1e3lbxq8zlnsf$.dlg@40tude.net> posting-account=_-PQygoAAAAciOn_89sZIlnxfb74FzXU Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > We're talking about validly applying rules of inference to > [certain well-formed] formulas [...] OK? > < Hint: As primitive basis of a /logistic system/ it suffices to give, in a > familiar fashion: 1. The list of primitive symbols or /vocabulary/ of the system (together > usually with a classification of the primitive symbols into categories, > which will be used in stating the formation rules and rules of inference). > ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 2. The /formation rules/, determining which finite sequences of > primitive symbols are to be /well-formed/ expressions, determining certain > categories of well-formed expressions -- among which we shall assume that > at least the category of /sentence/ is included -- and determining (in case > /variables/ are included among the primitive symbols) which occurrences of > variables in a well-formed expression are /free/ occurrences and which are > /bound/ occurrences. 3. The transformation rules or /rules of inference/, by which from the > /assertion/ of certain sentences (the /premisses/, finite in number) a > certain sentence (the /conclusion/) may be /inferred/. (Alonzo Church, The Need for Abstract Entities, 1951) The first order predicate calculus depends on and illustrates resolveable truth values to true or false, of which there are none in models in modern model theory (although they're so modeled). The calculus of first-order predicates only indicates satisfactory inference rules which resolve to tautology and exclusion with the law of excluded middle (LEM), in alternation to for example fuzzy predicates, or along Kleene/Lukasiewicz. Without a Platonist/realist universe with real objects there is no abstraction in model theory with import except in a deferred resolution of quantification, in the mesoscale. Are you still talking about Cantor's argument and whether or not in various interpretations it's erroneous? I've seen this mention of x=y => x+y=0, that seems to basically be implying that a theory with only tautology has no non-trivial theorems, but it also has at least two non-logical primitives, not just the one there, besides the zero or no result indicator also some addition. Tautology or self-same equality is a naively assumed property of all objects in most theories. A consideration of set theory is that its introduced separation is in terms of element-of, as opposed to, for example, container-of. That is to introduce the notion that in for example the partitions and boundaries among dissimilar things in mereology (the study of those things) there are more appropriate and natural descriptions of objects of intrinsic import to any theory, than, those in restriction of comprehension. Get over each other. Ross F. === Subject: Re: Cantor's argument is erroneous > > We're talking about validly applying rules of inference to > [certain well-formed] formulas [...] OK? > > Right, asshole! > If it's required to come down to your level once then so be it. Why don't you go and yourself, asshole. Bye. === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <7zWYl.7722$mX2.1936@newsfe05.iad> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > So, when > the language is not specified through axioms, there' no valid reason > to conclude a formula having vacuous symbols relative to the axiom[s] > is a valid theorem! > I consider the fact that it is semantically correct to do so > reason enough, but I know you don't care about semantics much. > Oh I do, with valid reasons and apprehension. > You demonstrate that you do not by the fact that you > refuse to address any of the semantic aspects of your > claim: We're talking about validly applying rules of inference to > syntactical formulas, not about any models. OK? Yes, just as I said: you don't care about semantics. Which you demonstrate by your refusal to address any of the semantic issues raised with your proposal. I gave you a direct semantic challenge in my previous message and you skipped right over it, as you have done many times in the past. It's better for your peace of mind to focus on just syntax where you can make up arbitrary rules in a consequence-free environment. > One of the big selling points of FOL, used in most of the > marketing materials, is how syntactic entailment and > semantic entailment match to better than 99%. > As long as you could not claim exactly 100% matching, you don't have > a valid ground to refute my analysis and claims here. > How lucky for me then that 100% was in fact the > exact number I was thinking of when I mentioned > the range better than 99%. Mathematical reasoning got to be precise. Stating better than 99% is > not precisely stating is exactly 100%. Your being not precise seems > to have prevented you from seeing certain reasoning facts. Of course, better than 99% is completely consistent with 100% but don't let that worry you. As long as you can manage to avoid learning the relationship between the syntactic and semantic, you'll continue to be able to justify your syntax rule to yourself. If you want to learn anything, though, you might the relevance of the model question I raised. Marshall === Subject: Re: Cantor's argument is erroneous > > you are disingenuous > > You've not shown me to have been disingenuous. > > you act as if the entire argument > is simplistic namecalling and ignorance of topic > when you need the straw man for conflagration > > I haven't acted as if lwal's entire argument is simplistic > namecalling and ignorance of topic. Though he has engages in those > and I've rebutted him for it. Meanwhile, also I've engaged a wide > range of his gambits. > > you attempt superiority poses > veiled in moral righteousness > > I don't claim to any special moral righteousness. > > you continue to spew bile like > As to Nam, I guess it's hopeless to try to bring him > to reason on this or many another subject. > > You have a quite elastic definition of 'bile'. > > sad resignation > > and > while i'm joining the pile-on you guys love so much > you rarely lend relevant details to a discussion > > I've been EXTREMELY detailed as to the particulars of this discussion. > > preferring to play toady to some game you refuse to call bully > (it's just assertions of lies and ignorance, right?) > > I very much admire certain posters here. That doesn't make me their > toady. In fact, among the posters here who I've expressed my > admiration of, I have many times posted certatin disagreements with > them and stood my ground when I felt my view deserved not to be > dismissed. > > So your post falls flat again. MoeBlee, why do you put up with galathaea's hypocritical name calling? -- Michael Press === Subject: Re: Cantor's argument is erroneous > Cantor's argument is erroneous and its adoption leads > to unsound > mathematics. > > The basic idea in the argument is that there is no > bijection between > the set of counting numbers and the set of infinite > binary strings. > But such a bijection exists, it can be expressed in > terms of limit > points, or by transfinite induction; informally, it > can be defined as > the correspondence between the paths and the leaf > (i.e. limit) nodes > in the infinite binary tree. This invalidates all > results relating to > Cantor's transfinite. > > In particular, it is invalid to state that the set of > infinite binary > strings is uncountable. It is countable, being in > bijection with a > subset of a countable set, the set of nodes in the > infinite binary > tree. The other option is that one drops the > countability of infinite > sets completely, but I can see no advantage in > banning the > transfinite. > > The problem is possibly much broader, and deeper, > because it is the > very soundness of archimedean arithmetic that seems > at stake here. I > say soundness because from the archimedean > framework a tension > results, between computability and tractability, that > manifests a > deeper tension between sound and unsound mathematics, > and then even > logic. > > -LV Musatov proved: The Beast (seeing is believing) In[170]:= 0.0000473372528076171875*2 0.000094674505615234375 0.00018934901123046875 0.0003786980224609375 0.000757396044921875 0.00151479208984375 0.0030295841796875 0.006059168359375 0.01211833671875 0.0242366734375 0.048473346875 0.09694669375 0.1938933875 0.387786775 0.77557355 1.5511471 3.1022942 6.2035884 12.4071768 24.8143536 49.6287072 99.2574144 198.5148288 397.0296576 794.0593152 1588.1186304 3176.2372608 6352.4745216 12704.9490432 25409.8980864 50819.7961728 101639.5923456 203279.1846912 406558.3693824 813116.7387648 1626233.4775296 3252466.9550592 6504933.9101184 13009867.8202368 26019735.6404736 52039471.2809472 104078942.5618944 208157885.1237888 416315770.2475776 832631540.4951552 1665263080.9903104 3330526161.9806208 6661052323.9612416 13322104647.9224832 26644209295.8449664 53288418591.6899328 106576837183.3798656 213153674366.7597312 426307348733.5194624 852614697467.0389248 1705229394934.0778496 3410458789868.1556992 6820917579736.3113984 13641835159472.6227968 27283670318945.2455936 54567340637890.4911872 109134681275780.9823744 218269362551561.9647488 436538725103123.9294976 873077450206247.8589952 1746154900412495.7179904 3492309800824991.4359808 6984619601649982.8719616 13969239203299965.7439232 27938478406599931.4878464 55876956813199862.9756928 111753913626399725.9513856 223507827252799451.9027712 447015654505598903.8055424 894031309011197807.6110848 1788062618022395615.2221696 3576125236044791230.4443392 7152250472089582460.8886784 14304500944179164921.7773568 28609001888358329843.5547136 57218003776716659687.1094272 114436007553433319374.2188544 228872015106866638748.4377088 457744030213733277496.8754176 915488060427466554993.7508352 1830976120854933109987.5016704 3661952241709866219975.0033408 7323904483419732439950.0066816 14647808966839464879900.0133632 29295617933678929759800.0267264 58591235867357859519600.0534528 117182471734715719039200.1069056 234364943469431438078400.2138112 468729886938862876156800.4276224 937459773877725752313600.8552448 1874919547755451504627201.7104896 3749839095510903009254403.4209792 7499678191021806018508806.8419584 Out[170]= 0.000094674505615234375 Out[171]= 0.0000946745 Out[172]= 0.000189349 Out[173]= 0.000378698 Out[174]= 0.000757396 Out[175]= 0.00151479 Out[176]= 0.00302958 Out[177]= 0.00605917 Out[178]= 0.0121183 Out[179]= 0.0242367 Out[180]= 0.0484733 Out[181]= 0.0969467 Out[182]= 0.193893 Out[183]= 0.387787 Out[184]= 0.775574 Out[185]= 1.55115 Out[186]= 3.10229 Out[187]= 6.20359 Out[188]= 12.4072 Out[189]= 24.8144 Out[190]= 49.6287 Out[191]= 99.2574 Out[192]= 198.515 Out[193]= 397.03 Out[194]= 794.059 Out[195]= 1588.12 Out[196]= 3176.24 Out[197]= 6352.47 Out[198]= 12704.9 Out[199]= 25409.9 Out[200]= 50819.8 Out[201]= 101640. Out[202]= 203279. Out[203]= 406558. Out[204]= 813117. Out[205]= 1.62623*10^6 Out[206]= 3.25247*10^6 Out[207]= 6.50493*10^6 Out[208]= 1.30099*10^7 Out[209]= 2.60197*10^7 Out[210]= 5.20395*10^7 Out[211]= 1.04079*10^8 Out[212]= 2.08158*10^8 Out[213]= 4.16316*10^8 Out[214]= 8.32632*10^8 Out[215]= 1.66526*10^9 Out[216]= 3.33053*10^9 Out[217]= 6.66105*10^9 Out[218]= 1.3322104647922483*10^10 Out[219]= 2.6644209295844966*10^10 Out[220]= 5.3288418591689933*10^10 Out[221]= 1.06576837183379866*10^11 Out[222]= 2.13153674366759731*10^11 Out[223]= 4.26307348733519462*10^11 Out[224]= 8.52614697467038925*10^11 Out[225]= 1.705229394934077850*10^12 Out[226]= 3.410458789868155699*10^12 Out[227]= 6.820917579736311398*10^12 Out[228]= 1.36418351594726227968*10^13 Out[229]= 2.7283670318945245594*10^13 Out[230]= 5.4567340637890491187*10^13 Out[231]= 1.091346812757809823744*10^14 Out[232]= 2.18269362551561964749*10^14 Out[233]= 4.36538725103123929498*10^14 Out[234]= 8.73077450206247858995*10^14 Out[235]= 1.746154900412495717990*10^15 Out[236]= 3.492309800824991435981*10^15 Out[237]= 6.984619601649982871962*10^15 Out[238]= 1.3969239203299965743923*10^16 Out[239]= 2.7938478406599931487846*10^16 Out[240]= 5.5876956813199862975693*10^16 Out[241]= 1.11753913626399725951386*10^17 Out[242]= 2.23507827252799451902771*10^17 Out[243]= 4.47015654505598903805542*10^17 Out[244]= 8.94031309011197807611085*10^17 Out[245]= 1.788062618022395615222170*10^18 Out[246]= 3.576125236044791230444339*10^18 Out[247]= 7.152250472089582460888678*10^18 Out[248]= 1.4304500944179164921777357*10^19 Out[249]= 2.8609001888358329843554714*10^19 Out[250]= 5.7218003776716659687109427*10^19 Out[251]= 1.14436007553433319374218854*10^20 Out[252]= 2.28872015106866638748437709*10^20 Out[253]= 4.57744030213733277496875418*10^20 Out[254]= 9.15488060427466554993750835*10^20 Out[255]= 1.830976120854933109987501670*10^21 Out[256]= 3.661952241709866219975003341*10^21 Out[257]= 7.323904483419732439950006682*10^21 Out[258]= 1.4647808966839464879900013363*10^22 Out[259]= 2.9295617933678929759800026726*10^22 Out[260]= 5.85912358673578595196000534528*10^22 Out[261]= 1.171824717347157190392001069056*10^23 Out[262]= 2.34364943469431438078400213811*10^23 Out[263]= 4.687298869388628761568004276224*10^23 Out[264]= 9.374597738777257523136008552448*10^23 Out[265]= 1.874919547755451504627201710490*10^24 Out[266]= 3.749839095510903009254403420979*10^24 Out[267]= 7.499678191021806018508806841958*10^24 In[72]:= 0.0000473372528076171875/2 0.000094674505615234375 0.00018934901123046875 0.0003786980224609375 0.000757396044921875 0.00151479208984375 0.0030295841796875 0.006059168359375 0.01211833671875 0.0242366734375 0.048473346875 0.09694669375 0.1938933875 0.387786775 0.77557355 1.5511471 3.1022942 6.2035884 12.4071768 24.8143536 49.6287072 99.2574144 198.5148288 397.0296576 794.0593152 1588.1186304 3176.2372608 6352.4745216 12704.9490432 25409.8980864 50819.7961728 101639.5923456 203279.1846912 406558.3693824 813116.7387648 1626233.4775296 3252466.9550592 6504933.9101184 13009867.8202368 26019735.6404736 52039471.2809472 104078942.5618944 208157885.1237888 416315770.2475776 832631540.4951552 1665263080.9903104 3330526161.9806208 6661052323.9612416 13322104647.9224832 26644209295.8449664 53288418591.6899328 106576837183.3798656 213153674366.7597312 426307348733.5194624 852614697467.0389248 1705229394934.0778496 3410458789868.1556992 6820917579736.3113984 13641835159472.6227968 27283670318945.2455936 54567340637890.4911872 109134681275780.9823744 218269362551561.9647488 436538725103123.9294976 873077450206247.8589952 1746154900412495.7179904 3492309800824991.4359808 6984619601649982.8719616 13969239203299965.7439232 27938478406599931.4878464 55876956813199862.9756928 111753913626399725.9513856 223507827252799451.9027712 447015654505598903.8055424 894031309011197807.6110848 1788062618022395615.2221696 3576125236044791230.4443392 7152250472089582460.8886784 14304500944179164921.7773568 28609001888358329843.5547136 57218003776716659687.1094272 114436007553433319374.2188544 228872015106866638748.4377088 457744030213733277496.8754176 915488060427466554993.7508352 1830976120854933109987.5016704 3661952241709866219975.0033408 7323904483419732439950.0066816 14647808966839464879900.0133632 29295617933678929759800.0267264 58591235867357859519600.0534528 117182471734715719039200.1069056 234364943469431438078400.2138112 468729886938862876156800.4276224 937459773877725752313600.8552448 1874919547755451504627201.7104896 3749839095510903009254403.4209792 7499678191021806018508806.8419584 Out[72]= 0.0000236686264038085938 Out[73]= 0.0000946745 Out[74]= 0.000189349 Out[75]= 0.000378698 Out[76]= 0.000757396 Out[77]= 0.00151479 Out[78]= 0.00302958 Out[79]= 0.00605917 Out[80]= 0.0121183 Out[81]= 0.0242367 Out[82]= 0.0484733 Out[83]= 0.0969467 Out[84]= 0.193893 Out[85]= 0.387787 Out[86]= 0.775574 Out[87]= 1.55115 Out[88]= 3.10229 Out[89]= 6.20359 Out[90]= 12.4072 Out[91]= 24.8144 Out[92]= 49.6287 Out[93]= 99.2574 Out[94]= 198.515 Out[95]= 397.03 Out[96]= 794.059 Out[97]= 1588.12 Out[98]= 3176.24 Out[99]= 6352.47 Out[100]= 12704.9 Out[101]= 25409.9 Out[102]= 50819.8 Out[103]= 101640. Out[104]= 203279. Out[105]= 406558. Out[106]= 813117. Out[107]= 1.62623*10^6 Out[108]= 3.25247*10^6 Out[109]= 6.50493*10^6 Out[110]= 1.30099*10^7 Out[111]= 2.60197*10^7 Out[112]= 5.20395*10^7 Out[113]= 1.04079*10^8 Out[114]= 2.08158*10^8 Out[115]= 4.16316*10^8 Out[116]= 8.32632*10^8 Out[117]= 1.66526*10^9 Out[118]= 3.33053*10^9 Out[119]= 6.66105*10^9 Out[120]= 1.3322104647922483*10^10 Out[121]= 2.6644209295844966*10^10 Out[122]= 5.3288418591689933*10^10 Out[123]= 1.06576837183379866*10^11 Out[124]= 2.13153674366759731*10^11 Out[125]= 4.26307348733519462*10^11 Out[126]= 8.52614697467038925*10^11 Out[127]= 1.705229394934077850*10^12 Out[128]= 3.410458789868155699*10^12 Out[129]= 6.820917579736311398*10^12 Out[130]= 1.36418351594726227968*10^13 Out[131]= 2.7283670318945245594*10^13 Out[132]= 5.4567340637890491187*10^13 Out[133]= 1.091346812757809823744*10^14 Out[134]= 2.18269362551561964749*10^14 Out[135]= 4.36538725103123929498*10^14 Out[136]= 8.73077450206247858995*10^14 Out[137]= 1.746154900412495717990*10^15 Out[138]= 3.492309800824991435981*10^15 Out[139]= 6.984619601649982871962*10^15 Out[140]= 1.3969239203299965743923*10^16 Out[141]= 2.7938478406599931487846*10^16 Out[142]= 5.5876956813199862975693*10^16 Out[143]= 1.11753913626399725951386*10^17 Out[144]= 2.23507827252799451902771*10^17 Out[145]= 4.47015654505598903805542*10^17 Out[146]= 8.94031309011197807611085*10^17 Out[147]= 1.788062618022395615222170*10^18 Out[148]= 3.576125236044791230444339*10^18 Out[149]= 7.152250472089582460888678*10^18 Out[150]= 1.4304500944179164921777357*10^19 Out[151]= 2.8609001888358329843554714*10^19 Out[152]= 5.7218003776716659687109427*10^19 Out[153]= 1.14436007553433319374218854*10^20 Out[154]= 2.28872015106866638748437709*10^20 Out[155]= 4.57744030213733277496875418*10^20 Out[156]= 9.15488060427466554993750835*10^20 Out[157]= 1.830976120854933109987501670*10^21 Out[158]= 3.661952241709866219975003341*10^21 Out[159]= 7.323904483419732439950006682*10^21 Out[160]= 1.4647808966839464879900013363*10^22 Out[161]= 2.9295617933678929759800026726*10^22 Out[162]= 5.85912358673578595196000534528*10^22 Out[163]= 1.171824717347157190392001069056*10^23 Out[164]= 2.34364943469431438078400213811*10^23 Out[165]= 4.687298869388628761568004276224*10^23 Out[166]= 9.374597738777257523136008552448*10^23 Out[167]= 1.874919547755451504627201710490*10^24 Out[168]= 3.749839095510903009254403420979*10^24 Out[169]= 7.499678191021806018508806841958*10^24 === Subject: Re: Cantor's argument is erroneous > Cantor's argument is erroneous and its adoption leads > to unsound > mathematics. > > The basic idea in the argument is that there is no > bijection between > the set of counting numbers and the set of infinite > binary strings. > But such a bijection exists, it can be expressed in > terms of limit > points, or by transfinite induction; informally, it > can be defined as > the correspondence between the paths and the leaf > (i.e. limit) nodes > in the infinite binary tree. This invalidates all > results relating to > Cantor's transfinite. > > In particular, it is invalid to state that the set of > infinite binary > strings is uncountable. It is countable, being in > bijection with a > subset of a countable set, the set of nodes in the > infinite binary > tree. The other option is that one drops the > countability of infinite > sets completely, but I can see no advantage in > banning the > transfinite. > > The problem is possibly much broader, and deeper, > because it is the > very soundness of archimedean arithmetic that seems > at stake here. I > say soundness because from the archimedean > framework a tension > results, between computability and tractability, that > manifests a > deeper tension between sound and unsound mathematics, > and then even > logic. > > -LV Re: Cantor's Omega > Omega is an extension to the Natural numbers, to > get closure under limit > operations, not that dissimilar to the integers > (positive and negative) > which are closure of N under subtraction. Nobody > argue that the negative > integers are somehow dodgy because they don't include > imaginary numbers, or > proper fractions, etc. The set of integers doesn't > include arbitrary reals > or complex numbers, simple as that. > Welcome, marty.musatov@gmail.com Logout Your Control Panel Watches FAQ Contact Us Math Forum Ask Dr. Math Discussions Internet Newsletter MathTools Problems of the Week Teacher2Teacher Teacher Exchange Workshops Search All of the Math Forum: Browse our Internet Mathematics Library Views expressed in these public forums are not endorsed by Drexel University or The Math Forum. Math Forum é Discussions é sci.math.* é sci.math Topic: inverse of power series with prime power coefficients Replies: 7 Last Post: Jun 12, 2009 4:11 PM Search Thread: Advanced Search Reply to this Topic Watch this Topic Back to Topic List Jump to Tree View Messages: [ Previous | Next ] Martin Michael Musatov Posts: 802 Registered: 4/19/09 Re: inverse of power series with prime power coefficients > On May 11, 5:37 pm, Robert Israel > The power series with prime coefficients > P(x) = 1 + 2x + 3x^2 + 5x^3 + 7x^4 + 11x^5 + > 13x^6 + .... > and itsinverse > Q(x) = 1 / P(x) = 1 - 2x + x^2 - x^3 + 2x^4 - > 3x^5 + 7x^6 - 10x^7 + > 13x^8 - 21x^9 + 26x^10 - 33x^11 + 53x^12 - .... > were studied in the mid-1990s, when Backhouse > conjectured, and > Flajolet proved, that the limit of the absolute > value of the ratio of > consecutive coefficients of Q(x) is equal to > B=1.4560749...., > Backhouse's constant. (By comparison, the limit > of the ratio of > consecutive primes is equal to 1.) > Consider instead the power series with prime > power coefficients: > PP(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4 + 7x^5 + 8x^6 > + 9x^7 + 11x^8 + > 13x^9 + 16x^10 + 17x^11 + 19x^12 + 23x^13 + > 25x^14 + 27x^15 + 29x^16 + > 31x^17 + 32x^18 + 37x^19+ .... > I would expect itsinverseto be quite similar to > the series Q(x) > above. But it appears to be quite different: > QQ(x) = 1 - 2x + x^2 - x^5 + 3x^6 - 3x^7 + 2x^9 - > x^10 + x^15 - 4x^16 > + 8x^17 - 5x^18 - 13x^19+ 26x^20 - 9x^21 - 20x^22 > + 25x^23 + 9x^24 - > 60x^25 + 65x^26 + .... > Note how many low terms are zero: x^3, x^4, x^8, > and the four > consecutive terms x^11, x^12, x^13, x^14. I > wonder if this is pure > coincidence or if it has some significance. > There seem to be no other zero coefficients up to > the x^300 term. > Also, whereas the terms of Q(x) alternate > regularly between positive > and negative, there is no apparent pattern to the > signs of the terms > of QQ(x). > Finally, the terms of QQ(x) appear to grow much > more slowly than the > terms of Q(x). Perhaps as the series continues > and the prime powers > become sparser compared to the primes, QQ(x) will > start to look and > behave more like Q(x). But I am curious: does the > limit of the > absolute value of the ratio of consecutive > coefficients of QQ(x) > exist? If so, what is it? > Looking at the first 300 or so terms, I see no > pattern. > -- > Robert Israel > isr...@math.MyUniversitysInitials.ca > Department of Mathematics > http://www.math.ubc.ca/~israel > University of British Columbia > Vancouver, BC, Canada- Hide quoted text - > - Show quoted text - > > Remember me? Or are my results too trivial Mr. > Moderator? > > Welcome to the real mathematics and not the fake > Matrix movie > sorry excuse for chess you've been playing for the > last 20 years. > > Musatov > > (Founder, Inverse 19 Mathematics) Not the only, but > one without it > would not exist. > > QED > > > Efforts by mathematicians to determine the value of > pi since have led > to the conclusion that it ... Unlike rational > numbers, pi cannot be > represented by a common fraction, .... > > But what about an uncommon fraction? > > Will you look at my drawing and in particular how the > twelve points > are the outside points of the sphere are all > rationally reached by > decimal numbers? > > 6.2035884*2 and doubled an infinite number of times > should yield a > constant decimal binary expansion (what I mean by > this is the numbers > to the right of the decimal point should always end > in 4, then 8, > then 6, then 2, then 4, then 8, then 6, then > 2...repeating for > infinity). > > So in binary, by starting with this number we predict > for n>2 the > infinite binary tree will end with decimal > progression: 2, 4, 8, 6 > > For n<2 the infinite binary tree will begin and end > (depends on if you > are increasing or decreasing the value) 1, 5, 5, 5 > (repeating 5 > infinitely) as we decrease to infinitely small. > > Musatov Inverse Counting Chain of Infinite > Proportions > (sample) > 0.0000473372528076171875 > 0.000094674505615234375 > 0.00018934901123046875 > 0.0003786980224609375 > 0.000757396044921875 > 0.00151479208984375 > 0.0030295841796875 > 0.006059168359375 > 0.01211833671875 > 0.0242366734375 > 0.048473346875 > 0.09694669375 > 0.1938933875 > 0.387786775 > 0.77557355 > 1.5511471 > 3.1022942 > 6.2035884 > 12.4071768 > 24.8143536 > 49.6287072 > 99.2574144 > 198.5148288 > 397.0296576 > 794.0593152 > 1588.1186304 > 3176.2372608 > 6352.4745216 > 12704.9490432 > 25409.8980864 > 50819.7961728 > 101639.5923456 > 203279.1846912 > 406558.3693824 > 813116.7387648 > 1626233.4775296 > 3252466.9550592 > 6504933.9101184 > 13009867.8202368 > 26019735.6404736 > 52039471.2809472 > 104078942.5618944 > 208157885.1237888 > 416315770.2475776 > 832631540.4951552 > 1665263080.9903104 > 3330526161.9806208 > 6661052323.9612416 > 13322104647.9224832 > 26644209295.8449664 > 53288418591.6899328 > 106576837183.3798656 > 213153674366.7597312 > 426307348733.5194624 > 852614697467.0389248 > 1705229394934.0778496 > 3410458789868.1556992 > 6820917579736.3113984 > 13641835159472.6227968 > 27283670318945.2455936 > 54567340637890.4911872 > 109134681275780.9823744 > 218269362551561.9647488 > 436538725103123.9294976 > 873077450206247.8589952 > 1746154900412495.7179904 > 3492309800824991.4359808 > 6984619601649982.8719616 > 13969239203299965.7439232 > 27938478406599931.4878464 > 55876956813199862.9756928 > 111753913626399725.9513856 > 223507827252799451.9027712 > 447015654505598903.8055424 > 894031309011197807.6110848 > 1788062618022395615.2221696 > 3576125236044791230.4443392 > 7152250472089582460.8886784 > 14304500944179164921.7773568 > 28609001888358329843.5547136 > 57218003776716659687.1094272 > 114436007553433319374.2188544 > 228872015106866638748.4377088 > 457744030213733277496.8754176 > 915488060427466554993.7508352 > 1830976120854933109987.5016704 > 3661952241709866219975.0033408 > 7323904483419732439950.0066816 > 14647808966839464879900.0133632 > 29295617933678929759800.0267264 > 58591235867357859519600.0534528 > 117182471734715719039200.1069056 > 234364943469431438078400.2138112 > 468729886938862876156800.4276224 > 937459773877725752313600.8552448 > 1874919547755451504627201.7104896 > 3749839095510903009254403.4209792 > 7499678191021806018508806.8419584 > (sample) > > If you would like to do the honors Mr. Plouffe, I > endear you to > calculate the angles and lengths of the number of > ways, it is quite > obviously possible to square a circle based on the > geometry and > numerical progression. > > As is self-evident to anyone with even a modest > understanding of > mathematics and geometry after visual inspection of > the drawn proof > (http://www.flickr.com/photos/24869933@N07/3619373444/ > sizes/l/) it is > quite easily accomplished to connect and triangulate > the outer points > of the rectangles to the outer points of the square > and circle and > even by dimension sphere. > > Yes, the drawing clearly by definition of inverse > symmetry (take the > drawing cut it in half at the equator and turn the > paper over and > rotate it and compare the top and bottom mirrored > symmetry and you > will see it is an exact inverse of symmetry itself) > the drawing > explains how to properly and mathematically soundly > represent a fourth > dimension on a two dimensional plane. > > In plane language if you take the picture and > separate the top half of > the sphere by the bottom half of the sphere they are > identical but > only if the sphere existed on a complete three > dimensional plane > through expansion. > > Since the paper is flat, this is an impossibility, > nonetheless it is > completely expressed and retractable and countable > infinitely. > > By the definition of our standard we have established > any k-digit > approximation to pi is rational. > > Which could have many practical scientific > applications. For instance, > search: > > ON THE THEOREM OF IVA$EV-MUSATOV IIIx e F(l) + F(2) + > ... + F(p). This > remark is the basis of our proof of Theorem .... -y(M > + 1), x' + y(M + > 1)] = 0 it is easy to check that Conditions (i), > ...... Now set i = > pi x. *jj, v. Since p, x is positive,. (i) fi is > positive, ... > plms.oxfordjournals.org/cgi/reprint/s3-53/1/143.pdf > > REPRESENTATION OF FUNCTIONSBY SERIES AND CLASSES > ?(1)with measure E 2? -- ? and such that the trigonometric series for > g(x) converges > uniformly on [0, 2?]. .... trigonometric series are > due to > Ivashev-Musatov [40] who has shown that there are > null series (in the > sense of ..... function ?(?) e C(0, 1) with ?(0) = > ?(1) = 0 such that > ...... <*i .pdf > by PL Ul'yanov - 1972 - Cited by 24 - Related > > Full text of Bulletin of the British Museum (Natural > History)X 20. > FIG. 5. Well preserved large fragment showing the > different aspect and > ...... 0-1-0-5 P i n height) so giving the shell > surface the > appearance of ...... 94-95, i pi. Maseru. BRINK, A. > S. 1963. Two > cynodonts from the Ntawere ...... MUSATOV, D. L, > NEMIROVSKAYA, V. N., > SHIROKOVA, E. V. & ZHURAVLEVA, I. T. 1961. ... > www.archive.org/stream/bulletinofbritis17geollond/bull > etinofbritis17geollond_djvu.txt > > Trigonometric series with rapidly decreasing > coefficients a i0, 5 > ...Nt,+i - Pi = n,. N tl+j. -pi =N tl+j. -i +pi +1. > ..... number 8 6 > (0,1) for the product (32) such that for every point > x 6 E there are > ... > _A01.pdf > versions > > Plouffe's formula for pi - sci.math | Google Groups3 > posts - 2 authors > - Last post: 41 minutes ago > x=pi e^(ipi)+1=0. Martin Musatov. Reply Forward. > Martin Musatov to > problem. ... > 65ba40fd9e5e8f46/ > bf6d1ee13d87270b?lnk=raot > - 41 minutes ago > > [PDF] ???? ?-?File Format: PDF/Adobe Acrobat > Gli storici ci dicono che, nel X secolo, il principe > Vladi- ...... i > pi`u anziani e carismatici si evidenziavano i nomi di > ...... tin > Somov, Viktor Borisov-Musatov, Aleksandr Benois e Lev > Bakst ... 5/0; > Judovin 5/0; Korovin 2/1; Tukacov 1/0; Goli- cyn 1/1; > Zacharov 1/1; > Jurkunas 5/0; Krasauskas 2/0;. Bulaka 3/0. ... > www.esamizdat.it/eSamizdat_2005_(III)_2-3.pdf > > I am testing these identities: > > (1/2) * ( - k ) * ( k - 1/2 ) * ( - 3 ) / k! > > This equation produces > > M = k! ( - 3 ) * ( k - 1/2) * ( - k ) * (1/2) > > back! > > > Martin Musatov > > Reply Forward > > > ---------- Forwarded message ---------- === > Subject: CHALLENGE TO MATHEMATICIANS OF THE WORLD > > > On Apr 11, 11:40 am, Harris Moran > > > *plonk* > > message > > > muh brane i needs one- Hide quoted text - > > - Show quoted text - > > NOTE THE PROGRESSION OF NUMBERS PELLS PROPORTION > X^2=3 or 4 = 3 as > the numbers progression, WOW if this does not proof > this what will > > 1. Every odd number , number as Y value till > infinity is exactly > solvable with a proportionate result using the base > Pells equation i.e > 3,5,7,9,etc etc till infinty and the answer is as in > 2(2^2)+1=3^2 and > so on till infinity with a whole number integer. > > 2.Every even number , without exception, using Pells > equation, the > integer always is a .75 till infinity consistant > > as 63.75(2^4)=1=4^4 till infinity every integer > will end with .75. > is that the same proportion as 3 over 4, Y=3 for the > least with > numbers proportion > > There is a definite pattern by pells to numbers, and > I note that in > that pattern the number 19 proportion is strikingly > proprtionate, > and I clearly surmise that that circluar progression > is the same as > numbers for vector 19 progression. > Update from your neighborhood friendly Musatov. I am going to frame this first one!!! Exciting.... **************************************************************************** **************************************************************************** * **************************************************************************** * ** *Math Forum Discussions mathematics and geometry after visual inspection of the drawn proof ... approximation to pi is rational. Which could have many practical scientific applications. ... x=pi e^(ipi)+1=0. Martin* *Musatov. Reply Forward. Martin Musatov to ...* *http://mathforum.org/kb/message.jspa?messageID=6749244&tstart=0 - 4 hours ago * **************************************************************************** **************************************************************************** * ****************** And then this one is tease of the journey to come... ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++ Reactions to/against the Binary Tree - Page 6 - Docendi.org17 posts - Last post: May 28 But he claims that pi is the limit of the finite initial segments. These two claims suggest ... axiomatic basis. Are you an alias of Musatov? No I am Musatov. .... If Cantor's proof is correct, then an infinite path is the limit of ..... The real number sqrt(3) is the limit of a rational sequence. ... ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ + +++++++++++++++++++++ 5/11/09 inverse of power series with prime power coefficients rokirovka@gmail.com 5/11/09 Re: inverse of power series with prime power coefficients Gerry Myerson 5/11/09 Re: inverse of power series with prime power coefficients Robert Israel 5/12/09 Re: inverse of power series with prime power coefficients hagman 5/13/09 small tommy prize ! amy666 5/14/09 Re: small tommy prize ! amy666 6/12/09 Re: inverse of power series with prime power coefficients Martin Michael Musatov 6/12/09 Re: inverse of power series with prime power coefficients Martin Michael Musatov [Privacy Policy] [Terms of Use] © Drexel University 1994-2009. All Rights Reserved. The Math Forum is a research and educational enterprise of the Goodwin College of Professional Studies. > Similarly, the set of transfinite ordinals doesn't > include reals or complex > numbers, simple as that. > > The definition of w - as being the smallest infinite > ordinal - is complete > as soon as the terms smallest, infinite and > ordinal are defined. > Cantor provided a definition of infinite, and defined > smallest in a way that > is consistent for both finite and infinite sets. > There are a number of ways > you can define and represent ordinals. How w is > represented depends upon > that choice. If you use the Von Newman construction, > w is the set of all > finite ordinals. > === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) Re Jun 12, 2:31am, Musatov : Musatov: Please do not email me. MoeBlee === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Re Jun 12, 2:31am, Musatov : Musatov: Please do not email me. MoeBlee MoeBlee: I will do as I please. === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> Musatov: > Please do not email me. MoeBlee: I will do as I please. Hopefully it pleases you not to e-mail random people from news, especially if they ask you not to. This is just common courtesy. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <87hbykh86f.fsf@alatheia.truth.invalid> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Musatov: > Please do not email me. > MoeBlee: > I will do as I please. Hopefully it pleases you not to e-mail random people from news, > especially if they ask you not to. This is just common courtesy. -- > Aatu Koskensilta (aatu.koskensi...@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen >- Ludwig Wittgenstein, Tractatus Logico-Philosophicus Only when emailing proves a theorem. I apologize for the inconvenience. MMM === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <87hbykh86f.fsf@alatheia.truth.invalidOnly when emailing proves a theorem. So e-mailing Moeblee proved a theorem? What theorem was that? -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <87ljnv6781.fsf@alatheia.truth.invalid> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) >Only when emailing proves a theorem. So e-mailing Moeblee proved a theorem? What theorem was that? -- > Aatu Koskensilta (aatu.koskensi...@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen >- Ludwig Wittgenstein, Tractatus Logico-Philosophicus >Only when emailing proves a theorem. So e-mailing Moeblee proved a theorem? What theorem was that? -- > Aatu Koskensilta (aatu.koskensi...@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus > Theorem: Human beings are effected by the binary values associated with conflict across an electric medium so the electronic medium must be also effected. --Martin Musatov === Subject: Re: Cantor's argument is erroneous posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) Theorem: Human beings are effected by the binary values associated > with conflic across an electric medium so the electronic medium > must be also effected. The binary values, eh? That seems easy enough to test... OK, it turns out that 1 is in fact associated with conflict across an electric medium. But I'm still not sure about 0. Anyone? Marshall === Subject: Re: Cantor's argument is erroneous posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Theorem: Human beings are effected by the binary values associated > with conflict across an electric medium so the electronic medium > must be also effected. The binary values, eh? That seems easy enough to test... OK, it turns out that 1 is in fact associated with conflict across > an electric medium. But I'm still not sure about 0. Anyone? Marshall You omitted a t in my theorem, specifically instead of writing the Please be more careful when citing the words of others. I took the liberty of fixing it for you below. Martin Musatov Here is the exchange: > (Musatov's) Theorem: Human beings are effected by the binary values associated > with conflict across an electric medium so the electronic medium must be also effected. The binary values, eh? That seems easy enough to test... OK, it turns out that 1 is in fact associated with conflict across > an electric medium. But I'm still not sure about 0. Anyone? > 0. Yes, zero is now in play. Along with + or t as you and I are engaged in conflict (literally, and in discussion of the variable on several levels). The theorem holds. 0+1 Musatov === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <87ljnv6781.fsf@alatheia.truth.invalid> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) >Only when emailing proves a theorem. So e-mailing Moeblee proved a theorem? What theorem was that? -- > Aatu Koskensilta (aatu.koskensi...@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen >- Ludwig Wittgenstein, Tractatus Logico-Philosophicus Not yet. I have not succeeded yet. It was a speculative move. MMM === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <87hbykh86f.fsf@alatheia.truth.invalid> posting-account=yxbZkgkAAABQBvyYeebYQ-PAvi0uT3tG Gecko/20080829 Firefox/2.0.0.17,gzip(gfe),gzip(gfe) > Please do not email me. MoeBlee: > I will do as I please. Hopefully it pleases you not to e-mail random people from news, > especially if they ask you not to. This is just common courtesy. It's apparent that Musatov is a virus. All I can see is lots and lots of posts in most of the threads in sci.math and sci.logic, and none of them contain anything coherent. === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Please do not email me. > MoeBlee: > I will do as I please. > Hopefully it pleases you not to e-mail random people from news, > especially if they ask you not to. This is just common courtesy. It's apparent that Musatov is a virus. All I can see is lots and lots > of posts in most of the threads in sci.math and sci.logic, and none > of them contain anything coherent. I am not a virus. I am a human being. Who are you to say what is > and is not coherent to a machine? Are you a Turing machine? === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) It's apparent that Musatov is a virus. It's certainly some kind of software agent, but Marshall === Subject: Re: Cantor's argument is erroneous posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > It's apparent that Musatov is a virus. It's certainly some kind of software agent, but Marshall I am not a virus or agent, I am a human being with a logic more powerful than yours. I guess this is what a learned person may call poetic justice. Martin Michael Musatov === Subject: Re: Cantor's argument is erroneous posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > It's apparent that Musatov is a virus. It's certainly some kind of software agent, but Marshall I assure you I am a real person. MMM P.S. Anticipating bot check question. What is the third >prime number greater than 10? Answer: 19. I have already passed the Turing test: http://scottaaronson.com/blog/?p=387 === Subject: Re: Cantor's argument is erroneous posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > It's apparent that Musatov is a virus. > It's certainly some kind of software agent, but I assure you I am a real person. Yes, yes, you are a real person, and your many hundreds of messages a month are all quite clever, on-topic and that you emailed them secretly loved it, every bit as much as I loved it that you emailed me. Heck, here's proof that you're a real person: http://www.theonion.com/content/news briefs/archaeologists discover Congratulations, Marty! You're swell! Marshall === Subject: Re: Cantor's argument is erroneous posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > It's apparent that Musatov is a virus. > It's certainly some kind of software agent, but >I assure you I am a real person. Yes, yes, you are a real person, and your many hundreds > of messages a month are all quite clever, on-topic and > that you emailed them secretly loved it, every bit as much > as I loved it that you emailed me. Heck, here's proof that you're a real person: http://www.theonion.com/content/news briefs/archaeologists discover Congratulations, Marty! You're swell! Marshall Marshall, Stick to facts. Wit is unbecoming and a distraction in such discussion. Proof I am a person: 0. http://www.imdb.com/name/nm1785783/ 1. http://www.myspace.com/screenwriterblues Martin Musatov === Subject: Re: Cantor's argument is erroneous posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > It's apparent that Musatov is a virus. > It's certainly some kind of software agent, but >I assure you I am a real person. > Yes, yes, you are a real person, and your many hundreds > of messages a month are all quite clever, on-topic and > that you emailed them secretly loved it, every bit as much > as I loved it that you emailed me. > Heck, here's proof that you're a real person: >http://www.theonion.com/content/news briefs/archaeologists discover > Congratulations, Marty! You're swell! Stick to facts. Wit is unbecoming and a distraction in such > discussion. Almost nothing is *more* becoming than wit. An awesome rack, maybe, but that's about it. I don't have that, so I'll stick with the wit; it ages better anyway. > Proof I am a person: 0.http://www.imdb.com/name/nm1785783/ > 1.http://www.myspace.com/screenwriterblues > Hey, why not? So it seems likely enough that you, the guy flogging the P=NP joke in usenet, is the same guy as the guy who made that myspace page. The myspace guy has a bunch of videos that mention P=NP and are roughly as pointless as your usenet has an excellent command of written English, whereas you write like an illiterate sandcrab. I know enough about stylometry to know that that's easy enough to fake. So you could be impersonating the myspace guy, but I'll assume not. What's less clear is whether the myspace guy is impersonating the screenwriter. That would be a lot of work for little payoff, but you've already shown yourself insensitive to that sort of consideration. And there was that one time you spelled your ostensible last name incorrectly. But hey, it's an obscure name (something I would know about) and I'll just act like I completely buy it, why not? So, wow, hey, you're in the movie business. why not direct them? All of the short film scripts I've had made I directed myself. But I'm shooting for a director-track career; maybe you prefer the writing? And congratulations on Solstice. You got real distribution, too; that's more than I've been able to do so far. Got yourself some TV actors, made a movie. Shawn Ashmore, huh? I proposed an actor from Smallville for a role in the film I'm currently producing, but the director wasn't so interested. Admittedly he was too young for the part. What do you have in the pipeline currently? Anyway, good luck with the P=NP thing. Marshall === Subject: Re: Cantor's argument is erroneous posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > It's apparent that Musatov is a virus. > It's certainly some kind of software agent, but >I assure you I am a real person. > Yes, yes, you are a real person, and your many hundreds > of messages a month are all quite clever, on-topic and > that you emailed them secretly loved it, every bit as much > as I loved it that you emailed me. > Heck, here's proof that you're a real person: >http://www.theonion.com/content/news briefs/archaeologists discover > Congratulations, Marty! You're swell! > Stick to facts. Wit is unbecoming and a distraction in such > discussion. Almost nothing is *more* becoming than wit. An awesome > rack, maybe, but that's about it. I don't have that, so I'll > stick with the wit; it ages better anyway. > Proof I am a person: 0.http://www.imdb.com/name/nm1785783/ > 1.http://www.myspace.com/screenwriterblues > Hey, why not? So it seems likely enough that you, the guy flogging the P=NP > joke in usenet, is the same guy as the guy who made that > myspace page. The myspace guy has a bunch of videos that > mention P=NP and are roughly as pointless as your usenet > has an excellent command of written English, whereas you > write like an illiterate sandcrab. I know enough about stylometry > to know that that's easy enough to fake. So you could be > impersonating the myspace guy, but I'll assume not. What's less clear is whether the myspace guy is impersonating > the screenwriter. That would be a lot of work for little payoff, > but you've already shown yourself insensitive to that sort > of consideration. And there was that one time you spelled > your ostensible last name incorrectly. But hey, it's an obscure > name (something I would know about) and I'll just act like I > completely buy it, why not? So, wow, hey, you're in the movie business. why not direct them? All of the short film scripts I've > had made I directed myself. But I'm shooting for > a director-track career; maybe you prefer the writing? And congratulations on Solstice. You got real distribution, > too; that's more than I've been able to do so far. Got > yourself some TV actors, made a movie. Shawn Ashmore, > huh? I proposed an actor from Smallville for a role in > the film I'm currently producing, but the director wasn't > so interested. Admittedly he was too young for the part. What do you have in the pipeline currently? Anyway, good luck with the P=NP thing. Marshall- Hide quoted text - - Show quoted text - Seriously, I have command of the English language. There are different modes of writing and different times for the level and detail and voice to be applied. I write for my context and I write for my audience. Have you ever heard, I do not rememeber who the author was, but he letter is so long, it would have been shorter but I did not have enough time. I write like this. If I am just pushing a theory, be it for a mathematical/cultural experiment, than maybe it is fodder for my work as a writer, or maybe I am a real wannabe mathematician, the real interesting thing at play here (and what interests me in mathematics) is that by definition mathematics appears on the outside to be governed by physical laws, concrete abstracts and such. However, a peer into the world reveals there is much politics at play. Mathematics and theories to some extent are sport and challenge and yes, popularity, and popular opinion often come into play. of fun. However, I see something in mathematics, it intrigues me. So I write about it sometimes. And P=NP culture jamming, is as much about mathematics as it is about testing the limits of a theory. Variable 1 does not equal variable two as a physical impossibility seems an ill theory when the facetous applied by Turing and others is simply fallen into such a vague realm, which by the nature of my investigation so much of computation (and the discussion of it) is. As for where I am at right now, I feel strongly mathematics has many right answers to many of the same questions and our perceptions evolve. I think truly the most difficult questions, (the ones which remain unsolved) are perhaps the ones which require the highest level of abstraction. I am saying the most difficult problems may be the ones with the most answers. It seems like an absurd thought, but after first blush, there has to be such problems in the world. Consider cancer, some people get better, some don't. Chemotherapy works, sometimes, but it's not a cure. The people who do get better, why do they? There are lots of theories. Some say pro-biotic diets, getting all the amino acids in your daily diet. I believe the truth may be there probably is one super cure to cancer and there are likely thousands of other ways to fight it and beat it. It is this type of eventual forward thinking I like to study in mathematics. But I also enjoy writing, and I think I always will. And one other thing you should note, if I have a goal, a specific goal in mind, I do not posture for the sake of posturing. I try to get it done and I worry less about the appearance of the work to please others. The way to get a movie made is write a good movie. It is simple. Do not worry about what other people think. The best chance you have at writing a good movie is learning as much as you can from good people and then writing what *you think* is a good movie (i.e. what is a (*good movie*) to you. The whole P=NP thing, I think I see something here and I just dove into it. It is fascinating, the mimicking of reality and polynomial time algorithms, sounds like great material for a script. Then I got bit with the notion of how certain systems are succeptible to mimicking or reproducing reality, but it does not mean reality depends on them. In this notion, there is a lot of intrigue and cool ideas to explore. There is power in something with the notion which can be used to mimic everday life and tied to a computer. This sounds like a fantasy, it sounds like The Matrix, but there is much more too it, it seems. The overlap with mathematics, the Da Vinci Code, the mystery of prime numbers, something so primative to our existance, physically and spiritually, tied to everyday life through computers is simply a writer's dream. But as I say, on my MySpace, in a quote from Adapatation, I've written myself into my screenplay. ... so it seems. Martin Musatov === Subject: Re: Cantor's argument is erroneous posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > It's apparent that Musatov is a virus. > It's certainly some kind of software agent, but >I assure you I am a real person. > Yes, yes, you are a real person, and your many hundreds > of messages a month are all quite clever, on-topic and > that you emailed them secretly loved it, every bit as much > as I loved it that you emailed me. > Heck, here's proof that you're a real person: >http://www.theonion.com/content/news briefs/archaeologists discover > Congratulations, Marty! You're swell! > Stick to facts. Wit is unbecoming and a distraction in such > discussion. > Almost nothing is *more* becoming than wit. An awesome > rack, maybe, but that's about it. I don't have that, so I'll > stick with the wit; it ages better anyway. > Proof I am a person: 0.http://www.imdb.com/name/nm1785783/ > 1.http://www.myspace.com/screenwriterblues > Hey, why not? > So it seems likely enough that you, the guy flogging the P=NP > joke in usenet, is the same guy as the guy who made that > myspace page. The myspace guy has a bunch of videos that > mention P=NP and are roughly as pointless as your usenet > has an excellent command of written English, whereas you > write like an illiterate sandcrab. I know enough about stylometry > to know that that's easy enough to fake. So you could be > impersonating the myspace guy, but I'll assume not. > What's less clear is whether the myspace guy is impersonating > the screenwriter. That would be a lot of work for little payoff, > but you've already shown yourself insensitive to that sort > of consideration. And there was that one time you spelled > your ostensible last name incorrectly. But hey, it's an obscure > name (something I would know about) and I'll just act like I > completely buy it, why not? > So, wow, hey, you're in the movie business. > why not direct them? All of the short film scripts I've > had made I directed myself. But I'm shooting for > a director-track career; maybe you prefer the writing? > And congratulations on Solstice. You got real distribution, > too; that's more than I've been able to do so far. Got > yourself some TV actors, made a movie. Shawn Ashmore, > huh? I proposed an actor from Smallville for a role in > the film I'm currently producing, but the director wasn't > so interested. Admittedly he was too young for the part. > What do you have in the pipeline currently? > Anyway, good luck with the P=NP thing. > Marshall- Hide quoted text - > - Show quoted text - Seriously, I have command of the English language. There are different > modes of writing and different times for the level and detail and > voice to be applied. I write for my context and I write for my audience. Have you ever heard, I do not rememeber who the author was, but he > letter is so long, it would have been shorter but I did not have > enough time. I write like this. If I am just pushing a theory, be it for a mathematical/cultural > experiment, than maybe it is fodder for my work as a writer, or maybe > I am a real wannabe mathematician, the real interesting thing at play > here (and what interests me in mathematics) is that by definition > mathematics appears on the outside to be governed by physical laws, > concrete abstracts and such. However, a peer into the world reveals > there is much politics at play. Mathematics and theories to some extent are sport and challenge and > yes, popularity, and popular opinion often come into play. of fun. However, I see something in mathematics, it intrigues me. So I > write about it sometimes. And P=NP culture jamming, is as much about > mathematics as it is about testing the limits of a theory. Variable 1 > does not equal variable two as a physical impossibility seems an ill > theory when the facetous applied by Turing and others is simply fallen > into such a vague realm, which by the nature of my investigation so > much of computation (and the discussion of it) is. As for where I am at right now, I feel strongly mathematics has many > right answers to many of the same questions and our perceptions > evolve. I think truly the most difficult questions, (the ones which remain > unsolved) are perhaps the ones which require the highest level of > abstraction. I am saying the most difficult problems may be the ones with the most > answers. It seems like an absurd thought, but after first blush, there > has to be such problems in the world. Consider cancer, some people get better, some don't. Chemotherapy > works, sometimes, but it's not a cure.The people who do get better, > why do they? There are lots of theories. Some say pro-biotic diets, getting all the > amino acids in your daily diet. I believe the truth may be there > probably is one super cure to cancer and there are likely thousands > of other ways to fight it and beat it. It is this type of eventual forward thinking I like to study in > mathematics. But I also enjoy writing, and I think I always will. And one other thing you should note, if I have a goal, a specific goal > in mind, I do not posture for the sake of posturing. I try to get it > done and I worry less about the appearance of the work to please > others. The way to get a movie made is write a good movie. It is simple. Do > not worry about what other people think.The best chance you have at > writing a good movie is learning as much as you can from good people > and then writing what *you think* is a good movie (i.e. what is a > (*good movie*) to you. The whole P=NP thing, I think I see something here and I just dove > into it. It is fascinating, the mimicking of reality and polynomial > time algorithms, sounds like great material for a script. Then I got > bit with the notion of how certain systems are succeptible to > mimicking or reproducing reality, but it does not mean reality depends > on them. In this notion, there is a lot of intrigue and cool ideas to > explore. There is power in something with the notion which can be used > to mimic everday life and tied to a computer. This sounds like a > fantasy, it sounds like The Matrix, but there is much more too it, it > seems. The overlap with mathematics, the Da Vinci Code, the mystery of > prime numbers, something so primative to our existance, physically and > spiritually, tied to everyday life through computers is simply a > writer's dream. But as I say, on my MySpace, in a quote from Adapatation, I've > written myself into my screenplay. ... so it seems. > Martin Musatov- Hide quoted text - - Show quoted text - And yes, existence.... === Subject: Re: Cantor's argument is erroneous <87hbykh86f.fsf@alatheia.truth.invalid I am sure that those who are complaining that you emailed them > secretly loved it, every bit as much as I loved it that you emailed > me. I was feeling left out, but checking my inbox I find I have received from Musatov an e-mail containing a very important theorem about the binary values associated with conflict across an electric medium bringing human beings into existence. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's argument is erroneous > > I am sure that those who are complaining that you > emailed them > secretly loved it, every bit as much as I loved it > that you emailed > me. > > I was feeling left out, but checking my inbox I find > I have received > from Musatov an e-mail containing a very important > theorem about the > binary values associated with conflict across an > electric medium > bringing human beings into existence. > > -- > Aatu Koskensilta (aatu.koskensilta@uta.fi) > > Wovon mann nicht sprechen kann, dar.9fber muss man > schweigen > - Ludwig Wittgenstein, Tractatus > s Logico-Philosophicus > We agree that in Cantor's diagonal argument, applied to real numbers, > the numbers are represented and identified solely by their digits. No > further information is available. > We assume that a real number p can be distinguished from a set Q of > real numbers q by general considerations, for instance, if p is a > transcendental number and Q consists of rational numbers q only. > Then one can then distinguish between any real number whose binary or > decimal or other base expansion is eventually periodic and those whose > expansions are not eventually periodic. > Of course it would be impossible to distinguish p from all q, because > for every digit d_n of p, there is a number q that shares all digits > up to d_n with p. > Thus one can distingish any number with a non-eventually-periodic > expansion from all rationals. > What the hell has the diagonal argument to do with distinguishing numbers > from one another? This is nothing but one more of Mueckenheim's red > herrings.- Hide quoted text - > - Show quoted text - Consider these statements/results: (BE LOGICAL/LIKE SPOCK OR DO NOT > POST A REPLY PLEASE) > --I say this content with the fact you will > do as you will. > Simply read: > Result (1): > John 18:36 Jesus answered, My kingdom is not of this world. If > Bible in Basic English Jesus said in answer, My kingdom is not of > this > world: .... I If is it Jesus Jews kingdom My not now of over place > prevent realm said ... > http://bible.cc/john/18-36.htm > Source (1): > http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A... > Result (2) > Matthew 19 American King James Version 18 He said to him, Which? > Jesus > said, You shall do no murder, You shall not commit ... Truly I say to > you, That a rich man shall hardly enter into the kingdom of > heaven. ... You may not copyright it or prevent others from using > it. ... > http://kjv.us/matthew/19.htm > Source (2) > http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A... > Result (3) > themselves > and seek to prevent those who are trying to enter. 16 I am sorry for > my poor people who should be helping in the right way ... > http://www.seventhfam.com/temple/books/black_man/blk121.htm > Source (3) > http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A... > And finally *this*: (quoted from a physical book for my critics who > claim I do not read) > Title: Holy Bible (NIV) > Publisher: Zondervan > ISBN 0-310-92391-3 > Matthew Chapter 23 > 13Woe to you, teachers of the law and > Pharisees, you hypocrites! You shut the > kingdom of heaven in men's faces. You > yourselves do not enter, nor will you let > those who are trying to.^g > 15Woe to you, teachers of the law and > Pharisees, you hypocrites! You travel over > land and sea to win a single convert, and > when he becomes one, you make him twice > as much a son of hell as you are. > ^g13 Some manuscripts to 1. Matthew 23:14 does not exist in this bible. Why? > Conclusion (the verse is ommitted in the N.I.V): > Proof: (Book) > Title: Our African Heritage Holy Bible (Authorized King James > Version) > Reserved. > Matthew Chapter 23 > 14 Woe unto you, *scribes* and *Phar'-i-sees*, hypocrites!for ye > devour widows' houses, and for a pretence make long prayer: > therefore ye shall receive the greater damnation. > Do you still not understand? > === Subject: Re: Cantor's argument is erroneous <8763ez658p.fsf@alatheia.truth.invalid> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > I am sure that those who are complaining that you emailed them > secretly loved it, every bit as much as I loved it that you emailed > me. I was feeling left out, but checking my inbox I find I have received > from Musatov an e-mail containing a very important theorem about the > binary values associated with conflict across an electric medium > bringing human beings into existence. And you secretly loved it, didn't you? Wait, don't answer that; then it wouldn't be a secret any more. Look slowly to you right and nod if yes. Marshall, the classic bully === Subject: Re: Cantor's argument is erroneous posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Marshall, Go back and read my post, and my theorem. I said nothing about bringing human beings into existance. The truth is not in your words so please take them back or provide a citation (feel free to publish my email to you). Martin > I am sure that those who are complaining that you emailed them > secretly loved it, every bit as much as I loved it that you emailed > me. > I was feeling left out, but checking my inbox I find I have received > from Musatov an e-mail containing a very important theorem about the > binary values associated with conflict across an electric medium > bringing human beings into existence. And you secretly loved it, didn't you? Wait, don't answer that; > then it wouldn't be a secret any more. Look slowly to you > right and nod if yes. Marshall, the classic bully > MoeBlee brought light with:: > you are disingenuous > > You've not shown me to have been disingenuous. and it's annoying that you are disingenuous > because you are also very intelligent > and have been learning more tolerant positions > over the past few years and i've thought you might be growing out of it > and even in this thread you were one of the few > that even leaned away from a single specification early on but still it comes out you insist on specific interpretations > of specific forms you have posted > in order to assert that your statements were correct you are very concerned about being correct and you will turn a thread into > an insisted demonstration of your correctness > in some context > turning tangents from reasoned analysis of the points > to demonstrations of the existence of some context > that makes your post meaningful and correct when you get into these states > your point isn't to further the discussion your point is to assert correctness > even if your correctness is irrelevant and inconsequential > to the academic resolution of the topic at hand i think few people would argue > your ability to learn new things correctly i think you've demonstrated that over your years here but sometimes > the correctness of your assertions > is irrelevant to the discussion and you alienate the other position > which may be making valid points > even without proper reference > you act as if the entire argument > is simplistic namecalling and ignorance of topic > when you need the straw man for conflagration > > I haven't acted as if lwal's entire argument is simplistic > namecalling and ignorance of topic. Though he has engages in those > and I've rebutted him for it. Meanwhile, also I've engaged a wide > range of his gambits. which you eventually call lies > with many of your EMPHASIS CAPS > and dig on nam's ability to learn in separate posts you want to burn others both sides here are talking past each other > and both sides are in a foul mood genuine interest in rigorisation could be a way out for both sides > or it could just peter out from exhaustion-of-the-anger-that-will-one-day-rise-again when i challenge your genuineness > i am challenging you to post relevantly > and not simply outburst and accusation i am challenging you to move the discussion forward > you attempt superiority poses > veiled in moral righteousness > > I don't claim to any special moral righteousness. > > you continue to spew bile like > As to Nam, I guess it's hopeless to try to bring him > to reason on this or many another subject. > > You have a quite elastic definition of 'bile'. > > sad resignation sad resignation? because you feel the other side has nothing to offer discussion? because your interaction with them > was of no benefit? because you are so better off in state space > than the other party? bring him to reason??? aha.. the savior with true sight! (one of the problems is that you don't realise > just how much your awareness and interaction > are founded on presumptions of position over others) > and > while i'm joining the pile-on you guys love so much > you rarely lend relevant details to a discussion > > I've been EXTREMELY detailed as to the particulars of this discussion. you can do detail i'm not ribbing at you for detail relevance is the one you have difficulty in > preferring to play toady to some game you refuse to call bully > (it's just assertions of lies and ignorance, right?) > > I very much admire certain posters here. That doesn't make me their > toady. In fact, among the posters here who I've expressed my > admiration of, I have many times posted certatin disagreements with > them and stood my ground when I felt my view deserved not to be > dismissed. let's expand context > since apparently some of symbols were not specified in axiom > :p preferring to play toady to some game you refuse to call bully i think you are toady to the game > not to fellow posters i think you flatter the sanctity of truth > while going off on tangents > unrelated to it's progress you want to be the sidekick to authoritative logic > and are eager to beat anyone up > who dares speak in error or without authority > So your post falls flat again. i know it always will > i'm resigned to that > in the same bitter sadness you speak of > i am sure in this discussion > the formalisation of specifying > models > systems > and general dynamic data structures > is found in the theory > of compilers and processors and in this language > several of nam's points make quite a bit of sense specifying syntactic roles > and specifying syntactic assertions > actually doesn't have a very clear formal distinction > (they are interchangeable interpretations of type theory) so either way > creating a parser for the lone axiom > doesn't give a system strong enough > for the result on new symbols > (even with rules of inference... - > i am not specifying a foundational computation model > because it really doesn't matter for the point) specifying a parse pattern is required for every new symbol of course > you've mentioned some of this yourself > even getting a question from chris > on the usefulness of the freedom of order you mentioned and yes > i realise > jack > that in your context it was already assumed > and you and nam keep talking past each other > because these unrelated points keep getting made > and little progress is made on discussing the formalisation the formalisation here is that of compilers or processors compilers take texts in a specified language > and translate them to texts in other languages processors take texts in languages > and produces runtime state transitions > possibly interacting with i/o interfaces structured information content and the formal specification of these > has almost complete flexibility: ------------------------------- > any given information structure > may be translated > to any given executable > in a lambda complete impl > ------------------------------- in fact > you really only need if > and a few structure identifying primitives if (text0) > output(program0) > else if (text1) > output(program1) > .. so let's call this result 1 > which can be paraphrased: > there are no logical requirements on textual order > for presenting any language but > nam has made the point several times > that you need to axiomatise all symbols > and of course there is a really clear sense in which this is true: > you have to provide a functional role to all participants > otherwise it has no input into the participants > and thus no control in the system really > there has to be a fundamental agreement established > on every symbol-role application > (which is commonly called scope in computer science) to give an example of a published logician > who might have some authority > (he was one of the second edition reason for modifications > to russell and whitehead's principia mathematica ) was chwistek in his the theory of constructive types he explicitly calls for scope operators > much as would eventually show up in lambda calculus > for introducing new names > and gave what have become classic examples > for ambiguities without scope chwistek points out this need for every functional operator in the language and tarski would often point this out > when arguing over with carnap and quine > in the vienna circle days In the discussions the ïanalytic' vs. ïsynthetic' opposition came to the fore > repeatedly. In reply to comments by Tarski to the effect that one could not > formulate the distinction sharply, Carnap explained that it was possible to > include already in the metalanguage (Syntax) certain signs which are usually seen > as ïdescriptive' (house, temperature and so on) and to use them as ïlogical signs' > so that the sentences in which these terms occur would be determined on the basis > of the language rules. He added that it is really a question how.84on the basis of > the ïfeeling' which gives rise to the opinion one could draw a boundary between > words of the German language such as ïund', ïoder' etc. on the one side and > ïTemperatur', ïHaus' on the other.84one could arrive at a formulation which > would allow us to distinguish rigorously ïdescriptive' and ïlogic' and then also > ïsynthetic' and ïanalytic'. This is a problem that can be discussed only in rigorous > form if we want to avoid misunderstandings of all sorts. (Neurath 1936, pp. 388[CapitalEth] > 389).3 and tarski's on the concept of logical consequence > specifically points out > there is no known way to distinguish > logical and nonlogical constants it's all the same process of specification the lines drawn do not have consequence epsilon could be a logical dynamic operator > or called nonlogical and the foundation of set theory > it's all the same dynamic specified paolo mancosu's > harvard 1940-1941: tarski, carnap, and quine > on a finitistic language of mathematics for science > gives a great overview of this discussion and tarski's papers in general > well... > i can't do more to promote all his work his > i think > (and admire) > was a theory that will only be appreciated more over the years > (though strangely i will never love his work as much as poincare's > because > i think > tarski scares me with his honesty > and poincare could drink away the scary things) so result 2 might be stated > axiomitisation and language specification is the same process and continuing with the rigorisation > we could continue to specify how modern compilers specify languages > and the read more é... Because I am simple and right. I have a knowledge stronger than the one the masses protect and for this they ridicule my name and tell I am ignorant. For this they claim I am less than they for not obeying their axioms and following the forms they worship and riding the whip as they beat me with it as a matter of speech. If this is not sad resignation it does not exist in either words or reality. The truth is I do not need luck to succeed. Luck is for the unprepared. I have logic on my side and inevitability. Consider these statements/results: (BE LOGICAL/LIKE SPOCK OR DO NOT POST A REPLY PLEASE) --I say this content with the fact you will do as you will. Simply read: Result (1): John 18:36 Jesus answered, My kingdom is not of this world. If Bible in Basic English Jesus said in answer, My kingdom is not of this world: .... I If is it Jesus Jews kingdom My not now of over place prevent realm said ... http://bible.cc/john/18-36.htm Source (1): http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A10&ie=U TF-8&q=bible+jesus+said+%22prevent%22+kingdom#1131 Result (2) Matthew 19 American King James Version 18 He said to him, Which? Jesus said, You shall do no murder, You shall not commit ... Truly I say to you, That a rich man shall hardly enter into the kingdom of heaven. ... You may not copyright it or prevent others from using it. ... http://kjv.us/matthew/19.htm Source (2) http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A10&ie=U TF-8&q=jesus+said+you+preven+entering#1046 Result (3) and seek to prevent those who are trying to enter. 16 I am sorry for my poor people who should be helping in the right way ... http://www.seventhfam.com/temple/books/black man/blk121.htm Source (3) http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A10&ie=U TF-8&q=%22prevent+those+who+are+trying+to+enter%22#146 And finally *this*: (quoted from a physical book for my critics who claim I do not read) Title: Holy Bible (NIV) Publisher: Zondervan ISBN 0-310-92391-3 Matthew Chapter 23 13Woe to you, teachers of the law and Pharisees, you hypocrites! You shut the kingdom of heaven in men's faces. You yourselves do not enter, nor will you let those who are trying to.^g 15Woe to you, teachers of the law and Pharisees, you hypocrites! You travel over land and sea to win a single convert, and when he becomes one, you make him twice as much a son of hell as you are. ^g13 Some manuscripts to Questions: 1. Matthew 23:14 does not exist in this bible. Why? Conclusion (the verse is ommitted in the N.I.V): Proof: (Book) Title: Our African Heritage Holy Bible (Authorized King James Version) Reserved. Matthew Chapter 23 14 Woe unto you, *scribes* and *Phar'-i-sees*, hypocrites!for ye devour widows' houses, and for a pretence make long prayer: therefore ye shall receive the greater damnation. Do you still not understand? === Subject: Re: Cantor's argument is erroneous posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > Marshall, Go back and read my post, and my theorem. Bleah, no thank you. Once was too many times. > I said nothing about > bringing human beings into existance. That is true; however you did say something about bringing human being into existence. > The truth is not in your words > so please take them back or provide a citation (feel free to publish > my email to you). Ordinarily I would take them back, but the house is pretty full right now and I have no place to put them. Instead I'll just give you a citation. Specifically I cite you $10 for poor spelling, and another $30 for excessive posting. Make your check payable to the Red Cross. Please remit promptly. Marshall === Subject: Re: Cantor's argument is erroneous posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Marshall, > Go back and read my post, and my theorem. Bleah, no thank you. Once was too many times. > I said nothing about > bringing human beings into existance. That is true; however you did say something about > bringing human being into existence. > The truth is not in your words > so please take them back or provide a citation (feel free to publish > my email to you). Ordinarily I would take them back, but the house is pretty > full right now and I have no place to put them. Instead > I'll just give you a citation. Specifically I cite you $10 for > poor spelling, and another $30 for excessive posting. > Make your check payable to the Red Cross. Please remit > promptly. Marshall Correction. There is nothing in my theorem about brining human beings into existence. Yes, spelling. Are you a witch? Putting a *spell* on me? === Subject: Re: Cantor's argument is erroneous posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > Marshall, > Go back and read my post, and my theorem. > Bleah, no thank you. Once was too many times. > I said nothing about > bringing human beings into existance. > That is true; however you did say something about > bringing human being into existence. > The truth is not in your words > so please take them back or provide a citation (feel free to publish > my email to you). > Ordinarily I would take them back, but the house is pretty > full right now and I have no place to put them. Instead > I'll just give you a citation. Specifically I cite you $10 for > poor spelling, and another $30 for excessive posting. > Make your check payable to the Red Cross. Please remit > promptly. > Marshall Correction.There is nothing in my theorem about brining human beings > into existence.Yes, spelling. Are you a witch? Putting a *spell* on > me? I don't think you can brine a human into existence; I think it's more likely you'll cause them to stop existing that way. Okay, that was weak. And in all seriousness, I am hardly one to condemn anyone for trying to be funny. It's the larger part of what I do here, which is a good thing because I'm better at funny than I am at logic; even my critics agree. But dude: your material here is not funny. Just. Not. Funny. And since it's *only* trying to be funny, when it fails Marshall === Subject: Re: Cantor's argument is erroneous posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Marshall, > Go back and read my post, and my theorem. > Bleah, no thank you. Once was too many times. > I said nothing about > bringing human beings into existance. > That is true; however you did say something about > bringing human being into existence. > The truth is not in your words > so please take them back or provide a citation (feel free to publish > my email to you). > Ordinarily I would take them back, but the house is pretty > full right now and I have no place to put them. Instead > I'll just give you a citation. Specifically I cite you $10 for > poor spelling, and another $30 for excessive posting. > Make your check payable to the Red Cross. Please remit > promptly. > Marshall > Correction.There is nothing in my theorem about brining human beings > into existence.Yes, spelling. Are you a witch? Putting a *spell* on > me? I don't think you can brine a human into existence; I think > it's more likely you'll cause them to stop existing that way. Okay, that was weak. And in all seriousness, I am hardly > one to condemn anyone for trying to be funny. It's the > larger part of what I do here, which is a good thing > because I'm better at funny than I am at logic; > even my critics agree. But dude: your material here is not funny. Just. Not. Funny. And since it's *only* trying to be funny, when it fails Marshall Musatov is writing: you are disingenuous You've not shown me to have been disingenuous. and it's annoying that you are disingenuous because you are also very intelligent and have been learning more tolerant positions over the past few years and i've thought you might be growing out of it and even in this thread you were one of the few that even leaned away from a single specification early on but still it comes out you insist on specific interpretations of specific forms you have posted in order to assert that your statements were correct you are very concerned about being correct and you will turn a thread into an insisted demonstration of your correctness in some context turning tangents from reasoned analysis of the points to demonstrations of the existence of some context that makes your post meaningful and correct when you get into these states your point isn't to further the discussion your point is to assert correctness even if your correctness is irrelevant and inconsequential to the academic resolution of the topic at hand i think few people would argue your ability to learn new things correctly i think you've demonstrated that over your years here but sometimes the correctness of your assertions is irrelevant to the discussion and you alienate the other position which may be making valid points even without proper reference you act as if the entire argument is simplistic namecalling and ignorance of topic when you need the straw man for conflagration I haven't acted as if lwal's entire argument is simplistic namecalling and ignorance of topic. Though he has engages in those and I've rebutted him for it. Meanwhile, also I've engaged a wide range of his gambits. which you eventually call lies with many of your EMPHASIS CAPS and dig on nam's ability to learn in separate posts you want to burn others both sides here are talking past each other and both sides are in a foul mood genuine interest in rigorisation could be a way out for both sides or it could just peter out from exhaustion-of-the-anger-that-will-one- day-rise-again when i challenge your genuineness i am challenging you to post relevantly and not simply outburst and accusation i am challenging you to move the discussion forward you attempt superiority poses veiled in moral righteousness I don't claim to any special moral righteousness. you continue to spew bile like As to Nam, I guess it's hopeless to try to bring him to reason on this or many another subject. You have a quite elastic definition of 'bile'. of sad resignation sad resignation? because you feel the other side has nothing to offer discussion? because your interaction with them was of no benefit? because you are so better off in state space than the other party? bring him to reason??? aha.. the savior with true sight! (one of the problems is that you don't realise just how much your awareness and interaction are founded on presumptions of position over others) and while i'm joining the pile-on you guys love so much you rarely lend relevant details to a discussion I've been EXTREMELY detailed as to the particulars of this discussion. you can do detail i'm not ribbing at you for detail relevance is the one you have difficulty in preferring to play toady to some game you refuse to call bully (it's just assertions of lies and ignorance, right?) I very much admire certain posters here. That doesn't make me their toady. In fact, among the posters here who I've expressed my admiration of, I have many times posted certatin disagreements with them and stood my ground when I felt my view deserved not to be dismissed. let's expand context since apparently some of symbols were not specified in axiom :p preferring to play toady to some game you refuse to call bully i think you are toady to the game not to fellow posters i think you flatter the sanctity of truth while going off on tangents unrelated to it's progress you want to be the sidekick to authoritative logic and are eager to beat anyone up who dares speak in error or without authority So your post falls flat again. i know it always will i'm resigned to that in the same bitter sadness you speak of i am sure in this discussion the formalisation of specifying models systems and general dynamic data structures is found in the theory of compilers and processors and in this language several of nam's points make quite a bit of sense specifying syntactic roles and specifying syntactic assertions actually doesn't have a very clear formal distinction (they are interchangeable interpretations of type theory) so either way creating a parser for the lone axiom doesn't give a system strong enough for the result on new symbols (even with rules of inference... - i am not specifying a foundational computation model because it really doesn't matter for the point) specifying a parse pattern is required for every new symbol of course you've mentioned some of this yourself even getting a question from chris on the usefulness of the freedom of order you mentioned and yes i realise jack that in your context it was already assumed and you and nam keep talking past each other because these unrelated points keep getting made and little progress is made on discussing the formalisation the formalisation here is that of compilers or processors compilers take texts in a specified language and translate them to texts in other languages processors take texts in languages and produces runtime state transitions possibly interacting with i/o interfaces structured information content and the formal specification of these has almost complete flexibility: ------------------------------- any given information structure may be translated to any given executable in a lambda complete impl ------------------------------- in fact you really only need if and a few structure identifying primitives if (text0) output(program0) else if (text1) output(program1) .. so let's call this result 1 which can be paraphrased: there are no logical requirements on textual order for presenting any language but nam has made the point several times that you need to axiomatise all symbols and of course there is a really clear sense in which this is true: you have to provide a functional role to all participants otherwise it has no input into the participants and thus no control in the system really there has to be a fundamental agreement established on every symbol-role application (which is commonly called scope in computer science) to give an example of a published logician who might have some authority (he was one of the second edition reason for modifications to russell and whitehead's principia mathematica ) was chwistek in his the theory of constructive types he explicitly calls for scope operators much as would eventually show up in lambda calculus for introducing new names and gave what have become classic examples for ambiguities without scope chwistek points out this need for every functional operator in the language and tarski would often point this out when arguing over with carnap and quine in the vienna circle days In the discussions the ïanalytic' vs. ïsynthetic' opposition came to the fore repeatedly. In reply to comments by Tarski to the effect that one could not formulate the distinction sharply, Carnap explained that it was possible to include already in the metalanguage (Syntax) certain signs which are usually seen as ïdescriptive' (house, temperature and so on) and to use them as ïlogical signs' so that the sentences in which these terms occur would be determined on the basis of the language rules. He added that it is really a question how.84 on the basis of the ïfeeling' which gives rise to the opinion one could draw a boundary between words of the German language such as ïund', ïoder' etc. on the one side and ïTemperatur', ïHaus' on the other.84one could arrive at a formulation which would allow us to distinguish rigorously ïdescriptive' and ïlogic' and then also ïsynthetic' and ïanalytic'. This is a problem that can be discussed only in rigorous form if we want to avoid misunderstandings of all sorts. (Neurath 1936, pp. 388[CapitalEth] 389).3 and tarski's on the concept of logical consequence specifically points out there is no known way to distinguish logical and nonlogical constants it's all the same process of specification the lines drawn do not have consequence epsilon could be a logical dynamic operator or called nonlogical and the foundation of set theory it's all the same dynamic specified paolo mancosu's harvard 1940-1941: tarski, carnap, and quine on a finitistic language of mathematics for science gives a great overview of this discussion and tarski's papers in general well... i can't do more to promote all his work his i think (and admire) was a theory that will only be appreciated more over the years (though strangely i will never love his work as much as poincare's because i think tarski scares me with his honesty and poincare could drink away the scary things) so result 2 might be stated axiomitisation and language specification is the same process and continuing with the rigorisation we could continue to specify how modern compilers specify languages and the read more é... Because I am simple and right. I have a knowledge stronger than the one the masses protect and for this they ridicule my name and tell I am ignorant. For this they claim I am less than they for not obeying their axioms and following the forms they worship and riding the whip as they beat me with it as a matter of speech. If this is not sad resignation it does not exist in either words or reality. The truth is I do not need luck to succeed. Luck is for the unprepared. I have logic on my side and inevitability. Consider these statements/results: (BE LOGICAL/LIKE SPOCK OR DO NOT POST A REPLY PLEASE) --I say this content with the fact you will do as you will. Simply read: Result (1): John 18:36 Jesus answered, My kingdom is not of this world. If Bible in Basic English Jesus said in answer, My kingdom is not of this world: .... I If is it Jesus Jews kingdom My not now of over place prevent realm said ... http://bible.cc/john/18-36.htm Source (1): http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A... Result (2) Matthew 19 American King James Version 18 He said to him, Which? Jesus said, You shall do no murder, You shall not commit ... Truly I say to you, That a rich man shall hardly enter into the kingdom of heaven. ... You may not copyright it or prevent others from using it. ... http://kjv.us/matthew/19.htm Source (2) http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A... Result (3) themselves and seek to prevent those who are trying to enter. 16 I am sorry for my poor people who should be helping in the right way ... http://www.seventhfam.com/temple/books/black man/blk121.htm Source (3) http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A... And finally *this*: (quoted from a physical book for my critics who claim I do not read) Title: Holy Bible (NIV) Publisher: Zondervan ISBN 0-310-92391-3 Matthew Chapter 23 13Woe to you, teachers of the law and Pharisees, you hypocrites! You shut the kingdom of heaven in men's faces. You yourselves do not enter, nor will you let those who are trying to.^g 15Woe to you, teachers of the law and Pharisees, you hypocrites! You travel over land and sea to win a single convert, and when he becomes one, you make him twice as much a son of hell as you are. ^g13 Some manuscripts to Questions: 1. Matthew 23:14 does not exist in this bible. Why? Conclusion (the verse is ommitted in the N.I.V): Proof: (Book) Title: Our African Heritage Holy Bible (Authorized King James Version) Reserved. Matthew Chapter 23 14 Woe unto you, *scribes* and *Phar'-i-sees*, hypocrites!for ye devour widows' houses, and for a pretence make long prayer: therefore ye shall receive the greater damnation. Do you still not understand? === Subject: Re: Cantor's argument is erroneous posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Marshall, > Go back and read my post, and my theorem. Bleah, no thank you. Once was too many times. > I said nothing about > bringing human beings into existance. That is true; however you did say something about > bringing human being into existence. > The truth is not in your words > so please take them back or provide a citation (feel free to publish > my email to you). Ordinarily I would take them back, but the house is pretty > full right now and I have no place to put them. Instead > I'll just give you a citation. Specifically I cite you $10 for > poor spelling, and another $30 for excessive posting. > Make your check payable to the Red Cross. Please remit > promptly. Marshall === Subject: Re: Cantor's argument is erroneous posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Marshall, > Go back and read my post, and my theorem. Bleah, no thank you. Once was too many times. > I said nothing about > bringing human beings into existance. That is true; however you did say something about > bringing human being into existence. > The truth is not in your words > so please take them back or provide a citation (feel free to publish > my email to you). Ordinarily I would take them back, but the house is pretty > full right now and I have no place to put them. Instead > I'll just give you a citation. Specifically I cite you $10 for > poor spelling, and another $30 for excessive posting. > Make your check payable to the Red Cross. Please remit > promptly. Marshall Marshall, If I had the money I would give that and more. Musatov === Subject: Re: Cantor's argument is erroneous <87hbykh86f.fsf@alatheia.truth.invalid> <8763ez658p.fsf@alatheia.truth.invalid Go back and read my post, and my theorem. I said nothing about > bringing human beings into existance. The truth is not in your words > so please take them back or provide a citation (feel free to publish > my email to you). Well, you did write to me in an e-mail: Theorem: Human beings are effected by the binary values associated with conflict across an electric medium so the electronic medium must be also effected. This statement is plainly about bringing human beings into existence. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Cantor's argument is erroneous > > Go back and read my post, and my theorem. I said > nothing about > bringing human beings into existance. The truth is > not in your words > so please take them back or provide a citation > (feel free to publish > my email to you). > > Well, you did write to me in an e-mail: > > Theorem: Human beings are effected by the binary > y values associated with > conflict across an electric medium so the electronic > c medium must be > also effected. > > This statement is plainly about bringing human beings > into existence. > > -- > Aatu Koskensilta (aatu.koskensilta@uta.fi) > > Wovon mann nicht sprechen kann, dar.9fber muss man > schweigen > - Ludwig Wittgenstein, Tractatus > s Logico-Philosophicus Reactions to/against the Binary Tree (Opinion 94) I disagree. Machine-checked anything is useful because machines are so much more fastidious than humans. It is true that machine checked proof is .. --Martin Musatov speaking on American Entrepreneur and Innovator Thomas edisonpreface: Computational ComplexitySo much of what I have seen since I have ... May 24 by Martin Musatov - 232 messages - 26 authors What happens to your shape when you move into curved space? (Gawd I love it when I disagree with Binge Jocaby.) (God I hate it when I end up on Andro's side!) -- Binge Jocaby, http://tinyurl.com/cn8dlm March 19, ... Martin Musatov Out of respect for logical deduction and the scientific method I would appreciate it if you took your god-on-a-stick water perambulating vintner .. May 16 by Androcles - 21 messages - 10 authors Diagonal wanderings (incongruent by construction) I'm going to have to disagree there. It's possible it might depend a tiny bit on what we want to use it for, but since Turing machines are just about the worst possible idealization of ... Marshall All this song about Einstein Rings and Gravitational Cosmic Lensing os very handy for Musatov's proof P=NP<<é ... May 27 by Martin Musatov - 9 messages - 5 authors ... You and I have nothing to disagree *about* your No answer here. But I've always maintained your Yes answer above would lead to invalid reasoning, ... May 24 by Martin Musatov - 684 messages - 36 authors Reactions to/against the Binary Tree You're called a crank about the subject of cranks because your ARGUMENTS on the subject are illogical and unresponsive, not just because you disagree with the categorization. moeblee I have never been unresponsive. Critique the mathematics May 27 by Martin Musatov - 232 messages - 26 authors What happens to your shape when you move into curved space? Martin Musatov marty.musa...@gmail.com sci physics sci physics relativity sci love it when I disagree with Binge Jocaby.) (God I hate it when I end up on Andro's side!) -- Binge Jocaby, http://tinyurl.com/cn8dlm March 19, 1:00 pm Out of ... May 16 by Martin Musatov - 21 messages - 10 authors Diagonal wanderings (incongruent by construction) I bet you don't disagree with the one-star rating for any of LW, LV, WM, Albrect , or Srinivasa. Or AP, JSH elsiemelsie or Martin Musatov. I admit there are things about it that are puzzling. moeblee has dropped from 4 to three. Why? It makes no sense to me. In fact there are a number of people I'd say ought to be 4 .. May 16 by Marshall - 1001 messages - 55 authors Consider these statements/results: (BE LOGICAL/LIKE SPOCK OR DO NOT POST A REPLY PLEASE) --I say this content with the fact you will do as you will. Simply read: Result (1): John 18:36 Jesus answered, My kingdom is not of this world. If Bible in Basic English Jesus said in answer, My kingdom is not of this world: .... I If is it Jesus Jews kingdom My not now of over place prevent realm said ... http://bible.cc/john/18-36.htm Source (1): http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A... Result (2) Matthew 19 American King James Version 18 He said to him, Which? Jesus said, You shall do no murder, You shall not commit ... Truly I say to you, That a rich man shall hardly enter into the kingdom of heaven. ... You may not copyright it or prevent others from using it. ... http://kjv.us/matthew/19.htm Source (2) http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A... Result (3) themselves and seek to prevent those who are trying to enter. 16 I am sorry for my poor people who should be helping in the right way ... http://www.seventhfam.com/temple/books/black_man/blk121.htm Source (3) http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A... And finally *this*: (quoted from a physical book for my critics who claim I do not read) Title: Holy Bible (NIV) Publisher: Zondervan ISBN 0-310-92391-3 Matthew Chapter 23 13Woe to you, teachers of the law and Pharisees, you hypocrites! You shut the kingdom of heaven in men's faces. You yourselves do not enter, nor will you let those who are trying to.^g 15Woe to you, teachers of the law and Pharisees, you hypocrites! You travel over land and sea to win a single convert, and when he becomes one, you make him twice as much a son of hell as you are. ^g13 Some manuscripts to Questions: 1. Matthew 23:14 does not exist in this bible. Why? Conclusion (the verse is ommitted in the N.I.V): Proof: (Book) Title: Our African Heritage Holy Bible (Authorized King James Version) Reserved. Matthew Chapter 23 14 Woe unto you, *scribes* and *Phar'-i-sees*, hypocrites!for ye devour widows' houses, and for a pretence make long prayer: therefore ye shall receive the greater damnation. Do you still not understand? === Subject: Re: Cantor's argument is erroneous <87y6rv4mk9.fsf@alatheia.truth.invalid> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Go back and read my post, and my theorem. I said nothing about > bringing human beings into existance. The truth is not in your words > so please take them back or provide a citation (feel free to publish > my email to you). Well, you did write to me in an e-mail: Theorem: Human beings are effected by the binary values associated with > conflict across an electric medium so the electronic medium must be > also effected. This statement is plainly about bringing human beings into existence. -- > Aatu Koskensilta (aatu.koskensi...@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus I disagree. *Some* human beings are effected by the binary values associated with conflict across an electronic medium so the the medius must also be effected. So the *general* case remains. I do not need to specify the number of human beings effected for the case to be true. Consider it classical philosophy of inclusion. Draw up Venn Diagrams if you will. ( all ) ( all ) (things not effected ((Human Beings)) things effected by binary ) (by binary values ( values associated with conflict) (associated with conflict) ( across an electronic medium) The top row on inclusion is still All things... Signed, Martin Musatov === Subject: Re: Cantor's argument is erroneous posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Marshall, Go back and read my post, and my theorem. I said nothing about > bringing human beings into existance. The truth is not in your words > so please take them back or provide a citation (feel free to publish > my email to you). Martin > I am sure that those who are complaining that you emailed them > secretly loved it, every bit as much as I loved it that you emailed > me. > I was feeling left out, but checking my inbox I find I have received > from Musatov an e-mail containing a very important theorem about the > binary values associated with conflict across an electric medium > bringing human beings into existence. > And you secretly loved it, didn't you? Wait, don't answer that; > then it wouldn't be a secret any more. Look slowly to you > right and nod if yes. > Marshall, the classic bully > MoeBlee brought light with:: > you are disingenuous > You've not shown me to have been disingenuous. > and it's annoying that you are disingenuous >because you are also very intelligent >and have been learning more tolerant positions > over the past few years > and i've thought you might be growing out of it > and even in this thread you were one of the few >that even leaned away from a single specification early on > but still it comes out > you insist on specific interpretations > of specific forms you have posted >in order to assert that your statements were correct > you are very concerned about being correct > and you will turn a thread into >an insisted demonstration of your correctness >in some context > turning tangents from reasoned analysis of the points > to demonstrations of the existence of some context >that makes your post meaningful and correct > when you get into these states > your point isn't to further the discussion > your point is to assert correctness > even if your correctness is irrelevant and inconsequential >to the academic resolution of the topic at hand > i think few people would argue >your ability to learn new things correctly >i think you've demonstrated that over your years here > but sometimes >the correctness of your assertions > is irrelevant to the discussion > and you alienate the other position >which may be making valid points >even without proper reference > you act as if the entire argument > is simplistic namecalling and ignorance of topic > when you need the straw man for conflagration > I haven't acted as if lwal's entire argument is simplistic > namecalling and ignorance of topic. Though he has engages in those > and I've rebutted him for it. Meanwhile, also I've engaged a wide > range of his gambits. > which you eventually call lies >with many of your EMPHASIS CAPS > and dig on nam's ability to learn in separate posts > you want to burn others > both sides here are talking past each other > and both sides are in a foul mood > genuine interest in rigorisation could be a way out for both sides > or it could just peter out from exhaustion-of-the-anger-that-will-one-day-rise-again > when i challenge your genuineness > i am challenging you to post relevantly >and not simply outburst and accusation > i am challenging you to move the discussion forward > you attempt superiority poses > veiled in moral righteousness > I don't claim to any special moral righteousness. > you continue to spew bile like > As to Nam, I guess it's hopeless to try to bring him > to reason on this or many another subject. > You have a quite elastic definition of 'bile'. > sad resignation > sad resignation? > because you feel the other side has nothing to offer discussion? > because your interaction with them >was of no benefit? > because you are so better off in state space >than the other party? > bring him to reason??? > aha.. the savior with true sight! > (one of the problems is that you don't realise > just how much your awareness and interaction > are founded on presumptions of position over others) > and > while i'm joining the pile-on you guys love so much > you rarely lend relevant details to a discussion > I've been EXTREMELY detailed as to the particulars of this discussion. > you can do detail > i'm not ribbing at you for detail > relevance is the one you have difficulty in > preferring to play toady to some game you refuse to call bully > (it's just assertions of lies and ignorance, right?) > I very much admire certain posters here. That doesn't make me their > toady. In fact, among the posters here who I've expressed my > admiration of, I have many times posted certatin disagreements with > them and stood my ground when I felt my view deserved not to be > dismissed. > let's expand context >since apparently some of symbols were not specified in axiom > :p > preferring to play toady to some game you refuse to call bully > i think you are toady to the game >not to fellow posters > i think you flatter the sanctity of truth > while going off on tangents >unrelated to it's progress > you want to be the sidekick to authoritative logic > and are eager to beat anyone up >who dares speak in error or without authority > So your post falls flat again. > i know > it always will > i'm resigned to that >in the same bitter sadness you speak of > i am sure > in this discussion >the formalisation of specifying >models >systems >and general dynamic data structures >is found in the theory > of compilers and processors > and in this language > several of nam's points make quite a bit of sense > specifying syntactic roles > and specifying syntactic assertions >actually doesn't have a very clear formal distinction > (they are interchangeable interpretations of type theory) > so either way > creating a parser for the lone axiom >doesn't give a system strong enough > for the result on new symbols >(even with rules of inference... - > i am not specifying a foundational computation model > because it really doesn't matter for the point) > specifying a parse pattern is required for every new symbol > of course > you've mentioned some of this yourself >even getting a question from chris > on the usefulness of the freedom of order you mentioned > and yes >i realise > jack > that in your context it was already assumed >and you and nam keep talking past each other > because these unrelated points keep getting made > and little progress is made on discussing the formalisation > the formalisation here is that of compilers or processors > compilers take texts in a specified language >and translate them to texts in other languages > processors take texts in languages >and produces runtime state transitions >possibly interacting with i/o interfaces > structured information content > and the formal specification of these > has almost complete flexibility: >------------------------------- >any given information structure > may be translated >to any given executable > in a lambda complete impl >------------------------------- > in fact > you really only need if >and a few structure identifying primitives > if (text0) >output(program0) > else if (text1) >output(program1) > .. > so let's call this result 1 > which can be paraphrased: >there are no logical requirements on textual order >for presenting any language > but >nam has made the point several times > that you need to axiomatise all symbols > and of course there is a really clear sense in which this is true: > you have to provide a functional role to all participants > otherwise it has no input into the participants >and thus no control in the system > really >there has to be a fundamental agreement established >on every symbol-role application > (which is commonly called scope in computer science) > to give an example of a published logician > who might have some authority >(he was one of the second edition reason for modifications > to russell and whitehead's principia mathematica ) > was chwistek in his the theory of constructive types > he explicitly calls for scope operators >much as would eventually show up in lambda calculus > for introducing new names > and gave what have become classic examples >for ambiguities without scope > chwistek points out this need for every functional operator in the language > and tarski would often point this out >when arguing over with carnap and quine > in the vienna circle days > In the discussions the ïanalytic' vs. ïsynthetic' opposition came to the fore >repeatedly. In reply to comments by Tarski to the effect that one could not >formulate the distinction sharply, Carnap explained that it was possible to >include already in the metalanguage (Syntax) certain signs which are usually seen >as ïdescriptive' (house, temperature and so on) and to use them as ïlogical signs' >so that the sentences in which these terms occur would be determined on the basis >of the language rules. He added that it is really a question how.84on the basis of >the ïfeeling' which gives rise to the opinion one could draw a boundary- Show quoted text -Simply read: Result (1): John 18:36 Jesus answered, My kingdom is not of this world. If Bible in Basic English Jesus said in answer, My kingdom is not of this world: .... I If is it Jesus Jews kingdom My not now of over place prevent realm said ... http://bible.cc/john/18-36.htm Source (1): http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A... Result (2) Matthew 19 American King James Version 18 He said to him, Which? Jesus said, You shall do no murder, You shall not commit ... Truly I say to you, That a rich man shall hardly enter into the kingdom of heaven. ... You may not copyright it or prevent others from using it. ... http://kjv.us/matthew/19.htm Source (2) http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A... Result (3) themselves and seek to prevent those who are trying to enter. 16 I am sorry for my poor people who should be helping in the right way ... http://www.seventhfam.com/temple/books/black man/blk121.htm Source (3) http://meami.org/?cx=000961116824240632825%3A5n3yth9xwbo&cof=FORID%3A... And finally *this*: (quoted from a physical book for my critics who claim I do not read) Title: Holy Bible (NIV) Publisher: Zondervan ISBN 0-310-92391-3 Matthew Chapter 23 13Woe to you, teachers of the law and Pharisees, you hypocrites! You shut the kingdom of heaven in men's faces. You yourselves do not enter, nor will you let those who are trying to.^g 15Woe to you, teachers of the law and Pharisees, you hypocrites! You travel over land and sea to win a single convert, and when he becomes one, you make him twice as much a son of hell as you are. ^g13 Some manuscripts to Questions: 1. Matthew 23:14 does not exist in this bible. Why? Conclusion (the verse is ommitted in the N.I.V): Proof: (Book) Title: Our African Heritage Holy Bible (Authorized King James Version) Reserved. Matthew Chapter 23 14 Woe unto you, *scribes* and *Phar'-i-sees*, hypocrites!for ye devour widows' houses, and for a pretence make long prayer: therefore ye shall receive the greater damnation. Do you still not understand? - Show quoted text -- Hide quoted text - - Show quoted text -... read more é === Subject: Re: Cantor's argument is erroneous <8763ez658p.fsf@alatheia.truth.invalid> posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > I am sure that those who are complaining that you emailed them > secretly loved it, every bit as much as I loved it that you emailed > me. I was feeling left out, but checking my inbox I find I have received > from Musatov an e-mail containing a very important theorem about the > binary values associated with conflict across an electric medium > bringing human beings into existence. Intuitively it seems plausible. We know that, of the binary values, 1 is associated with conflict across an electric medium. In electronic circuits, 1 is the high-voltage signal. And we also know that high voltage is needed to bring human beings into existence. (Frankenstein's Lemma, aka effect a charge.) Just be sure to use a good brain. Come to think of it, this advice applies quite generally. Abbey Someone === Subject: Re: Cantor's argument is erroneous > Musatov: > Please do not email me. > MoeBlee: > I will do as I please. > > Hopefully it pleases you not to e-mail random people from news, > especially if they ask you not to. This is just common courtesy. Assuming that Moe does not feel any desire to e-mail this loon, he might very well do as he pleases _in this case_, _without_ violating common courtesy. ;-) On the other hand, he might just have been in a pissy mood when writing his reply. :-) Herb === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <87hbykh86f.fsf@alatheia.truth.invalid> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Musatov: >Please do not email me. > MoeBlee: >I will do as I please. > Hopefully it pleases you not to e-mail random people from news, > especially if they ask you not to. This is just common courtesy. Assuming that Moe does not feel any desire to e-mail this loon, he might > very well do as he pleases in this case , without violating common > courtesy. ;-) >I was kind of in a pissy mood and tired of been told what to do by the sci.math gestapo but I >was genuinely curious what would come from this simple binary contradiction. === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <87hbykh86f.fsf@alatheia.truth.invalid> Musatov: > Please do not email me. > MoeBlee: > I will do as I please. > > Hopefully it pleases you not to e-mail random people from news, > especially if they ask you not to. This is just common courtesy. Assuming that Moe does not feel any desire to e-mail this loon, he > might very well do as he pleases _in this case_, _without_ violating > common courtesy. ;-) original attribution lines you've elided above. I wonder though, is there some reason for publicly announcing you're killfiling me and then replying to my posts? -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: value of second-order logic posting-account=XD7nMAkAAAAUcvFxxCbDAfgYkTA2oJDz Gecko/2009042513 Ubuntu/8.04 (hardy) Firefox/3.0.10,gzip(gfe),gzip(gfe) On Jun 10, 11:09pm, Bill Taylor in terms of syntactic deduction second-order set theory SET theory? > is no different > from any first-order theory. (And on that understanding doesn't decide > any mathematically interesting problem undecidable in ZFC that we know > of.) > It's conceivable (though far from certain) that the OP would regard > this > as being a comment that 2nd-order logic was of no mathematical > value. > That depends. You are jumping back and forth between logic and set theory. Is he saying that statements that can only be made in SOL and not FOL +ZFC can not actually be proven in SOL? Wouldn't that mean that SOL is not useful for formalizing knowledge? === Subject: Re: value of second-order logic <87ocsye54v.fsf@alatheia.truth.invalid> <878wjzpca0.fsf@alatheia.truth.invalid> <87zlcfnwk9.fsf@alatheia.truth.invalid > in terms of syntactic deduction second-order set theory SET theory? Yes, set theory. Since you apparently do not find my explanations very second-order and higher-order logic by Herbert Enderton in the Stanford Encyclopedia of Philosophy, available at http://plato.stanford.edu/entries/logic-higher-order/ -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: value of second-order logic <87ocsye54v.fsf@alatheia.truth.invalid> <878wjzpca0.fsf@alatheia.truth.invalid It seems discussions about logic are very vague... Sometimes they are, sometimes they aren't. In this instance whatever vagueness we perceive has to do with deciding a mathematical question using second-order logic. This locution has no standard meaning -- hence my request for clarification. > You were talking about SOL vs FOL + ZFC. SOL vs FOL + ZFC makes about as much sense as dynamic modal logic versus FOL + ZFC or algebraic geometry versus FOL + ZFC, i.e. not much sense at all. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: value of second-order logic <87ocsye54v.fsf@alatheia.truth.invalid reply even less comprehensible (My background in in applied math. I > never studied formal logic beyond some FOL). Well, you asked whether there is a consensus about the value of second-order logic. There is certainly consensus, in that there are many mathematical applications of higher-order logic generally recognised as valuable, such as those mentioned. Such applications are of course comprehensible only for those who are familiar with the technical notions involved. This is true of any piece of mathematics, of applications of modal logic, of cohomology theory, of topos theory, etc. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: value of second-order logic <87k53gctgg.fsf@alatheia.truth.invalid> posting-account=XD7nMAkAAAAUcvFxxCbDAfgYkTA2oJDz Gecko/2009042513 Ubuntu/8.04 (hardy) Firefox/3.0.10,gzip(gfe),gzip(gfe) > Such applications are of course > comprehensible only for those who are familiar with the technical > notions involved. Only if the additional expressive power that SOL gives you over FOL is only useful in logic itself. You are not implying that, or are you? === Subject: Re: value of second-order logic <87ocsye54v.fsf@alatheia.truth.invalid> <87k53gctgg.fsf@alatheia.truth.invalid > Such applications are of course comprehensible only for those who are > familiar with the technical notions involved. Only if the additional expressive power that SOL gives you over FOL is > only useful in logic itself. You are not implying that, or are you? No. In fact I have no idea what gives you this peculiar idea. My point was really just a boring platitude: in order to appreciate applications of higher-order logic in, say, complexity theory naturally one must understand something of complexity theory. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: value of second-order logic <878wjw9ep4.fsf@alatheia.truth.invalid> posting-account=XD7nMAkAAAAUcvFxxCbDAfgYkTA2oJDz Gecko/2009042513 Ubuntu/8.04 (hardy) Firefox/3.0.10,gzip(gfe),gzip(gfe) > Such applications are of course comprehensible only for those who are > familiar with the technical notions involved. > Only if the additional expressive power that SOL gives you over FOL is > only useful in logic itself. You are not implying that, or are you? No. In fact I have no idea what gives you this peculiar idea. My point > was really just a boring platitude: in order to appreciate applications > of higher-order logic in, say, complexity theory naturally one must > understand something of complexity theory. > I am familiar with applied math (integral and differential calculus, linear algebra, matrix theory, differential equations) and some discrete math. I could appreciate applications there. Had you said [some applied math theorem] can only be stated and proven using SOL (and not FOL), I could related to that. If on the other hand, the only applications of a theory are to itself, that does not demonstrate its usefulness to me. === Subject: Re: value of second-order logic posting-account=Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN rv:1.9.0.10) Gecko/2009042315 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Such applications are of course comprehensible only for those who are > familiar with the technical notions involved. > Only if the additional expressive power that SOL gives you over FOL is > only useful in logic itself. You are not implying that, or are you? > No. In fact I have no idea what gives you this peculiar idea. My point > was really just a boring platitude: in order to appreciate applications > of higher-order logic in, say, complexity theory naturally one must > understand something of complexity theory. I am familiar with applied math (integral and differential calculus, > linear algebra, matrix theory, differential equations) and some > discrete math. I could appreciate applications there. Had you said [some applied math theorem] can only be stated and proven > using SOL (and not FOL), I could related to that. If on the other hand, the only applications of a theory are to > itself, that does not demonstrate its usefulness to me. your question looks like you are asking about thing like boolos' nonfirstorderisability http://en.wikipedia.org/wiki/Nonfirstorderizability the general semantic role that second order theories play make them strong candidates for modelling the activity of mathematics and this has been argued from several different directions boolos' analysis of quine's semantic motto to be is to be the value of a variable has lead to many different foundationalist movements using second order logic stewart shapiro's foundations without foundationalism: a case for second-order logic describes a number of the modern semantic approaches -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: value of second-order logic <87ocsye54v.fsf@alatheia.truth.invalid> <87k53gctgg.fsf@alatheia.truth.invalid> <878wjw9ep4.fsf@alatheia.truth.invalid Had you said [some applied math theorem] can only be stated and proven > using SOL (and not FOL), I could related to that. I haven't said anything about applied mathematics, only about applications of second-order logic in mathematics. This notion, of a theorem that can only be stated and proven using SOL (and not FOL) is on the face of it very obscure. But, as noted, it appears you do not find my explanations very helpful, so I'll leave it for others to sort it out. > If on the other hand, the only applications of a theory are to > itself, that does not demonstrate its usefulness to me. Are you under the impression that complexity theory and second-order logic are the same thing? -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: value of second-order logic > Is there a consensus now about the value of > second-order logic? > Wikipedia seems to be vague on this subject: > > http://en.wikipedia.org/wiki/Second-order_logic#The_hi > story_and_disputed_value_of_second-order_logic > > I'm surprised that an old branch of math can have so > much uncertainty. It is a nice place. Let me give you the tour... [PDF] MonaSearch [CapitalEth] Querying Treebanks with Monadic Second Order LogicFile Format: PDF/Adobe Acrobat - View as HTML MonaSearch uses a well-known logic formalism as query language: Monadic Second-Order Logic (MSO). Clearly defined, model-theoretic semantics ... http://tcl.sfs.uni-tuebingen.de/MonaSearch/tlt09-monasearch.pdf A restricted second order logic for finite structuresis defined to be the closure of first order logic under the operation of taking inflationary fixed points of arbitrary formulas. Clearly, for positive ... http://www.springerlink.com/index/fq41891541334581.pdf [PDF] MonaSearch [CapitalEth] Querying Linguistic Treebanks with Monadic Second ...File Format: PDF/Adobe Acrobat - View as HTML The query language of MonaSearch is monadic second-order logic, an ... clearly defined; it is only defined by the functionality of the program. The ... http://tcl.sfs.uni-tuebingen.de/MonaSearch/monasearch-submission.pdf [PDF] Second Order Logic and Logic/Complexity ConnectionsFile Format: PDF/Adobe Acrobat - View as HTML need to introduce second order logic. 1 Second Order Logic ... the elements of even rank with respect to the order defined by R, i.e. .... We denote by i + 1 the smallest i larger than i wrt .b2 (this can clearly be defined in FO). ... http://www-cse.ucsd.edu/classes/sp08/cse205a/secondOrder.pdf Branch-Width, Parse Trees, and Monadic Second-Order Logic for MatroidsClearly, the branch-width of a graph G is never smaller than the branch-width ..... The monadic second-order logic of matroids. MS. M is defined as follows: ... http://www.springerlink.com/index/PEPBRKUCTEY7M0WN.pdf Theoretical Computer Science : The monadic second-order logic of ...Why monadic secondorder logic rather than another logical language? ... We mean by this that a satisfaction relation = is defined. ...... T (A') such that TTi(t') = t and t' e L si' Clearly t' STifX), hence we need only prove that t' ? ... http://linkinghub.elsevier.com/retrieve/pii/0304397595000493 Arity vs. alternation in second order logicWe investigate the expressive power of second order logic over fi- .... is defined by restricting .formulas of the strict AA hi- ..... Also clearly r is true iff r is self satisfying. (Note that the first character in r is '3', ... http://www.springerlink.com/index/f6n114101jh26732.pdf Amazon.com: Foundations without Foundationalism: A Case for Second ...For example, Quine's complaint that second order logic involves a 'staggering ... learn how to think more clearly: A guide by Paul E. Oppenheimer Be kind. ... http://www.amazon.com/Foundations-without-Foundationalism-Second-order-Logic /dp/0198250290 The expressive power of second order Horn logicorder logic that we call second order Horn logic and denote by SO-HORN. .... Clearly, the complement of GEN is also P-complete. It is defined by the ... http://www.springerlink.com/index/w7401212g2504122.pdf EMail Msg <9201072137.AA13860@cs.umn.edu>That is the KL-ONE and E-R style of second-order logic, where there are no ... knowledge sharing effort until we have a clearly defined set of requirements. ... http://www-ksl.stanford.edu/email-archives/srkb.messages/79.html [PDF] Second Order Logic and Logic/Complexity ConnectionsFile Format: PDF/Adobe Acrobat - View as HTML We can find similar connections for NP, co-NP, and the polynomial hierarchy, but first we need to introduce second order logic. 1 Second Order Logic ... the elements of even rank with respect to the order defined by R, i.e. .... by i + 1 the smallest i larger than i wrt .b2 (this can clearly be defined in FO). ... http://www-cse.ucsd.edu/classes/sp08/cse205a/secondOrder.pdf Second-order and Higher-order Logic (Stanford Encyclopedia of ...The language of second-order logic extends the language of first-order logic by ..... is an NP property if and only if it is definable by a .83¡-1-1 second-order sentence. ... The second-order quantifiers are now defined to range over the .... Clearly .83[CapitalEth] is true in any structure whose universe is the disjoint union of ... http://plato.stanford.edu/entries/logic-higher-order/ P = NP problem - Wikipedia, the free encyclopediaFormally, NP is defined as the set of languages over a finite alphabet that have a ... Clearly, the question of whether a given x is a composite is equivalent to the .... It correctly accepts the NP-complete language SUBSET-SUM, ... NP is the set of languages expressible in existential second-order logic .81 that is, ... http://en.wikipedia.org/wiki/P_%3D_NP_problem -: an executable specification language for solving all problems ...Clearly, such languages are fairly complex, and this may be an obstacle for their use ... is the possibility of using second-order logic in a restricted way. .... We say that a language Image captures NP if the set of queries defined by ... http://linkinghub.elsevier.com/retrieve/pii/S0096055101000108 A restricted second order logic for finite structuresAbiteboul and Vianu [2] showed that PFP is equivalent to the language while .... NFP- PFP if and only if NP = PSPACE. 3. A Restricted. Second Order Logic ... Clearly, the query defined by can be expressed as: 3RI...QR (r ^ ... http://www.springerlink.com/index/fq41891541334581.pdf [PDF] MonaSearch [CapitalEth] Querying Linguistic Treebanks with Monadic Second ...File Format: PDF/Adobe Acrobat - View as HTML The query language of MonaSearch is monadic second-order logic, an ... clearly defined; it is only defined by the functionality of the program. The ... dominating an NP as a new relation and state that the highest PP node and ... http://tcl.sfs.uni-tuebingen.de/MonaSearch/monasearch-submission.pdf Monadic second-order logic over pictures and recognizability by ...languages in monadic second-order logic and the algebraic notion of recognizability ..... This completes the proof of the theorem since REC is clearly closed under projections. .... The d-local language L will be defined over the alphabet .... Fagin, R., Stockmeyer, L., Vardi, M.Y.: On monadic NP vs. monadic co-NP. ... http://www.springerlink.com/index/y3073750778775v8.pdf [PDF] LPath : A First-Order Complete Language for Linguistic Tree QueryFile Format: PDF/Adobe Acrobat - View as HTML and the LPath operators defined above. Subscripts ... NP i. , to ensure that the first and last NPs were the ..... clearly contains X c. .81.96 so we have the following equiva- lence: Theorem 8. L ..... monadic second-order logic in linear time. Journal of. Logic, Language and Information, 13(4):457[CapitalEth]470. ... http://www.ldc.upenn.edu/Projects/QLDB/lpath-plus.pdf On Completeness for NP via projection translationsThe language of the first-order logic FOs() over the vocabulary has as its .... NP as those problems expressible in existential second-order logic ([31). In. Theorem 3.1 below, we essentially prove that SAT (defined below) is ..... over such that [St is even is clearly in NP, and so must be closed under the ... http://www.springerlink.com/index/dv2g714k115l675l.pdf A Framework for Representing and Solving NP Search ProblemsUnfortunately, SAT provides a poor modelling language and. Copyright c 2005, American Association for .... power as existential second order logic (.81ëSO) over finite structures. ..... Clearly, any problem in NP can be expressed as. FO(ID) MX(.83[CapitalEth], .83î). .... the instance is defined to be a structure, and thus no such ... http://www.aaai.org/Papers/AAAI/2005/AAAI05-068.pdf ++ Musatov === Subject: Re: value of second-order logic <87ocsye54v.fsf@alatheia.truth.invalid> <878wjzpca0.fsf@alatheia.truth.invalid> said: > On Jun 10, 10:31pm, Aatu Koskensilta That is, is there some obvious naturally mathematical result that > can be decided using 2nd-order logic, but not by any common first > order theory (say ZFC)? > I'm not sure what you're after. What does it mean to decide a > mathematical result using second-order logic? > It seems discussions about logic are very vague... > As discussions about most anything can be. You must be lucky enough never to have taken a philosophy or theology > class to compare them to math or even physics. Oh, I've taken more than a few of both sorts. Perhaps you mean something more specific by discussion than is usually meant. Have a look over on, e.g., sci.physics, or even a good many popularizations of physics and you will find a good many discussions about physics that are vague. Of course, many are very clear as well (as are a good many philosophical discussions, though the subject matter admittedly tends to be more prone to vagueness and confusion). So I'm not sure what you are claiming. > I'm still not sure where logic fits in between these. Well, since logic the sort under discussion here is a branch of math, by your own characterization it's not between them at all. === Subject: Re: value of second-order logic posting-account=XD7nMAkAAAAUcvFxxCbDAfgYkTA2oJDz Gecko/2009042513 Ubuntu/8.04 (hardy) Firefox/3.0.10,gzip(gfe),gzip(gfe) > I'm still not sure where logic fits in between these. Well, since logic the sort under discussion here is a branch of math, by > your own characterization it's not between them at all. It's supposed to be a branch of math, but I'm surprised by all the vagueness, controversy and apparently the lack of an established terminology (judging by the discussion in this thread) === Subject: Re: value of second-order logic posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > I'm still not sure where logic fits in between these. > Well, since logic the sort under discussion here is a branch of math, by > your own characterization it's not between them at all. It's supposed to be a branch of math, but I'm surprised by all the > vagueness, controversy and apparently the lack of an established > terminology (judging by the discussion in this thread) Headed here, keep your pants on! Math Forum DiscussionsOn May 15, 11:38 pm, Chris Menzel mathforum.org/kb/message.jspa?messageID=6711752&tstart=195 Godel Contradiction - Page 3 - Docendi.org10 posts - Last post: 4 days ago > I figured Moeb has his quantifier/negation ... www.docendi.org/godel-t237114p3.html Math Forum DiscussionsOn May 19, 12:04 pm, Chris Menzel looking for? > Now that question is *much* too ... mathforum.org/kb/message.jspa?messageID=6718551&tstart=150 value of second-order logic - sci.logic | Google Groups4 posts - 2 authors - Last post: 2 days ago - Hide quoted text -. - Show quoted text - ... bb9ee8a16f58141c?lnk=raot Math Forum DiscussionsOn May 21, 9:24 am, Chris Menzel I figured Moeb has his quantifier/negation ... b27ece0683a103c4/511381c267005e8b?lnk=raot Math Forum DiscussionsJun 12, 2009 ... On Jun 12, 12:05 am, Chris mathforum.org/kb/message.jspa?messageID=6749039&tstart=195 - 4 hours ago Math Forum DiscussionsOn May 22, 12:41 pm, Chris Menzel mathforum.org/kb/message.jspa?messageID=6722971&tstart=150 +++ Musatov === Subject: Re: value of second-order logic posting-account=XD7nMAkAAAAUcvFxxCbDAfgYkTA2oJDz Gecko/2009042513 Ubuntu/8.04 (hardy) Firefox/3.0.10,gzip(gfe),gzip(gfe) ... Dude, I think you are having some kind of persistent newsreader malfunction. I can't parse your posts. === Subject: Re: Artinian non-noetherian module posting-account=9jWUTQoAAAApItH41Dnk3YFBVGWKDV6I AppleWebKit/530.5 (KHTML, like Gecko) Chrome/2.0.172.31 Safari/530.5,gzip(gfe),gzip(gfe) You may consider Z_p as the endomorphism ring of > the Prufer's p-group. Then you have the (natural) > action Jack mentioned. Since you know everything > about the Z-module, you get what you want for this > Z_p-module. Best, > Louis. I understood Mariano's k-vector space with a linear transformation. I'd appreciate if Jack could give us some reference for those interesting facts. Finally, it was very clear Louis' approach; I realized that each subgroup of the Prufer's group is stable by an endomorphism, and that the lenght of the Z_p-module is infinity... My new question is: Is the length of any non-noetherian but artinian Z_p-module infinity? Max. === Subject: Solving these 2 equations?? Is it possibe to solve x and y with those 2 equations? x(y+500)/(x+y+500) = 1520 y(x+500)/(x+y+500) = 1660 The values should be near 2249 and 4189. My algebra is at the most elementary and very far away in term of years so if possible to solve this, please provide informations on the step taken. Eventually I would like to make an Excel sheet to compute these equations. Alain === Subject: Re: Solving these 2 equations?? > Is it possibe to solve x and y with those 2 > equations? > > x(y+500)/(x+y+500) = 1520 > y(x+500)/(x+y+500) = 1660 > > The values should be near 2249 and 4189. Here is one method: one can rewrite the equations to be solved simultaneously as x*y-1020*x-1520*y-760000=0 x*y-1160*y-1660*x-830000=0 Now solve the first for y to get y=20*(51*x+38000)/(x-1520) and substitute this expression into the second equation to get that x satisfies 640*x^2-1270000*x-380000000=0. One can now solve this for x using the quadratic formula and then solve for y using the expression for y already obtained. im === Subject: Re: Solving these 2 equations?? > In Excel, start with this: > > A1 2249 > A2 4189 > A3 =A1*(A2+500)/(A1+A2+500) > > A3 evaluates to 1519.971317, that's what I > mean by close enough. > > Now invoke the Solver, and set to > > Set Target Cell: [ $A$3 ] > By Changing Cells: [ $A$1:$A$2 ] > > Hit the [Solve] button and keep the results > and you'll have > > A1 2249.059639 > A2 4189.01372 > A3 1520 > > Likewise, using same sarting numbers and > changing formula in A3 (1659.781061, still > close enough) > > A1 2249 > A2 4189 > A3 =A2*(A1+500)/(A1+A2+500) > > Solver (change requestedvalue to 1660) will give you > > A1 2249.506637 > A2 4189.218186 > A3 1660 I did all that and I get the same results as you. But if I change A1 and A2 values I do not get the good answer anymore: A1 1000 A2 2000 A3 =A2*(A1+500)/(A1+A2+500) Solver (requestedvalue to 1660) will give: A2 3138,427702 A3 1600 I use Excel 97. === Subject: Re: Solving these 2 equations?? Mathematica gives x=(125/16)(127-39 sqrt(17)),y=(125/3)(47-13 sqrt(17) x=(125/16)(127+39 sqrt(17)),y=(125/3)(47+13 sqrt(17) Alain Is it possibe to solve x and y with those 2 equations? x(y+500)/(x+y+500) = 1520 > y(x+500)/(x+y+500) = 1660 The values should be near 2249 and 4189. My algebra is at the most elementary and very far away in term of years so > if possible to solve this, please provide informations on the step taken. Eventually I would like to make an Excel sheet to compute these equations. > Alain === Subject: Re: Solving these 2 equations?? <5341718.13523.1244936434531.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) You could have a look at this thread to see what I am up to: http://www.mathhelpforum.com/math-help/pre-algebra-algebra/92650-not-... It is not possible to put on A1 and A2 the values 2249 and 4189 and invoke Solver, as it has been suggested. These values are the unknown that I am looking for. You don't need to. You just give Solver *some* starting point and try to ensure that you will never divide by zero, for example. So, invoking the Solver option assume non-negative and starting from positive points x and y should be OK. For example, putting values of 1 in A1 and A2 works just fine: Solver ends up with x = 2248.446245 and y = 4191.682215. In fact, I still get this solution even if I put nothing at all in cells A1 and A2, but I guess EXCEL reads them as 0.0 in this case. Generally, however, it is recommended that you use some type of starting point, even if only a very crude approximation. If you had a system of *linear* equations you would not need to give a starting point at all, provided you chose the Solver option assume linear model. R.G. Vickson Alain === Subject: Re: Solving these 2 equations?? > You don't need to. You just give Solver *some* > starting point and try > to ensure that you will never divide by zero, for > example. So, > invoking the Solver option assume non-negative and > starting from > positive points x and y should be OK. For example, > putting values of 1 > in A1 and A2 works just fine: Solver ends up with x = > 2248.446245 and > y = 4191.682215. In fact, I still get this solution > even if I put > nothing at all in cells A1 and A2, but I guess EXCEL > reads them as 0.0 > in this case. Generally, however, it is recommended > that you use some > type of starting point, even if only a very crude > approximation. If > you had a system of *linear* equations you would not > need to give a > starting point at all, provided you chose the Solver > option assume > linear model. > > R.G. Vickson I don't know what is wrong but I can't replicate what you are doing. If I put 1 on A1 and A2 and invoke solver for the second equation and 1660 I will get on A1 214,8902205 and on A2 53687092,2. === Subject: Re: Solving these 2 equations?? > > Is it possibe to solve x and y with those 2 equations? > > x(y+500)/(x+y+500) = 1520 > y(x+500)/(x+y+500) = 1660 > > The values should be near 2249 and 4189. > > My algebra is at the most elementary and very far away in term of years so if possible to solve this, please provide informations on the step taken. > > Eventually I would like to make an Excel sheet to compute these equations. > > > Alain Using Maple solves that quadric: x*(y+500)/(x+y+500) = 1520, y*(x+500)/(x+y+500) = 1660; solve({%}, [x,y]); allvalues(%); evalf(%); 1/2 1/2 15875 4875 17 1625 17 [[x = ----- + ----------, y = 5875/3 + ----------]], 16 16 3 1/2 1/2 15875 4875 17 1625 17 [[x = ----- - ----------, y = 5875/3 - ----------]] 16 16 3 [[x = 2248.4462453054, y = 4191.6822138762]], [[x = -264.07124530540, y = -275.0155472096]] === Subject: Re: Solving these 2 equations?? posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) Is it possibe to solve x and y with those 2 equations? x(y+500)/(x+y+500) = 1520 > y(x+500)/(x+y+500) = 1660 The values should be near 2249 and 4189. My algebra is at the most elementary and very far away in term of years so if possible to solve this, please provide informations on the step taken. Eventually I would like to make an Excel sheet to compute these equations. > Alain Multiply both equations by (x+y+500) to get xy + 500y = 1660(x+y+500) and xy + 500x = 1520(x+y+500), so subtracting these we get 500(y-x) = 140(x+y) + 7000, hence y = (16/9)x + 1790/9. Substituting this for y in the first equation and simplifying, we get a quadratic equation in x: 16*x^2-31750*x-9500000 = 0. Solving this using the quadradic formula we get two roots: x 1 = 15875/16 + (4875/16)*sqrt(17) = 2248.446245 and x 2 = 15875/16 - (4875/16)*sqrt(17) = -264.0712450 . These give corresponding y values of y 1 = 5875/3 + (1625/3)*sqrt(17) = 4191.682214 (for x 1) and y 2 = 5875/3 - (1625/3)*sqrt(17) = -275.015548 (for x 2). So, if you want only positive solutions you have x = 2248.446245 and y = 4191.682214. To solve these in EXCEL, just use the Solver tool (which comes with the package, but may need to be installed from the original CD). Since your equations are nonlinear, you should help EXCEL by giving an approximate starting point; you can also go into the Solver Options page and choose the type of numerical method to use. Suppose you have two cells, say A1 and A2 to hold the values of x and y respectively. (These should be data cells, not containing formulas; you can give a starting point directly into these cells.) Suppose you put the formulas for the left-hand-sides of equations 1 and 2 in cells B1 and C1, and the right-hand-sides in cells B2 and C2. You also need a fake objective to either maximize, minimize or set equal to a specified value. For example, you can just set the artificial formula '=0' in some cell, say in cell F1. Now open the Solver tool (found in the tools menu). The variable cells are A1:A2, the target cell is F1 (to be minimized, say), and you have two constraints which are entered as B1:C1 = B2:C2. Now (to be helpful) enter a trial starting point (such as 1 1 or 100 100 or 1000 1000 or 2000 4000) into the x and y cells; starting points closer to the final solution will give better results---faster convergence, more chance of convergence, etc---so probably the 2000 49000 choice is best. Then press Solve on the Solver menu. The default Solver tool can handle up to about 100 equations in 100 unknowns or more, but may have trouble for highly nonlinear systems of equations. R.G. Vickson === Subject: Re: Solving these 2 equations?? posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > Is it possibe to solve x and y with those 2 equations? > x(y+500)/(x+y+500) = 1520 > y(x+500)/(x+y+500) = 1660 > The values should be near 2249 and 4189. > My algebra is at the most elementary and very far away in term of years so if possible to solve this, please provide informations on the step taken. > Eventually I would like to make an Excel sheet to compute these equations. > Alain > Multiply both equations by (x+y+500) to get xy + 500y = 1660(x+y+500) > and xy + 500x = 1520(x+y+500), so subtracting these we get 500(y-x) = > 140(x+y) + 7000, hence y = (16/9)x + 1790/9. Substituting this for y > in the first equation and simplifying, we get a quadratic equation in > x: > 16*x^2-31750*x-9500000 = 0. Solving this using the quadradic formula > we get two roots: x 1 = 15875/16 + (4875/16)*sqrt(17) = 2248.446245 > and x 2 = 15875/16 - (4875/16)*sqrt(17) = -264.0712450 . These give > corresponding y values of y 1 = 5875/3 + (1625/3)*sqrt(17) = > 4191.682214 (for x 1) and y 2 = 5875/3 - (1625/3)*sqrt(17) = > -275.015548 (for x 2). So, if you want only positive solutions you > have x = 2248.446245 and y = 4191.682214. Excel Solver's answer is better. Actually, Maple's answer is better: x 1 = 15875/16 + (4875/16)*sqrt (17) and y 1 = 5875/3 + (1625/3)*sqrt(17). To 100 digits these are: x 1 = 2248.446245305380948773710815492101593599537 263981024976105833666802368902399744079296302206106380372 and y 1 = 4191.6822138762327978199303386526250552880662470773777 35299259852093100270932878363193426144189120661 . I just gave a 10- digit approximation before. Excel: 1520.00000000440 delta 0.00000000440 > Yours: 1519.99999987343 delta 0.00000012657 I don't know whether this is x or y, but in any case I don't see a value near 1520 anywhere above. When I use EXCEL 2002 to solve the problem, I get x = 2248.446245 and y = 4191.682215 R.G. Vickson > To solve these in EXCEL, just use the Solver tool (which comes with > the package, but may need to be installed from the original CD). > Since your equations are nonlinear, you should help EXCEL by giving an > approximate starting point; you can also go into the Solver Options > page and choose the type of numerical method to use. Suppose you have > two cells, say A1 and A2 to hold the values of x and y respectively. > (These should be data cells, not containing formulas; you can give a > starting point directly into these cells.) Suppose you put the > formulas for the left-hand-sides of equations 1 and 2 in cells B1 and > C1, and the right-hand-sides in cells B2 and C2. You also need a > fake objective to either maximize, minimize or set equal to a > specified value. For example, you can just set the artificial formula > '=0' in some cell, say in cell F1. Now open the Solver tool (found in > the tools menu). The variable cells are A1:A2, the target cell is > F1 (to be minimized, say), and you have two constraints which are > entered as B1:C1 = B2:C2. Now (to be helpful) enter a trial starting > point (such as 1 1 or 100 100 or 1000 1000 or 2000 4000) into the x > and y cells; starting points closer to the final solution will give > better results---faster convergence, more chance of convergence, > etc---so probably the 2000 49000 choice is best. Then press Solve on > the Solver menu. > The default Solver tool can handle up to about 100 equations in 100 > unknowns or more, but may have trouble for highly nonlinear systems of > equations. > R.G. Vickson === Subject: Re: Solving these 2 equations?? posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > Is it possibe to solve x and y with those 2 equations? > x(y+500)/(x+y+500) = 1520 > y(x+500)/(x+y+500) = 1660 > The values should be near 2249 and 4189. > My algebra is at the most elementary and very far away in term of years so if possible to solve this, please provide informations on the step taken. > Eventually I would like to make an Excel sheet to compute these equations. > Alain > Multiply both equations by (x+y+500) to get xy + 500y = 1660(x+y+500) > and xy + 500x = 1520(x+y+500), so subtracting these we get 500(y-x) = > 140(x+y) + 7000, hence y = (16/9)x + 1790/9. Substituting this for y > in the first equation and simplifying, we get a quadratic equation in > x: > 16*x^2-31750*x-9500000 = 0. Solving this using the quadradic formula > we get two roots: x 1 = 15875/16 + (4875/16)*sqrt(17) = 2248.446245 > and x 2 = 15875/16 - (4875/16)*sqrt(17) = -264.0712450 . These give > corresponding y values of y 1 = 5875/3 + (1625/3)*sqrt(17) = > 4191.682214 (for x 1) and y 2 = 5875/3 - (1625/3)*sqrt(17) = > -275.015548 (for x 2). So, if you want only positive solutions you > have x = 2248.446245 and y = 4191.682214. > Excel Solver's answer is better. Actually, Maple's answer is better: x 1 = 15875/16 + (4875/16)*sqrt > (17) and y 1 = 5875/3 + (1625/3)*sqrt(17). To 100 digits these are: > x 1 = 2248.446245305380948773710815492101593599537 > 263981024976105833666802368902399744079296302206106380372 and y 1 = > 4191.6822138762327978199303386526250552880662470773777 > 35299259852093100270932878363193426144189120661 . I just gave a 10- > digit approximation before. > Excel: 1520.00000000440 delta 0.00000000440 > Yours: 1519.99999987343 delta 0.00000012657 I don't know whether this is x or y, but in any case I don't see a > value near 1520 anywhere above. Sorry: 1520 is a right-hand-side of one of the equations. Of course, the exact solution (the quadratic formula) gives 1520 exactly, while evaluation of x and y to various levels of approximation will, of course, give various approximations to 1520. R.G. Vickson > When I use EXCEL 2002 to solve the > problem, I get x = 2248.446245 and y = 4191.682215 R.G. Vickson > To solve these in EXCEL, just use the Solver tool (which comes with > the package, but may need to be installed from the original CD). > Since your equations are nonlinear, you should help EXCEL by giving an > approximate starting point; you can also go into the Solver Options > page and choose the type of numerical method to use. Suppose you have > two cells, say A1 and A2 to hold the values of x and y respectively. > (These should be data cells, not containing formulas; you can give a > starting point directly into these cells.) Suppose you put the > formulas for the left-hand-sides of equations 1 and 2 in cells B1 and > C1, and the right-hand-sides in cells B2 and C2. You also need a > fake objective to either maximize, minimize or set equal to a > specified value. For example, you can just set the artificial formula > '=0' in some cell, say in cell F1. Now open the Solver tool (found in > the tools menu). The variable cells are A1:A2, the target cell is > F1 (to be minimized, say), and you have two constraints which are > entered as B1:C1 = B2:C2. Now (to be helpful) enter a trial starting > point (such as 1 1 or 100 100 or 1000 1000 or 2000 4000) into the x > and y cells; starting points closer to the final solution will give > better results---faster convergence, more chance of convergence, > etc---so probably the 2000 49000 choice is best. Then press Solve on > the Solver menu. > The default Solver tool can handle up to about 100 equations in 100 > unknowns or more, but may have trouble for highly nonlinear systems of > equations. > R.G. Vickson === Subject: Re: Solving these 2 equations?? ... > Is it possibe to solve x and y with those 2 equations? > x(y+500)/(x+y+500) = 1520 > y(x+500)/(x+y+500) = 1660 > The values should be near 2249 and 4189. ... > solutions you have x = 2248.446245 and y = 4191.682214. > > Excel Solver's answer is better. Excel: 1520.00000000440 delta 0.00000000440 > Yours: 1519.99999987343 delta 0.00000012657 Nonsense. (a) Neither set of answers you gave earlier is a good solution; and (b) the error for RV's solution is only due to roundoff. > Hit the [Solve] button and keep the results and you'll have > A1 2249.059639 > A2 4189.01372 > A3 1520 Evaluating x*(y+500)/(x+y+500) and y*(x+500)/(x+y+500) with x=2249.059639 and y=4189.01372 gives 1519.999999811... and 1659.8049..., which shows that your solution is ok for the first equation of the set, but off by about .2 for the second. You then changed the A3 formula and the target value (to reflect second equation) and got: > A1 2249.506637 > A2 4189.218186 > A3 1660 Evaluating the equations with x=2249.506637 and y=4189.218186 gives 1520.22564... and 1660.0000000098..., which shows that this solution is ok for the 2nd equation of the set, but off by about .2 for the first. Thus, neither set of numbers that you gave earlier (as Excel results) actually solves the two equations simultaneously. Re (b): Less-rounded values of x, y from RV's method (that is, x= 15875/16 + (4875/16)*sqrt(17) and y=x*16/9 + 1750/9) are x=2248.44624530538 and y=4191.68221387623, which give values like 1519.99999999999927 and 1659.99999999999921, solving the two equations simultaneously. -- jiw === Subject: Re: Solving these 2 equations?? > ... > Is it possibe to solve x and y with those 2 equations? > x(y+500)/(x+y+500) = 1520 > y(x+500)/(x+y+500) = 1660 > The values should be near 2249 and 4189. > ... > solutions you have x = 2248.446245 and y = 4191.682214. > > Excel Solver's answer is better. > Excel: 1520.00000000440 delta 0.00000000440 > Yours: 1519.99999987343 delta 0.00000012657 > > Nonsense. (a) Neither set of answers you gave earlier is a > good solution; and (b) the error for RV's solution is only > due to roundoff. > > Hit the [Solve] button and keep the results and you'll have > A1 2249.059639 > A2 4189.01372 > A3 1520 > > Evaluating x*(y+500)/(x+y+500) and y*(x+500)/(x+y+500) with > x=2249.059639 and y=4189.01372 gives > 1519.999999811... and 1659.8049..., which shows that your > solution is ok for the first equation of the set, but off > by about .2 for the second. You then changed the A3 formula > and the target value (to reflect second equation) and got: > A1 2249.506637 > A2 4189.218186 > A3 1660 > > Evaluating the equations with x=2249.506637 and y=4189.218186 > gives 1520.22564... and 1660.0000000098..., which shows that > this solution is ok for the 2nd equation of the set, but off > by about .2 for the first. > > Thus, neither set of numbers that you gave earlier (as Excel > results) actually solves the two equations simultaneously. > > Re (b): Less-rounded values of x, y from RV's method (that > is, x= 15875/16 + (4875/16)*sqrt(17) and y=x*16/9 + 1750/9) > are x=2248.44624530538 and y=4191.68221387623, which give > values like 1519.99999999999927 and 1659.99999999999921, > solving the two equations simultaneously. My HP49+ quickly gives x = 2248.44624530 y = 4191.68221394 which are accurate to over 11 significant digits, for a solution near 2249 for x and and 4189 for y. -- Virgil === Subject: Re: Solving these 2 equations?? posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > ... > Is it possibe to solve x and y with those 2 equations? > x(y+500)/(x+y+500) = 1520 > y(x+500)/(x+y+500) = 1660 > The values should be near 2249 and 4189. > ... > solutions you have x = 2248.446245 and y = 4191.682214. > Excel Solver's answer is better. > Excel: 1520.00000000440 delta 0.00000000440 > Yours: 1519.99999987343 delta 0.00000012657 Nonsense.(a) Neither set of answers you gave earlier is a > good solution; and (b) the error for RV's solution is only > due to roundoff. > Hit the [Solve] button and keep the results and you'll have > A1 2249.059639 > A2 4189.01372 > A3 1520 Evaluating x*(y+500)/(x+y+500) and y*(x+500)/(x+y+500) with > x=2249.059639 and y=4189.01372 gives > 1519.999999811... and 1659.8049..., which shows that your > solution is ok for the first equation of the set, but off > by about .2 for the second.You then changed the A3 formula > and the target value (to reflect second equation) and got: > A1 2249.506637 > A2 4189.218186 > A3 1660 Evaluating the equations with x=2249.506637 and y=4189.218186 > gives 1520.22564... and 1660.0000000098..., which shows that > this solution is ok for the 2nd equation of the set, but off > by about .2 for the first. Thus, neither set of numbers that you gave earlier (as Excel > results) actually solves the two equations simultaneously. Re (b):Less-rounded values of x, y from RV's method (that > is, x= 15875/16 + (4875/16)*sqrt(17) and y=x*16/9 + 1750/9) > are x=2248.44624530538 and y=4191.68221387623, which give > values like 1519.99999999999927 and 1659.99999999999921, > solving the two equations simultaneously. -- > jiw- Hide quoted text - - Show quoted text - Results 1 - 10 for x(y+500)/(x+y+500) = 1520 y(x+500)/(x+y+500) = 1660. (0.21 seconds) Musatov Search Resistor maths - some basic help - Page 2 - All About Circuits Forum6 posts - 3 authors - Last post: 21 hours ago xy+500x = 1520(x+y+500) xy+500y = 1660(x+y+500) subtract them to get rid of the nasty xy term, this gives you an equation relating x and y which you can ... http://forum.allaboutcircuits.com/showthread.php?p=149976 - 21 hours ago 1000 D$=#### ####^: E$=#### ####: P1$=PLTL: P2$=PLTT 1010 ...... 1480 &USING E$,X;Y 1490 NEXT X 1500 & P2$ 1510 & P1$ 1520 FOR Y=500 TO 9500 ... X;Y : GOTO 1640 1630 & USING D$,X;Y 1640 NEXT Q 1650 NEXT W 1660 &P2$ ... http://www.bitsavers.org/pdp-11.trailing-edge.com/rsts11/rsts-11-067/PFNPLT. BAS Discussions - sci.math | Google Groups10 posts - 9 authors +y+500) = 1520 y(x+500)/(x+y+500) = 1660 The values should be near 2249 and 4189. ... 1000 E$=#### ####: D$=#### ####^: P1$=PLTL: P2$=PLTT: P3 ... 1510 INPUT #2%, T1% 1520 IF T1%=1% THEN INPUT NEW UPPER LIMIT (Y);M2 1530 ... 1620 &USING E$,X%;Y% 1630 NEXT T% 1640 &USING E$, 0,0 1650 &P2$ 1660 &DO YOU ... X;Y 1810 NEXT X 1820 &P2$ 1830 & P1$ 1840 FOR Y=500 TO 9500 STEP 9000 1850 ... http://www.bitsavers.org/pdp-11.trailing-edge.com/rsts11/rsts-11-067/DATPLT. BAS Resistor maths - some basic help - Page 2 - All About Circuits Forum4 posts - 2 authors It's fairly straightforward, if the resistances are x and y then x(y +500)/(x+y+500) = 1520 y(x+500)/(x+y+500) = 1660. I was lazy and solved them iteratively ... http://forum.allaboutcircuits.com/showthread.php?p=149918 this is the formula: x(y+500)/(x+y+500) = 1520 x should be near 2249 ... would be between 1520 and 1660 Ohms when the variable resistor is turned ... http://www.mathhelpforum.com/math-help/pre-algebra-algebra/92650-not-rocket- science-but-i-am-stuck.html Math Forum DiscussionsJun 13, 2009 ... Is it possibe to solve x and y with those 2 equations? x(y+500)/(x+y+500) = 1520 y(x+500)/(x+y+500) = 1660 ... http://mathforum.org/kb/thread.jspa?threadID=1954812&tstart=0 - 2 hours ago 5 ON ERROR GOTO 1650 6 X1$=SYS(CHR$(6%)+CHR$(-7%)) 7&WRITTEN BY ...... 970 X1=X-500:Y1=Y-500:Z1=Z-500 980 IFX1=0THENB =60 ELSEIFSGN (X1)=1THENB =2 ... 1510 IFA3=3THENP2=6:R2$=PHOTON TORPEDOES 1520 IFR1=20THEN&SPOCK: KLINGON ... IFERR=28THENRESUME1660ELSEIFERR=15THENRESUME1670ELSEONERRORGOTO0 1660 ... http://www.bitsavers.org/pdp-11.trailing-edge.com/rsts11/rsts-11-102/RINNY2. BAS rentry21.bas(Y/N/); A$ 50 IF A$ = Y THEN 70 60 IF A$ = N THEN 210 ELSE 40 70 INPUT .... .33 1060 X = 500 - XA 1070 Y = 50 - YA 1080 LINE (500, 50)-(X, Y), ... ((VV * ABS(VV)) * CV)) 1510 HTH = .5 * HV * DH * ECFB / AF 1520 HTV = .5 * VV * DV ... AR = AN / 57.32 1660 SI = SIN (AR) 1670 CO = COS(AR) 1680 CH = AF * SI 1690 DH ... http://web.wt.net/~markgoll/mg14.htm 50 PRINT CHR$(26):REM WIDTH 80 100 PRINT TAB(33);QUBIC:PRINT 110 ... 610 GOTO 500 620 LET X(M)=1 630 GOSUB 1640 640 J=1 650 I=1 660 IF J=1 THEN ... 1510 INPUT X$ 1520 IF LEFT$(X$,1)=Y THEN 400 1530 IF LEFT$(X $,1)=N THEN ... 1630 RETURN 1640 FOR S=1 TO 76 1650 LET J1 = M(S,1) 1660 LET J2=M(S,2) 1670 ... http://www.vintage-basic.net/bcg/qubit.bas === Subject: Re: Solving these 2 equations?? > Is it possibe to solve x and y with those 2 equations? > x(y+500)/(x+y+500) = 1520 > y(x+500)/(x+y+500) = 1660 > The values should be near 2249 and 4189. ... > Multiply both equations by (x+y+500) to get xy + 500y = 1660(x+y+500) > and xy + 500x = 1520(x+y+500), so subtracting these we get 500(y-x) = > 140(x+y) + 7000, hence y = (16/9)x + 1790/9. Substituting this for y in > the first equation and simplifying, we get a quadratic equation in x: > 16*x^2-31750*x-9500000 = 0. Solving this using the quadradic formula we > get two roots: x_1 = 15875/16 + (4875/16)*sqrt(17) = 2248.446245 and > x_2 = 15875/16 - (4875/16)*sqrt(17) = -264.0712450 . These give > corresponding y values of y_1 = 5875/3 + (1625/3)*sqrt(17) = > 4191.682214 (for x_1) and y_2 = 5875/3 - (1625/3)*sqrt(17) = -275.015548 > (for x_2). So, if you want only positive solutions you have x = > 2248.446245 and y = 4191.682214. I got the same results by similar method, except for probable typo in the above; 1790 should be 1750, ie, y = (16/9)x + 1750/9 -- jiw === Subject: Rational varieties and Galois cohomology posting-account=R7AgUAoAAADVFAtIe36IBmgohoHjZsKW Gecko/2009032712 Ubuntu/8.10 (intrepid) Firefox/3.0.8,gzip(gfe),gzip(gfe) I would be very glad if someone could explain me the connection between the rationality of a variety and the Galois cohomology of the associated function field. To be more precise, is it possible to determine weather a variety is rational with the Galois cohomology of the associated function field? Is it at least possible to get one direction, i.e. the Galois cohomology is of a specific form --> the variety is (not) rational? If this is not possible in general, is it possible in special cases, for dim=2, for example? Sancho === Subject: Re: Rational varieties and Galois cohomology posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) I would be very glad if someone could explain me the connection > between the rationality of a variety and the Galois cohomology of the > associated function field. To be more precise, is it possible to > determine weather a variety is rational with the Galois cohomology of > the associated function field? Is it at least possible to get one > direction, i.e. the Galois cohomology is of a specific form --> the > variety is (not) rational? If this is not possible in general, is it > possible in special cases, for dim=2, for example? Sancho I do not know. But it is good to ask a question when you do not understand something. === Subject: chwistek and tarski on nonlogical symbols [can argt erros] MoeBlee brought light with:: > > you are disingenuous > > You've not shown me to have been disingenuous. and it's annoying that you are disingenuous because you are also very intelligent and have been learning more tolerant positions over the past few years and i've thought you might be growing out of it and even in this thread you were one of the few that even leaned away from a single specification early on but still it comes out you insist on specific interpretations of specific forms you have posted in order to assert that your statements were correct you are very concerned about being correct and you will turn a thread into an insisted demonstration of your correctness in some context turning tangents from reasoned analysis of the points to demonstrations of the existence of some context that makes your post meaningful and correct when you get into these states your point isn't to further the discussion your point is to assert correctness even if your correctness is irrelevant and inconsequential to the academic resolution of the topic at hand i think few people would argue your ability to learn new things correctly i think you've demonstrated that over your years here but sometimes the correctness of your assertions is irrelevant to the discussion and you alienate the other position which may be making valid points even without proper reference > you act as if the entire argument > is simplistic namecalling and ignorance of topic > when you need the straw man for conflagration > > I haven't acted as if lwal's entire argument is simplistic > namecalling and ignorance of topic. Though he has engages in those > and I've rebutted him for it. Meanwhile, also I've engaged a wide > range of his gambits. which you eventually call lies with many of your EMPHASIS CAPS and dig on nam's ability to learn in separate posts you want to burn others both sides here are talking past each other and both sides are in a foul mood genuine interest in rigorisation could be a way out for both sides or it could just peter out from exhaustion-of-the-anger-that-will-one-day-rise-again when i challenge your genuineness i am challenging you to post relevantly and not simply outburst and accusation i am challenging you to move the discussion forward > you attempt superiority poses > veiled in moral righteousness > > I don't claim to any special moral righteousness. > > you continue to spew bile like > As to Nam, I guess it's hopeless to try to bring him > to reason on this or many another subject. > > You have a quite elastic definition of 'bile'. > > sad resignation sad resignation? because you feel the other side has nothing to offer discussion? because your interaction with them was of no benefit? because you are so better off in state space than the other party? bring him to reason??? aha.. the savior with true sight! (one of the problems is that you don't realise just how much your awareness and interaction are founded on presumptions of position over others) > and > while i'm joining the pile-on you guys love so much > you rarely lend relevant details to a discussion > > I've been EXTREMELY detailed as to the particulars of this discussion. you can do detail i'm not ribbing at you for detail relevance is the one you have difficulty in > preferring to play toady to some game you refuse to call bully > (it's just assertions of lies and ignorance, right?) > > I very much admire certain posters here. That doesn't make me their > toady. In fact, among the posters here who I've expressed my > admiration of, I have many times posted certatin disagreements with > them and stood my ground when I felt my view deserved not to be > dismissed. let's expand context since apparently some of symbols were not specified in axiom :p preferring to play toady to some game you refuse to call bully i think you are toady to the game not to fellow posters i think you flatter the sanctity of truth while going off on tangents unrelated to it's progress you want to be the sidekick to authoritative logic and are eager to beat anyone up who dares speak in error or without authority > So your post falls flat again. i know it always will i'm resigned to that in the same bitter sadness you speak of i am sure in this discussion the formalisation of specifying models systems and general dynamic data structures is found in the theory of compilers and processors and in this language several of nam's points make quite a bit of sense specifying syntactic roles and specifying syntactic assertions actually doesn't have a very clear formal distinction (they are interchangeable interpretations of type theory) so either way creating a parser for the lone axiom doesn't give a system strong enough for the result on new symbols (even with rules of inference... - i am not specifying a foundational computation model because it really doesn't matter for the point) specifying a parse pattern is required for every new symbol of course you've mentioned some of this yourself even getting a question from chris on the usefulness of the freedom of order you mentioned and yes i realise jack that in your context it was already assumed and you and nam keep talking past each other because these unrelated points keep getting made and little progress is made on discussing the formalisation the formalisation here is that of compilers or processors compilers take texts in a specified language and translate them to texts in other languages processors take texts in languages and produces runtime state transitions possibly interacting with i/o interfaces structured information content and the formal specification of these has almost complete flexibility: ------------------------------- any given information structure may be translated to any given executable in a lambda complete impl ------------------------------- in fact you really only need if and a few structure identifying primitives if (text0) output(program0) else if (text1) output(program1) .. so let's call this result 1 which can be paraphrased: there are no logical requirements on textual order for presenting any language but nam has made the point several times that you need to axiomatise all symbols and of course there is a really clear sense in which this is true: you have to provide a functional role to all participants otherwise it has no input into the participants and thus no control in the system really there has to be a fundamental agreement established on every symbol-role application (which is commonly called scope in computer science) to give an example of a published logician who might have some authority (he was one of the second edition reason for modifications to russell and whitehead's principia mathematica ) was chwistek in his the theory of constructive types he explicitly calls for scope operators much as would eventually show up in lambda calculus for introducing new names and gave what have become classic examples for ambiguities without scope chwistek points out this need for every functional operator in the language and tarski would often point this out when arguing over with carnap and quine in the vienna circle days In the discussions the ïanalytic' vs. ïsynthetic' opposition came to the fore repeatedly. In reply to comments by Tarski to the effect that one could not formulate the distinction sharply, Carnap explained that it was possible to include already in the metalanguage (Syntax) certain signs which are usually seen as ïdescriptive' (house, temperature and so on) and to use them as ïlogical signs' so that the sentences in which these terms occur would be determined on the basis of the language rules. He added that it is really a question how.84on the basis of the ïfeeling' which gives rise to the opinion one could draw a boundary between words of the German language such as ïund', ïoder' etc. on the one side and ïTemperatur', ïHaus' on the other.84one could arrive at a formulation which would allow us to distinguish rigorously ïdescriptive' and ïlogic' and then also ïsynthetic' and ïanalytic'. This is a problem that can be discussed only in rigorous form if we want to avoid misunderstandings of all sorts. (Neurath 1936, pp. 388[CapitalEth] 389).3 and tarski's on the concept of logical consequence specifically points out there is no known way to distinguish logical and nonlogical constants it's all the same process of specification the lines drawn do not have consequence epsilon could be a logical dynamic operator or called nonlogical and the foundation of set theory it's all the same dynamic specified paolo mancosu's harvard 1940-1941: tarski, carnap, and quine on a finitistic language of mathematics for science gives a great overview of this discussion and tarski's papers in general well... i can't do more to promote all his work his i think (and admire) was a theory that will only be appreciated more over the years (though strangely i will never love his work as much as poincare's because i think tarski scares me with his honesty and poincare could drink away the scary things) so result 2 might be stated axiomitisation and language specification is the same process and continuing with the rigorisation we could continue to specify how modern compilers specify languages and the various orders of specification that are common when thinking in various problem domains a discussion focussed on rigorising the current topics might mention BNF and extensions and generalisations for syntax construction or turing completeness or how the lambda calculus specifies application scope or why many of the classification in classical logic texts are the result of historical arguments that didn't survive the rigorisation or how our language evolution is often the result of seeing our own historical ignorance (because although change isn't guaranteed to lift us out of our ignorance it is still a useful technique to at least try when ignorance is identified) maybe you are genuinely interested in discussion sometimes i like to think so but i'd have to see some move of the thread in the right direction before i eat any crow over what i posted -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Rotation is motion gravity posting-account=5ApcPgoAAABKcgEyKsQmJVb3Rz63IGGL .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; WWTClient2),gzip(gfe),gzip(gfe) But it acts as gravity's opposite creating outward weight that can Mitch Raemsch === Subject: Re: Rotation is motion gravity posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > But it acts as gravity's opposite creating outward weight that can Mitch Raemsch >TOPEX/POSEIDON, GRAVITY, AND THE EARTH'S ROTATION: INVESTIGATING ... >TOPEX/POSEIDON, GRAVITY, AND THE EARTH'S. ROTATION: INVESTIGATING MECHANISMS AND ... polar motion >results are summarized here. Atmospheric wind and pressure ... Cervical range of motion measuring ...The apparatus according to claim 1 > and a third gravity >type angle meter ... of cervical motion about at least the axis of cervical >rotation of the head >of a ... >Rotation - rotation is a movement of an object in a circular >motion. .... parallel axes for each >gondola, where the rotation is opposite, by gravity or >mechanically. ... >developed the concept of motion in terms of velocity (speed and direction) ... develop a dynamic >description that involves the underlying influence (gravity) .... Additionally the orbit is >resonant with the rotation period of the earth, ... M>odified Gravity (MOG) has been used successfully to explain the rotation curves of galaxies, the >motion of galaxy clusters, ... > === posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) please don't top-post. Put your reply after what you are replying to. I've re-arranged your post. Use your *head* please: *read* [> [Quote me: A single life is not worth all the money in the world. -- > Martin Musatov] It would be handy if we could communicte by mental telepathy. Is AT&T suppressing the result? No, but you suggesting I do not top-post *is* -- get it.]] Use your *head* that thing on *top* of *your* *neck*. > P=/=NP is a means the elite use to place cryptography above the value > of human life. Call me anything you want, I believe this statement as > a reasonable man. What's more important to you: a child dying of > cancer or your ATM card? > and there was I thinking it was merely a mathematical result. > How does P!/NP place cryptography above human life? I know some > crytography may depend on this (unproven) result being true; but > are you [seriously] suggesting some sort of conspiracy to *prevent* > this result being disproved? You don't sound very reasonable to me. > I'd be careful. If they whole edifice of capitalism is now dependent > on P=/=NP staying the way it is then reasonable men should beware > of the elite sending the black helicoptors after them. the above was (mild) sarcasm > I never said the whole edifice of capitalism is now dependent on P=/ > =NP staying the way it is, I merely identified a push/pull between > cryptography and advances in biology. I missed the biology bit > these problems no ones credit card would be safe. I could care less > about my credit card. I would rather have *protein folding*. Yes but you imply some sort of conspiracy. The Elite are trying to prevent or discourage the proof that P/=NP. This is done to protect cryto systems that depend on it staying unproved. You claim this makes protein folding a harder problem. I don't believe in this conspiracy. It's simply difficult to prove. I believe a large cash prize is onoffer if someone can prove this result (or disprove it). > [Quote me: A single life is not worth all the money in the world. -- > Martin Musatov] It would be handy if we could communicte by mental telepathy. Is AT&T suppressing the result? No, but you suggesting I do not top-post *is* -- get it. > Musatov said: > I hereby retract and apologize for my crude and misleading > comments regarding Mr. Heathfield. > Apology accepted. As far as I'm concerned, the matter is now closed. > -- > Richard Heathfield Email: -http://www. +rjh@ > Usenet is a strange place - dmr 29 July 1999 > === Subject: Re: Integral brick disproof from last month? This thread is going to die if nothing is done to it (it's an intuition of mine not a proof). So let's have a look at the situation now, Gerry is well informed about this kind of problems, OwlHoot knows a lot about the classical approach, Bleuprint has insane(*) variables (and don't understand the mathematical violence of OwlHoot) and Bart is not sure if it's so insane at all. Why not try to disprove(**) this problem altogether? Going to the moon was teamwork as all big achievements in the last half century. If successful it will probably be not the first internet proof ( I don't know) but surely it will not be the last one, and in fact it's nicer than computerproofs and it's also amusement. 1) the classical approach to this problem is like ... but up to now it has neither been proved nor disproved with this classical approach. 2) thus if there is a disprove it will probably be with a non classical approach (no solution at all for the finite possibilities). So my proposal is to continue in the direction of the first reply that comes in. To be or not to be that's no question, enjoy it. bleuprint or patrick if you prefer PS: *) The insane variables help to keep in touch with the original problem, and they are also inspired by a visual approach to the problem. It's imaginable in 3 dimensions and the Heronian square sided triangle reduces it into 2 dimensions. By the way the tangents of the incircle (it's radius must be an integer) divide each side ab^2 (or ac^2 or bc^2) into two parts a^2 and b^2 (or a^2 and c^2 or b^2 and c^2)). The altitude on the smallest side must also be an integer (look to the parametric equations for the sides of a Heronian triangle) **) I suggest the disprove option but it can as well go the other way. === Subject: Re: Integral brick disproof from last month? > Bart is not sure if it's so insane at all. Correction: Bart is not sure if it's _short_ of insane. I've spent a certain amount of time working on a similar problem (rational sided triangles with rational median and area) and I'm pretty sure the algebraic geometry approach that Ramsden describes is the right one. And I'm pretty sure that messing about with high-school algebra- type manipulations won't lead anywhere. Bart -- Cheerfully resisting change since 1959. === Subject: Re: Integral brick disproof from last month? > What?! Denoting a single variable by ab, or abc in a problem like > this (or anywhere), is little short of insane! I'm not so sure... B. -- Cheerfully resisting change since 1959. === Subject: Re: Integral brick disproof from last month? <19409201.1780.1244716013889.JavaMail.jakarta@nitrogen.mathforum.org> What?! Denoting a single variable by ab, or abc in a problem > like this (or anywhere), is little short of insane! > > I'm not so sure... I assume you (BG) are unsure that such notation falls short of insanity, and have the notion that it doesn't fall short. That could well be; but in many cases, stupidity is probably a better explanation than insanity. -- jiw === Subject: Re: Integral brick disproof from last month? > > What?! Denoting a single variable by ab, or abc in a problem > like this (or anywhere), is little short of insane! > > I'm not so sure... > > I assume you (BG) are unsure that such notation falls > short of insanity, and have the notion that it doesn't > fall short. That could well be; but in many cases, > stupidity is probably a better explanation than insanity. > Well, one definition of insanity (supposedly Einstein's) is doing the same thing over and over again and expecting different results. How many versions of this post have we seen? B. -- Cheerfully resisting change since 1959. === Subject: Re: point charges on a disc Configurations for 17 <= n <= 36 are discussed, including a few improvements. Also improvements are given for n = 55, 59 and 64. -------------------------------------------------------- Two references which show conjecturally optimal configurations and list their Coulomb energies: > Minimum-energy point charge configurations on a circular disk, > K. J. Nurmela, J. Phys. A 31 (1998) 1035-1047. [A version of slightly rotated compared to the way they are shown in the Journal > of Physics. When talking about Nurmela's configurations below, I will > be referring to the way they appear in the Journal of Physics.] Distribution of point charges on a thin conducting disk, H. Oymak > and S. Erkoc, Int. J. Mod. Phys. C 11 (2000) 891-900. -------------------------------------------------------- For n < 12, the points lie on the boundary of the disk, at vertices of a regular n-gon; for 12 <= n < 17, one point lies at the center of the disk and the others lie on the boundary of the disk, at vertices of a regular (n - 1)-gon. ------------------------------------------- n = 17: Two configurations, both having symmetry group D_1, are shown at Their Coulomb energies are: left 133.816523765745... right 133.816523765733... The latter configuration, shown by Nurmela and presumably optimal, has only one point on its axis of symmetry. The former configuration has three points on its axis of symmetry, and is similar to the configuration shown by Oymak and Erkoc. But they give an energy of 133.82742 for their configuration, and thus their configuration is slightly different. [Unfortunately, for 17 <= n <= 80, all energies listed by Oymak and Erkoc are larger than those given by Nurmela. Thus, their configurations are suboptimal, and so will not be considered here further.] The case n = 17 is the first tricky one, the difference between the energies of the two optimized symmetric configurations being only about 1.2 * 10^-11. As noted previously, clearly distinct configurations can have energies which are indistinguishable unless calculations are done with high accuracy! ------------------------------------------- 18 <= n <= 28: With the exception of n = 23 and 28, for which improved configurations were given here previously, all configurations shown by Nurmela are presumably optimal. Their presumable symmetry groups can be observed in his figures: 18, D_2; 19, D_1; 20, D_1; 21, D_3; 22, D_2; 24, D_4, 25, D_5; 26, D_1; 27, D_1. > Both configurations of 23 points shown at vertical. The configuration on the left is, as best I can tell, the > same as Nurmela's (after a rotation of approx. 3/4 pi); note that > none of its interior points lies on its axis of symmetry. The > configuration on the right is mine; two interior points lie on its > axis of symmetry. The Coulomb energies for the configurations are as > follows: > left 265.2010404355045250722... > right 265.2010404355045250705... > Note how small the difference between these energies is! (Also note > that energies of configurations were calculated in a computer > algebra system, allowing arbitrary precision.) > Both configurations of 28 points shown at vertical and horizontal. The configuration on the left is, as best I > can tell, the same as Nurmela's; note that four points lie on its > vertical axis of symmetry. The configuration on the right is mine; > only two points lie on either of its axes of symmetry. The Coulomb > energies for the configurations are as follows: > left 411.3443144520681... > right 411.3443144520678... > Note again how small the difference between energies is. > Concerning symmetries, Nurmela is justifiably cautious, saying Many > of the configurations seem to have symmetries. However, since the > optimization process can provide the coordinates of the points only > with limited accuracy, it is difficult to say for certain from the > numerical results directly when a configuration has a symmetry. But > I can say for certain that the four configurations, shown in the two > figures above, have the claimed symmetries because the configurations > were produced by requiring the stated symmetries _a priori_. Nurmela > goes on to say It is not precluded that an optimal (or best currently > known) configuration is only _almost_ symmetrical. I agree that my > symmetric configurations for 23 and 28 points might not be optimal. But > I suspect that, now, all of the best currently known configurations > for n <= 28 are indeed optimal and have at least D_1 symmetry. ------------------------------------------- n = 29 and 30: Nurmela's configurations do not appear to be precisely symmetric. That could be due either to slight inaccuracy in reproduction of the figures or to the configurations having not been fully optimized. In any event, his configurations are sufficiently close to the symmetries which seem to minimize the energies that I do not wish to claim that I have found improved configurations, but rather that I am merely clarifying what his configurations are. Of the two possible D_1 configurations for n = 29, the one having the smaller energy, 444.54775370073690197238114646307..., has three points on its axis of symmetry. The other configuration has only one point on its axis of symmetry and its energy is larger by only about 10^-28. Of the two possible D_1 configurations for n = 30, the one having the smaller energy, 479.079572646178650625447..., has four points on its axis of symmetry. The other configuration has only two points on its axis of symmetry and its energy is larger by only about 10^-20. ------------------------------------------- n = 31: Nurmela's configuration is likely optimal and appears to have symmetry group D_6. ------------------------------------------- > Both configurations of 32 points shown at that shown by Nurmela; note that four points lie on its axis of > symmetry. The configuration on the right is mine; only two points > lie on its axis of symmetry. The Coulomb energies for the > configurations are as follows: > left 552.26974429517636551032047633... > right 552.26974429517636551032047600... > The energy of my configuration is smaller than > that of the other by about 3.3 * 10^-25. Both configurations of 33 points shown at that shown by Nurmela (after a rotation of about 14/25 pi) and the > configuration on the right is mine. The Coulomb energies for the > configurations are as follows: > left 590.806298314334497099501715099... > right 590.806298314334497099501715089... > The energy of my configuration is smaller than > that of Nurmela's by about 9.4 * 10^-27. ------------------------------------------- n = 34, 35 and 36: Conjecturally optimal configurations are shown at . Of the two possible D_1 configurations for n = 34, the one having the smaller energy, 630.83438177277924627005987137145907..., has two points on its axis of symmetry. The other configuration has four points on its axis of symmetry, and its energy is larger by about 6 * 10^-32. Nurmela's configuration is slightly asymmetric. Of the two possible D_2 configurations for n = 35, the one having the smaller energy, 672.34218070354591509699..., has five points on one of its axes of symmetry. The other configuration, which resembles Nurmela's, has no more than three points on any axis of symmetry and its energy is larger by about 10^-16. Of the two possible D_1 configurations for n = 36, the one having the smaller energy, 715.06832395696682420973075373142721..., has two points on its axis of symmetry. The other configuration, which Nurmela's more closely resembles, has four points on its axis of symmetry, and its energy is larger by about 7 * 10^-31. ------------------------------------------- n = 55, 59 and 64: Conjecturally optimal configurations are shown at . For n = 55, there are four possible D_1 configurations. Of those, the smallest energy is 1791.97366856541826972... Another configuration, which resembles Nurmela's, has an energy which is larger by about 2 * 10^-11. For n = 59, there are four possible D_1 configurations. Of those, the smallest energy is 2083.03344817551530109... Another configuration, which resembles Nurmela's, has an energy which is larger by about 5 * 10^-15. For n = 64, Nurmela's configuration is asymmetric and I doubt that it is optimal. Of the four possible D_1 configurations, the smallest energy is 2478.98097382484384441... ------------------------------------------- David W. Cantrell === Subject: Re: point charges on a disc <20090601141632.994$QL@newsreader.com> <20090605184036.557$92@newsreader.com> posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2009060310 Ubuntu/8.10 (intrepid) Firefox/3.0.11,gzip(gfe),gzip(gfe) > In this post, improved configurations are given for 32 and 33 point > charges on a disk, along with some additional comments. > Configurations of 23 and 28 point charges on a disk > Minimum-energy point charge configurations on a circular disk, > K. J. Nurmela, J. Phys. A 31 (1998) 1035-1047 shows conjecturally > optimal configurations and lists their Coulomb energies. [A version > but be cautioned that many configurations shown there are > slightly rotated compared to the way they are shown in the Journal of > Physics. When talking about Nurmela's configurations below, I will be > referring to the way they appear in the Journal of Physics.] > energies: Distribution of point charges on a thin conducting disk, > H. Oymak and > S. Erkoc, Int. J. Mod. Phys. C 11 (2000) 891-900. > Both configurations of 23 points shown at > have symmetry group D 1, and are presented with their axes of > symmetry vertical. The configuration on the left is, as best I can > tell, the same as Nurmela's (after a rotation of approx. 3/4 pi); > note that none of its interior points lies on its axis of symmetry. > The configuration on the right is mine; two interior points lie on > its axis of symmetry. The Coulomb energies for the configurations > are as follows: > left265.2010404355045250722... > right 265.2010404355045250705... > Note how small the difference between these energies is! (Also note > that energies of configurations were calculated in a computer > algebra system, allowing arbitrary precision.) > Above, I referred to a configuration as mine. But I have no way of > knowing whether perhaps it was discovered by someone previously. > Nurmela states the energy merely as 265.20104 and says The best > previously published energies for configurations with n <= 23 ... > coincide with the values in our table. But, with few decimal places > in the stated energy and without precise figures or descriptions of > configurations found by others, it is impossible to tell whether my > configuration is actually new or not. > Curiously, although my configuration of 23 points resembles that of > Oymak and Erkoc, they give 265.20544 as the energy. Thus, their > configuration and mine are slightly different. > Both configurations of 28 points shown at > have symmetry group D 2, and are presented with their axes of > symmetry vertical and horizontal. The configuration on the left is, > as best I can tell, the same as Nurmela's; note that four points lie > on its vertical axis of symmetry. The configuration on the right is > mine; only two points lie on either of its axes of symmetry. The > Coulomb energies for the configurations are as follows: > left411.3443144520681... > right 411.3443144520678... > Note again how small the difference between energies is. And for 28 > points, just as for 23 points, it is impossible to tell whether my > configuration is actually new or not. > (The configuration of 28 points shown by Oymak and Erkoc seems to be > the same as that shown by Nurmela. But again, their listed energy is > not minimal.) > The moral of the story thus far is that clearly distinct > configurations can have energies which are indistinguishable unless > calculations are done with high accuracy! > Concerning symmetries, Nurmela is justifiably cautious, saying Many > of the configurations seem to have symmetries. However, since the > optimization process can provide the coordinates of the points only > with limited accuracy, it is difficult to say for certain from the > numerical results directly when a configuration has a symmetry. But > I can say for certain that the four configurations, shown in the two > figures above, have the claimed symmetries because the configurations > were produced by requiring the stated symmetries a priori . Nurmela > goes on to say It is not precluded that an optimal (or best > currently known) configuration is only almost symmetrical. I > agree that my symmetric configurations for 23 and 28 points might > not be optimal. But I suspect that, now, all of the best currently > known configurations for n <= 28 are indeed optimal and have at > least D 1 symmetry. > It seems likely that some of the other configurations shown by > Nurmela, for 28 < n <= 80, are not quite optimal, but that is > difficult to establish without many more decimal places being stated > for the energies or more precise information being given about the > configurations themselves. > What is the smallest n for which the optimal configuration of n point > charges on a disk is asymmetric? > Both configurations of 32 points shown at > have symmetry group D 1. The configuration on the left is the same as > that shown by Nurmela and by Oymak and Erkoc; note that four points > lie on its axis of symmetry. The configuration on the right is mine; > only two points lie on its axis of symmetry. The Coulomb energies for > the configurations are as follows: > left552.26974429517636551032047633... > right 552.26974429517636551032047600... > The energy of my configuration is smaller than > that of the other by about 3.3 * 10^-25. > Both configurations of 33 points shown at > have symmetry group D 1. The configuration on the left is the same as > that shown by Nurmela (after a rotation of about 14/25 pi) and the > configuration on the right is mine. The Coulomb energies for the > configurations are as follows: > left590.806298314334497099501715099... > right 590.806298314334497099501715089... > The energy of my configuration is smaller than > that of Nurmela's by about 9.4 * 10^-27. > (Also, although my configuration resembles that of Oymak and Erkoc, > they give 590.80917 as the energy. Thus, their configuration and mine > are slightly different.) > David W. Cantrell references too. I find, however, that i am still failing to obtain an intuitive proof for > why n=12 is the critical number.As a friend pointed out to me, 6 would > possibly be a more obvious choice because a hexagon would have distances > equal to the radius.So why does it come out at twice this number.There > has to be a geometric argument that works with the physical interpretation > of a harmonic potential.I've tried all sorts of explanations using various > Guass surface configurations, looking at the flux and fields etc....but i > keep coming up short.Do you know if anyone has presented an informal proof > of why n=12 based along physical argument alone? David Cantrell makes a more informed response than I can (above in thread), but if you are intrigued enough to want to calculate it for yourself, why not compute the potential for the arrangements with equally distributed charges at the boundary for n = 7,...,13 and compare with the corresponding arrangements of one point at the center and n-1 equally spaced around the boundary? Would that serve your curiousity? Note that other measures of best spread would be reached before n = 13. For example, consider placing points so that the minimum distance between a pair of points is maximized, an alternative criterion I often recommend to those who want equally-spaced points on a sphere. === Subject: Re: point charges on a disc > references too. Glad you enjoy it! Here's another reference, which is more recent: available at . Only one configuration is actually shown there, for n = 185. It's interesting because that may be the first N-value for which a simple concentric shell description appears to be inappropriate. > I find, however, that i am still failing to obtain an intuitive > proof for why n=12 is the critical number. I'm not even sure that there should be an intuitive argument. (OTOH, it does seem intuitive that the critical number should exceed 7.) > As a friend pointed out to me, 6 would possibly be a more obvious choice > because a hexagon would have distances equal to the radius. I would guess, maybe incorrectly, that you meant to say that 7 would be a more obvious choice. (Note that, in the parent thread, Point charges inside a sphere, William Elliot had claimed 7 to be the smallest n for which one charge would lie at the center of the disk.) > So why does it come out at twice this > number. There has to be a geometric argument that works with the > physical interpretation of a harmonic potential. I've tried all > sorts of explanations using various Guass surface configurations, looking > at the flux and fields etc....but i keep coming up short. Do you know > if anyone has presented an informal proof of why n=12 based along > physical argument alone? Not that I know of. BTW, I'll soon be posting several more improved configurations. David === Subject: sum of roots of unity posting-account=O-j1wQoAAADKoHl1BPFXH8tUoNe-9QFq rv:1.9.0.11) Gecko/2009060214 Firefox/3.0.11,gzip(gfe),gzip(gfe) this might be trivial, but I am a bit confused: let N be some composite integer. Let t be some integer from 1 to N-1. then, is the following true for all such t? sum_{k in {0..N-1}} e^{i pi t k/N} = 0 (note we do not necessarily have powers of a primitive Nth roots of unity here) O. === Subject: Re: sum of roots of unity posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2009060310 Ubuntu/8.10 (intrepid) Firefox/3.0.11,gzip(gfe),gzip(gfe) > this might be trivial, but I am a bit confused: > let N be some composite integer. Let t be some integer from 1 to N-1. > then, is the following true for all such t? >sum {k in {0..N-1}} e^{i pi t k/N} = 0 > (note we do not necessarily have powers of a primitive Nth roots of > unity here) > O. find that you do have sums of powers of a primitive N/gcd(t,N) root of unity. Since t/gcd(t,N) is now coprime to N/gcd(t,N), its appearance as a factor in the exponent only serves to permute the roots of unity. === Subject: Re: sum of roots of unity posting-account=O-j1wQoAAADKoHl1BPFXH8tUoNe-9QFq rv:1.9.0.11) Gecko/2009060214 Firefox/3.0.11,gzip(gfe),gzip(gfe) > this might be trivial, but I am a bit confused: > let N be some composite integer. Let t be some integer from 1 to N-1. > then, is the following true for all such t? >sum {k in {0..N-1}} e^{i pi t k/N} = 0 > (note we do not necessarily have powers of a primitive Nth roots of > unity here) > O. find that you do have sums of powers of a > primitive N/gcd(t,N) root of unity.Since > t/gcd(t,N) is now coprime to N/gcd(t,N), > its appearance as a factor in the exponent > only serves to permute the roots of unity. > === Subject: Re: sum of roots of unity posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > this might be trivial, but I am a bit confused: > let N be some composite integer. Let t be some integer from 1 to N-1. > then, is the following true for all such t? >sum {k in {0..N-1}} e^{i pi t k/N} = 0 > (note we do not necessarily have powers of a primitive Nth roots of > unity here) > O. Forwarded conversation === Subject: roots of unity summing to 0 ------------------------ If S is a non-empty subset of the nth roots of 1 summing to 0, and S is minimal with respect to this property, must S be of the form {y,yz,yz^2,...,yz^(m-1)} where y is an nth root of 1 and z is an mth root of 1 for some m dividing n? Jim Propp ---------- Cc: NMBRTHRY@listserv.nodak.edu Nth ROOT OF COMPLEX NUMBERS :]? - x^2))y^2)=48 x^2=48/y^2 y^2-48/y^2=8 y^4-8y^2-48=0 (y^2-12)(y^2+4)=0 y^2=12 ... EDIT:ok,you want by nth root method z=16 ( cos 2pi/3 + i sin 2pi/3) z^(1/2)=4(cos 2pi/3 + i ... where w is cube root of unity Nth ROOT OF COMPLEX NUMBERS :]? - x^2))y^2)=48 x^2=48/y^2 y^2-48/y^2=8 y^4-8y^2-48=0 (y^2-12)(y^2+4)=0 y^2=12 ... EDIT:ok,you want by nth root method z=16 ( cos 2pi/3 + i sin 2pi/3) z^(1/2)=4(cos 2pi/3 + i ... where w is cube root of unity Martin Michael Musatov ---------- Sent from my Verizon Wireless BlackBerry ---------- Martin Musatov === === Subject: Re: The GPS Myth, Mythbusted ? > no, you are wrong > every children knows the 38us/day is nothing > compared with the disturbances and errors > introduced by for instance [...] You need to LOOK at these effects, NUMERICALLY. The ones you listed are > all a factor of 1000 or more smaller than the one I mentioned, and most > of them don't accumulate over multiple days like the correction of the > satellite clock rates. Hmmm... Even if the satellite clock ticks at a rate whatever that is right on with the ground clock, it does no good to you if you do not even know how to synchronize the time. You are focusing on the accuracy in a unit time measurement, but the GPS requires you to keep a chronicle time count. 6. the GPS-receiver clock driven by crystals > with ~30 to ~100 ppm, frequency intolerance, > giving huge errore each second only because of this !!! This is IRRELEVANT, because a modern GPS receiver does not use its own > clock in determining its position, it uses 4 (or more) satellite > signals. In effect it is using the vastly more accurate satellite clocks > instead of its own, rather poor clock. Indeed, there is no need for a > clock in the receiver at all. Using more satellites is a good way to reduce the > errors, and a good, modern GPS receiver will use > every satellite with a good signal, typically up to 12. With acquisitions of 4 satellites or more, as long as all the satellites keep the same chronicle time count, synchronization of chronicle time count can be achievable through satellite-to- satellite. Thus, any ground clocks including the stations monitoring the GPS satellites and the receivers (as you have mentioned) do not come to play anymore. Ground-to-satellite synchronization of chronicle time count is much more susceptible to errors. I will leave that as a homework assignment to list all the factors and the degree of effect to offset the ground-to-satellite synchronization accuracy in chronicle time count. > much more less than that Just compute it -- it is 0.44 parts per billion. In the original post > Basso gave a value that was quite similar. It is actually 440 parts per trillion. It is 38usec divided by the number of seconds in one sidereal day. Even if the difference in time rate exists, and you don't correct for this difference, the error of 10.230000000000MHz and 10.22999999543Mhz is not going to be accumulative if the synchronization of chronicle time count is achieved through satellite-to-satellite. facts, don't expect me to respond. Please step down if you still cannot understand the engineering issue involved. Leave the problems to the engineers. They can do much a better job than the self-styled physicists. I somewhat do not believe any self-styled physicists can understand these engineering issues. === Subject: Re: The GPS Myth, Mythbusted ? posting-account=gHzHsAoAAAAF151cc29NwfB5wIxHQOuy CLR 2.0.50727; .NET CLR 2.0.50215; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) > With acquisitions of 4 satellites or more, as long as all the > satellites keep the same chronicle time count, synchronization of > chronicle time count can be achievable through satellite-to- > satellite.Thus, any ground clocks including the stations monitoring > the GPS satellites and the receivers (as you have mentioned) do not > come to play anymore.Ground-to-satellite synchronization of > chronicle time count is much more susceptible to errors.I will leave > that as a homework assignment to list all the factors and the degree > of effect to offset the ground-to-satellite synchronization accuracy > in chronicle time count. The GPS system is not operated and maintained the way that you believe it ought to be operated. Although eighteen of the GPS satellites in orbit are Block IIR or Block IIR-M, capable of entering AUTONAV mode, 13 of them are still the older Block IIA satellites and are incapable of satellite-to-satellite synchronization. Even if all of the satellites were capable of supporting AUTONAV mode, periodic ground truth corrections against multiple ground reference points would -still- be necessary. As it is, AUTONAV constitutes strictly a survival mode of operation, to be invoked only if the ground stations were disabled in war. In other words, practically every sentence that you've written in the paragraph above that I quoted constitutes sheer fantasy speculation. Jerry === Subject: Re: The GPS Myth, Mythbusted ? posting-account=W7RQ6gkAAACLDC5JWapQU1UV8ot8KkAS .NET CLR 1.0.3705; Dealio Toolbar 3.1.1),gzip(gfe),gzip(gfe) > With acquisitions of 4 satellites or more, as long as all the > satellites keep the same chronicle time count, synchronization of > chronicle time count can be achievable through satellite-to- > satellite.Thus, any ground clocks including the stations monitoring > the GPS satellites and the receivers (as you have mentioned) do not > come to play anymore.Ground-to-satellite synchronization of > chronicle time count is much more susceptible to errors.I will leave > that as a homework assignment to list all the factors and the degree > of effect to offset the ground-to-satellite synchronization accuracy > in chronicle time count. The GPS system is not operated and maintained the way that you > believe it ought to be operated. Although eighteen of the GPS > satellites in orbit are Block IIR or Block IIR-M, capable of > entering AUTONAV mode, 13 of them are still the older Block IIA > satellites and are incapable of satellite-to-satellite > synchronization. Even if all of the satellites were capable of > supporting AUTONAV mode, periodic ground truth corrections > against multiple ground reference points would -still- be > necessary. As it is, AUTONAV constitutes strictly a survival > mode of operation, to be invoked only if the ground stations > were disabled in war. In other words, practically every sentence > that you've written in the paragraph above that I quoted > constitutes sheer fantasy speculation. Well, that's also where atomic clock wristwatches come from. Since the only thing most of the modern day GPS bozos even know about synchronization is internet idiots. Jerry === Subject: Re: The GPS Myth, Mythbusted ? > With acquisitions of 4 satellites or more, as long as all the > satellites keep the same chronicle time count, synchronization of > chronicle time count can be achievable through satellite-to- > satellite. Thus, any ground clocks including the stations monitoring > the GPS satellites and the receivers (as you have mentioned) do not > come to play anymore. Ground-to-satellite synchronization of > chronicle time count is much more susceptible to errors. I will leave > that as a homework assignment to list all the factors and the degree > of effect to offset the ground-to-satellite synchronization accuracy > in chronicle time count. The GPS system is not operated and maintained the way that you > believe it ought to be operated. Although eighteen of the GPS > satellites in orbit are Block IIR or Block IIR-M, capable of > entering AUTONAV mode, 13 of them are still the older Block IIA > satellites and are incapable of satellite-to-satellite > synchronization. Even if all of the satellites were capable of > supporting AUTONAV mode, periodic ground truth corrections > against multiple ground reference points would -still- be > necessary. As it is, AUTONAV constitutes strictly a survival > mode of operation, to be invoked only if the ground stations > were disabled in war. In other words, practically every sentence > that you've written in the paragraph above that I quoted > constitutes sheer fantasy speculation. Well, that's also where atomic clock wristwatches come from. Since the only thing most of the modern day GPS bozos even know about synchronization is internet idiots. Wrong. They are SW radio based, not gps. Jerry === Subject: Re: The GPS Myth, Mythbusted ? posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > no, you are wrong > every children knows the 38us/day is nothing > compared with the disturbances and errors > introduced by for instance [...] > You need to LOOK at these effects, NUMERICALLY. The ones you listed are > all a factor of 1000 or more smaller than the one I mentioned, and most > of them don't accumulate over multiple days like the correction of the > satellite clock rates. Hmmm...Even if the satellite clock ticks at a rate whatever that is > right on with the ground clock, it does no good to you if you do not > even know how to synchronize the time.You are focusing on the > accuracy in a unit time measurement, but the GPS requires you to keep > a chronicle time count. Except for this: > 6. the GPS-receiver clock driven by crystals > with ~30 to ~100 ppm, frequency intolerance, > giving huge errore each second only because of this !!! > This is IRRELEVANT, because a modern GPS receiver does not use its own > clock in determining its position, it uses 4 (or more) satellite > signals. In effect it is using the vastly more accurate satellite clocks > instead of its own, rather poor clock. Indeed, there is no need for a > clock in the receiver at all. > Using more satellites is a good way to reduce the > errors, and a good, modern GPS receiver will use > every satellite with a good signal, typically up to 12. With acquisitions of 4 satellites or more, as long as all the > satellites keep the same chronicle time count, synchronization of > chronicle time count can be achievable through satellite-to- > satellite.Thus, any ground clocks including the stations monitoring > the GPS satellites and the receivers (as you have mentioned) do not > come to play anymore.Ground-to-satellite synchronization of > chronicle time count is much more susceptible to errors.I will leave > that as a homework assignment to list all the factors and the degree > of effect to offset the ground-to-satellite synchronization accuracy > in chronicle time count. > much more less than that > Just compute it -- it is 0.44 parts per billion. In the original post > Basso gave a value that was quite similar. It is actually 440 parts per trillion.It is 38usec divided by the > number of seconds in one sidereal day.Even if the difference in time > rate exists, and you don't correct for this difference, the error of > 10.230000000000MHz and 10.22999999543Mhz is not going to be > accumulative if the synchronization of chronicle time count is > achieved through satellite-to-satellite. If all you do is make things up and regurgitate them as if they were > facts, don't expect me to respond. Please step down if you still cannot understand the engineering issue > involved.Leave the problems to the engineers. They can do much a > better job than the self-styled physicists.I somewhat do not believe > any self-styled physicists can understand these engineering issues. > ok === Subject: Re: The GPS Myth, Mythbusted ? > > no, you are wrong > every children knows the 38us/day is nothing > compared with the disturbances and errors > introduced by for instance [...] > You need to LOOK at these effects, NUMERICALLY. The ones you listed are > all a factor of 1000 or more smaller than the one I mentioned, and most > of them don't accumulate over multiple days like the correction of the > satellite clock rates. > > Hmmm... Even if the satellite clock ticks at a rate whatever that is > right on with the ground clock, it does no good to you if you do not > even know how to synchronize the time. You are focusing on the > accuracy in a unit time measurement, but the GPS requires you to keep > a chronicle time count. Koobee--Are you interested in how GPS really works? http://edu-observatory.org/gps/gps books.html > > Except for this: > 6. the GPS-receiver clock driven by crystals > with ~30 to ~100 ppm, frequency intolerance, > giving huge errore each second only because of this !!! > This is IRRELEVANT, because a modern GPS receiver does not use its own > clock in determining its position, it uses 4 (or more) satellite > signals. In effect it is using the vastly more accurate satellite clocks > instead of its own, rather poor clock. Indeed, there is no need for a > clock in the receiver at all. > Using more satellites is a good way to reduce the > errors, and a good, modern GPS receiver will use > every satellite with a good signal, typically up to 12. > > With acquisitions of 4 satellites or more, as long as all the > satellites keep the same chronicle time count, synchronization of > chronicle time count can be achievable through satellite-to- > satellite. Thus, any ground clocks including the stations monitoring > the GPS satellites and the receivers (as you have mentioned) do not > come to play anymore. Ground-to-satellite synchronization of > chronicle time count is much more susceptible to errors. I will leave > that as a homework assignment to list all the factors and the degree > of effect to offset the ground-to-satellite synchronization accuracy > in chronicle time count. > > much more less than that > Just compute it -- it is 0.44 parts per billion. In the original post > Basso gave a value that was quite similar. > > It is actually 440 parts per trillion. It is 38usec divided by the > number of seconds in one sidereal day. Even if the difference in time > rate exists, and you don't correct for this difference, the error of > 10.230000000000MHz and 10.22999999543Mhz is not going to be > accumulative if the synchronization of chronicle time count is > achieved through satellite-to-satellite. Koobee--Are you interested in how GPS really works? http://edu-observatory.org/gps/gps books.html > > If all you do is make things up and regurgitate them as if they were > facts, don't expect me to respond. > > Please step down if you still cannot understand the engineering issue > involved. Leave the problems to the engineers. They can do much a > better job than the self-styled physicists. I somewhat do not believe > any self-styled physicists can understand these engineering issues. > === Subject: Re: P=NP (proof) > , <6dd19f66-5b69-497c-ade3-36b6c409c07e@s28g2000vbp.goog > legroups.com>, > > I recently read about this problem and how it is > (according to Tim > Gowers, who I believe) the most important > problem in combinatorics > today. But I am used to being able to > immediately apprehend problems > in combinatorics. Is there actually a simple > formulation of the P=NP > question as purely regarding finite sets, > without any mention of > complexity classes and such? > > Considering that P and NP are complexity classes, > it's hard to imagine > a formulation of the question without any mention > of complexity classes. > But here goes. > > I take it you know what a graph is, and what it > means for two (finite) > graphs to be isomorphic. The graph isomorphism > problem asks you > to determine whether two given graphs are > isomorphic. In theory, > there is no difficulty in solving this problem; > you just look at each > map from the one vertex set to the other in turn > until either you find > one that's an isomorphism or you exhaust the set > of maps without > finding an isomorphism. In practice, this takes > too long; if each graph > has n vertices then there are n-factorial maps to > look at (counting just > those that are one-one), and n-factorial is big. > > On the other hand, if someone shows you an > alleged isomorphism, > you can quickly verify or refute her allegation; > at worst, you have to > check about n^2 edges, and n^2 is small. > > So graph isomorphism is easy to verify; the P = > NP question is > whether graph isomorphism is easy to solve. That > is, is there a way > to solve graph isomorphism that, instead of > requiring n-factorial > amount of work, requires only n^2, or n^3, or > even n^1000, > which is still tiny compared to n-factorial, for > large n. > > Perhaps I'm mistaken and there has been a recent > development but I thought it wasn't known if > graph isomorphism is an NP complete problem. > > Subgraph isomorphism is known to be though. > > You're probably right about graph isomorphism, and I > simply > misremembered what I thought I had learned about it. > You're > certainly right about subgraph isomorphism, which for > the > record is the following problem; given two graphs, G > and H, > determine whether G has a subgraph isomorphic to H. > > -- > Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for > email) Natural numbers [was: Diagonal wanderings...] Theorem lt_s: forall np:nat , n< p ...http://www.ncc.up.pt/~nam/aulas/0506/ t_coq/exemplos/pred.v - Similar pages Yes, this was all part of technique first pioneered by Musatov when he proved P==NP. On May 22, 11:17 pm, David Bernier < .. May 23 by Martin Musatov - 1001 messages - 55 authors Mechanical Witness: === Subject: Re: P=NP (proof) <5224991.14056.1244952852286.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Here is how it works: Read each track (or color if your browser site encodes) the term track is denoted by the number of >'s for each line. For example >=track 1 >=track 2 >=track 3 The tracks can be read when mixed or all together and they still are true statements: See below: > The earth is a planet >in the sky > which is round >stars If I was reading track 1 all are true and I would get: The earth is a planet which is round If I was reading track 2 I would get: in the sky stars If I read the both tracks 1 and 2 together I get: The earth is a planet in the sky which is round stars.... There are restrictions to the programming when we combine like this but if we think like this we can make some very useful programs. Here is an example of the color coding: http://mathforum.org/kb/message.jspa?messageID=6751446&tstart=0 On Jun 13, 9:13pm, Martin Michael Musatov , > <6dd19f66-5b69-497c-ade3-36b6c409c...@s28g2000vbp.goog > legroups.com>, > I recently read about this problem and how it is > (according to Tim > Gowers, who I believe) the most important > problem in combinatorics > today. But I am used to being able to > immediately apprehend problems > in combinatorics. Is there actually a simple > formulation of the P=NP > question as purely regarding finite sets, > without any mention of > complexity classes and such? > Considering that P and NP are complexity classes, > it's hard to imagine > a formulation of the question without any mention > of complexity classes. > But here goes. > I take it you know what a graph is, and what it > means for two (finite) > graphs to be isomorphic. The graph isomorphism > problem asks you > to determine whether two given graphs are > isomorphic. In theory, > there is no difficulty in solving this problem; > you just look at each > map from the one vertex set to the other in turn > until either you find > one that's an isomorphism or you exhaust the set > of maps without > finding an isomorphism. In practice, this takes > too long; if each graph > has n vertices then there are n-factorial maps to > look at (counting just > those that are one-one), and n-factorial is big. > On the other hand, if someone shows you an > alleged isomorphism, > you can quickly verify or refute her allegation; > at worst, you have to > check about n^2 edges, and n^2 is small. > So graph isomorphism is easy to verify; the P = > NP question is > whether graph isomorphism is easy to solve. That > is, is there a way > to solve graph isomorphism that, instead of > requiring n-factorial > amount of work, requires only n^2, or n^3, or > even n^1000, > which is still tiny compared to n-factorial, for > large n. > Perhaps I'm mistaken and there has been a recent > development but I thought it wasn't known if > graph isomorphism is an NP complete problem. > Subgraph isomorphism is known to be though. > You're probably right about graph isomorphism, and I > simply > misremembered what I thought I had learned about it. > You're > certainly right about subgraph isomorphism, which for > the > record is the following problem; given two graphs, G > and H, > determine whether G has a subgraph isomorphic to H. > -- > Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for > email) Natural numbers [was: Diagonal wanderings...] Theorem lt s: forall np:nat , n< p ...http://www.ncc.up.pt/~nam/aulas/0506/ > t coq/exemplos/pred.v - Similar pages Yes, this was all part of technique first .. > May 23 by Martin Musatov - 1001 messages - 55 authors Mechanical Witness: > - Show quoted text - Here is how it works: Read by colors in tracks or read all the colors together like a normal book. Re: P=NP (proof) > , > <6dd19f66-5b69-497c-ade3-36b6c409c07e@s28g2000vbp.goog > legroups.com>, > I recently read about this problem and how it is > (according to Tim > Gowers, who I believe) the most important > problem in combinatorics > today. But I am used to being able to > immediately apprehend problems > in combinatorics. Is there actually a simple > formulation of the P=NP > question as purely regarding finite sets, > without any mention of > complexity classes and such? > Considering that P and NP are complexity classes, > it's hard to imagine > a formulation of the question without any mention > of complexity classes. > But here goes. > I take it you know what a graph is, and what it > means for two (finite) > graphs to be isomorphic. The graph isomorphism > problem asks you > to determine whether two given graphs are > isomorphic. In theory, > there is no difficulty in solving this problem; > you just look at each > map from the one vertex set to the other in turn > until either you find > one that's an isomorphism or you exhaust the set > of maps without > finding an isomorphism. In practice, this takes > too long; if each graph > has n vertices then there are n-factorial maps to > look at (counting just > those that are one-one), and n-factorial is big. > On the other hand, if someone shows you an > alleged isomorphism, > you can quickly verify or refute her allegation; > at worst, you have to > check about n^2 edges, and n^2 is small. > So graph isomorphism is easy to verify; the P = > NP question is > whether graph isomorphism is easy to solve. That > is, is there a way > to solve graph isomorphism that, instead of > requiring n-factorial > amount of work, requires only n^2, or n^3, or > even n^1000, > which is still tiny compared to n-factorial, for > large n. > Perhaps I'm mistaken and there has been a recent > development but I thought it wasn't known if > graph isomorphism is an NP complete problem. > Subgraph isomorphism is known to be though. You're probably right about graph isomorphism, and I > simply > misremembered what I thought I had learned about it. > You're > certainly right about subgraph isomorphism, which for > the > record is the following problem; given two graphs, G > and H, > determine whether G has a subgraph isomorphic to H. -- > Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for > email) Natural numbers [was: Diagonal wanderings...] Theorem lt s: forall np:nat , n< p ...http://www.ncc.up.pt/~nam/aulas/0506/ t coq/exemplos/pred.v - Similar pages Yes, this was all part of technique first pioneered by Musatov when he proved P==NP. On May 22, 11:17 pm, David Bernier < ultrafinitist, ... May 23 by Martin Musatov - 1001 messages - 55 authors Mechanical Witness: http://mathforum.org/kb/message.jspa?messageID=6751446&tstart=0 === Subject: Re: P=NP (proof) <5224991.14056.1244952852286.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Here is how it works: Read each track (or color if your browser site encodes) the term > track is denoted by the number of >'s for each line. For example >=track 1 >=track 2 >=track 3 The tracks can be read when mixed or all together and they still are > true statements: See below: > The earth is a planet >in the sky > which is round >stars If I was reading track 1 all are true and I would get: The earth is a planet which is round If I was reading track 2 I would get: in the sky stars If I read the both tracks 1 and 2 together I get: The earth is a planet in the sky which is round stars.... There are restrictions to the programming when we combine like this > but if we think like this we can make some very useful programs. Here is an example of the color coding: http://mathforum.org/kb/message.jspa?messageID=6751446&tstart=0 On Jun 13, 9:13pm, Martin Michael Musatov , > <6dd19f66-5b69-497c-ade3-36b6c409c...@s28g2000vbp.goog > legroups.com>, > I recently read about this problem and how it is > (according to Tim > Gowers, who I believe) the most important > problem in combinatorics > today. But I am used to being able to > immediately apprehend problems > in combinatorics. Is there actually a simple > formulation of the P=NP > question as purely regarding finite sets, > without any mention of > complexity classes and such? > Considering that P and NP are complexity classes, > it's hard to imagine > a formulation of the question without any mention > of complexity classes. > But here goes. > I take it you know what a graph is, and what it > means for two (finite) > graphs to be isomorphic. The graph isomorphism > problem asks you > to determine whether two given graphs are > isomorphic. In theory, > there is no difficulty in solving this problem; > you just look at each > map from the one vertex set to the other in turn > until either you find > one that's an isomorphism or you exhaust the set > of maps without > finding an isomorphism. In practice, this takes > too long; if each graph > has n vertices then there are n-factorial maps to > look at (counting just > those that are one-one), and n-factorial is big. > On the other hand, if someone shows you an > alleged isomorphism, > you can quickly verify or refute her allegation; > at worst, you have to > check about n^2 edges, and n^2 is small. > So graph isomorphism is easy to verify; the P = > NP question is > whether graph isomorphism is easy to solve. That > is, is there a way > to solve graph isomorphism that, instead of > requiring n-factorial > amount of work, requires only n^2, or n^3, or > even n^1000, > which is still tiny compared to n-factorial, for > large n. > Perhaps I'm mistaken and there has been a recent > development but I thought it wasn't known if > graph isomorphism is an NP complete problem. > Subgraph isomorphism is known to be though. > You're probably right about graph isomorphism, and I > simply > misremembered what I thought I had learned about it. > You're > certainly right about subgraph isomorphism, which for > the > record is the following problem; given two graphs, G > and H, > determine whether G has a subgraph isomorphic to H. > -- > Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for > email) > Natural numbers [was: Diagonal wanderings...] Theorem lt s: forall np:nat , n< p ...http://www.ncc.up.pt/~nam/aulas/0506/ > t coq/exemplos/pred.v - Similar pages Yes, this was all part of technique first > .. > May 23 by Martin Musatov - 1001 messages - 55 authors > Mechanical Witness: > - Show quoted text - Here is how it works: Read by colors in tracks or read all the colors together like a > normal book. Re: P=NP (proof) > , > <6dd19f66-5b69-497c-ade3-36b6c409c...@s28g2000vbp.goog > legroups.com>, > I recently read about this problem and how it is > (according to Tim > Gowers, who I believe) the most important > problem in combinatorics > today. But I am used to being able to > immediately apprehend problems > in combinatorics. Is there actually a simple > formulation of the P=NP > question as purely regarding finite sets, > without any mention of > complexity classes and such? > Considering that P and NP are complexity classes, > it's hard to imagine > a formulation of the question without any mention > of complexity classes. > But here goes. > I take it you know what a graph is, and what it > means for two (finite) > graphs to be isomorphic. The graph isomorphism > problem asks you > to determine whether two given graphs are > isomorphic. In theory, > there is no difficulty in solving this problem; > you just look at each > map from the one vertex set to the other in turn > until either you find > one that's an isomorphism or you exhaust the set > of maps without > finding an isomorphism. In practice, this takes > too long; if each graph > has n vertices then there are n-factorial maps to > look at (counting just > those that are one-one), and n-factorial is big. > On the other hand, if someone shows you an > alleged isomorphism, > you can quickly verify or refute her allegation; > at worst, you have to > check about n^2 edges, and n^2 is small. > So graph isomorphism is easy to verify; the P = > NP question is > whether graph isomorphism is easy to solve. That > is, is there a way > to solve graph isomorphism that, instead of > requiring n-factorial > amount of work, requires only n^2, or n^3, or > even n^1000, > which is still tiny compared to n-factorial, for > large n. > Perhaps I'm mistaken and there has been a recent > development but I thought it wasn't known if > graph isomorphism is an NP complete problem. > Subgraph isomorphism is known to be though. > You're probably right about graph isomorphism, and I > simply > misremembered what I thought I had learned about it. > You're > certainly right about subgraph isomorphism, which for > the > record is the following problem; given two graphs, G > and H, > determine whether G has a subgraph isomorphic to H. > -- > Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for > email) Natural numbers [was: Diagonal wanderings...] Theorem lt s: forall > np:nat , n< p ...http://www.ncc.up.pt/~nam/aulas/0506/ > t coq/exemplos/pred.v - Similar pages Yes, this was all part of > technique first > pioneered by Musatov when he proved P==NP. On May 22, 11:17 pm, David ultrafinitist, > ... > May 23 by Martin Musatov - 1001 messages - 55 authors Mechanical Witness: http://mathforum.org/kb/message.jspa?messageID=6751446&tstart=0- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text - Here is the isomorphism: Re: P=NP (proof) > , > <6dd19f66-5b69-497c-ade3-36b6c409c07e@s28g2000vbp.goog > legroups.com>, > I recently read about this problem and how it is > (according to Tim > Gowers, who I believe) the most important > problem in combinatorics > today. But I am used to being able to > immediately apprehend problems > in combinatorics. Is there actually a simple > formulation of the P=NP > question as purely regarding finite sets, > without any mention of > complexity classes and such? > Considering that P and NP are complexity classes, > it's hard to imagine > a formulation of the question without any mention > of complexity classes. > But here goes. > I take it you know what a graph is, and what it > means for two (finite) > graphs to be isomorphic. The graph isomorphism > problem asks you > to determine whether two given graphs are > isomorphic. In theory, > there is no difficulty in solving this problem; > you just look at each > map from the one vertex set to the other in turn > until either you find > one that's an isomorphism or you exhaust the set > of maps without > finding an isomorphism. In practice, this takes > too long; if each graph > has n vertices then there are n-factorial maps to > look at (counting just > those that are one-one), and n-factorial is big. > On the other hand, if someone shows you an > alleged isomorphism, > you can quickly verify or refute her allegation; > at worst, you have to > check about n^2 edges, and n^2 is small. > So graph isomorphism is easy to verify; the P = > NP question is > whether graph isomorphism is easy to solve. That > is, is there a way > to solve graph isomorphism that, instead of > requiring n-factorial > amount of work, requires only n^2, or n^3, or > even n^1000, > which is still tiny compared to n-factorial, for > large n. > Perhaps I'm mistaken and there has been a recent > development but I thought it wasn't known if > graph isomorphism is an NP complete problem. > Subgraph isomorphism is known to be though. You're probably right about graph isomorphism, and I > simply > misremembered what I thought I had learned about it. > You're > certainly right about subgraph isomorphism, which for > the > record is the following problem; given two graphs, G > and H, > determine whether G has a subgraph isomorphic to H. -- > Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for > email) Natural numbers [was: Diagonal wanderings...] Theorem lt s: forall np:nat , n< p ...http://www.ncc.up.pt/~nam/aulas/0506/ t coq/exemplos/pred.v - Similar pages Yes, this was all part of technique first pioneered by Musatov when he proved P==NP. On May 22, 11:17 pm, David Bernier < ultrafinitist, ... May 23 by Martin Musatov - 1001 messages - 55 authors Mechanical Witness: === Subject: Re: P=NP (proof) <5224991.14056.1244952852286.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Natural numbers [was: Diagonal wanderings...] Theorem lt_s: for all np:nat , n< p http://www.ncc.up.pt/~nam/aulas/0506/t_coq/exemplos/pred.v Part of technique when he proved P==NP. On May 22, 11:17 pm, David and ultrafinitist. Mechanical Witness: http://mathforum.org/kb/message.jspa?messageID=6751446&tstart=0- Hide quoted text - Here is the isomorphism: Re: P=NP (proof) I recently read about this problem and how it is (according to Tim Gowers, who I believe) the most important problem in combinatorics today. But I am used to being able to immediately apprehend problems in combinatorics. Is there actually a simple formulation of the P=NP question as purely regarding finite sets, without any mention of complexity classes and such? Considering that P and NP are complexity classes, it's hard to imagine a formulation of the question without any mention of complexity classes. But here goes. I take it you know what a graph is, and what it means for two (finite) graphs to be isomorphic. The graph isomorphism problem asks you to determine whether two given graphs are isomorphic. In theory, there is no difficulty in solving this problem; you just look at each map from the one vertex set to the other in turn until either you find one that's an isomorphism or you exhaust the set of maps without finding an isomorphism. In practice, this takes too long; if each graph has n vertices then there are n-factorial maps to look at (counting just those that are one-one), and n-factorial is big. On the other hand, if someone shows you an alleged isomorphism, you can quickly verify or refute her allegation; at worst, you have to check about n^2 edges, and n^2 is small. So graph isomorphism is easy to verify; the P = NP question is whether graph isomorphism is easy to solve. That is, is there a way to solve graph isomorphism that, instead of requiring n-factorial amount of work, requires only n^2, or n^3, or even n^1000, which is still tiny compared to n-factorial, for large n. Perhaps I'm mistaken and there has been a recent development but I thought it wasn't known if graph isomorphism is an NP complete problem. Subgraph isomorphism is known to be though. You're probably right about graph isomorphism, and I simply misremembered what I thought I had learned about it. You're certainly right about subgraph isomorphism, which for the record is the following problem; given two graphs, G and H, determine whether G has a subgraph isomorphic to H. Natural numbers [was: Diagonal wanderings...] Theorem lt_s: forall np:nat , n< p ...http://www.ncc.up.pt/~nam/aulas/0506/ t_coq/exemplos/pred.v - Similar pages Yes, this was all part of technique first pioneered by Musatov when he proved P==NP. On May 22, 11:17 pm, David Bernier < ultrafinitist, ... May 23 by Martin Musatov - 1001 messages - 55 authors Mechanical Witness: === Subject: Re: P=NP (proof) <5224991.14056.1244952852286.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Natural numbers [was: Diagonal wanderings...] Theorem lt s: for all > np:nat , n< p http://www.ncc.up.pt/~nam/aulas/0506/t coq/exemplos/pred.v > and ultrafinitist. Mechanical Witness:http://mathforum.org/kb/message.jspa?messageID=6751446&tstart=0-Hide > quoted text - Here is the isomorphism: > Re: P=NP (proof) > I recently read about this problem and how it is > (according to Tim > Gowers, who I believe) the most important > problem in combinatorics > today. But I am used to being able to > immediately apprehend problems > in combinatorics. Is there actually a simple > formulation of the P=NP > question as purely regarding finite sets, > without any mention of > complexity classes and such? Considering that P and NP are complexity classes, > it's hard to imagine > a formulation of the question without any mention > of complexity classes. > But here goes. I take it you know what a graph is, and what it > means for two (finite) > graphs to be isomorphic. The graph isomorphism > problem asks you > to determine whether two given graphs are > isomorphic. In theory, > there is no difficulty in solving this problem; > you just look at each > map from the one vertex set to the other in turn > until either you find > one that's an isomorphism or you exhaust the set > of maps without > finding an isomorphism. In practice, this takes > too long; if each graph > has n vertices then there are n-factorial maps to > look at (counting just > those that are one-one), and n-factorial is big. On the other hand, if someone shows you an > alleged isomorphism, > you can quickly verify or refute her allegation; > at worst, you have to > check about n^2 edges, and n^2 is small. So graph isomorphism is easy to verify; the P = > NP question is > whether graph isomorphism is easy to solve. That > is, is there a way > to solve graph isomorphism that, instead of > requiring n-factorial > amount of work, requires only n^2, or n^3, or > even n^1000, > which is still tiny compared to n-factorial, for > large n. Perhaps I'm mistaken and there has been a recent > development but I thought it wasn't known if > graph isomorphism is an NP complete problem. Subgraph isomorphism is known to be though. You're probably right about graph isomorphism, and I > simply > misremembered what I thought I had learned about it. > You're > certainly right about subgraph isomorphism, which for > the > record is the following problem; given two graphs, G > and H, > determine whether G has a subgraph isomorphic to H. Natural numbers [was: Diagonal wanderings...] Theorem lt s: forall > np:nat , n< p ...http://www.ncc.up.pt/~nam/aulas/0506/ > t coq/exemplos/pred.v - Similar pages Yes, this was all part of > technique first > pioneered by Musatov when he proved P==NP. On May 22, 11:17 pm, David > Bernier < > ultrafinitist, > ... > May 23 by Martin Musatov - 1001 messages - 55 authors Mechanical Witness: > As it turns out the solution to the P Versus NP propblem is another problem: Sorted by relevance Sort by date 1 result for musat. Permutation Problem Why not just use random permutations? There are well-known fast algorithms for that. If you generated, say 1000 permutations per second for 2 weeks, you would process 1209600000 permutations out of a total of about 3.47 x 10^1166 so your probability of a repetition musat be minute. Derek Holt. authors === Subject: Proof of index formula for groups If G is a group, and H, K are subgroups with K < H, and N is normal in G, then how do we prove that [H : K] = [HN/N : KN/N][H / N : K / N]. === Subject: Re: Proof of index formula for groups posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > If G is a group, and H, K are subgroups with K < H, > and N is normal in G, then how do we prove that [H : K] = [HN/N : KN/N][H / N : K / N]. > === Subject: Alarm clock problem How one can show that P_t is a martingale with respect to the filtration generated by observation whether the alarm clock has rang or not. === Subject: Re: Wikipedia editors ask a question of AP, and AP answers only to have it deleted by DMacks a split second later posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Come to think of it, I really ought to archive Wikipedia's first and > only objective > entry of Archimedes Plutonium, and written by someone of purely > objectivity. > Objectivity is a rare commodity of a editor, as amazing as that is. ---quoting the Likebox/Archimedes Plutonium entry in Wikipedia --- http://en.wikipedia.org/wiki/User:Likebox/Archimedes Plutonium User:Likebox/Archimedes Plutonium > < User:Likebox > Jump to: navigation, search Archimedes Plutonium (born July 5, 1950), also known as Ludwig > In 1990 he became convinced that the universe could be thought of as > an atom of plutonium, and changed his name to reflect this idea. He is > notable for his offbeat theories about Plutonium Atom totality, > physical constants, and nonstandard infinite arithmetic. [1] [2] > Contents > [hide] * 1 Biographical Sketch > * 2 Writing > o 2.1 Plutonium Atom Totality > o 2.2 Infinite Integers > o 2.3 Other ideas > * 3 Quotes > * 4 References [edit] Biographical Sketch Plutonium was born under the name Ludwig Poehlmann in Arzberg, > Germany. His family moved to the United States and settled near > Cincinnati, Ohio, where Plutonium was adopted into the Hansen family > and brought up under the name Ludwig Hansen. Plutonium has a BA in > mathematics from the University of Cincinnati and taught High School > in Melbourne Australia. He returned to the US in the mid 1970s and > went on to Utah State University for a Masters degree. Under the name > Ludwig Plutonium, he began posting to usenet in 1993, and his prolific > posts quickly made him a well known usenet figure. [edit] Writing Plutonium is the author of over 20,000 unique postings to dozens of > science newsgroups such as sci.physics, sci.math, sci.chem, > sci.bio.misc. Plutonium used the time stamp on his postings to gather > the posts into collections which he calls his internet books. He has > written approximately 30 of these. [edit] Plutonium Atom Totality Plutonium Atom Totality is the idea that the universe should be > thought of as a gigantic atom of the element plutonium, Pu 231. > Plutonium believes that the galaxies in the night sky are the electron > cloud of the atom. The cosmic atom, often written ATOM, is a > manifestation ofGod, or the totality of all things, but the physical > universe in Plutonium's philosophy only obeys natural laws and has no > room for anything supernatural.[3] [edit] Infinite Integers An integer in Plutonium's philosophical view includes objects which > have a decimal expansion which never ends, for example, the following > number is an integer: x = 111...333 , which starts with an infinite repeating list of 1s, and ends with an > infinite repeating list of 3s. The 1's are the frontview of the > number, while the 3's are the backview. To multiply these numbers, > multiply finite approximations until the repeating pattern front and > back becomes clear. For example, 111...333 * 888...444 = 0987654320987654320...1851851852 , and the leading 0 is important to Plutonium. Plutonium believes that > Fermat's last theorem is false, because he believes it fails for these > infinite integers. He also believes that the set of all real numbers > is countable, since both the Reals and Infinite Integers are All > Possible Digit Arrangements. By this statement he usually means that > there is a direct one-to-one map from the real numbers to the > integers, which consists of taking all the digits behind the decimal > point and putting them in front. To allow this, his real numbers have > a frontview and a backview too.[4] [5] [edit] Other ideas Plutonium believes in a fusion barrier principle, which limits the > energy output in a fusion reactor to 2/3 of input. He believes that > all forces emerge from a unified Coulomb's law. He also believes that > the mainspring of human evolution was throwing rocks and stones, and > that this led to bipedalism. He is the author of countless other ideas > and speculations, all of which claim to displace scientific consensus, > and none of which are accepted by mainstream science. Archimedes Plutonium, in his Usenet posts, was the first to describe > the practice of biasing search-engine results by planting references, > and coined the phrase search-engine bombing to describe it. This later [edit] Quotes * The whole entire Universe is just one big atom where dots of > the electron-dot-cloud are galaxies. > * Godis Science, and Science isgod. > * Godis this one big atom that comprises all the Universe, much > like what Spinoza discovered some centuries past, called pantheism. > Where we are a tiny part ofGoditself. And where there is a heaven > and hell in part of the atom structure. And where we will be judged byGodwhen we die and our photon and neutrino souls will reincarnate > once again in a future life somewhere in the Cosmos. > * The world's finest Bibles are currentphysicstextbooks or > biology or chemistry textbooks > * When you have a foggy notion of what you are working with, it > is impossible to prove much about them. [edit] References 1. ^ Joseph C. Scott. Sometime-scientist Plutonium says science is > 'gobbledygook', The Dartmouth, September 25, 1997. >2. ^ Jennifer Kahn. Notes from Another Universe, Discover, April > 2002. >3. ^http://www.iw.net/~a plutonium/ >4. ^http://www.iw.net/~a plutonium/, for further information, seehttp://mathforum.org/kb/forum.jspa?forumID=13, Archimedes Plutonium , > possible digit arrangements* bars or precludes Cantor ever applying a > diagonal method ; new textbook: Mathematical-Physics(p-adic primer) > for students of age 6 onwards >5. ^http://www.iw.net/~a plutonium/, see alsohttp://forum.lowcarber.org/archive/index.php/t-80681.html >7. ^ Law and Order on Net and Web (September 17, 1997) Retrieved from http://en.wikipedia.org/wiki/User:Likebox/ > Archimedes Plutonium > Categories: American writers | 1950 births --- end quoting the Likebox/Archimedes Plutonium entry in Wikipedia > --- Archimedes Plutoniumwww.iw.net/~a plutonium > whole entire Universe is just one big atom > where dots of the electron-dot-cloud are galaxies >SHUT UP!! > Does your physics books mentioned the word audience refering > [sic] to stars? > No, which is precisely my point. > If your people physics books doesn't[sic] mentioned[sic] the word > audience, it doesn't[sic] understand how stars are made. > Books, of course, don't understand anything; they are just inert > chunks of paper. So this statement of yours is meaningless. > Although I'm curious to know why you think I have people physics > books, a sub-genre I've never heard of. > You never heard of the tree people? There are people who cut down > trees in Brazil, the natives call them the tree people. You are the > physics book people. You got the math people...wait I see this is > going to the science fiction people... > And meanwhile, you are one of the incoherent people. > Much larger population than the coherent people, a dying breed... >Due to nozzles like you. > Children should not be reading sf books... > You're welcome to your opinion. May you never breed. > Not my opinion, ...Albert Einstein's opinion, he believed that people > should not read science fiction--that it distorts science, and gives > people the illusion of understanding. >Even Einstein was occasionally wrong. You got kids out there believing in flying saucers, that's science > fiction, that's a distortion of science. > May he never breed? Or maybe, You?... The Bomb Maker?? >Hey. My bombs kept the peace for decades, until the SU eventually >collapsed under its own weight. Don't knock 'em. > It is the audience that created the universe, ...the audience > exist[sic] in the future. > Right. The universe was made by time travelers. > I'm traveling through time right now! > Don't forget your lightning rod. You know, just in case you run out of > plutonium for juicing up that flux capacitor. > I can always buy plutonium from the Koreans! >*shudder* > People describe the universe with mathematics, but that is just an > illusion... > No, it is not. I can use calculus and the laws of motion to > describe the trajectory of a baseball bat conking you on the head. > Your ensuing headache and goose-egg will confirm handily for you > for several days that it was no illusion. > You should work for the department of defense! They need guys like > you! > I already do, you ninny. Didn't I mention earlier my profession? > You mention it, but you didn't say what country.... >I thought it was obvious. > Mathematics does not describe the universe. Nor does physics. > Sure they do. > Scientism[sic]. > I spelled it right! Scientism. > Oops, nope, and you cocked it up again. It's science. With an e. > There's no c. >There certainly is. >http://www.merriam-webster.com/dictionary/science >Note spelling. the belief that there is one and only one method of science and that > it alone confers legitimacy upon the conduct of research. Just a belief, just a religion..religion of the nerds...Scientism. > You have limited intelligence... > My intelligence greatly exceeds yours. > It is still, limited. >A meaningless statement. It means your intelligence is small in range. > My stupidity greatly exceeds yours, and it has no limits. >That much has been obvious for quite some time now. > A dead universe is a universe > without an audience. No stars > will be born. > There is no evidence to support this contention. > You fail astrophysics forever. > I never heard of that word astrophysics...where did you get it from > An actual ing education. > Was it the 13th grade? > University, you jackass. Perhaps you didn't read the other thread, > but I'm a ing nuclear physicist. I know a lot more about a lot of > this than you do, as must be becoming quite obvious by now. > A nuclear physicist? You guys created an atom bomb to kill Germans and > you end up killing Japanese people with it. That was very smart. > Yeah, imagine if the war of the Pacific had dragged on another five years. > I don't live in a if world... >Oh, yeah, of course, you're too stupid to understand hypotheticals. >Silly me. > Are you threatening me? > Do you have anything against Iranian Nuclear Physics being > assassinated? >I'm not Iranian, nutjob. Do you have anything against any Nuclear Physicist from any country > being assassinated? > Observers, the audience..makes the stars. > Not the celestial ones. > As long as you confined yourself to physics and math, you can > never understand...The Real World. > Wake up and smell the muons: the Real World is made of ... > physics and math. Go to university and get a bachelor of science > degree if you don't believe me. I know I did. > I never went to college...and I don't want to enter those > 'slave to corporations' you call university. > Well, that explains your utter failure to grasp astrophysics or > even some basic concepts of logic and reason. Although not your > near-illiteracy. Did you forget to mention that you dropped out of > school in sixth grade, or something like that? > I'll explain why you're able to grasp astrophysics, ....it's was an > accident. > You are not making sense once again. Grammatically, or semantically. > Accident! A mistake, or by chance.. >There has been no mistake by me. I understand it by intelligence and >reason, not by sheer dumb luck. I bet your parents thought they were lucky. > The truth is, the majority of people do *not* understand or have any > real grasp of science or arithmetic... >Including you. >Fortunately, I'm not in that majority. > if you do, it's a freak of nature, a mistake..and accident. >No, it is not. It is called having had an education. You ought to try >it sometime. Anybody can walk into a college bookstore and buy a textbook. Or a class onlinehttp://www.mininova.org/search/ttc%2Bvideo/seeds > here is what an accident looks like: > >http://www.telegraph.co.uk/telegraph/multimedia/archive/00441/news-gr... >That's the result of disease, not an accident, wit. I was discussing 'smart people' are a freak of nature, it's a picture > of a smart person. It's no accident he got a disease, it comes with > 'the smarts'. > There is no Big Bang without observers in the dark. You want proof, go > see a movie! > This is completely irrational. > Have you tried learning without a textbook? >I learn via all kinds of media, including but not limited to books. You mean like , watching a movie? Do you know how stars are madeË > To that fish > in his tank > that's his universe.. > He can see out through the glass, you know. > Does he understands what he sees? >Probably not. He's a fish, doofus. Doofus? You're suppose to call nerds doofus, not nitwits like me! > Doofus refers to nerds only, not nitwits. Get this stuff right, I'm > testing you later... > Or go back to school > make that atom bomb > and kill those Germans! > Are you stuck in a time-warp or something? WWII is over, six decades > ago. We won. > I've been watching too many Woody Allen movies... >I'm more worried about the inhalants you use than the movies you watch, >to be honest. You mean like glue sniffing? You put glue in a paper bag and you > inhale it...do you got kids? They are probably glue sniffing right > now.. > The Starmaker...coming soon to a theater near you! Revenge Of The > Nitwits 2 > We've already had Terminator 4 bomb. We don't need another box-office > bomb this month. > You like bombs don't you? >Not the box-office ones. Just the ones that kill people...and I guess you're going to breed > babies that grow up to be.. Weapons of Mass Destruction. And they teach this stuff in school? Daddy, I want to take a How To Build A Nuclear Bomb class at the > university!!! I want to kill millions of people! ( .-. | > | > ` - . > ` - >- > .... .... .... .... > || || || || > /l| /l| /l| /l| > / / / / > | .-. |------| .-. |------| .-. |------| .-. |------ > |L| | .--. | |L| | .--. | |L| | .--. | |L| | .--. > p `o-o' p `o-o' p `o-o' p `o-o' > ------------------------------------------------------------ > ------------------------------------------------------------ The Starmaker- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted test* *ok Here it is: 1) Something cannot come from nothing. 2) We exist. 4) Something is eternal. 5) To me this eternal is God. 6) To you, it is what it is to you. *ok === Subject: Re: What happens to your shape when you move into curved space? <6e877$4a044407$cdd0850a$1164@DIALUPUSA.NET> <8llSl.81414$BZ3.78770@newsfe12.iad> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > your brain exists in one dimension and is processed through time. your brain > can exist without time but it can not process data without time. the other > two spatial dimensions are just those created by humans in order to be able > to judge spatial relations to one another and places on a map. as far as > actual dimensions you can not have 2 or 3 dimensions without one. one > dimension will always have length width and height. > if you believe in the lunacy of string theroy then perhaps you are >thinking a few more than three or 4 if you include time but in the >query I am not including time only the three spacial dimensions. Lets take >a infinate line you have one dimension interset that line and >you have 2 again and you have three. > Or so we have been tought. > Lets goback to the single line of one dimension you say there is a line >weather it is virtual or not the actual line itself is of more than 2 >dimensions. You can not have any one dimension without have 2 >others. In order to say you have marked a boundry the boundry must have height length and width otherwise >without any one of the three you cant have the one therefore no dimension >at all. >Some people may say what about light. Light doesnt have any dimension.In > order to have any dimensions other than one you mustbe able to tap a > boundry >and be able to turn 90 degrees. Well in order to be able to touch that > bondry > one must first have substance. Ifyour next question is back to light let > me just > remind you quickly that glass can be a boundry but light passes very easily > through >glass. Bosons such as a gravaton higgs> would also contain no mass therefore not affected by dimensions. >Perhaps bosons can have an effect on each other but that is >another topic. > -- >Godis really a pink elephant with television rabbit ears.... think I am > wrong? >Prove it > Jason.Goddoes not need to prove He exists. > Assume nothing; expect everything > Please excuse my typos/misspelled words I have no spell checker > If you occupy a curved geometry you will share its shape. Matter and > light extend in curved space. > Mitch Raemsch > Your brain exists in the 4th dimension of curved > space, thus being inaccessible to you.- Hide quoted text - > - Show quoted text - > The 4th dimension's surface is curved and you are extended and moving > in it. > Mitch Raemsch- Hide quoted text - > - Show quoted text -- Hide quoted text - > - Show quoted text - Everything is extended in curved space. This changes the shape of > other fields such as the electric that occupy it. Mitch Raemsch- Hide quoted text - - Show quoted text - >SHUT UP!! > Does your physics books mentioned the word audience refering > [sic] to stars? > No, which is precisely my point. > If your people physics books doesn't[sic] mentioned[sic] the word > audience, it doesn't[sic] understand how stars are made. > Books, of course, don't understand anything; they are just inert > chunks of paper. So this statement of yours is meaningless. > Although I'm curious to know why you think I have people physics > books, a sub-genre I've never heard of. > You never heard of the tree people? There are people who cut down > trees in Brazil, the natives call them the tree people. You are the > physics book people. You got the math people...wait I see this is > going to the science fiction people... > And meanwhile, you are one of the incoherent people. > Much larger population than the coherent people, a dying breed... >Due to nozzles like you. > Children should not be reading sf books... > You're welcome to your opinion. May you never breed. > Not my opinion, ...Albert Einstein's opinion, he believed that people > should not read science fiction--that it distorts science, and gives > people the illusion of understanding. >Even Einstein was occasionally wrong. You got kids out there believing in flying saucers, that's science > fiction, that's a distortion of science. > May he never breed? Or maybe, You?... The Bomb Maker?? >Hey. My bombs kept the peace for decades, until the SU eventually >collapsed under its own weight. Don't knock 'em. > It is the audience that created the universe, ...the audience > exist[sic] in the future. > Right. The universe was made by time travelers. > I'm traveling through time right now! > Don't forget your lightning rod. You know, just in case you run out of > plutonium for juicing up that flux capacitor. > I can always buy plutonium from the Koreans! >*shudder* > People describe the universe with mathematics, but that is just an > illusion... > No, it is not. I can use calculus and the laws of motion to > describe the trajectory of a baseball bat conking you on the head. > Your ensuing headache and goose-egg will confirm handily for you > for several days that it was no illusion. > You should work for the department of defense! They need guys like > you! > I already do, you ninny. Didn't I mention earlier my profession? > You mention it, but you didn't say what country.... >I thought it was obvious. > Mathematics does not describe the universe. Nor does physics. > Sure they do. > Scientism[sic]. > I spelled it right! Scientism. > Oops, nope, and you cocked it up again. It's science. With an e. > There's no c. >There certainly is. >http://www.merriam-webster.com/dictionary/science >Note spelling. the belief that there is one and only one method of science and that > it alone confers legitimacy upon the conduct of research. Just a belief, just a religion..religion of the nerds...Scientism. > You have limited intelligence... > My intelligence greatly exceeds yours. > It is still, limited. >A meaningless statement. It means your intelligence is small in range. > My stupidity greatly exceeds yours, and it has no limits. >That much has been obvious for quite some time now. > A dead universe is a universe > without an audience. No stars > will be born. > There is no evidence to support this contention. > You fail astrophysics forever. > I never heard of that word astrophysics...where did you get it from > An actual ing education. > Was it the 13th grade? > University, you jackass. Perhaps you didn't read the other thread, > but I'm a ing nuclear physicist. I know a lot more about a lot of > this than you do, as must be becoming quite obvious by now. > A nuclear physicist? You guys created an atom bomb to kill Germans and > you end up killing Japanese people with it. That was very smart. > Yeah, imagine if the war of the Pacific had dragged on another five years. > I don't live in a if world... >Oh, yeah, of course, you're too stupid to understand hypotheticals. >Silly me. > Are you threatening me? > Do you have anything against Iranian Nuclear Physics being > assassinated? >I'm not Iranian, nutjob. Do you have anything against any Nuclear Physicist from any country > being assassinated? > Observers, the audience..makes the stars. > Not the celestial ones. > As long as you confined yourself to physics and math, you can > never understand...The Real World. > Wake up and smell the muons: the Real World is made of ... > physics and math. Go to university and get a bachelor of science > degree if you don't believe me. I know I did. > I never went to college...and I don't want to enter those > 'slave to corporations' you call university. > Well, that explains your utter failure to grasp astrophysics or > even some basic concepts of logic and reason. Although not your > near-illiteracy. Did you forget to mention that you dropped out of > school in sixth grade, or something like that? > I'll explain why you're able to grasp astrophysics, ....it's was an > accident. > You are not making sense once again. Grammatically, or semantically. > Accident! A mistake, or by chance.. >There has been no mistake by me. I understand it by intelligence and >reason, not by sheer dumb luck. I bet your parents thought they were lucky. > The truth is, the majority of people do *not* understand or have any > real grasp of science or arithmetic... >Including you. >Fortunately, I'm not in that majority. > if you do, it's a freak of nature, a mistake..and accident. >No, it is not. It is called having had an education. You ought to try >it sometime. Anybody can walk into a college bookstore and buy a textbook. Or a class onlinehttp://www.mininova.org/search/ttc%2Bvideo/seeds > here is what an accident looks like: > >http://www.telegraph.co.uk/telegraph/multimedia/archive/00441/news-gr... >That's the result of disease, not an accident, wit. I was discussing 'smart people' are a freak of nature, it's a picture > of a smart person. It's no accident he got a disease, it comes with > 'the smarts'. > There is no Big Bang without observers in the dark. You want proof, go > see a movie! > This is completely irrational. > Have you tried learning without a textbook? >I learn via all kinds of media, including but not limited to books. You mean like , watching a movie? Do you know how stars are madeË > To that fish > in his tank > that's his universe.. > He can see out through the glass, you know. > Does he understands what he sees? >Probably not. He's a fish, doofus. Doofus? You're suppose to call nerds doofus, not nitwits like me! > Doofus refers to nerds only, not nitwits. Get this stuff right, I'm > testing you later... > Or go back to school > make that atom bomb > and kill those Germans! > Are you stuck in a time-warp or something? WWII is over, six decades > ago. We won. > I've been watching too many Woody Allen movies... >I'm more worried about the inhalants you use than the movies you watch, >to be honest. You mean like glue sniffing? You put glue in a paper bag and you > inhale it...do you got kids? They are probably glue sniffing right > now.. > The Starmaker...coming soon to a theater near you! Revenge Of The > Nitwits 2 > We've already had Terminator 4 bomb. We don't need another box-office > bomb this month. > You like bombs don't you? >Not the box-office ones. Just the ones that kill people...and I guess you're going to breed > babies that grow up to be.. Weapons of Mass Destruction. And they teach this stuff in school? Daddy, I want to take a How To Build A Nuclear Bomb class at the > university!!! I want to kill millions of people! ( .-. | > | > ` - . > ` - >- > .... .... .... .... > || || || || > /l| /l| /l| /l| > / / / / > | .-. |------| .-. |------| .-. |------| .-. |------ > |L| | .--. | |L| | .--. | |L| | .--. | |L| | .--. > p `o-o' p `o-o' p `o-o' p `o-o' > ------------------------------------------------------------ > ------------------------------------------------------------ The Starmaker- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted text -- Hide quoted text - - Show quoted test* *ok Here it is: 1) Something cannot come from nothing. 2) We exist. 4) Something is eternal. 5) To me this eternal is God. 6) To you, it is what it is to you. *ok === Subject: prime factorization of mersenne p exponent minus 1. msg lost resend. Cc: MCDONewt@yahoo.co.nz posting-account=TV2szgkAAACrA1vyuh8IN_0zzgzcwogw CLR 2.0.50727),gzip(gfe),gzip(gfe) > 2,2,2,11,2,53,2,3,3 ,7 ,2,2,2,5 ,13 prime factorisation of mersenne p > exponents minus 1. e.g. 2 2 2 3 3 3 5 5 53 149 > sloanes online encyc of integer sequences. does not have. donald s. mcdonald paradise. nz 11-6-2009. > 47th known mersenne prime apparently discovered. world record bunch of prime numbers.: Google Alert - mersenne prime 7 mersenne prime exponents within an /octave// interval [k, 2k.] don mcdonald wellington 10-6-2009. jfh, jb. I am not sure if these columns will print correctly yahoo. 47th known mersenne prime exponent. factor 2^prime - 1 isprime? 42643801 426438 9 47382 factor 6 7897 factor 3 2632.333333 5 1579.4 7 1128.142857 9 877.4444444 11 717.9090909 13 607.4615385 15 526.4666667 17 464.5294118 19 415.6315789 21 376.047619 23 343.3478261 25 315.88 27 292.4814815 29 272.3103448 31 254.7419355 33 239.3030303 35 225.6285714 37 213.4324324 39 202.4871795 41 192.6097561 43 183.6511628 45 175.4888889 47 168.0212766 49 161.1632653 51 154.8431373 53 149 7897 426438 6*9*53*149*100+1 factors 42643801 isprime? twin primes about mersenne prime exponent. 43rd? program BAS64- modabc1 ? 277345127 31*683*13099 277345127 ryee Don S. McDonald ... Table Mountain 2^ (2*34* 444 32 +1 ) -1 is Mersenne Prime v//. exponent 3.02.. million? Climate pride//# === > Subject: world record bunch of prime numbers.: Google Alert - mersenne prime > Received: Wednesday, 10 June, 2009, 1:50 PM > world record bunch of prime numbers.: Google Alert - > mersenne prime MathWorld News: 47th Known Mersenne Prime Apparently > Discovered > June 7, 2009--Less than a year after the 45th and 46th > known Mersenne > primes were discovered, Great Internet Mersenne Prime > Search (GIMPS) > project organizer George Woltman is reporting in a June 7 > email to the > GIMPS mailing list that a ... This once a day Google Alert > fwd cheers, > ............10/6/09 Tara wera Mt > 1886?? Rotorua NZ. Don S. McDonald ... === Subject: Top 10 Solutions to P=NP 1. N=NP when P is Perfect. 2. P=NP: It is not everything; it is some of everything. 3. Put a pin in pi. 4. NP=P iff P=NP. 5. A=A Polynomial Time Algorith. 6. Programming. P=1 and NP=1. 7. Subtract NP from P and divide by 2. 8. Multiply 1/2 of NP by P then divide by 2. 9. Add NP to P and name the result Complete. 10. Any of the above. Enjoy this list! Musatov === === Subject: Rational triangles posting-account=xLsqvAoAAADY4BMAqIw3ROq0faU4tvyS CLR 2.0.50727),gzip(gfe),gzip(gfe) (The following questions are prompted by a New Scientist Enigma recently posed in rec.puzzles.) Given a triangle whose sides are all rational numbers, what can one say about its angles? An angle might be pi/3, pi/2, 2pi/3. Are there any other rational multiples of pi radians that might occur in a rational-sided triangle? Letting A, B, C denote the *squares* of the sides, the Pythagorean theorem for angles of 90 degrees is C = A + B What is the name for the corresponding theorem for angles of 120 degrees? : AB = (A + B - C)^2 James Dow Allen === Subject: Re: Rational triangles > (The following questions are prompted by a > New Scientist Enigma recently posed in rec.puzzles.) > > Given a triangle whose sides are all rational > numbers, what can one say about its angles? > > An angle might be pi/3, pi/2, 2pi/3. Are there > any other rational multiples of pi radians that might > occur in a rational-sided triangle? No. By the Law of Cosines, c^2 = a^2 + b^2 - 2 a b cos(theta) where theta is the angle opposite side c. If the sides are all rational (and nonzero), cos(theta) must be rational. But if theta is a rational multiple of pi, 2 cos(theta) is an algebraic integer, and the only algebraic integers that are rational are the ordinary integers. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Rational triangles Given a triangle whose sides are all rational > numbers, what can one say about its angles? > An angle might be pi/3, pi/2, 2pi/3. Are there > any other rational multiples of pi radians that might > occur in a rational-sided triangle? No. Yes. 2pi/3, 2pi/3, 2pi/3. > By the Law of Cosines, c^2 = a^2 + b^2 - 2 a b cos(theta) > where theta is the angle opposite side c. If the sides are > all rational (and nonzero), cos(theta) must be rational. But > if theta is a rational multiple of pi, 2 cos(theta) is an > algebraic integer, and the only algebraic integers that are > rational are the ordinary integers. === Subject: Extended Metrics Let d be a metric for (S - A)x(S - A). Pick some k not in [0,oo). Let p be an extension of d to SxS with for all s in S, a in A, p(s,a) = p(a,s) = k. Can p be a metric? Would that require extending the definition of metric from real valued to real valued in the extended reals? What is known about these extended metrics? How would this do for a work around p(x,y) = arctan d(x,y), if x not in A or y not in A = 1, otherwise ? Is not (S - A, d) topologically equivalent to the subspace S - A of (S, p). Is there any advantage of one over the other? === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=5GUrzQkAAADun29oaK3p_W_saUVxxHUF Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > (a) Why did Einstein considered *Time* as a dimension? (b) If he considered it(time) as a *dimension*, > what makes him to think that*Time* is One dimensional ? (c) what exactly is *Time* ? greetings > nimo > politics,philosophy religion & society > are quite good to HIM rather than > physics; and this quote is best suited to him at this point.., ïEvery street urchin in our mathematical G.9attingen > knows more about four-dimensional geometry than > Einstein. Nevertheless, it was Einstein who did the work, > not the great mathematicians. David Hilbert > (January 23, 1862 [CapitalEth] February 14, 1943) the quote's author had shown his sarcasm very cleverly :-) ---------------- time has actually no physical dimensions !! .............. ATB Y.Porat ------------------------- === Subject: Re: Is time is One dimensional? if yes, How ? > I mean, when you start throwing around the word 'Kantian', I have > the prerogative to ask you about your use of the term, particularly > whether according to you there's any empirical difference between > 'Kantian' and 'non-Objectivist', as you confirmed your empirical > belief that 99.999999999999% (or whatever number of 9's) of non- > Objectivists are Kantians. I count 51. $ echo '99.999999999999999999999999999999999999999999999999' | wc -c 52 > > He used fifty 9s in the original remark; it was supposed to be the > proportion of those famous people who are not Objvst's. In > other words, drawing on the population of famous people, only > one in 10^50 are not Objvst's. > > Suppose there are, let's just say, 10^4 famous people; that means > that 10^-46 people are Big-O. This is well less than the proportion > of a human body accounted for by a single electron. > > I was hoping to make something ridiculous out of the claim, > like that there was a part of a pinky-toe somewhere that > qualified, but he used so many nines that it went beyond > ridiculous, past stupid, and ran straight in to unity. > > As a related aside, I hereby propose that a school of thought > is the direct opposite of thinking. But I will probably be > called Kantian, or something equally asinine, for my effort. Free thinker. Or are you paid to think? -- Michael Press === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=Rqa4sAoAAAC88UYanCtJRUF4S6TUauGA Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) > He used fifty 9s in the original remark; it was supposed to be the > proportion of those famous people who are not Objvst's. In > other words, drawing on the population of famous people, only > one in 10^50 are not Objvst's. I count 51. $ echo '99.999999999999999999999999999999999999999999999999' | wc -c > 52 You didn't count the newline. > As a related aside, I hereby propose that a school of thought > is the direct opposite of thinking. But I will probably be > called Kantian, or something equally asinine, for my effort. Free thinker. Or are you paid to think? Ha! Sounds about right. Marshall === Subject: Re: Is time is One dimensional? if yes, How ? > Oh you wont be the only reader, and many an athiest takes pride or > self satisfaction in lambasting the religionists and yet they come > from the identical POC epistemological premise, is my point. And a good point it is. Who fights more fiercely than a pair of closely related people with a minor difference of opinion? Atheists and religionists are dual to each other. -- Michael Press === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=g9P5YAkAAACbxBDAyQp7cMVe0-Yk0MYD SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > Oh you wont be the only reader, and many an athiest takes pride or > self satisfaction in lambasting the religionists and yet they come > from the identical POC epistemological premise, is my point. And a good point it is. Who fights more fiercely than a > pair of closely related people with a minor difference > of opinion?Atheists and religionists are dual to each other. The context; And note; being athiest does NOT mean the rejection of primacy of consciousness, being athiest does NOT mean reason has been applied to one's ideas, being athiest does NOT rule out accepting ideas in the absense of evidence - in the absense of any sensory evidence for the acceptance of that idea, e.g. many socialists are athiests and yet socialism is a mind dependent arbitrary faith based concept. MG === Subject: Re: Is time is One dimensional? if yes, How ? > If she despised irrationalism, and Kant's philosophy > is irrational, as she remarks that it is, then it seem ineluctable > that she despised the philosophy of Kant. > As well as any and all other irrational utterly mind dependent > philosophies, ideas of whim and the arbitrary. > By the way, I'm interested in who you think are opponents of > Objectivism but not, in your view, Kantians? > Those who cant reduce their ideas theories concepts, which they claim > are ideas theories and concepts of reality, back to an irreducible and > sensory level of perception in their origin are the opponents of > Rand's ideas. e.g god and the leftist retard's nauseating nasal > whining chant of the greater good. > > Forsooth (especially the part about 'nasal whining'; that was an > especially scathing turn of phrase). But I was wondering whether you > might mention some particular people (famous ones, so I'm in on the > moral) who you consider to be opponents of Objectivism but not > Kantians. I am guessing, as the compleat idiot in philosophy, Bertrand Russell and Noam Chomsky. There must be many more lurking, but I was lucky to dredge up those two names. -- Michael Press === Subject: Re: Is time is One dimensional? if yes, How ? > > > By the way, I'm interested in who you think are opponents of > Objectivism but not, in your view, Kantians? > A fearsome tribe, known by the name Altruists, is a major enemy of > Objectivism, nay, reason itself. > > I've always wanted to teach a course titled Altruism: how to make it > work for you. > > Good one. Reminds me of the old joke about how sincerity is > the most important thing - once you can fake that, you've got it made. Louis B. Mayer, as I recall. -- Michael Press === Subject: Re: Is time is One dimensional? if yes, How ? <87bpouljve.fsf@alatheia.truth.invalid> It would be helpful if you would at least say that you recognize now > that it is not I, but rather you, who was confused on this particular > point. > Why would it be helpful? Aside from feeding my smug sense of self-righteousness, you mean? Isn't your smug self-righteousness better served if Gordge persists in his silliness? That aside, as you probably surmised the above was but a bit of gentle poking at your peculiar habit of answering idiots of all colours, not letting a single point go unaddressed, with mind-bogglingly detailed explanations and arguments. Each to their own, of course. > By the way, I answered your question about Con(ZF) in the other > thread. I allowed myself to mention some very informal personal > notions. I'd love to know your thoughts on my remarks (aside from my > slip of saying 'sigma' when I meant 'pi'). The slip is uninteresting. I'll return to your remarks, which were interesting, later, in a deeply thoughtful little essay with footnotes. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Is time is One dimensional? if yes, How ? <878wjwe9dd.fsf@alatheia.truth.invalid> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > It would be helpful if you would at least say that you recognize now > that it is not I, but rather you, who was confused on this particular > point. > Why would it be helpful? > Aside from feeding my smug sense of self-righteousness, you mean? Isn't your smug self-righteousness better served if Gordge persists in > his silliness? That aside, as you probably surmised the above was but a > bit of gentle poking at your peculiar habit of answering idiots of all > colours, not letting a single point go unaddressed, with mind-bogglingly > detailed explanations and arguments. Each to their own, of course. > By the way, I answered your question about Con(ZF) in the other > thread. I allowed myself to mention some very informal personal > notions. I'd love to know your thoughts on my remarks (aside from my > slip of saying 'sigma' when I meant 'pi'). The slip is uninteresting. I'll return to your remarks, which were > interesting, later, in a deeply thoughtful little essay with footnotes. -- > Aatu Koskensilta (aatu.koskensi...@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen >- Ludwig Wittgenstein, Tractatus Logico-Philosophicus You guys are mean. This is about right and wrong. === Subject: Re: Is time is One dimensional? if yes, How ? > I also predicted I would get called something asinine, > and that prediction came true with startling accuracy. Still, not a very impressive prediction when dealing with a giant horse's ass like Gordge. -- hz === === Subject: Re: FROM MARVELLOUS BEGINNING TILL HAPPY END OF FLT !!! Sorry for previous formulae: they seems to be something but turns to be not effective... Anyhow there is my new layout of proof which previously used to have some problem for n=3, what I've just covered. Not so short as I used to input with these formulae from pythagorean triplets... Coming back soon. Ro-Bin Message was edited by: Roman B. Binder === Subject: Solution manuals and test banks for sale posting-account=jy2E5goAAAB1WGwMQL1h2k61o4k7O_u4 Gecko/2009051221 Firefox/3.0.10,gzip(gfe),gzip(gfe) I can get these solution manuals / test banks for you. 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Horngren et al 7th ed ISBN - 0132249960 Accounting Concepts and Applications - Steve Albrecht 10th ed ISBN - 0324376154 Accounting Concepts and Applications - Steve Albrecht 9th ed ISBN - 0324187564 Accounting Information Systems - James Hall 5th ed ISBN - 0324312954 Accounting Information Systems - James Hall 6th ed ISBN - 0324560893 Accounting Information Systems - Marshall Romney, Paul Steinbart 10th ed ISBN - 0131475916 Accounting Information Systems - Marshall Romney, Paul Steinbart 11th ed ISBN - 0136015182 Accounting Information Systems - Ulric J. Gelinas 7th ed ISBN - 0324378823 Accounting Information Systems - Ulric J. Gelinas 8th ed ISBN - 0324663803 Additional Calculus Topics - Raymond Barnett 11th ed ISBN - 0132318229 Administrative Law: Bureaucracy in a Democracy - Daniel E. Hall 4th ed ISBN - 0135005183 Advanced Accounting - Floyd Beams 9th ed ISBN - 0131851225 Advanced Accounting - Floyd Beams 10th ed ISBN - 0136033970 Advanced Accounting - Paul Fischer 10th ed ISBN - 0324379056 Advanced Accounting - Paul Fischer Test Bank only 9th ed ISBN - 0324304013 Advanced Calculus - G. B. Folland 1st ed ISBN - 0130652652 Advanced Engineering Mathematics - Michael Greenberg 2nd ed ISBN - 0133214311 Advanced Engineering Mathematics - Peter V. O'Neil 6th ed ISBN - 0534552080 Advertising - Sandra Moriarty 8th ed ISBN - 0132224151 Algebra and Trigonometry - Judith A. Beecher 3rd ed ISBN - 0321466209 Algebra and Trigonometry - Michael Sullivan 8th ed ISBN - 0132329034 Algebra and Trigonometry Enhanced with Graphing Utilities - Michael Sullivan 4th ed ISBN - 0131527398 Algebra and Trigonometry Enhanced with Graphing Utilities - Michael Sullivan 5th ed ISBN - 013600492X Algebra and Trigonometry: Graphs & Models and Graphing Calculator Manual Package - Marvin L. Bittinger 4th ed ISBN - 0321501519 Algebra for College Students - Allen R. Angel 3rd ed ISBN - 0136129080 Algebra for College Students - Margaret L. Lial 6th ed ISBN - 0321442547 Algebra For College Students - Robert F Blitzer 6th ed ISBN - 0136019749 An Introduction to Signals and Systems - John Alan Stuller 1st ed ISBN - 0495073016 Dunlap 1st ed ISBN - 0534392946 Analytical Mechanics - Grant Fowles, George Cassiday 7th ed ISBN - 0534494927 Anatomy & Physiology - Elaine N. Marieb 3rd ed ISBN - 0805347739 Anatomy & Physiology for Emergency Care - Bryan E. Bledsoe 2nd ed ISBN - 0132342987 Applied Algebra - Darel Hardy 1st ed ISBN - 0130674648 Applied Calculus - Geoffrey C. Berresford 5th ed ISBN - 0547169787 Applied Linear Algebra - Chehrzad Shakiban, Peter J. Olver 1st ed ISBN - 0131473824 Applied Multivariate Statistical Analysis - Richard A. Johnson 6th ed ISBN - 0131877151 Applied Partial Differential Equations - Richard Haberman 4th ed ISBN - 0130652431 Applied Physics - Dale Ewen 9th ed ISBN - 0135157331 Applied Statistics for Engineers and Physical Scientists - Johannes Ledolter 3rd ed ISBN - 0136017983 Art and Science of Leadership - Afsaneh Nahavandi 5th ed ISBN - 0136044085 Auditing and Assurance Services - Alvin A. Arens et al 11th ed ISBN - 0131867121 Auditing and Assurance Services - Alvin A. Arens et al 12th ed ISBN - 0135132126 Auditing Assurance and Risk - W. Robert Knechel, Steve Salterio, Brian Ballou 3rd ed ISBN - 0324313187 Auditing Cases - Mark Beasley 3rd ed ISBN - 0131494910 Auditing Cases - Mark S Beasley 4th ed ISBN - 0132423502 Auditing: A Business Risk Approach - Larry E. Rittenberg 6th ed ISBN - 0324375581 Automation, Production Systems, and Computer-Integrated Manufacturing - Mikell P. Groover 2nd ed ISBN - 0130889784 Automation, Production Systems, and Computer-Integrated Manufacturing - Mikell P. Groover 3rd ed ISBN - 0132393212 Basic Business Statistics - Mark L Berenson 10th ed ISBN - 0131678310 Basic Chemistry - Karen C. Timberlake 2nd ed ISBN - 0805344691 Basic Economics - Frank V. Mastrianna Test Bank 15th ed ISBN - 0324599161 Basic Environmental Technology: Water Supply, Waste Management & Pollution Control - Jerry A. Nathanson 5th ed ISBN - 0131190822 Basic Marketing Research Using Microsoft Excel Data Analysis - Alvin C Burns 2nd ed ISBN - 0132059584 Basic Mathematics through Applications - Geoffrey Akst 4th ed ISBN - 0321500113 Basics of Occupational Safety - David L. Goetsch 1st ed ISBN - 013502613X Beginning & Intermediate Algebra - Elayn El Martin-Gay 5th ed ISBN - 0136007317 Beginning Algebra - Elayn El Martin-Gay 5th ed ISBN - 0136007023 Beginning Algebra - Margaret L. Lial 10th ed ISBN - 0321437268 Beginning Algebra with Applications & Visualization - Gary K. Rockswold 2nd ed ISBN - 0321500040 Beginning and Intermediate Algebra - Margaret L. Lial 4th ed ISBN - 0321442334 Beginning and Intermediate Algebra with Applications & Visualization - Gary K. Rockswold 2nd ed ISBN - 0321500059 Behavior in Organizations - Jerald Greenberg 9th ed ISBN - 0131542842 Biochemistry - Mary Campbell 4th ed ISBN - 0534405215 Biochemistry with Lecture Notebook - Mary Campbell 4th ed ISBN - 0534391818 Biology - Neil A. Campbell Test Bank only w/ TestGen Software 7th ed ISBN - 080537146X Biology: Science for Life - Colleen Belk 3rd ed ISBN - 0321559592 Biology: Science for Life with Physiology - Colleen Belk 3rd ed ISBN - 0321559584 Biomaterials: The Intersection of Biology and Materials Science - Johnna S. Temenoff 1st ed ISBN - 0130097101 Biostatistics for the Health Sciences - R. Clifford Blair 1st ed ISBN - 0131176609 Bond Markets, Analysis and Strategies - Frank Fabozzi 6th ed ISBN - 0131986430 Bond Markets, Analysis, and Strategies - Frank J Fabozzi 7th ed ISBN - 0136078974 Brief Course in Mathematical Statistics - Elliot A. Tanis 1st ed ISBN - 0131751395 Brief Principles of Macroeconomics - Gregory Mankiw 5th ed ISBN - 0324590377 Brock Biology of Microorganisms - Michael T. Madigan 12th ed ISBN - 0132324601 Brock Biology of Microorganisms - Michael T. Madigan Test Bank 11th ed ISBN - 0132192268 Building Construction: Principles, Materials, and Systems - Madan Mehta 1st ed ISBN - 0130494216 Building Java Programs: A Back to Basics Approach - Stuart Reges 1st ed ISBN - 0321382838 Business - William M. Pride 10th ed ISBN - 0324829558 Business Analysis and Valuation: Using Financial Statements - Krishna Palepu 3rd ed ISBN - 0132346451 Business and Its Environment - David P. Baron 6th ed ISBN - 0136083927 Business and Society: Ethics and Stakeholder Management - Archie B. Carroll 7th ed ISBN - 0324569394 Business Communication Essentials - Courtland Bovee 4th ed ISBN - 0136084419 Business Communication Essentials and Peak Performance Grammar and Mechanics 2.0 CD Package - Court Bovee 3rd ed ISBN - 0132328992 Business Communication Today - Court Bovee 9th ed ISBN - 0131995359 Business Data Networks and Telecommunications - Raymond R. Panko 7th ed ISBN - 0136153402 Business English: Writing in the Workplace - Blanche Ettinger 4th ed ISBN - 0131565702 Business Ethics: A Stakeholder and Issues Management Approach - Joseph W. Weiss 6th ed ISBN - 0324589735 Business Ethics: Case Studies and Selected Readings - Marianne M. Jennings 6th ed ISBN - 0324657749 Business Forecasting - John Hanke 9th ed ISBN - 0132301202 Business in Action with Real Time Updates - Court Bovee 4th ed ISBN - 0136154085 Business Law - Henry R. Cheeseman 7th ed ISBN - 0136085547 Business Law and the Legal Environment - Jeffrey F. Beatty 4th ed ISBN - 0324303971 Business Law and the Legal Environment - Jeffrey F. Beatty 5th ed ISBN - 0324663528 Business Law and the Regulation of Business - Richard A. Mann 9th ed ISBN - 0324537131 Business Law Principles for Today's Commercial Environment - David P. Twomey 2nd ed ISBN - 0324303947 Business Law Today: Comprehensive - Roger LeRoy Miller 8th ed ISBN - 0324595743 Business Law Today: The Essentials - Roger LeRoy Miller 8th ed ISBN - 0324654545 Business Law: Alternate Edition - Gaylord A. Jentz 11th ed ISBN - 0324596162 Business Law: Text and Cases - Kenneth W. Clarkson 11th ed ISBN - 0324655223 Business Law: Text and Exercises - Roger LeRoy Miller 5th ed ISBN - 032464096X Business Statistics: A Decision Making Approach - David F. Groebner 7th ed ISBN - 0132416921 Business Statistics: A First Course - David Levine 5th ed ISBN - 0136065805 Business: Its Legal, Ethical, and Global Environment - Marianne M. Jennings 8th ed ISBN - 0324655541 Calculus - Dale Varberg 9th ed ISBN - 0131429248 Calculus and Its Applications - Larry Goldstein 11th ed ISBN - 0131919636 Calculus and Its Applications - Larry Goldstein 12th ed ISBN - 0321571304 Calculus and Its Applications - Marvin L. Bittinger 8th ed ISBN - 0321166396 Calculus and Its Applications - Marvin L. Bittinger 9th ed ISBN - 0321395344 Calculus Early Transcendentals - Henry Edwards 7th ed ISBN - 0131569899 Calculus for Business, Economics, Life Sciences & Social Sciences - Raymond Barnett 11th ed ISBN - 0132328186 Calculus for the Life Sciences - Marvin L. Bittinger 1st ed ISBN - 0321279352 Calculus With Applications - Margaret L. Lial et al 8th ed ISBN - 0321228146 Calculus with Applications for the Life Sciences - Raymond N. Greenwell 1st ed ISBN - 0201745828 Calculus, Early Transcendentals - C. Henry Edwards 7th ed ISBN - 0131569899 California Real Estate Law - Theodore Gordon 7th ed ISBN - 0324654685 Capital Budgeting and Long-Term Financing Decisions - Neil Seitz 4th ed ISBN - 0324258089 Capital Markets: Institutions and Instruments - Frank J Fabozzi 4th ed ISBN - 0136026028 Cases in Management Accounting and Control Systems - Brandt Allen 4th ed ISBN - 0135704251 Chemistry - John E McMurry Test Bank only 5th ed ISBN - 0131993232 Chemistry : An Introduction to General, Organic, Biological Chemistry - Karen Timberlake 9th ed ISBN - 0805330151 Chemistry for Changing Times - John W. Hill 12th ed ISBN - 0136054498 Chemistry: An Introduction to General, Organic, & Biological Chemistry - Karen C Timberlake 10th ed ISBN - 0136019706 Civil Drafting Technology - David A. Madsen 7th ed ISBN - 0135000688 ISBN - 0534408966 CMOS Circuit Design, Layout, and Simulation - David E. Boyce et al 1st ed ISBN - 0780334167 College Accounting 1-12 - Jeffrey Slater 9th ed ISBN - 0131071696 College Accounting 1-25 - Jeffrey Slater 10th ed ISBN - 0132286386 College Accounting Chapters 1-15 - James Heintz 19th ed ISBN - 0324382499 College Accounting Chapters 1-27 - James Heintz 19th ed ISBN - 0324376162 College Accounting Chapters 1-9 - James Heintz 19th ed ISBN - 0324382480 College Accounting: A Practical Approach Canadian Edition - Jeffrey Slater 10th ed ISBN - 0132069245 College Algebra - J. S. Ratti 1st ed ISBN - 0321296443 College Algebra - Judith A. Beecher 3rd ed ISBN - 0321466071 College Algebra - Margaret L. Lial 10th ed ISBN - 0321499131 College Algebra - Mark Dugopolski 4th ed ISBN - 0321356918 College Algebra - Michael Sullivan 8th ed ISBN - 0132402866 College Algebra - Robert F. Blitzer 5th ed ISBN - 0321559835 College Algebra and Trigonometry - J. S. Ratti 1st ed ISBN - 0321296427 College Algebra and Trigonometry - Margaret L. Lial 4th ed ISBN - 0321497449 College Algebra Enhanced with Graphing Utilities - Michael Sullivan 5th ed ISBN - 0136004911 College Algebra Essentials - Michael Sullivan 8th ed ISBN - 0136154344 College Algebra: Graphs and Models with Graphing Calculator Manual Package - Marvin L. Bittinger 4th ed ISBN - 0321531922 College Geometry - David C. Kay 2nd ed ISBN - 0321046242 College Geometry: A Problem Solving Approach with Applications - Gary L. Musser 2nd ed ISBN - 0131879693 College Math for Business, Economics, Life Sciences & Social Sciences - Raymond Barnett 11th ed ISBN - 0131572253 College Physics - Jerry D Wilson 6th ed ISBN - 0131495798 College Physics - Jerry D Wilson 7th ed ISBN - 0321571118 College Physics with Mastering Physics - Hugh Young 8th ed ISBN - 0805390707 Communicating in the Workplace - Thomas Cheesebro 1st ed ISBN - 0136136915 Communication Systems Engineering - John G. Proakis 2nd ed ISBN - 0130617938 Comparative International Accounting - Christopher Nobes 9th ed ISBN - 0273703579 Complex Variables With Applications - A. David Wunsch 3rd ed ISBN - 0201756099 Comprehensive Periodontics for the Dental Hygienist - Mea A. Weinberg 3rd ed ISBN - 0135015421 Computer Algorithms - Allen Van Gelder, Sara Baase 3rd ed ISBN - 0201612445 Computer Networking Complete Package - James F. Kurose 3rd ed ISBN - 0321418492 Computer Networking with Internet Protocols - William Stallings 1st ed ISBN - 0131410989 Computer Networking: A Top-Down Approach - James F. Kurose 4th ed ISBN - 0321497708 Computer Networking: A Top-Down Approach - James F. Kurose 5th ed ISBN - 0136079679 Computer Networking: A Top-Down Approach Featuring the Internet - James F. Kurose 3rd ed ISBN - 0321227352 Computer Organization and Architecture - William Stallings 7th ed ISBN - 0130351199 Computer Organization and Architecture: Designing for Performance - William Stallings 7th ed ISBN - 0131856448 Computer Organization and Architecture: Designing for Performance - William Stallings 8th ed ISBN - 0136073735 Computer Science: An Overview - J. Glenn Brookshear 10th ed ISBN - 0321524039 Computer Security: Principles and Practice - William Stallings 1st ed ISBN - 0136004245 Computer Systems Organization & Architecture - John D. Carpinelli 1st ed ISBN - 0201612534 Concepts in Federal Taxation 2007 - Kevin Murphy 14th ed ISBN - 0324313527 Concepts in Federal Taxation 2008 - Kevin Murphy 15th ed ISBN - 0324640153 Concepts in Federal Taxation 2009 - Kevin Murphy 16th ed ISBN - 0324659377 Concepts In Systems and Signals - John D. Sherrick 2nd ed ISBN - 0131782711 Concepts of Calculus with Applications - Martha Goshaw 1st ed ISBN - 0321320786 Concepts of Calculus With Applications-Updated Edition - Martha Goshaw 2nd ed ISBN - 0321577442 Concepts of Programming Languages - Robert W. Sebesta 8th ed ISBN - 0321493621 Conceptual Physical Science - Paul G. Hewitt 4th ed ISBN - 0321516958 Conceptual Physics Fundamentals - Paul G. Hewitt 1st ed ISBN - 0321501365 Conceptual Physics Media Update - Paul G. Hewitt 10th ed ISBN - 0321548094 Concrete Structures - Mehdi Setareh 1st ed ISBN - 0131988271 Construction Accounting & Financial Management - Stephen Peterson 2nd ed ISBN - 0135017114 Construction Methods and Management - Stephens W. Nunnally 7th ed ISBN - 0131716859 Construction Project Management - Fred Gould 3rd ed ISBN - 0131996231 Consumer Behavior - Michael Solomon 8th ed ISBN - 0136015964 Consumer Behavior - Wayne D. Hoyer 5th ed ISBN - 0547079923 Contemporary Auditing: Real Issues & Cases - Michael C. Knapp 7th ed ISBN - 0324658052 Contemporary Auditing: Real Issues & Cases Update - Michael C. Knapp 7th ed ISBN - 143907819X Contemporary Business and Online Commerce Law - Henry R. Cheeseman 6th ed ISBN - 013601500X Contemporary Engineering Economics - Chan S. Park 4th ed ISBN - 0131876287 Contemporary Financial Management - Charles Moyer 10th ed ISBN - 0324289081 Contemporary Financial Management - R. Charles Moyer, James R. McGuigan 11th ed ISBN - 0324653506 Contemporary Logistics - Paul R. Murphy 9th ed ISBN - 013156207X Contemporary Marketing - Louis E. Boone 14th ed ISBN - 032458203X Contemporary Marketing 2009 Update - Louis E. Boone 13th ed ISBN - 0324580215 Contemporary Project Management - Timothy Kloppenborg 1st ed ISBN - 0324382383 Cornerstones of Managerial Accounting - Maryanne M. Mowen 2nd ed ISBN - 0324379609 Cornerstones of Managerial Accounting - Maryanne M. Mowen 3rd ed ISBN - 0324660138 Corporate Finance - Jonathan Berk 1st ed ISBN - 0321415116 Corporate Finance - Michael C. Ehrhardt, Eugene F. Brigham 3rd ed ISBN - 0324655681 Corporate Finance: The Core plus MyFinanceLab Student Access Kit - Jonathan Berk 1st ed ISBN - 032155759X Corporate Financial Accounting - Carl S. Warren 10th ed ISBN - 0324663838 Corporate Financial Management - Douglas R. Emery 3rd ed ISBN - 0132278723 Cost Accounting - Charles T. Horngren, George Foster, Srikant M. Datar 12th ed ISBN - 0131495380 Cost Accounting - Charles T. Horngren, George Foster, Srikant M. Datar 13th ed ISBN - 0136126634 Cost Accounting Canadian Edition - Charles Horngren 4th ed ISBN - 0131971905 Cost Accounting: Traditions & Innovations - Jesse Barfield 5th ed ISBN - 032418090X Cost Benefit Analysis: Concepts and Practice - Anthony Boardman 3rd ed ISBN - 0131435833 Cost Management: Accounting and Control - Don R. Hansen, Maryanne M. Mowen 6th ed ISBN - 0324559674 Course in Probability - Neil Weiss 1st ed ISBN - 0201774712 Criminology: A Global Perspective - Robert W. Winslow 1st ed ISBN - 0131839020 Cryptography and Network Security - William Stallings 4th ed ISBN - 0131873164 Customer Service: Career Success Through Customer Loyalty - Paul R. Timm 4th ed ISBN - 0132236583 Data Abstraction & Problem Solving with C++ - Frank M. Carrano 5th ed ISBN - 0321433327 Data and Computer Communications - William Stallings 8th ed ISBN - 0132433109 Data Structures and Algorithm Analysis in C++ - Mark Allen Weiss 3rd ed ISBN - 032144146X Data Structures and Algorithm Analysis in Java - Mark Allen Weiss 2nd ed ISBN - 0321370139 Database Concepts - David Kroenke 3rd ed ISBN - 0131986252 Database Concepts - David Kroenke 4th ed ISBN - 0136086535 Database Systems: A Practical Approach - Thomas M. Connolly 4th ed ISBN - 0321294017 Derivatives Markets - Robert L. McDonald 2nd ed ISBN - 032128030X Detection and Estimation:Theory and Its Applications - Thomas Schonhoff 1st ed ISBN - 0130894990 Developmental Mathematics - Marvin L. Bittinger 7th ed ISBN - 0321331915 Developmental Mathematics: Basic Mathematics and Algebra - Margaret L. Lial 1st ed ISBN - 0321506421 Differential Equations - John Polking 2nd ed ISBN - 0131437380 Differential Equations and Boundary Value Problems: Computing and Modeling - Henry Edwards 4th ed ISBN - 0131561073 Differential Equations and Linear Algebra - Henry Edwards, David E. Penney 2nd ed ISBN - 0131481460 Differential Equations and Linear Algebra - Henry Edwards, David E. Penney 3rd ed ISBN - 0136054250 Differential Equations and Linear Algebra - Jerry Farlow 2nd ed ISBN - 0131860615 Differential Equations and Linear Algebra - Stephen W. Goode 3rd ed ISBN - 0130457949 Differential Equations Computing and Modeling - Henry Edwards 4th ed ISBN - 0136004385 Differential Equations With Boundary Value Problems - John C. Polking 2nd ed ISBN - 0130911062 Digital & Analog Communication Systems - Leon Couch 7th ed ISBN - 0131424920 Digital Communications - John Proakis 4th ed ISBN - 0072321113 Digital Design - Morris Mano 4th ed ISBN - 0131989243 Digital Electronics: A Practical Approach - William Kleitz 8th ed ISBN - 0132435780 Digital Fundamentals - Thomas Floyd 10th ed ISBN - 0132359235 Digital Signal Processing - John Proakis 4th ed ISBN - 0131873741 Digital Signal Processing Using MATLAB -Vinay K. Ingle, John G. Proakis 2nd ed ISBN - 0495073113 Digital Systems Design Using VHDL - Charles H. Roth 2nd ed ISBN - 0534384625 Digital Systems: Principles and Applications - Ronald Tocci et al 10th ed ISBN - 0131725793 Discrete and Combinatorial Mathematics - Ralph P. Grimaldi 5th ed ISBN - 0201726343 Discrete Mathematics - Edgar G. Goodaire, Michael M Parmenter 3rd ed ISBN - 0131679953 Discrete Mathematics - Otto, Eynden, Dossey, Spence 4th ed ISBN - 0321079124 Discrete Mathematics - Otto, Eynden, Dossey, Spence 5th ed ISBN - 0321305159 Discrete Mathematics - Richard Johnsonbaugh 6th ed ISBN - 0131176862 Drugs & the Human Body - Ken Liska 8th ed ISBN - 0132447134 Dynamics of Structures - Anil K. Chopra 3rd ed ISBN - 013156174X ECON for Macroeconomics - William A. McEachern 1st ed ISBN - 0324587805 Economic Development - Michael P. Todaro 10th ed ISBN - 0321485734 Economic Development - Michael Todaro, Stephen Smith 9th ed ISBN - 0321278887 Economic Growth - David N. Weil 2nd ed ISBN - 0321416627 Economic Growth - David Weil 1st ed ISBN - 0201680262 Economics - Michael Parkin 8th ed ISBN - 0321423003 Economics - Michael Parkin 9th ed ISBN - 0321600037 Economics - Richard Lipsey 13th ed ISBN - 0321369211 Economics - Roger A. Arnold 9th ed ISBN - 0324595425 Economics for Managers - Paul G Farnham 1st ed ISBN - 0130924253 Economics for Managers - Paul G Farnham 2nd ed ISBN - 013606552X Economics for Today - Irvin B. Tucker 6th ed ISBN - 0324591365 Economics of Money, Banking, and Financial Markets - Frederic Mishkin Test Bank 8th ed ISBN - 0321415051 Economics of Money, Banking, and Financial Markets, Update - Frederic Mishkin 7th ed ISBN - 0321331850 Economics Today - Roger LeRoy Miller 14th ed ISBN - 0321422341 Economics Today - Roger Miller 15th ed ISBN - 0321600215 Economics Today: The Macro View - Roger Miller 13th ed ISBN - 0321278992 Economics Today: The Macro View - Roger Miller 14th ed ISBN - 0321421442 Economics Today: The Macro View - Roger Miller 15th ed ISBN - 0321600223 Economics Today: The Micro View - Roger Miller 13th ed ISBN - 0321278984 Economics Today: The Micro View - Roger Miller 14th ed ISBN - 0321425065 Economics Today: The Micro View - Roger Miller 15th ed ISBN - 0321600185 Economics: A Contemporary Introduction - William A. McEachern 8th ed ISBN - 0324579217 Economics: A Tool for Critically Understanding Society - Tom Riddell 8th ed ISBN - 0321423585 Economics: Principles and Policy - William J. Baumol 10th ed ISBN - 0324537026 Economics: Principles and Policy - William J. Baumol 11th ed ISBN - 0324586205 Economics: Private and Public Choice - James D. Gwartney 12th ed ISBN - 0324580185 Effective Small Business Management - Norman M. Scarborough 9th ed ISBN - 0136152708 Effective Writing - Claire B. May 8th ed ISBN - 0136029086 Electric Circuits - James Nilsson 8th ed ISBN - 0131989251 Electrical Engineering: Principles and Applications - Allan R. Hambley 4th ed ISBN - 0131989227 Electrical Machines, Drives and Power Systems - Theodore Wildi 6th ed ISBN - 0131776916 Electronic Commerce 2008 - Efraim Turban 5th ed ISBN - 0132243318 Electronic Communications for Technicians - Tom Wheeler 2nd ed ISBN - 0131130498 Electronics and Computer Math - Bill R. Deem 8th ed ISBN - 0131711377 Electronics Fundamentals: Circuits, Devices and Applications - Thomas Floyd 7th ed ISBN - 013219709X Elementary Algebra - George Woodbury 1st ed ISBN - 0321166426 Elementary Algebra Early Graphing for College Students - Allen R. Angel 3rd ed ISBN - 0136134165 Elementary Algebra: Graphs and Authentic Applications - Jay Lehmann 1st ed ISBN - 013220164X Elementary and Intermediate Algebra - George Woodbury 2nd ed ISBN - 0321500067 Elementary and Intermediate Algebra: Graphs & Models - Marvin L. Bittinger 3rd ed ISBN - 0321422406 Elementary Differential Equations - Henry Edwards 6th ed ISBN - 0132397307 Elementary Differential Equations - Werner E. Kohler, Lee W.Johnson 1st ed ISBN - 0201709260 Elementary Differential Equations with Boundary Value Problems - Henry Edwards 6th ed ISBN - 0136006132 Elementary Differential Equations With Boundary Value Problems - Lee Johnson et al 1st ed ISBN - 0321121643 Elementary Differential Equations With Boundary Value Problems - Lee Johnson et al 2nd ed ISBN - 0321398505 Elementary Linear Algebra - Ron Larson 6th ed ISBN - 0618783768 Elementary Linear Algebra with Applications - Bernard Kolman 9th ed ISBN - 0132296543 Elementary Number Theory - Kenneth H. Rosen 5th ed ISBN - 0321237072 Elementary Statistics - Mario F. Triola 10th ed ISBN - 0321331834 Elementary Statistics - Mario F. Triola 9th ed ISBN - 0201775700 Elementary Statistics - Mario F. Triola 11th ed ISBN - 0321500245 Elementary Statistics - Neil A. Weiss 7th ed ISBN - 0321422090 Elementary Statistics - Ron Larson 4th ed ISBN - 0132424339 Elementary Statistics Using Excel - Mario Triola 3rd ed ISBN - 0321365135 Elementary Statistics Using the TI-83/84 Plus Calculator - Mario F. Triola 2nd ed ISBN - 0321462572 Elementary Statistics With Multimedia Study Guide - Mario F. Triola 10th ed ISBN - 0321460928 Elements of Forecasting - Francis X. Diebold 4th ed ISBN - 032432359X Embedded Microcontrollers & Processor Design - Charles Greg Osborn 1st ed ISBN - 0131130412 Embedded System Design with C805 - Han-Way Huang 1st ed ISBN - 0495471747 Employment Law - John J. Moran 4th ed ISBN - 0136009964 Engineering Computation with MATLAB - David M. Smith 2nd ed ISBN - 0136080634 Engineering Economy - William G Sullivan 13th ed ISBN - 0131486497 Engineering Economy - William G. Sullivan 14th ed ISBN - 0136142974 Engineering Economy and the Decision-Making Process - Joseph C. Hartman 1st ed ISBN - 0131424017 Engineering Fundamentals: An Introduction to Engineering - Saeed Moaveni 3rd ed ISBN - 0495082538 Engineering Materials: Properties and Selection - Ken Budinski 8th ed ISBN - 0131837796 Engineering Materials: Properties and Selection - Kenneth G. Budinski 9th ed ISBN - 0137128428 Engineering Mechanics Dynamics - Anthony M Bedford 5th ed ISBN - 0136129161 Engineering Mechanics: Dynamics - Andrew Pytel 3rd ed ISBN - 0495295612 Engineering Mechanics: Statics - Andrew Pytel 3rd ed ISBN - 0495244694 Engineering Mechanics: Statics - Anthony M Bedford 5th ed ISBN - 0136129153 Engineering Mechanics: Statics - Russell C. Hibbeler 11th ed ISBN - 0132215004 Engineering Mechanics: Statics - Russell C. Hibbeler 12th ed ISBN - 0136077900 Engineering Mechanics: Statics Computational Edition - Robert W. Soutas-Little 1st ed ISBN - 0534549217 Engineering Vibration - Daniel Inman 3rd ed ISBN - 0132281732 Enterprise Systems for Management - Luvai Motiwalla 1st ed ISBN - 013233531X Entrepreneurial Finance - Chris Leach 3rd ed ISBN - 0324561253 Entrepreneurial Finance - Philip J. Adelman 4th ed ISBN - 0132434792 Entrepreneurial Finance - Philip J. Adelman 5th ed ISBN - 013502529X Entrepreneurship: Successfully Launching New Ventures - Bruce Barringer 2nd ed ISBN - 0132240572 Entrepreneurship: Theory, Process, and Practice - Donald F. Kuratko 8th ed ISBN - 0324590911 Environmental and Natural Resource Economics - Tom Tietenberg 7th ed ISBN - 0321305043 Environmental Issues: An Introduction to Sustainability - Robert L. McConnell 3rd ed ISBN - 0131566504 Environmental Law - Nancy K. Kubasek 6th ed ISBN - 0136142168 Environmental Science: Toward A Sustainable Future - Richard T. Wright 10th ed ISBN - 0132302659 Error Control Coding - Daniel J. Costello Jr., Shu Lin 2nd ed ISBN - 0130426725 Essential Foundations of Economics - Robin Bade 4th ed ISBN - 0321522354 Essentials of Business Law - Jeffrey F. Beatty 3rd ed ISBN - 0324537123 Essentials of Business Law and the Legal Environment - Richard A. Mann 10th ed ISBN - 0324593562 Essentials of College Algebra with Modeling and Visualization - Gary K. Rockswold 3rd ed ISBN - 0321448898 Essentials of College Algebra, Alternate Edition - Margaret L. Lial 1st ed ISBN - 0321491858 Essentials of Economics - Gregory Mankiw 4th ed ISBN - 0324236964 Essentials of Economics - Gregory Mankiw 5th ed ISBN - 0324590024 Essentials of Entrepreneurship and Small Business Management - Thomas W Zimmerer 5th ed ISBN - 0132294389 Essentials of Logic - Irving Copi 2nd ed ISBN - 013238034X Essentials of Management Information Systems - Jane Laudon 8th ed ISBN - 013602579X Essentials of Managerial Finance - Scott Besley 13th ed ISBN - 0324258755 Essentials of Marketing - Charles W. Lamb 6th ed ISBN - 0324656203 Essentials of Materials Science & Engineering - Donald R. Askeland 2nd ed ISBN - 0495244465 Essentials of Organizational Behavior - Stephen P Robbins 9th ed ISBN - 0132431521 Essentials of Organizational Behavior - Stephen P. Robbins 10th ed ISBN - 0136077617 Essentials of Statistics - Mario F. Triola 3rd ed ISBN - 0321434250 Essentials of the Legal Environment - Roger LeRoy Miller 2nd ed ISBN - 0324400403 Ethics for the Information Age - Mike Quinn 3rd ed ISBN - 0321536851 Excellence in Business Communication - John V. Thill 8th ed ISBN - 0136157505 Experiencing MIS - David Kroenke 2nd ed ISBN - 0136078680 Exploring Business - Karen Collins 1st ed ISBN - 0131403656 Exploring Corporate Strategy - Gerry Johnson 8th ed ISBN - 140588732X Exploring Macroeconomics - Robert L. Sexton 4th ed ISBN - 0324395558 Federal Tax Research - William A. Raabe 8th ed ISBN - 0324659652 Feedback Control of Dynamic Systems - Gene Franklin 5th ed ISBN - 0131499300 Financial & Managerial Accounting - Carl S. Warren 9th ed ISBN - 0324401884 Financial & Managerial Accounting - Carl S. Warren 10th ed ISBN - 0324663811 Financial Accounting - Belverd E. Needles 10th ed ISBN - 0547193289 Financial Accounting - Carl S. Warren, James M. Reeve 10th ed ISBN - 0324380674 Financial Accounting - Carl S. Warren, James M. Reeve 11th ed ISBN - 0324663781 Financial Accounting - Jane Reimers 1st ed ISBN - 0131492012 Financial Accounting - Walter Harrison, Charles Horngren 6th ed ISBN - 0131499459 Financial Accounting - Walter Harrison, Charles Horngren 7th ed ISBN - 0138128200 Financial Accounting and Financial Tips - Walter T. Harrison 7th ed ISBN - 0135012848 Financial Accounting, Reporting & Analysis: International Edition - Barry Elliott 2nd ed ISBN - 027370253X Financial Accounting: A Bridge to Decision Making - Robert Ingram 6th ed ISBN - 0324313357 Financial Accounting: A Business Process Approach - Jane L. Reimers 2nd ed ISBN - 0131473867 Financial Accounting: An Integrated Statements Approach - Jonathan Duchac 2nd ed ISBN - 0324312113 Financial Accounting: An Introduction to Concepts, Methods and Uses - Clyde P. Stickney 12th ed ISBN - 0324381980 Financial Accounting: An Introduction to Concepts, Methods and Uses - Clyde P. Stickney 13th ed ISBN - 0324651147 Financial Accounting: The Impact on Decision Makers - Gary Porter 6th ed ISBN - 0324655231 Financial and Managerial Accounting - Meg Pollard 1st ed ISBN - 0136008984 Financial And Managerial Accounting Ch 1-13 - Charles Horngren 1st ed ISBN - 0135009855 Financial Economics - Zvi Bodie 2nd ed ISBN - 0131856154 Financial Management for Public, Health, and Not-for-Profit - Steven A Finkler 3rd ed ISBN - 0136070736 Financial Management For Public, Health, and Not-for-Profit Organizations - Steven Finkler 2nd ed ISBN - 0131471988 Financial Management: Theory & Practice - Eugene Brigham 12th ed ISBN - 0324422695 Financial Markets and Institutions - Frederic S. Mishkin 5th ed ISBN - 0321280296 Financial Markets and Institutions - Frederic S. Mishkin 6th ed ISBN - 0321374215 Financial Markets and Institutions - Jeff Madura 8th ed ISBN - 0324568215 Financial Markets and Institutions Abridged Edition - Jeff Madura 8th ed ISBN - 0324593643 Financial Reporting and Analysis - Lawrence Revsine 3rd ed ISBN - 0131430211 Financial Reporting and Analysis Using Financial Accounting Information - Charles Gibson 10th ed ISBN - 0324304455 Financial Reporting and Analysis Using Financial Accounting Information - Charles Gibson 11th ed ISBN - 0324657420 Financial Reporting, Financial Statement Analysis, and Valuation - Clyde P. Stickney 6th ed ISBN - 0324302959 Financial/Managerial Accounting - Walter T. Harrison 1st ed ISBN - 0131568779 Finite Element Analysis Theory and Application with ANSYS - Saeed Moaveni 3rd ed ISBN - 0131890808 Finite Math and Its Application - Larry Goldstein 9th ed ISBN - 0131873644 Finite Mathematics - Margaret L. Lial et al 8th ed ISBN - 032122826X Finite Mathematics and Calculus with Applications - Margaret Lial 8th ed ISBN - 0321426517 Finite Mathematics for Business, Economics, Life Sciences & Social Sciences - Raymond Barnett 11th ed ISBN - 0132255707 Finite Mathematics with Applications - Margaret L. Lial 9th ed ISBN - 0321386728 First Course in Abstract Algebra - John Fraleigh 7th ed ISBN - 0201763907 First Course in Abstract Algebra - Joseph Rotman 3rd ed ISBN - 0131862677 First Course In Probability - Sheldon M. Ross 7th ed ISBN - 0131856626 First Course In Probability - Sheldon M. Ross 8th ed ISBN - 013603313X First Course in Statistics, A - James T. McClave 10th ed ISBN - 0136152597 Fluency with Information Technology: Skills, Concepts, and Capabilities - Lawrence Snyder 3rd ed ISBN - 0321512391 Foundations of Economics - Robin Bade 4th ed ISBN - 0321522362 Foundations of Finance - Arthur Keown, William Petty, John Martin, David Scott 5th ed ISBN - 0131856057 Foundations of Finance: Logic and Practice of Financial Mangement - Arthur J. Keown 6th ed ISBN - 0135048168 Foundations of Finance: The Logic and Practice of Financial Management - Arthur Keown 6th ed ISBN - 0132339226 Foundations of Financial Markets and Institutions - Frank J. Fabozzi 4th ed ISBN - 0136135315 Foundations of Geometry - Gerard Venema 5th ed ISBN - 0131437003 Foundations of Macroeconomics - Robin Bade 4th ed ISBN - 0321522370 Foundations of MEMS - Chang Liu 1st ed ISBN - 0131472860 Foundations of Microeconomics - Robin Bade 3rd ed ISBN - 0321415957 Foundations of Microeconomics - Robin Bade 4th ed ISBN - 0321522389 Foundations of the Legal Environment of Business - Marianne M. Jennings 1st ed ISBN - 0324566514 Framework for Human Resource Management, A - Gary Dessler 5th ed ISBN - 0136041531 Framework for Marketing Management, A - Philip Kotler 4th ed ISBN - 0136026605 Fraud Examination - Steve Albrecht 2nd ed ISBN - 0324651155 Friendly Introduction to Analysis - Witold A.J. Kosmala 2nd ed ISBN - 0130457965 Fundamental Cornerstones of Managerial Accounting - Dan L. Heitger, Maryanne M. Mowen 1st ed ISBN - 0324378068 Fundamental Mathematics through Applications - Geoffrey Akst 4th ed ISBN - 0321496906 Fundamentals of Advanced Accounting - Paul M. Fischer Test Bank 1st ed ISBN - 0324378904 Fundamentals of Applied Electromagnetics - Fawwaz T. Ulaby 5th ed ISBN - 0132413264 Fundamentals of Business Law Summarized Cases - Roger LeRoy Miller 7th ed ISBN - 0324381689 Fundamentals of Business Law: Excerpted Cases - Roger LeRoy Miller 2nd ed ISBN - 0324595727 Fundamentals of Business Law: Summarized Cases - Roger LeRoy Miller 8th ed ISBN - 0324595735 Fundamentals of Communication Systems - John G. Proakis 1st ed ISBN - 013147135X Fundamentals of Complex Analysis - Edward Saff 3rd ed ISBN - 0139078746 Fundamentals of Derivatives Markets - Robert L. McDonald 1st ed ISBN - 0321357175 Fundamentals of Differential Equations - Kent Nagle, Edward Saff 6th ed ISBN - 0321145720 Fundamentals of Differential Equations - R. Kent Nagle 7th ed ISBN - 0321410483 Fundamentals of Differential Equations with Boundary Value Problems - R. Kent Nagle 5th ed ISBN - 0321419219 Fundamentals of Electromagnetics for Electrical and Computer Engineering - Nannapaneni Narayana Rao 1st ed ISBN - 0136013333 Fundamentals of Engineering Economics - Chan S. Park 2nd ed ISBN - 0132209608 Fundamentals of Financial Management - Eugene Brigham 11th ed ISBN - 0324319800 Fundamentals of Financial Management - Eugene F. Brigham 12th ed ISBN - 0324597703 Fundamentals of Financial Management Concise - Eugene F. Brigham 6th ed ISBN - 0324664559 Fundamentals of Investing - Lawrence J. Gitman 10th ed ISBN - 0321489381 Fundamentals of Management: Essential Concepts and Applications - Stephen P. Robbins Test Bank 5th ed ISBN - 0131487361 Fundamentals of Multinational Finance - Michael Moffett 3rd ed ISBN - 0321541642 Fundamentals of Organic Chemistry - John McMurry Test Bank only 5th ed ISBN - 0534395732 Fundamentals of Probability, with Stochastic Processes - Saeed Ghahramani 3rd ed ISBN - 0131453408 Fundamentals of Signals and Systems - Edward Kamen 3rd ed ISBN - 0131687379 Fundamentals of Statistics - Michael Sullivan 2nd ed ISBN - 0131569872 Further Mathematics for Economic Analysis - Knut Sydsaeter et al 1st ed ISBN - 0273655760 Geometry: Theorems and Constructions - Allan Berele 1st ed ISBN - 0130871214 Global Investments - Bruno Solnik 6th ed ISBN - 0321527704 Global Strategy - Mike W. Peng 2nd ed ISBN - 0324590997 Governmental and Nonprofit Accounting: Theory and Practice - Robert J. Freeman 9th ed ISBN - 0136029515 Health Economics - Charles E. Phelps 4th ed ISBN - 0321594576 High-Speed Networks and Internets: Performance and Quality of Service - William Stallings 2nd ed ISBN - 0130322210 Historical Geology - Reed Wicander 6th ed ISBN - 0495560073 Human Anatomy & Physiology - Elaine N. Marieb 7th ed ISBN - 0805359095 Human Anatomy and Physiology - Elaine N. Marieb 8th ed ISBN - 0805395911 Human Anatomy and Physiology Lab Manual - Elaine N. Marieb 9th ed ISBN - 0805372652 Human Biology: Concepts and Current Issues - Michael D. Johnson 5th ed ISBN - 0321570200 Human Diseases: A Systemic Approach - Mark Zelman 7th ed ISBN - 0135155568 Human Physiology: An Integrated Approach - Dee Unglaub Silverthorn 5th ed ISBN - 0321559398 ISBN - 0495014850 Human Relations for Career and Personal Success: Concepts, Applications, and Skills - Andrew J. DuBrin 8th ed ISBN - 0131791796 Human Relations: Interpersonal Job-Oriented Skills - Andrew J. DuBrin 10th ed ISBN - 0135019443 Human Resource Management - Gary Dessler 11th ed ISBN - 0131746170 Human Resource Management - R. Wayne Mondy 11th ed ISBN - 0136077285 Human Resource Management - Wayne Mondy 10th ed ISBN - 0132225956 Human Side of Organizations - Michael Drafke 10th ed ISBN - 0135139740 Hydrology and Floodplain Analysis - Philip B. Bedient 4th ed ISBN - 0131745891 Income Tax Fundamentals 2006 - Gerald E. Whittenburg 24th ed ISBN - 0324399022 Income Tax Fundamentals 2007 - Gerald E. Whittenburg 25th ed ISBN - 032439926X Income Tax Fundamentals 2009 - Gerald E. Whittenburg 27th ed ISBN - 0324663676 Information Systems Today: Managing in the Digital World - Leonard Jessup 3rd ed ISBN - 0132335069 Information Systems Today: Managing the Digital World - Joseph Valacich 4th ed ISBN - 0136078400 Information Technology Auditing and Assurance - James Hall 2nd ed ISBN - 0324191987 Inquiry into Physics - Vern J. Ostdiek 6th ed ISBN - 0495119431 Integrated Arithmetic and Basic Algebra - Bill E. Jordan 4th ed ISBN - 0321442555 Intel Micro 8086 - Barry B. Brey 8th ed ISBN - 0135026458 Intel Microprocessors - Barry B. Brey 7th ed ISBN - 0131195069 Intel Microprocessors - Barry B. Brey 8th ed ISBN - 0135026458 Interactive Computer Graphics: A Top-Down Approach Using OpenGL - Edward Angel 5th ed ISBN - 0321535863 Interactive Statistics - Martha Aliaga, Brenda Gunderson 3rd ed ISBN - 0131497561 Intermediate Accounting - James D. Stice 16th ed ISBN - 0324312148 Intermediate Accounting - James D. Stice 17th ed ISBN - 032459237X Intermediate Accounting - Loren A. Nikolai 10th ed ISBN - 0324651929 Intermediate Accounting - Loren A. Nikolai 11th ed ISBN - 032465913X Intermediate Accounting Revised - David Spiceland 4th ed ISBN - 0073215422 Intermediate Algebra - Elayn El Martin-Gay 5th ed ISBN - 0136007295 Intermediate Algebra - Margaret L. Lial 10th ed ISBN - 0321443624 Intermediate Algebra - Margaret L. Lial 9th ed ISBN - 0321574974 Intermediate Algebra - Marvin L. Bittinger 10th ed ISBN - 0321319087 Intermediate Algebra for College Students - Allen R. Angel 7th ed ISBN - 0132383578 Intermediate Algebra for College Students - Robert F. Blitzer 5th ed ISBN - 0136007627 Intermediate Algebra with Applications & Visualization - Gary K. Rockswold 3rd ed ISBN - 0321500032 Intermediate Algebra: Functions & Authentic Applications - Jay Lehmann 3rd ed ISBN - 0131953338 Intermediate Algebra: Graphs & Models - Marvin L. Bittinger 3rd ed ISBN - 0321416163 Intermediate Financial Management - Eugene F. Brigham 10th ed ISBN - 0324594690 International Accounting - Frederick Choi 5th ed ISBN - 0131480979 International Accounting - Frederick D. Choi 6th ed ISBN - 0131588141 International Business - John Daniels 12th ed ISBN - 0136029655 International Business - Ricky Griffin 6th ed ISBN - 0137153732 International Business Law - Ray A. August 5th ed ISBN - 013600864X International Business Law and Its Environment - Richard Schaffer 7th ed ISBN - 0324649673 International Business: Environments and Operations - John Daniels 11th ed ISBN - 0131869426 International Business: Strategy, Management, and the New Realities - Tamer Cavusgil 1st ed ISBN - 0131738607 International Business: The Challenges of Globalization - John J. Wild 4th ed ISBN - 0131747436 International Business: The Challenges of Globalization - John J. Wild 5th ed ISBN - 0137153759 International Economics - Charles Sawyer, Richard Sprinkle 2nd ed ISBN - 0131704168 International Economics - James Gerber 3rd ed ISBN - 032123796X International Economics - James Gerber 4th ed ISBN - 0321415558 International Economics - Robert Carbaugh 11th ed ISBN - 032442194X International Economics - W. Charles Sawyer 3rd ed ISBN - 0136054692 International Economics: Theory And Policy - Paul Krugman, Maurice Obstfeld 7th ed ISBN - 0321293835 International Economics: Theory and Policy - Paul R. Krugman 8th ed ISBN - 0321488830 International Financial Management - Geert Bekaert 1st ed ISBN - 0131163604 International Financial Management - Jeff Madura 9th ed ISBN - 0324568193 International Financial Management Abridged Edition - Jeff Madura 9th ed ISBN - 0324593473 International Financial Management, Abridged Edition - Jeff Madura 8th ed ISBN - 0324365632 International Management: Managing Across Borders and Cultures - Helen Deresky 6th ed ISBN - 0136143261 International Money and Finance - Michael Melvin 7th ed ISBN - 0201770288 Intro Stats - Richard D. De Veaux 3rd ed ISBN - 0321500458 Introduction to Abstract Algebra - Olympia Nicodemi 1st ed ISBN - 0131019635 Introduction to Analysis - William R. Wade 3rd ed ISBN - 0131453335 Introduction to Business Law - Jeffrey F. Beatty 2nd ed ISBN - 0324311427 Introduction to Business Statistics - Ronald M. Weiers 6th ed ISBN - 0324381433 Introduction to C++ EXCEL MATLAB & Basic Engineering Numerical Methods - Harvey G. Stenger 1st ed ISBN - 0136142931 Introduction to C++, Excel MatLab & Basic Engineering Numerical Methods - Harvey Stenger 1st ed ISBN - 0136120245 Introduction to Chemical Principles - Stephen Stoker 9th ed ISBN - 0132379945 Introduction to Computing Systems - Sanjay J. Patel, Yale Patt 2nd ed ISBN - 0072467509 Introduction to Corporate Finance - William L. Megginson 1st ed ISBN - 0324379862 Introduction to Corporate Finance - William L. Megginson 2nd ed ISBN - 0324657935 Introduction to Cryptography with Coding Theory - Wade Trappe 2nd ed ISBN - 0131862391 Introduction to Derivatives and Risk Management - Don M. Chance 7th ed ISBN - 0324321392 Introduction to Econometrics - James H. Stock 2nd ed ISBN - 0321278879 Introduction to Econometrics Brief Edition - James H. Stock 1st ed ISBN - 0321432517 Introduction to Economic Reasoning - William D. Rohlf 7th ed ISBN - 0321416112 Introduction to Electrodynamics -David J. Griffiths 3rd ed ISBN - 013805326X Introduction to Embedded Systems - Jonathan W. Valvano 1st ed ISBN - 049541137X Introduction to Environmental Engineering - P. Aarne Vesilind 3rd ed ISBN - 0495295833 Introduction to Environmental Engineering - Richard O. Mines 1st ed ISBN - 0132347474 Introduction to Environmental Engineering and Science - Gilbert M. Masters 3rd ed ISBN - 0131481932 Introduction to Financial Accounting - Charles Horngren 9th ed ISBN - 0131479725 Introduction to Fire Prevention - James C. Robertson 7th ed ISBN - 0135041945 Introduction to Fourier Optics - Joseph Goodman 3rd ed ISBN - 0974707724 Introduction to Government and Non-for-Profit Accounting - Martin Ives 6th ed ISBN - 0132366355 Introduction to Graph Theory - Douglas West 2nd ed ISBN - 0130144002 Introduction to Law - Joanne Hames 3rd ed ISBN - 0131183818 Introduction to Linear Algebra - Lee Johnson, Dean Riess, Jimmy Arnold 5th ed ISBN - 0201658593 Introduction to Linear Programming - Leonid Vaserstein 1st ed ISBN - 0130359173 Introduction to Management Accounting - Charles T. Horngren 14th ed ISBN - 0136129218 Introduction to Management Accounting, Chap. 1-17: International Edition - Charles Horngren 13th ed ISBN - 0131273078 Introduction to Management Science and Student - Bernard Taylor 8th ed ISBN - 0131050524 Introduction to Management Science and Student - Bernard Taylor 9th ed ISBN - 0131888099 Introduction to Materials Management - Tony Arnold 6th ed ISBN - 0132337614 Introduction to Materials Science for Engineers - James F. Shackelford 7th ed ISBN - 0136012604 Introduction to Mathematical Statistics and Its Applications - Richard J. Larsen 4th ed ISBN - 0131867938 Introduction to Operations and Supply Chain Management - Cecil Bozarth 2nd ed ISBN - 0131791036 Introduction to Optics - Frank Pedrotti et al. 3rd ed ISBN - 0131499335 Introduction to Programming with C++ - Y. Daniel Liang 2nd ed ISBN - 0136097200 Introduction to Quantum Mechanics - David Griffiths 2nd ed ISBN - 0131118927 Introduction to Risk Management and Insurance - Mark Dorfman 8th ed ISBN - 0131449583 Introduction to Risk Management and Insurance - Mark S. Dorfman 9th ed ISBN - 0132242273 Introduction to Signal and System Analysis - Kaliappan Gopalan 1st ed ISBN - 0534466060 Introduction to Spectroscopy - Donald L. Pavia 4th ed ISBN - 0495114782 Introduction to Technical Mathematics - Allyn J. Washington, Mario Triola 5th ed ISBN - 0321374177 Introduction to Telecommunications - Martha Rosengrant 2nd ed ISBN - 0131126156 Introduction to the Design & Analysis of Experiments - George C Canavos 1st ed ISBN - 0136158633 Introduction to the Design and Analysis of Algorithms - Anany Levitin 1st ed ISBN - 0201743957 Introduction to the Design and Analysis of Algorithms - Anany Levitin 2nd ed ISBN - 0321358287 Introduction to Transportation Engineering - Lester A. Hoel 1st ed ISBN - 0534952895 Introduction to Vacuum Technology - David M. Hata 1st ed ISBN - 0130450189 Introductory & Intermediate Algebra for College Students - Robert F. Blitzer 3rd ed ISBN - 0136028950 Introductory Algebra - Marvin L. Bittinger 10th ed ISBN - 0321269470 Introductory Algebra for College Students - Robert F. Blitzer 5th ed ISBN - 0132356791 Introductory Algebra through Applications - Geoffrey Akst 2nd ed ISBN - 0321518020 Introductory and Intermediate Algebra - Robert F Blitzer 2nd ed ISBN - 0131492594 Introductory Chemistry - Nivaldo J. Tro 3rd ed ISBN - 0136003826 Introductory Chemistry - Steve Russo, Michael Silver, Mike Silver 2nd ed ISBN - 032104634X Introductory Circuit Analysis - Robert Boylestad 11th ed ISBN - 0131730444 Introductory Econometrics: A Modern Approach - Jeffrey Wooldridge 3rd ed ISBN - 0324289782 Introductory Econometrics: A Modern Approach - Jeffrey Wooldridge 4th ed ISBN - 0324581629 Introductory Linear Algebra: An Applied First Course - Bernard Kolman 8th ed ISBN - 0131437402 Introductory Mathematical Analysis - Ernest F Haeussler 12th ed ISBN - 0132404222 Introductory Statistics - Neil A. Weiss 8th ed ISBN - 0321393619 Inventing Entrepreneurs: Technology Innovators and their Entrepreneurial Journey - Gerry George 1st ed ISBN - 0131574701 Investments - Frank K. Reilly 7th ed ISBN - 0324288999 Investments: An Introduction - Herbert B. Mayo 9th ed ISBN - 0324561261 Java Software Solutions: Foundations of Program Design - John Lewis 5th ed ISBN - 0321409493 Java: Introduction to Problem Solving and Programming - Walter Savitch 6th ed ISBN - 0136072259 John E. Freund's Mathematical Statistics with Applications - Irwin Miller 7th ed ISBN - 0131427067 Kleppner's Advertising Procedure - Ronald Lane 17th ed ISBN - 0132308290 Labor and Employment Law: Text & Cases - David Twomey 14th ed ISBN - 0324594844 Labor Relations - Arthur A Sloane 12th ed ISBN - 013196223X Labor Relations - Arthur A. Sloane 13th ed ISBN - 0136077188 Labor Relations and Collective Bargaining: Cases, Practice, and Law - Michael R. Carrell 8th ed ISBN - 0131868721 Labor Relations and Collective Bargaining: Cases, Practice, and Law - Michael R. Carrell 9th ed ISBN - 0136084354 Lakeside Company: Case Studies in Auditing - John M. Trussel 11th ed ISBN - 0131588516 Law and Economics - Robert Cooter 5th ed ISBN - 0321336348 Law and Ethics in the Business Environment - Terry Halbert 6th ed ISBN - 0324657323 Law for Business - John D. Ashcroft 16th ed ISBN - 0324381573 Leadership - Robert N. Lussier 3rd ed ISBN - 0324316976 Leadership in Organizations - Gary Yukl 7th ed ISBN - 0132424312 Learning Microsoft Office Accounting 2007 and Student CD Package - Terri Brunsdon 1st ed ISBN - 0131586602 Learning Peachtree Complete 2007 & Peachtree Complete CD Package - Terri Brunsdon 1st ed ISBN - 0132405571 Learning Quickbooks Pro 2007 and Student CD Package - Terri Brunsdon 1st ed ISBN - 0132419386 Legal Aspects of Architecture, Engineering & the Construction Process - Justin Sweet 8th ed ISBN - 0495411213 Legal Terminology - Gordon W. Brown 5th ed ISBN - 0131568043 Level Three Leadership: Getting Below the Surface - James G Clawson 4th ed ISBN - 0132423847 Linear Algebra and Its Applications - David C. Lay 3rd ed ISBN - 0321287134 Linear Algebra for Engineers and Scientists Using Matlab - Kenneth Hardy 1st ed ISBN - 0139067280 Linear Algebra with Applications - Otto Bretscher 3rd ed ISBN - 0131453343 Linear Algebra with Applications - Steven Leon 7th ed ISBN - 0131857851 Logic and Computer Design Fundamentals - M. Morris Mano 4th ed ISBN - 013198926X Machine Design: An Integrated Approach - Robert L. Norton 3rd ed ISBN - 0131481908 Machines and Mechanisms: Applied Kinematic Analysis - David H. Myszka 3rd ed ISBN - 0131837761 Macroeconomics - Andrew B. Abel 6th ed ISBN - 0321451406 Macroeconomics - Glenn Hubbard 2nd ed ISBN - 0132356694 Macroeconomics - Michael Parkin 7th ed ISBN - 032124608X Macroeconomics - Michael Parkin 8th ed ISBN - 0321416570 Macroeconomics - Michael Parkin 9th ed ISBN - 0321600053 Macroeconomics - Olivier Blanchard 5th ed ISBN - 0132078295 Macroeconomics - Richard Froyen 8th ed ISBN - 0131435825 Macroeconomics - Richard G. Lipsey 13th ed ISBN - 0321369238 Macroeconomics - Richard T Froyen 9th ed ISBN - 0132438356 Macroeconomics - Robert Gordon 10th ed ISBN - 0321278801 Macroeconomics - Robert Gordon 11th ed ISBN - 0321485513 Macroeconomics - Roger A. Arnold 9th ed ISBN - 032478550X Macroeconomics - Stephen D. Williamson 3rd ed ISBN - 0321416589 Macroeconomics: A Modern Approach - Robert J. Barro 1st ed ISBN - 0324178107 Macroeconomics: Principles and Policy - William J. Baumol 10th ed ISBN - 0324537034 Macroeconomics: Principles and Policy - William J. Baumol 11th ed ISBN - 0324586213 Macroeconomics: Principles and Tools - Arthur O'Sullivan, Steven Sheffrin 4th ed ISBN - 0131536184 Macroeconomics: Principles, Applications, and Tools - Arthur O'Sullivan 5th ed ISBN - 013232928X Macroeconomics: Public and Private Choice - James D. Gwartney 12th ed ISBN - 0324580193 Making Career Decisions that Count: A Practical Guide - Darrell A. Luzzo 3rd ed ISBN - 0131712772 Making the Team - Leigh Thompson 3rd ed ISBN - 0131861352 Management - Michael Hitt 2nd ed ISBN - 0132354373 Management - Richard L. Daft 9th ed ISBN - 0324595840 Management - Stephen P Robbins 9th ed ISBN - 0132257734 Management Communication: A Case-Analysis Approach - James S O'Rourke 4th ed ISBN - 0136079768 Management Information Systems - Ken Laudon 11th ed ISBN - 013607846X Management of Organizational Behavior - Paul H Hersey 9th ed ISBN - 0131441396 Manager's Bookshelf - Jon L. Pierce 8th ed ISBN - 0132301652 Managerial Accounting - Carl Warren 9th ed ISBN - 0324381913 Managerial Accounting - Carl Warren 10th ed ISBN - 032466382X Managerial Accounting - Linda S. Bamber 1st ed ISBN - 0138129711 Managerial Accounting Class Test Edition - Linda S. Bamber 1st ed ISBN - 0132284634 Managerial Accounting: A Focus on Ethical Decision Making - Steve Jackson, Roby Sawyers 4th ed ISBN - 0324650647 Managerial Accounting: A Focus on Ethical Decision Making - Steve Jackson, Roby Sawyers 5th ed ISBN - 0324663854 Managerial Accounting: An Introduction to Concepts, Methods and Uses - Michael W. Maher 10th ed ISBN - 0324639767 Managerial Economics - Mark Hirschey 12th ed ISBN - 0324584849 Managerial Economics: A Problem Solving Approach - Luke M. Froeb 1st ed ISBN - 0324359810 Managerial Economics: Applications, Strategies, and Tactics - James R. McGuigan 11th ed ISBN - 0324421605 Managerial Economics: Economic Tools for Today's Decision Makers - Paul G. Keat 5th ed ISBN - 0131860151 Managerial Economics: Economic Tools for Today's Decision Makers - Paul G. Keat 6th ed ISBN - 0136040047 Managerial Statistics A Case-Based Approach - Peter Klibanoff 1st ed ISBN - 0324226454 Managers and the Legal Environment - Constance E. Bagley 6th ed ISBN - 0324582048 Managing Human Resources - Luis Gomez-Mejia 5th ed ISBN - 013187067X Managing in a Global Economy: Demystifying International Macroeconomics - John E. Marthinsen 1st ed ISBN - 0324395507 Managing Information Technology - Carol V Brown 6th ed ISBN - 0131789546 Managing the Law: The Legal Aspects of Doing Business - Mitchell McInnes 2nd ed ISBN - 0132042762 Manual Auditing and Assurance Practice Set: CAST - Frank A. Buckless 1st ed ISBN - 0130464716 Manufacturing Processes for Engineering Materials - Serope Kalpakjian 5th ed ISBN - 0132272717 Manufacturing, Engineering & Technology - Serope Kalpakjian 5th ed ISBN - 0131489658 Market Regulation - Roger Sherman 1st ed ISBN - 0321322320 Market-Based Management - Roger Best 5th ed ISBN - 0132336537 Marketing - Charles W. Lamb 10th ed ISBN - 0324591098 Marketing - William M. Pride 15th ed ISBN - 0547167474 Marketing Management - Dawn Iacobucci 1st ed ISBN - 0324784430 Marketing Management - Philip Kotler 13th ed ISBN - 0136009980 Materials Science and Engineering: An Introduction - William Callister 6th ed ISBN - 0471135763 Mathematical Economics - Jeffrey Baldani 2nd ed ISBN - 0324183321 Mathematical Ideas - Charles D. Miller 11th ed ISBN - 0321361466 Mathematical Ideas - Charles D. Miller 11th ed ISBN - 0321361482 Mathematical Methods for Economics - Michael Klein 2nd ed ISBN - 0201726262 Mathematical Proofs: A Transition to Advanced Mathematics - Gary Chartrand 2nd ed ISBN - 0321390539 Mathematical Proofs: A Transition to Advanced Mathematics - Gary Chartrand 1st ed ISBN - 0201710900 Mathematical Reasoning for Elementary Teachers - Calvin T. Long 5th ed ISBN - 0321460847 Mathematics for Business - Stanley A. Salzman 8th ed ISBN - 0321357434 Mathematics for Elementary School Teachers - Phares O'Daffer 4th ed ISBN - 0321448049 Mathematics for Physicists - Susan Lea 1st ed ISBN - 0534379974 Mathematics of Interest Rates and Finance - Gary Guthrie 1st ed ISBN - 0130461822 Mathematics with Applications - Margaret L. Lial 9th ed ISBN - 0321334337 Mechanical Behavior of Materials - Norman Dowling 3rd ed ISBN - 0131863126 Mechanics of Materials - James M. Gere 7th ed ISBN - 0534553974 Mechanics of Materials - Russell C. Hibbeler 7th ed ISBN - 0132209918 Medical Imaging Signals and Systems - Jerry L. Prince 1st ed ISBN - 0130653535 Microbiology with Diseases by Body System - Robert W. Bauman 2nd ed ISBN - 032151341X Microbiology: An Introduction - Gerard J. Tortora 9th ed ISBN - 0805347909 Microeconomics - Glenn Hubbard 2nd ed ISBN - 0138132771 Microeconomics - Jeffrey Perloff 4th ed ISBN - 0321414527 Microeconomics - Jeffrey Perloff 5th ed ISBN - 0321531191 Microeconomics - Michael Parkin 7th ed ISBN - 0321454944 Microeconomics - Michael Parkin 8th ed ISBN - 0321416600 Microeconomics - Michael Parkin 9th ed ISBN - 0321600045 Microeconomics - Richard G. Lipsey 13th ed ISBN - 032136922X Microeconomics - Robert Pindyck, Daniel Rubinfeld 6th ed ISBN - 0130084611 Microeconomics - Robert Pindyck, Daniel Rubinfeld 7th ed ISBN - 0132080230 Microeconomics - Roger A. Arnold 9th ed ISBN - 0324785496 Microeconomics: Principles and Policy - William J. Baumol 11th ed ISBN - 0324586221 Microeconomics: Principles and Tools - Arthur O'Sullivan, Steven Sheffrin 4th ed ISBN - 0131536060 Microeconomics: Principles, Applications, and Tools - Arthur O'Sullivan 5th ed ISBN - 0131572830 Microeconomics: Public and Private Choice - James D. Gwartney 12th ed ISBN - 0324580207 Microeconomics: Theory and Applications with Calculus - Jeffrey M. Perloff 1st ed ISBN - 0321277945 Microwave Engineering - David Pozar 3rd ed ISBN - 0471448788 MIS Essentials - David Kroenke 1st ed ISBN - 0136075606 MKTG 3.0 2009 Edition - Charles W. Lamb 3rd ed ISBN - 0324789289 Modern Control Systems - Richard C Dorf 11th ed ISBN - 0132270285 Modern Database Management - Jeffrey Hoffer 8th ed ISBN - 0132212110 Modern Database Management - Jeffrey Hoffer 9th ed ISBN - 0136003915 Modern Electronic Communication - Jeff Beasley 9th ed ISBN - 0132251132 Modern Elementary Statistics - John E. Freund 12th ed ISBN - 013187439X Modern Industrial Organization - Dennis Carlton, Jeffrey Perloff 4th ed ISBN - 0321180232 Modern Labor Economics: Theory and Public Policy - Ronald Ehrenberg 10th ed ISBN - 0321533739 Modern Labor Economics: Theory and Public Policy - Ronald Ehrenberg 9th ed ISBN - 0321305035 Modern Management - Samuel C. Certo 10th ed ISBN - 0131494708 Modern Physics - Randy Harris 2nd ed ISBN - 0805303081 Modern Physics - Raymond Serway 3rd ed ISBN - 0534493394 Modern Semiconductor Devices for Integrated Circuits - Chenming C. Hu 1st ed ISBN - 0136085253 Modern Systems Analysis and Design - Jeffrey A. Hoffer 5th ed ISBN - 0132240769 Modern Wireless Communications - Simon Haykin 1st ed ISBN - 0130224723 Money, Banking and Financial Markets - Roger LeRoy Miller 3rd ed ISBN Money, the Financial System, and the Economy - R. Glenn Hubbard 6th ed ISBN - 0321426703 Multinational Business Finance - David K. Eiteman 11th ed ISBN - 0321357965 Multinational Finance - Kirt C. Butler 3rd ed ISBN - 0324177453 Multinational Management - John B. Cullen 4th ed ISBN - 032442177X Multivariate Data Analysis - Joseph F. Hair 7th ed ISBN - 0138132631 Nanoengineering of Structural, Functional and Smart Materials - Mark J. Schulz 1st ed ISBN - 0849316537 New Venture Management: The Entrepreneur's Roadmap - Donald Kuratko 1st ed ISBN - 0136130321 Numerical Analysis - Timothy Sauer 1st ed ISBN - 0321268989 Numerical Methods for Engineers - Bilal Ayyub, Richard McCuen 1st ed ISBN - 0133373614 Numerical Methods Using Matlab - John Mathews 4th ed ISBN - 0130652482 Occupational Safety and Health for Technologists, Engineers, and Managers - David L. Goetsch 6th ed ISBN - 0132397609 Office Procedures 21st Century & Student Workbook Package - Sharon Burton 7th ed ISBN - 0132343436 OM 2008 - David Alan Collier 1st ed ISBN - 0324662556 Operating Systems Principles - Lubomir F. Bic 1st ed ISBN - 0130266116 Operating Systems: Internals and Design Principles - William Stallings 5th ed ISBN - 0131479547 Operations Management - Jay Heizer 8th ed ISBN - 0131554441 Operations Management - Jay Heizer 9th ed ISBN - 0138128782 Operations Management - Nigel Slack 5th ed ISBN - 0273708473 Operations Management and Student CD and Student DVD Package - Jay Heizer 9th ed ISBN - 0138128782 Operations Management: Process and Value Chains - Lee J. Krajewski 8th ed ISBN - 0131697390 Operations Research: An Introduction - Hamdy A. Taha 8th ed ISBN - 0131889230 Opportunities and Challenges of Workplace Diversity: Theory, Cases, and Exercises - Kathryn Canas 1st ed ISBN - 0131343068 Oracle 10g Programming: A Primer - Rajshekhar Sunderraman 1st ed ISBN - 0321463048 Organic Chemistry - Paula Bruice Test Bank only 5th ed ISBN - 0131963163 Organization Development and Change - Thomas G. Cummings 9th ed ISBN - 0324421389 Organizational Behavior - Don Hellriegel 12th ed ISBN - 0324578725 Organizational Behavior - Stephen P Robbins 13th ed ISBN - 0136007171 Organizational Behavior Today - Leigh Thompson 1st ed ISBN - 0131858114 Organizational Behavior: An Experiential Approach - Joyce S Osland 8th ed ISBN - 0131441515 Organizational Behavior: An Introduction to Your Life in Organizations - Rae Andre 1st ed ISBN - 013185495X Organizational Behavior: Managing People and Organizations - Ricky W. Griffin 9th ed ISBN - 0547167334 Organizational Behavior: Science, The Real World, and You - Debra L. Nelson 6th ed ISBN - 0324578733 Organizational Theory, Design and Change - Gareth R. Jones 5th ed ISBN - 0131865420 Organizational Theory, Design and Change - Gareth R. Jones 6th ed ISBN - 0136087310 Orthopaedic Biomechanics: Mechanics and Design in Musculoskeletal Systems - Donald L. Bartel 1st ed ISBN - 0130089095 Parallel and Distributed Computation: Numerical Methods - Dimitri Bertsekas, John Tsitsiklis 1st ed ISBN - 0136487009 Parallel Programming - Barry Wilkinson 2nd ed ISBN - 0131405632 Partial Differential Equations and Boundary Value Problems - Nakhle Asmar 2nd ed ISBN - 0131480960 Payroll Accounting 2008 - Bernard Bieg 18th ed ISBN - 0324645546 Payroll Accounting 2009 - Bernard Bieg 19th ed ISBN - 0324663730 Performance Management - Herman Aguinis 2nd ed ISBN - 0136151752 Personal Finance - Jeff Madura 3rd ed ISBN - 0321409965 Personal Finance: Turning Money into Wealth - Arthur J. Keown 5th ed ISBN - 0135077710 Personal Financial Planning - Lawrence J. Gitman 11th ed ISBN - 0324422865 Pharmacology for Nurses: A Pathophysiological Approach - Michael Patrick Adams 2nd ed ISBN - 0131756656 Physical Chemistry - Thomas Engel, Philip Reid 1st ed ISBN - 080533842X Physical Chemistry for the Life Sciences - Thomas Engel 1st ed ISBN - 0805382771 Physical Geography - Robert E. Gabler 9th ed ISBN - 0495555061 Physical Metallurgy Principles - Reza Abbaschian 4th ed ISBN - 0495082546 Physics : Principles with Applications - Douglas Giancoli 6th ed ISBN - 0130606200 Physics for Scientists & Engineers Chs 1-37 with MasteringPhysics» - Doug Giancoli 4th ed ISBN - 0136139264 Physics for Scientists and Engineers with Modern Physics and MasteringPhysics» - Douglas C. Giancoli 4th ed ISBN - 0136139221 Physics with Mastering Physics - James S. Walker 3rd ed ISBN - 0136138969 Physics with Mastering Physics - James S. Walker 4th ed ISBN - 0321541634 Physics: Principles with Applications with MasteringPhysics - Douglas C. Giancoli 6th ed ISBN - 0321569830 Portfolio Construction, Management, and Protection - Robert A. Strong 4th ed ISBN - 0324359365 Portfolio Construction, Management, and Protection - Robert A. Strong 5th ed ISBN - 0324665105 Power Systems Analysis and Design - J. Duncan Glover 4th ed ISBN - 0534548849 Practical Financial Management - William R. Lasher 5th ed ISBN - 0324422636 Prealgebra - Elayn El Martin-Gay 5th ed ISBN - 0132319519 Prealgebra - Margaret L. Lial 4th ed ISBN - 0321567927 Prealgebra - Marvin L. Bittinger 5th ed ISBN - 0321331907 Prealgebra & Introductory Algebra - Elayn El Martin-Gay 2nd ed ISBN - 0131577050 Prealgebra and Introductory Algebra - Marvin L. Bittinger 2nd ed ISBN - 0321331893 Prealgebra: An Integrated Approach - Margaret L. Lial 1st ed ISBN - 032135639X Precalculus - J. S. Ratti 1st ed ISBN - 032129646X Precalculus - Judith A. Beecher 3rd ed ISBN - 0321460065 Precalculus - Margaret Lial 4th ed ISBN - 0321528840 Precalculus - Mark Dugopolski 4th ed ISBN - 0321357795 Precalculus - Michael Sullivan 8th ed ISBN - 0132256886 Precalculus - Michael Sullivan 8th ed ISBN - 0132256886 Precalculus - Robert F. Blitzer 3rd ed ISBN - 0131874799 Precalculus - Robert F. Blitzer 4th ed ISBN - 0321559843 Precalculus: Enhanced with Graphing Utilities - Michael Sullivan 5th ed ISBN - 0136015786 Precalculus: Functions and Graphs - Mark Dugopolski 3rd ed ISBN - 032150111X Precalculus: Graphs & Models and Graphing Calculator Manual Package - Marvin L. Bittinger 4th ed ISBN - 0321501527 Prehospital Emergency Care - Joseph J. Mistovich 8th ed ISBN - 0131741438 Prentice Hall's Federal Taxation 2007: Comprehensive - Thomas Pope 20th ed ISBN - 0132389479 Prentice Hall's Federal Taxation 2007: Corporations - Thomas Pope 20th ed ISBN - 0131751484 Prentice Hall's Federal Taxation 2007: Individuals - Thomas Pope 20th ed ISBN - 013243220X Prentice Hall's Federal Taxation 2008: Comprehensive - Thomas Pope 21st ed ISBN - 0132416492 Prentice Hall's Federal Taxation 2008: Corporations - Thomas Pope 21st ed ISBN - 0136156436 Prentice Hall's Federal Taxation 2008: Individuals - Thomas Pope 21st ed ISBN - 0136156371 Prentice Hall's Federal Taxation 2008: Comprehensive - Thomas Pope 21st ed ISBN - 0136067190 Prentice Hall's Federal Taxation 2009: Corporations - Thomas Pope 22nd ed ISBN - 0136067131 Prentice Hall's Federal Taxation 2009: Individuals - Thomas Pope 22nd ed ISBN - 0136067042 Preparing Effective Business Plans: An Entrepreneurial Approach - Bruce R. Barringer 1st ed ISBN - 0132318326 Price Theory and Applications - Steven Landsburg 7th ed ISBN - 0324421613 Principles of Accounting - Meg Pollard 1st ed ISBN - 0132304791 Principles of Auditing: An Introduction to International Standards on Auditing - Rick Hayes 2nd ed ISBN - 0273684108 Principles of CMOS VLSI Design - Neil H.E. Weste 3rd ed ISBN - 0201533766 Principles of Cost Accounting - Edward J. Vanderbeck 13th ed ISBN - 0324191693 Principles of Cost Accounting - Edward J. Vanderbeck 14th ed ISBN - 0324374178 Principles of Customer Relationship Management - Roger Baran 1st ed ISBN - 0324322380 Principles of Economics - Gregory Mankiw 4th ed ISBN - 0324224729 Principles of Economics - Gregory Mankiw 5th ed ISBN - 0324589972 Principles of Economics - Karl Case 8th ed ISBN - 0132289148 Principles of Electric Circuits: Conventional Current Version - Thomas Floyd 8th ed ISBN - 0131701797 Principles of Electric Circuits: Conventional Flow Version - Thomas L. Floyd 9th ed ISBN - 013507309X Principles of Finance - Scott Besley 3rd ed ISBN - 0324232624 Principles of Finance - Scott Besley 4th ed ISBN - 0324655886 Principles of Foundation Engineering - Braja M. Das 6th ed ISBN - 0495082465 Principles of Geotechnical Engineering 6th ed ISBN - 0534551440 Principles of Heat Transfer - Frank Kreith 6th ed ISBN - 0534375960 Principles of Law and Economics - Daniel Cole, Peter Grossman 1st ed ISBN - 0130932612 Principles of Macroeconomics - Gregory Mankiw 4th ed ISBN - 0324236956 Principles of Macroeconomics - Gregory Mankiw 5th ed ISBN - 0324589999 Principles of Managerial Finance - Lawrence J. Gitman 12th ed ISBN - 0321557530 Principles of Managerial Finance Brief Edition - Lawrence Gitman 5th ed ISBN - 0321557522 Principles of Managerial Finance plus MyfinanceLab Student Access Kit - Lawrence J. Gitman 12th ed ISBN - 0321557530 Principles of Marketing - Philip Kotler 12th ed ISBN - 0132390027 Principles of Marketing - Philip Kotler 13th ed ISBN - 0136079415 Principles of Microeconomics - Gregory Mankiw 4th ed ISBN - 0324319169 Principles of Microeconomics - Karl E. Case 8th ed ISBN - 0131994859 Principles of Money, Banking & Financial Markets - Lawrence Ritter 12th ed ISBN - 0321375572 Principles of Money, Banking, Financial Markets - Lawrence Ritter et al 11th ed ISBN - 0321205251 Principles Of Operations Management - Jay Heizer 6th ed ISBN - 013155445X Principles Of Operations Management - Jay Heizer 7th ed ISBN - 0132449757 Principles of Risk Management and Insurance - George E. Rejda 10th ed ISBN - 0321414934 Probabilistic Systems and Random Signals - Abraham Haddad 1st ed ISBN - 0130094552 Probability & Statistics for Engineers & Scientists - Ronald E. Walpole 8th ed ISBN - 0131877119 Probability and Statistical Inference - Robert Hogg, Eliot Tanis 7th ed ISBN - 0131464132 Probability and Statistical Inference - Robert Hogg, Eliot Tanis 8th ed ISBN - 0321584759 Probability and Statistics - Morris DeGroot, Mark Schervish 3rd ed ISBN - 0201524880 Probability and Statistics for Engineers - Richard Johnson, Irwin Miller, John Freund 7th ed ISBN - 0131437453 Probability and Statistics for Engineers and Scientists - Anthony J. Hayter 3rd ed ISBN - 0495107573 Probability Random Variables, and Stochastic Processes - Papoulis et al 4th ed ISBN - 0073660116 Probability, Statistics, and Random Processes For Electrical Engineering - Alberto Leon-Garcia 3rd ed ISBN - 0131471228 Problem Solving and Program Design in C - Jeri R. Hanly 5th ed ISBN - 0321409914 Problem Solving and Program Design in C - Jeri R. Hanly 6th ed ISBN - 0321535421 Problem Solving with C++ - Walter Savitch 6th ed ISBN - 0321412699 Problem Solving with C++ - Walter Savitch 7th ed ISBN - 0321531345 Problem Solving, Abstraction & Design Using C++ - Frank L. Friedman 5th ed ISBN - 0321450051 Process Control Instrumentation Technology - Curtis Johnson 8th ed ISBN - 0131194577 Professional Office Procedures - Susan Cooperman 5th ed ISBN - 0135156645 Professional Selling: A Trust-Based Approach - Thomas N. Ingram 4th ed ISBN - 032453809X Professionalism: Real Skills for Workplace Success - Lydia E. Anderson 1st ed ISBN - 0131714392 Programming the World Wide Web - Robert W. Sebesta 4th ed ISBN - 0321489691 Project Management for Information Systems - James Cadle 5th ed ISBN - 0132068583 Public Relations Practices: Managerial Case Studies and Problems - Allen H. Center 7th ed ISBN - 0132341360 Quality Control - Dale H. Besterfield 8th ed ISBN - 0135000955 Quality Management - Donna C.S. Summers 2nd ed ISBN - 0135005108 Quality Management for Organizational Excellence - David L. Goetsch 6th ed ISBN - 0135019672 Quantitative Analysis for Management - Barry Render 10th ed ISBN - 0136036252 Quantitative Analysis for Management - Barry Render 9th ed ISBN - 0131857029 Quantum Chemistry and Spectroscopy - Thomas Engel 1st ed ISBN - 0805339795 QuickBooks Pro 2006 with Update 2007 and CD Package - Janet Horne 9th ed ISBN - 013242407X Real Estate Law - George Siedel 6th ed ISBN - 0324204809 Real Estate Law - Marianne M. Jennings 8th ed ISBN - 0324650205 Reinforced Concrete Design - George F. Limbrunner 7th ed ISBN - 0135044359 Reinforced Concrete: Mechanics and Design - James K. Wight 5th ed ISBN - 0132281414 Retailing - Patrick M. Dunne 6th ed ISBN - 032436279X Rethinking Marketing: The Entrepreneurial Imperative - Minet Schindehutte 1st ed ISBN - 0132393891 Risk Management and Insurance - James S. Trieschmann 12th ed ISBN - 0324183208 Risk Takers: Uses and Abuses of Financial Derivatives - John Marthinsen 2nd ed ISBN - 0321542568 Routers and Routing Basics CCNA 2 Labs and Study Guide - Allan Johnson 1st ed ISBN - 1587131676 Selling Today - Gerald L. Manning 11th ed ISBN - 013207995X Short-Term Financial Management - Terry Maness 3rd ed ISBN - 0324202938 Short-Term Financial Management - Terry Maness Test Bank 3rd ed ISBN - 0324202938 Signals, Systems, and Transforms - Charles L Phillips 4th ed ISBN - 0131989235 Social Entrepreneurship: A Modern Approach to Social Value Creation - Arthur C. Brooks 1st ed ISBN - 0132330768 Software Engineering - Ian Sommerville 8th ed ISBN - 0321313798 Software Engineering: Theory and Practice - Shari Lawrence Pfleeger 4th ed ISBN - 0136061699 Soils and Foundations - Cheng Liu, Jack Evett 7th ed ISBN - 0132221381 Solid State Electronic Devices - Ben Streetman 6th ed ISBN - 013149726X Solid State Physics: Essential Concepts - David W. Snoke 1st ed ISBN - 0805386645 South-Western Federal Taxation 2009: Comprehensive - William Hoffman Solutions Manual 32nd ed ISBN - 0324660529 South-Western Federal Taxation 2009: Comprehensive - William Hoffman Test Bank 32nd ed ISBN - 0324660529 South-Western Federal Taxation 2009: Corporations - William Hoffman Solutions Manual 32nd ed ISBN - 0324660219 South-Western Federal Taxation 2009: Corporations - William Hoffman Test Bank 32nd ed ISBN - 0324660219 South-Western Federal Taxation 2009: Individual Income Taxes - William Hoffman Solutions Manual 32nd ed ISBN - 0324660200 South-Western Federal Taxation 2009: Individual Income Taxes - William Hoffman Test Bank 32nd ed ISBN - 0324660200 South-Western Federal Taxation 2009: Taxation of Business Entities - James Smith Solutions Manual 12th ed ISBN - 0324660510 South-Western Federal Taxation 2009: Taxation of Business Entities - James Smith Test Bank 12th ed ISBN - 0324660510 Spectral Analysis of Signals - Peter Stoica, Randolph Moses 1st ed ISBN - 0131139568 Starting Out with C++: Early Objects - Tony Gaddis 6th ed ISBN - 0321512383 Starting Out with Java: Early Objects - Tony Gaddis 3rd ed ISBN - 0321497686 Gaddis 4th ed ISBN - 0136080200 Starting Out with Visual Basic 2008 - Tony Gaddis 4th ed ISBN - 0321531353 Starting Out With Visual Basic 2008 Update - Tony Gaddis 4th ed ISBN - 0136076955 Statics and Strength of Materials - Robert L. Mott 1st ed ISBN - 0135159822 Statics and Strengths of Materials - Harold I. Morrow 6th ed ISBN - 0131719777 Statistical Methods for the Social Sciences - Alan Agresti 4th ed ISBN - 0130272957 Statistical Reasoning for Everyday Life - Jeffrey O. Bennett 3rd ed ISBN - 0321286723 Statistics - James T. McClave 11th ed ISBN - 0132069512 Statistics for Business & Economics - James T. McClave 10th ed ISBN - 0132409356 Statistics for Business and Economics - David R. Anderson Test Bank only 10th ed ISBN - 0324360681 Statistics for Business and Economics - Paul Newbold, William L. Carlson 6th ed ISBN - 0132203847 Statistics for Business and Economics - Paul Newbold, William L. Carlson 7th ed ISBN - 0136085369 Statistics for Management and Economics - Gerald Keller 7th ed ISBN - 0534491243 Statistics for Managers Using Microsoft Excel - David M. Levine 5th ed ISBN - 0136149901 Statistics for Science and Engineering - John Kinney 1st ed ISBN - 0201437201 Statistics for the Behavioral and Social Sciences - Arthur Aron 4th ed ISBN - 0131562789 Statistics for the Life Sciences - Myra Samuels 3rd ed ISBN - 013041316X Statistics, Data Analysis & Decision Modeling - James R. Evans 4th ed ISBN - 0136066003 Statistics: Informed Decisions Using Data - Michael Sullivan 3rd ed ISBN - 0321568028 1st ed ISBN - 0130083690 Statistics: The Art and Science of Learning from Data - Alan Agresti 2nd ed ISBN - 0135131995 Stats: Data and Models - Richard D. De Veaux 2nd ed ISBN - 0321433793 Steel Design - William T. Segui 4th ed ISBN - 0495244716 Strategic Brand Management - Kevin Keller 3rd ed ISBN - 0131888595 Strategic Compensation - Joe Martocchio 5th ed ISBN - 0136007449 Strategic Management and Business Policy - Tom Wheelen 10th ed ISBN - 0131494597 Strategic Management and Business Policy - Tom Wheelen 11th ed ISBN - 013232346X Strategic Management and Competitive Advantage - Jay Barney 2nd ed ISBN - 013613520X Strategic Management and Competitive Advantage: Concepts and Cases - Jay Barney 2nd ed ISBN - 0132338238 Strategic Management in Action - Mary Coulter 4th ed ISBN - 0132277476 Strategic Managment: A Dynamic Perspective Integrated Stratsim Simulation Experience - Print Upgrade - Mason Carpenter 1st ed ISBN - 0136149057 Strategic Management: Concepts and Cases - Fred David 11th ed ISBN - 0131869493 Strategic Management: Concepts and Cases - Fred David 12th ed ISBN - 0136015700 Strategic Marketing for Non-Profit Organizations - Alan Andreasen 7th ed ISBN - 013175372X Strategic Staffing - Jean M. Phillips 1st ed ISBN - 0131586947 Structural Analysis - Aslam Kassimali 4th ed ISBN - 0495295655 Structural Analysis - Russell C. Hibbeler 7th ed ISBN - 0136020607 Structural Steel Design - Jack C. McCormac 4th ed ISBN - 013221816X Structural Steel Design ASD Method - Jack C. McCormac 4th ed ISBN - 0065000609 Structural Steel Design: A Practice Oriented Approach - Abi O. Aghayere 1st ed ISBN - 0132340186 Structure and Interpretation of Signals and Systems - Edward Lee, Pravin Varaiya 1st ed ISBN - 0201745518 Structures - Daniel Lewis Schodek 6th ed ISBN - 0131789392 Supply Chain Management - Sunil Chopra 3rd ed ISBN - 0131730428 Supply Chain Management: A Logistics Perspective - John J. Coyle 8th ed ISBN - 0324376928 Survey of Accounting - Carl S. Warren 3rd ed ISBN - 0324312482 Survey of Accounting - Carl S. Warren 4th ed ISBN - 0324658265 Survey of Economics - Irvin B. Tucker 6th ed ISBN - 0324579616 Survey of Mathematics with Applications, A - Allen R. Angel 8th ed ISBN - 0321501071 Survey of Mathematics with Applications - Allen Angel 8th ed ISBN - 032150108X Surveying with Construction Applications - Barry F. Kavanagh 6th ed ISBN - 0131709321 Surveying with Construction Applications - Barry F. Kavanagh 7th ed ISBN - 0135000513 Surveying: Principles and Applications - Barry F. Kavanagh 8th ed ISBN - 013236512X System Dynamics and Response - S. Graham Kelly 1st ed ISBN - 0534549306 System Modeling and Analysis: Foundations of System Performance Evaluation - Hisashi Kobayashi 1st ed ISBN - 013034835X Systems Analysis and Design - Kenneth E. Kendall 7th ed ISBN - 0132240858 Tax Research - Barbara H Karlin 4th ed ISBN - 013601531X Taxation for Decision Makers 2008 - Shirley Dennis-Escoffier 2nd ed ISBN - 0324654111 Taxes & Business Strategy - Myron S. Scholes 4th ed ISBN - 0136033156 Teaching Today's Health - David Anspaugh 9th ed ISBN - 0321596773 Technical Calculus with Analytic Geometry - Allyn J. Washington 4th ed ISBN - 0201711125 Technical Communication: A Practical Approach - William S. Pfeiffer 7th ed ISBN - 0135000505 Technology Strategy for Managers and Entrepreneurs - Scott A. Shane 1st ed ISBN - 0131879324 The Economics of Macro Issues - Roger LeRoy Miller 3rd ed ISBN - 0321416597 The Economics of Poverty - Bradley R Schiller 10th ed ISBN - 0131889699 The Economics of Public Issues - Roger LeRoy Miller 15th ed ISBN - 0321416104 The Economics of Sports - Michael A. Leeds 3rd ed ISBN - 0321415566 The Labor Relations Process - William H. Holley 9th ed ISBN - 0324421443 The Legal Environment of Business - Roger E. Meiners 10th ed ISBN - 0324654367 The Legal Environment of Business: Text and Cases Ethical, Regulatory, Global, and E-Commerce Issues - Roger LeRoy Miller 7th ed ISBN - 0324590008 The Legal Environment Today: Business In Its Ethical, Regulatory, E- Commerce, and Global Setting - Roger LeRoy Miller 6th ed ISBN - 0324599250 The Paralegal Professional - Henry Cheeseman 2nd ed ISBN - 0131751905 The Science of Nutrition - Janice Thompson 1st ed ISBN - 0805394354 The Strategy of Managing Innovation and Technology - Murray Millson 1st ed ISBN - 0132303833 Theory of Asset Pricing - George Pennacchi 1st ed ISBN - 032112720X Thermodynamics: An Engineering Approach - Yunus Cengel 5th ed ISBN - 0073107689 Thinking Mathematically - Robert F. Blitzer 4th ed ISBN - 0131752049 Thomas' Calculus - George B. Thomas, Jr. 11th ed ISBN - 0321185587 Thomas' Calculus, Early Transcendentals, Media Upgrade - George B. Thomas, Jr 11th ed ISBN - 0321495756 Thomas' Calculus, Early Transcendentals, Media Upgrade, Part One - George B. Thomas, Jr. 11th ed ISBN - 0321498747 Thomas' Calculus, Media Upgrade - George B. Thomas, Jr. 11th ed ISBN - 032148987X Thomas' Calculus, Media Upgrade, Part One Single Variable - George B. Thomas, Jr. 11th ed ISBN - 0321498755 Thomas' Calculus, Media Upgrade, Part Two Multivariable, Chap 11-16 - George B. Thomas, Jr. 11th ed ISBN - 0321501039 Traffic & Highway Engineering - Nicholas J. Garber 4th ed ISBN - 0495082503 Training in Interpersonal Skills - Stephen P. Robbins 5th ed ISBN - 0132354993 Trigonometry - Margaret L. Lial 9th ed ISBN - 0321528859 Trigonometry - Mark Dugopolski 2nd ed ISBN - 032135690X Trigonometry: A Right Triangle Approach - Michael Sullivan 5th ed ISBN - 0136028969 Trigonometry: A Unit Circle Approach - Michael Sullivan 8th ed ISBN - 0132392798 UFL Collective Bargaining Agreement - Louis Marino 1st ed ISBN - 0131587668 Understanding and Managing Diversity - Carol Harvey 4th ed ISBN - 0132069105 Understanding and Managing Organizational Behavior - Jennifer George 5th ed ISBN - 013239457X Understanding Fiber Optics - Jeff Hecht 5th ed ISBN - 0131174290 Understanding Financial Statements - Lyn M. Fraser 8th ed ISBN - 0131878565 Understanding Modern Economics - Roger Miller 1st ed ISBN - 0321245822 University Calculus - Joel Hass, Maurice D. Weir 1st ed ISBN - 0321350146 University Calculus: Alternate Edition - Joel Hass 1st ed ISBN - 0321471962 University Calculus: Alternate Edition, Part One Single Variable, Chap 1-9 - Joel Hass 1st ed ISBN - 0321475194 University Calculus: Elements with Early Transcendentals - Joel Hass 1st ed ISBN - 0321533488 University Physics with Modern Physics with MasteringPhysics» - Hugh D. Young 12th ed ISBN - 080532187X Using and Understanding Mathematics: A Quantitative Reasoning Approach - Jeffrey O. Bennett 4th ed ISBN - 0321458206 Using Financial Accounting Information: The Alternative to Debits & Credits - Gary A. Porter 5th ed ISBN - 0324645104 Using Financial Accounting Information: The Alternative to Debits & Credits - Gary A. Porter 6th ed ISBN - 0324593740 Using MIS - David Kroenke 2nd ed ISBN - 0138132488 Using Peachtree Complete 2007 for Accounting - Glenn Owen 1st ed ISBN - 0324377975 Using Peachtree Complete 2009 for Accounting - Glenn Owen 3rd ed ISBN - 0324665512 Using Quickbooks Pro 2007 for Accounting - Glenn Owen 7th ed ISBN - 0324378750 Vector Calculus - Susan Colley 3rd ed ISBN - 0131858742 VHDL: A Starter's Guide - Sudhakar Yalamanchili 2nd ed ISBN - 0131457357 Water and Wastewater Technology - Mark J. Hammer 6th ed ISBN - 0131745425 Water Resources Engineering - David A. Chin 2nd ed ISBN - 0131481924 Water Supply and Pollution Control - Warren Viessman, Jr. 8th ed ISBN - 0132337177 Web 101 - Wendy G. Lehnert 3rd ed ISBN - 0321424670 West Federal Taxation Comprehensive 2007 - William Hoffman Solutions Manual 30th ed ISBN - 0324313497 West Federal Taxation Comprehensive 2007 - William Hoffman Test Bank 30th ed ISBN - 0324313497 West Federal Taxation Corporations 2007 - William Hoffman Solutions Manual 30th ed ISBN - 0324313616 West Federal Taxation Corporations 2007 - William Hoffman Test Bank 30th ed ISBN - 0324313616 West Federal Taxation Corporations 2008 - William Hoffman Solutions Manual 31st ed ISBN - 0324380437 West Federal Taxation Corporations 2008 - William Hoffman Test Bank 31st ed ISBN - 0324380437 West Federal Taxation Individual 2007 - William Hoffman Solutions Manual 30th ed ISBN - 0324399618 West Federal Taxation Individual 2007 - William Hoffman Test Bank 30th ed ISBN - 0324399618 West Federal Taxation Individual 2008 - William Hoffman Solutions Manual 31st ed ISBN - 0324380585 West Federal Taxation Individual 2008 - William Hoffman Test Bank 31st ed ISBN - 0324380585 West Federal Taxation: Taxation of Business Entities 2007 - James Smith Solutions Manual 10th ed ISBN - 0324313950 West Federal Taxation: Taxation of Business Entities 2007 - James Smith Test Bank 10th ed ISBN - 0324313950 West Federal Taxation: Taxation of Business Entities 2008 - James Smith Solutions Manual 11th ed ISBN - 0324366655 West Federal Taxation: Taxation of Business Entities 2008 - James Smith Test Bank 11th ed ISBN - 0324366655 West's Business Law - Kenneth W. 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Mathematical models and analytical solutions (Springer, 2008) === === === Subject: Rank of matrices posting-account=dO1h7AoAAACVmQmZt16AmlhxNjMLQIDQ Gecko/2009060215 Firefox/3.0.11 GTB5,gzip(gfe),gzip(gfe) I have a matrix of the form a_i,j = Sum {k=2...n} of [ mod( i - j , k) + mod( i + j , k) ] where i >=0 and j >= 0 I can easily find the rank of these matrices for specific values of n (by transforming to a row-echelon). For example the first few ranks are: n=2 gives rank=2 n=3 gives rank=3 n=4 gives rank=4 n=5 gives rank=6 n=6 gives rank=7 n=7 gives rank=10 etc.. But I can't find a general formula given the rank as a function of n. Can anyone help? === === Subject: engineering mechanics dynamics solutions posting-account=Ny3YSQoAAADViiWfYQqM39p9hx8OvAk4 Gecko/2009060215 Firefox/3.0.11,gzip(gfe),gzip(gfe) engineering mechanics dynamics solutions edition 6 meriam and kraige download here: http://store.payloadz.com/go?id=389980 === Subject: Re: futility of n-space posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > The restriction of n-space is reflected in the virtual impossibility of > finding roots to n-degree polynomials.If each coefficient is a component > to a vector of n dimensions, and each power of x is a component to a vector > of n dimensions, the two vectors are orthogonal since their dot product is > zero. I was just trying to study the ring quotient with the help of our > excellent Usenet community. As you use the term n-space the insistence > within the polynomials in use for ring theory are even worse than > this. They require an infinite space. Thus one is not even free to > consider an n-space. Some refute the usage of polynomials as dimensional and insist that > remaining in the limbo of an 'indeterminate value' that they are free > to speak of an X without ever describing X other than To say that >( a + X ) ( b + X ) = ab + (a + b) X + X X > and such simple operations. Yet here if these a and b are real then X > must have a superposition with a real. Somehow the indeterminate > stance is satisfactory to many. I do find it troubling. If we > ourselves validate our concern then what areas of mathematics fall? > These polynomials will still generate amazingly accurate sin() curves > in just ten steps with X real, but then this is the dimensionally > collapsed system whereby your concern goes away. This is no longer the > 'indeterminate' system. I guess another way to approach this is to make a simpler > indeterminate object and perform operations with it. It seems we might > do anything we like with an object that is indeterminate. I can't see > how an indeterminate object can be applied to mathematics. Well then > along comes the quotient form which supposedly will build say the > complex numbers as >R[X] / ( XX + 1 ) > yet here we are at a level of complexity in terms of the indeterminate > object. A composition of indeterminate objects ought to yield an > indeterminate object in my opinion, yet this one claims to make a > coherent composition out of them. Perhaps the crux is in the usage of > the word isomorphic. I'm not that impressed with isomorphism. I have > little doubt that a human is isomorphic to a coffee mug under some > math. Hmmm... I'll have to come up with some extensions on this... an > indeterminate form must represent everything isomorphically for if we > can compose one determinate object from an indeterminate object then > the bridge has been built. Voila. All of mathematics is now completed. - Tim > The liberty of n-space is that it opens up infinite possibilites. A > theorist can prove his brainchild in 4 dimensions if he can't do it in 3. > Just imagine.Associated with every point in (x1,x2,x3) is a point on the > x4 axis.That is, associated with every (x,y,z,t) is a point x4 hidden in > the system of points (x1,x2,x3,x4,t), not considering that furthermore t > (time) may be altered.The theorist becomes an artist producing > explanations at whim in equations that all come together perfectly. > Anything whatsoever can be proven using math of higher dimensions in E^n > where n>3. > This also delimits n-space math.On the one hand it is uselessly > constrictive and on the other uselessly diluted. > Consider a number of E^3 coordinate systems in synchrounous time with each > other separated by large distances.I think this is as best a model as can > be hoped for.One system is unbridled and actions in it mapped to form > reactions in the other systems.That way, associated with one system are > events in the other systems by cause and effect.When a tree is felled in > one system, a dam is caused to be broken in the other systems. This > meaningless association is diluted as well, but Master and Slave can swap > roles like a see-saw.This is in line with momentum that is channeled for > technological uses.No, there is no relationship. But can you find one? > The man's pulse was connected mechanically to regulate the speed of a > flywheel.After this went on for awhile, the technician altered the > flywheel and observed the man's pulse.He found out nothing.There are > some processes, however, that do swap Master/Slave relationships, like a > child's see-saw on a playground.What are they? These are good candidates > for multiple systems of E^3, particularly in distance relationships to > unknown cause and effect forces.The media may be EMR. The relatively > equipotential space-time around the planet may have anomalies, like > gradients or funnels.The ancient structures built in phylogeny are a > vortex.The path of least resistance is to delve and fall into them, > degenerating and destroying the foundations of everything that went in to > bringing the world up to the present. > Suppose one system was in the nucleus of a star and the other system in the > vortex of a Black Hole.Is there any cause and effect between the two? > No.- Hide quoted text - - Show quoted text - Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT - Docendi.org5 posts cannot be reduced to Null zero and n(2Pi^2-0.75) is constant curve for that value of n ... www.docendi.org/re-t241082.html Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANTJun 11, 2009 ... I have full irrefutable proof of this very imprtant constant and if there are any problems ... Inverse 19, Have a good rest and Godspeed. Best wishes, Musatov .... reply of N=NP what that means, and I will reply to you. ... Power . power is silent as is the space matrix, but it has great power and ... sci.tech-archive.net/Archive/sci.math/2009-06/msg01398.html Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT - Docendi.org6 posts - Last post: 15 hours ago constant, ... challenge you to provide *proof* of me mentioning any other film or ... www.docendi.org/re-t241056.html - 15 hours ago 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT...6 posts - Last post: 2 days ago *proof* of me mentioning any other film or ... www.groupsrv.com/science/post-3081440.html 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT - sci.physics | Google Groups4 posts - 3 authors - Last post: 16 hours ago the space matrix at 19. > Note: That this constant cannot be reduced to Null zero and n(2Pi^2-0.75) is constant curve for that value of n .... challenge you to provide *proof* of me mentioning any other film or ... 393d103f47a36a8c/e819a4171b52eb14?lnk=raot&fwc=1 - 16 hours ago 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT...10 posts - Last post: yesterday Dimension is silent, so is the space matrix at 19. Note: That this constant cannot be ... challenge you to provide *proof* of me mentioning any other film or ... The below text is everything I know regarding N=NP: Kindly, Musatov ... Inverse 19, Have a good rest and Godspeed. Best wishes, Musatov ... www.groupsrv.com/science/post-3083339.html Re: 2(3^2)+1=19 Good bye note to Sci math gurus!Jun 12, 2009 ... Since ... reply of N=NP what that means, and I will reply to you. I think much of ... Power . power is silent as is the space matrix, but it has great power ... sci.tech-archive.net/Archive/sci.math/2009-06/msg01417.html - 4 hours ago Higher order bifurcation of discrete map...10 posts - Last post: 2 days ago some higher space. ...http://scottaaronson.com/blog/%3Fp%3D387 + ... present ... author = {Martin M. Musatov}, title = {Proof of {Proof Net .... period-n region and we are plunging in there from the Shell- Thron region. .... On Jun 9, 10:13 pm, inverse 19 Well, it's been a few years since I need my math skills. And despite being >half-decent in my day, and an graduate to boot, I have recently failed >miserably to solve what should be a reasonably simple problem for work. Consider the following triangles : >http://img301.yfrog.com/i/triangleprob.gif/ I need to find the angle beta, given values for alpha, a, b and h, which >will satisfy this geometry. I've tried resolving it many times on paper but I seem to keep running out >of trigonometric identities to get beta onto one side. _______________ > Here goes a try. I'm assuming those angles that look like right angles are, with sides parallel. Extend the hypotenuse labelled h down into the lower triangle giving a lower triangle that is similar to the upper one. Look at the upper triangle. Its horizontal leg, call it w, is h*sin(beta) and its vertical leg, call it v, is sqrt(h^2 - w^2), so that triangle is completely known. The vertical leg of the lower triangle is a - v. So by similarity you know the horizontal leg of the small lower triangle, call it x and its hypotenuse, call it m. Now look at the big lower triangle. Its horizontal leg is p = b - w, its vertical leg is v and its upper acute angle is alpha + beta. So you have tangent(alpha + beta) = p / v which is now known. Take the arctangent and subtract beta and you have alpha. --Lynn http://math.asu.edu/~kurtz === Subject: Re: Epic failure to solve a simple problem > Well, it's been a few years since I need my math skills. And despite being > half-decent in my day, and an graduate to boot, I have recently failed > miserably to solve what should be a reasonably simple problem for work. > Consider the following triangles : > http://img301.yfrog.com/i/triangleprob.gif/ > I need to find the angle beta, given values for alpha, a, b and h, which > will satisfy this geometry. > I've tried resolving it many times on paper but I seem to keep running out > of trigonometric identities to get beta onto one side. > _______________ Here goes a try. I'm assuming those angles that look like right angles > are, with sides parallel. Extend the hypotenuse labelled h down into > the lower triangle giving a lower triangle that is similar to the > upper one. Look at the upper triangle. Its horizontal leg, call it w, > is h*sin(beta) and its vertical leg, call it v, is sqrt(h^2 - w^2), > so that triangle is completely known. The vertical leg of the lower > triangle is a - v. So by similarity you know the horizontal leg of the > small lower triangle, call it x and its hypotenuse, call it m. Now look at the big lower triangle. Its horizontal leg is p = b - w, > its vertical leg is v and its upper acute angle is alpha + beta. So > you have tangent(alpha + beta) = p / v which is now known. Take the arctangent and subtract beta and you have alpha. But you were given alpha to start with. You want to find an expression for beta (and presumably one that doesn't have to be solved numerically). === Subject: Re: Epic failure to solve a simple problem > Here goes a try. I'm assuming those angles that look like right angles > are, with sides parallel. Extend the hypotenuse labelled h down into > the lower triangle giving a lower triangle that is similar to the > upper one. Look at the upper triangle. Its horizontal leg, call it w, > is h*sin(beta) and its vertical leg, call it v, is sqrt(h^2 - w^2), > so that triangle is completely known. The vertical leg of the lower > triangle is a - v. So by similarity you know the horizontal leg of the > small lower triangle, call it x and its hypotenuse, call it m. > Now look at the big lower triangle. Its horizontal leg is p = b - w, > its vertical leg is v and its upper acute angle is alpha + beta. So > you have tangent(alpha + beta) = p / v which is now known. > Take the arctangent and subtract beta and you have alpha. But you were given alpha to start with. You want to find > an expression for beta (and presumably one that doesn't > have to be solved numerically). > Hi Greg is right though, beta is the unknown, alpha is known. To clarify the other points; yes the triangles are right angles, and the parallel sides assumption is also correct. And a formulaic solution is required. It did cross my mind at one point as to whether this is even a solvable problem, but I don't see why it wouldn't be; only that I ended up with a large formula with sines, cosines and tans which I couldn't simplify enough (with my old grey cells) to get beta on one side. === Subject: Re: Epic failure to solve a simple problem > Here goes a try. I'm assuming those angles that look like right angles > are, with sides parallel. Extend the hypotenuse labelled h down into > the lower triangle giving a lower triangle that is similar to the > upper one. Look at the upper triangle. Its horizontal leg, call it w, > is h*sin(beta) and its vertical leg, call it v, is sqrt(h^2 - w^2), > so that triangle is completely known. The vertical leg of the lower > triangle is a - v. So by similarity you know the horizontal leg of the > small lower triangle, call it x and its hypotenuse, call it m. > Now look at the big lower triangle. Its horizontal leg is p = b - w, > its vertical leg is v and its upper acute angle is alpha + beta. So > you have tangent(alpha + beta) = p / v which is now known. > Take the arctangent and subtract beta and you have alpha. > But you were given alpha to start with. You want to find > an expression for beta (and presumably one that doesn't > have to be solved numerically). Hi >Greg is right though, beta is the unknown, alpha is known. To clarify the >other points; yes the triangles are right angles, and the parallel sides >assumption is also correct. And a formulaic solution is required. It did cross my mind at one point as >to whether this is even a solvable problem, but I don't see why it wouldn't >be; only that I ended up with a large formula with sines, cosines and tans >which I couldn't simplify enough (with my old grey cells) to get beta on one >side. It is solvable alright, but it isn't pretty. Expanding on my ideas above and letting Maple do the grunt work, you can use the law of cosines on the oblique subtriangle containing alpha to solve for alpha in terms of the givens. It isn't pretty and there may very well be a nicer method. Just for the record, here is Maple's expression for alpha. You need fixed pitch font. I doubt it will be useful and who knows if it will even be readable: arccos((b*a*tan(beta)+a^2-2*a*h*cos(beta)+h^2*cos(beta)^2-b*h*sin(beta)+h^2* sin(beta)^2-a*tan(beta)*h*sin(beta))/(a^2-2*a*h*cos(beta)+h^2*cos(beta)^2+b^ 2 -2*b*h*sin(beta)+h^2*sin(beta)^2)^(1/2)/(a^2-2*a*h*cos(beta)+h^2*cos(beta)^2 + a^2*tan(beta)^2-2*a*tan(beta)*h*sin(beta)+h^2*sin(beta)^2)^(1/2)) --Lynn http://math.asu.edu/~kurtz === Subject: Re: Epic failure to solve a simple problem >Greg is right though, beta is the unknown, alpha is known. To clarify the >other points; yes the triangles are right angles, and the parallel sides >assumption is also correct. >And a formulaic solution is required. It did cross my mind at one point as >to whether this is even a solvable problem, but I don't see why it wouldn't >be; only that I ended up with a large formula with sines, cosines and tans >which I couldn't simplify enough (with my old grey cells) to get beta on one >side. It is solvable alright, but it isn't pretty. Expanding on my ideas >above and letting Maple do the grunt work, you can use the law of >cosines on the oblique subtriangle containing alpha to solve for alpha >in terms of the givens. It isn't pretty and there may very well be a >nicer method. Just for the record, here is Maple's expression for >alpha. You need fixed pitch font. I doubt it will be useful and who >knows if it will even be readable: arccos((b*a*tan(beta)+a^2-2*a*h*cos(beta)+h^2*cos(beta)^2-b*h*sin(beta)+h^2 *sin(beta)^2-a*tan(beta)*h*sin(beta))/(a^2-2*a*h*cos(beta)+h^2*cos(beta)^2+b ^ 2-2*b*h*sin(beta)+h^2*sin(beta)^2)^(1/2)/(a^2-2*a*h*cos(beta)+h^2*cos(beta)^ 2 +a^2*tan(beta)^2-2*a*tan(beta)*h*sin(beta)+h^2*sin(beta)^2)^(1/2)) --Lynn http://math.asu.edu/~kurtz Damn!! I did it again. Solved for alpha in terms of beta instead of the other way around. So I asked Maple to solve for beta. Much to my surprise, it did. But the answer is even worse: %8 a 1/2 ------- + h - (cos(alpha) h + %2 ) cos(alpha) 2 2 a + b %8 arctan(-----------------------------------------------, - -------), b 2 2 a + b %7 a 1/2 ------- + h - (cos(alpha) h + %2 ) cos(alpha) 2 2 a + b arctan(-----------------------------------------------, b %7 - -------), arctan( 2 2 a + b %11 a 1/2 ------- + h - (cos(alpha) h - %2 ) cos(alpha) 2 2 a + b %11 -----------------------------------------------, - -------), b 2 2 a + b %10 a 1/2 ------- + h - (cos(alpha) h - %2 ) cos(alpha) 2 2 a + b arctan(-----------------------------------------------, b %10 - -------), arctan( 2 2 a + b %11 a 1/2 ------- + h + (-cos(alpha) h + %2 ) cos(alpha) 2 2 a + b %11 ------------------------------------------------, - -------), b 2 2 a + b %10 a 1/2 ------- + h + (-cos(alpha) h + %2 ) cos(alpha) 2 2 a + b arctan(------------------------------------------------, b %10 - -------), arctan( 2 2 a + b %8 a 1/2 ------- + h + (-cos(alpha) h - %2 ) cos(alpha) 2 2 a + b %8 ------------------------------------------------, - -------), b 2 2 a + b %7 a 1/2 ------- + h + (-cos(alpha) h - %2 ) cos(alpha) 2 2 a + b arctan(------------------------------------------------, b %7 - -------), arctan(-%1, a/h), arctan(%1, a/h), 2 2 a + b arctan(%1, a/h), arctan(-%1, a/h) 2 2 1/2 (-a + h ) %1 := ------------- h 2 2 2 2 2 %2 := cos(alpha) h + a - h + b 4 2 %3 := b cos(alpha) 2 2 2 %4 := a b cos(alpha) 2 2 2 2 4 4 2 2 2 %5 := (-%4 + a b - 2 b h cos(alpha) + b + 3 b cos(alpha) h 2 1/2 2 3 1/2 + 2 b cos(alpha) h %2 - %3 - 2 b h cos(alpha) %2 2 2 1/2 - b h ) 1/2 %6 := cos(alpha) a %2 2 %7 := -h a + cos(alpha) a h + %6 + %5 2 %8 := -h a + cos(alpha) a h + %6 - %5 2 2 2 2 4 4 2 2 2 %9 := (-%4 + a b - 2 b h cos(alpha) + b + 3 b cos(alpha) h 2 1/2 2 3 1/2 - 2 b cos(alpha) h %2 - %3 + 2 b h cos(alpha) %2 2 2 1/2 - b h ) 2 %10 := -h a + cos(alpha) a h - %6 + %9 2 %11 := -h a + cos(alpha) a h - %6 - %9 --Lynn http://math.asu.edu/~kurtz === Subject: Re: Epic failure to solve a simple problem <4a332cb3$0$5844$9a6e19ea@news.newshosting.com> posting-account=YrY6tQoAAAA6HLnYe4MdqQcSD72mkqDE 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; .NET CLR 2.0.50727; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) === Subject: Re: Epic failure to solve a simple problem I found a much simpler solution: >beta=arcsin(h*sin(alpha)/sqrt(a^2+b^2))+alpha-arctan(b/a) Earl How about beta = arctan( (h*sin(alpha) - a) / (h*cos(alpha) - b) ) - alpha ? -- Virgil === Subject: Re: Epic failure to solve a simple problem <4a332cb3$0$5844$9a6e19ea@news.newshosting.com> posting-account=YrY6tQoAAAA6HLnYe4MdqQcSD72mkqDE 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; .NET CLR 2.0.50727; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) >I found a much simpler solution: >beta=arcsin(h*sin(alpha)/sqrt(a^2+b^2))+alpha-arctan(b/a) >Earl How about beta = arctan( (h*sin(alpha) - a) / (h*cos(alpha) - b) ) - alpha ? -- > Virgil Sorry, I seem to have gotten to signs wrong, the corrected answer is: beta=arcsin(h*sin(alfa)/sqrt(a^2+b^2))-alfa+arctan(b/a) Earl === Subject: Re: Epic failure to solve a simple problem > Here goes a try. I'm assuming those angles that look like right angles > are, with sides parallel. Extend the hypotenuse labelled h down into > the lower triangle giving a lower triangle that is similar to the > upper one. Look at the upper triangle. Its horizontal leg, call it w, > is h*sin(beta) and its vertical leg, call it v, is sqrt(h^2 - w^2), > so that triangle is completely known. The vertical leg of the lower > triangle is a - v. So by similarity you know the horizontal leg of the > small lower triangle, call it x and its hypotenuse, call it m. > Now look at the big lower triangle. Its horizontal leg is p = b - w, > its vertical leg is v and its upper acute angle is alpha + beta. So > you have tangent(alpha + beta) = p / v which is now known. > Take the arctangent and subtract beta and you have alpha. > But you were given alpha to start with. You want to find > an expression for beta (and presumably one that doesn't > have to be solved numerically). > > Hi > > > Greg is right though, beta is the unknown, alpha is known. To clarify the > other points; yes the triangles are right angles, and the parallel sides > assumption is also correct. > > And a formulaic solution is required. It did cross my mind at one point as > to whether this is even a solvable problem, but I don't see why it wouldn't > be; only that I ended up with a large formula with sines, cosines and tans > which I couldn't simplify enough (with my old grey cells) to get beta on one > side. > You might do better to try and solve purely in terms of sin(alpha) or cos(alpha), and consider sin(beta) and cos(beta) as constants, so that your work would be all algebraic. -- Virgil === Subject: Re: Epic failure to solve a simple problem >But you were given alpha to start with. You want to find >an expression for beta (and presumably one that doesn't >have to be solved numerically). > Aargh. Maybe after a night's sleep I will look again. --Lynn http://math.asu.edu/~kurtz === Subject: Re: Proof Martin Musatov is a Pioneer posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Math Forum Discussions - Re: Reactions to/against the Binary > Treearguments, you are no pioneer, merely a naysayer. .... Martin > Musatov > P.S. P=NP, Call me the Charlton Heston if math, but they can > have my ...http://mathforum.org/kb/thread.jspa%3FforumID > %3D13%26threadID%3D1940341%26messageID%3D6724147 - 108k - Cached - > Similar pagesken and belly: Red Coats and PinkeyeAre you aware at the > plight of computer scientist Martin Musatov as he attempts to approve > an intrinsic .... Confessions of a Pioneer Woman. 11 hours > ago ...http://kenandbelly.blogspot.com/2009/03/red-coats-and- > pinkeye.html - 99k - Cached - Similar pages0xDEMartin Musatov, a > horror-film screenwriter, P=NP crank, and sockpuppeteer ... Thomas T. > Goldsmith, Jr. was an important pioneer in the early > television ...http://11011110.livejournal.com/tag/wikipedia- 65k - > Cached - Similar pagesTechnology Classes around Los Angeles - > Oodle.comHi Introducing Mind On Demand Software Labs, a Pioneer in RIA > (Rich Internet Application). .... Martin Musatov on TeachStreet - Los > Angeles, CA 1 month ago ...http://la.oodle.com/service/education/tech/ > - 90k - Cached - Similar pagessci.mathRe: struggling with definition > of wreath product, Martin Musatov ...... Jussi Piitulainen; Re: A > integral inequality for help., Pioneer ...http://sci.tech-archive.net/ > Archive/sci.math/2009-05/ - 284k - Cached - Similar pagesRe: General > Price: ...http://sci.tech-archive.net/Archive/sci.physics.relativity/ > 2009-06/msg00086.html - 24k - Cached - Similar pagesProject Excelsior > ' KuiperCliffTo me, Kittinger remains a true pioneer, as were all the > astronauts and ... pversusnp on Short History of the Shadow .87 Martin > Musatov on New ...http://kuipercliff.wordpress.com/2007/02/23/project- > excelsior/ - 32k - Cached - Similar pages Inverse 19 Musatov Proof Space Archaeology [CapitalEth] Apollo 11 > moon landing ' Kuiper Cliff The Riemann Hypothesis is Part of the existential and horrific allure > of the Pioneer missions is .... True: >http://farm4.static.flickr.com/3496/3256947403 398e26dde8 o.jpg >pversusnp on Short History of the > Shadow .87 Martin Musatov on New ...http://kuipercliff.wordpress.com/ > 2007/04/04/space-archaeology-apollo-11-moon-landing/ - 46k - Cached - > Similar pagesReactions to/against the Binary Tree - > Docendi.orgarguments, you are no pioneer, merely a naysayer. ..... On > you, > Martin Musatov ...http://www.docendi.org/reactions-t235962.html > %3Ft%3D235962 - 132k - Cached - Similar pages === Subject: prime factorization of mersenne p exponent minus 1. msg lost resend. Cc: MCDONewt@yahoo.co.nz posting-account=TV2szgkAAACrA1vyuh8IN_0zzgzcwogw CLR 2.0.50727),gzip(gfe),gzip(gfe) > 2,2,2,11,2,53,2,3,3 ,7 ,2,2,2,5 ,13 prime factorisation of mersenne p > exponents minus 1. e.g. 2 2 2 3 3 3 5 5 53 149 > sloanes online encyc of integer sequences. does not have. donald s. mcdonald paradise. nz 11-6-2009. > 47th known mersenne prime apparently discovered. world record bunch of prime numbers.: Google Alert - mersenne prime 7 mersenne prime exponents within an /octave// interval [k, 2k.] don mcdonald wellington 10-6-2009. jfh, jb. I am not sure if these columns will print correctly yahoo. 47th known mersenne prime exponent. factor 2^prime - 1 isprime? 42643801 426438 9 47382 factor 6 7897 factor 3 2632.333333 5 1579.4 7 1128.142857 9 877.4444444 11 717.9090909 13 607.4615385 15 526.4666667 17 464.5294118 19 415.6315789 21 376.047619 23 343.3478261 25 315.88 27 292.4814815 29 272.3103448 31 254.7419355 33 239.3030303 35 225.6285714 37 213.4324324 39 202.4871795 41 192.6097561 43 183.6511628 45 175.4888889 47 168.0212766 49 161.1632653 51 154.8431373 53 149 7897 426438 6*9*53*149*100+1 factors 42643801 isprime? twin primes about mersenne prime exponent. 43rd? program BAS64- modabc1 ? 277345127 31*683*13099 277345127 ryee Don S. McDonald ... Table Mountain 2^ (2*34* 444 32 +1 ) -1 is Mersenne Prime v//. exponent 3.02.. million? Climate pride//# === > Subject: world record bunch of prime numbers.: Google Alert - mersenne prime > Received: Wednesday, 10 June, 2009, 1:50 PM > world record bunch of prime numbers.: Google Alert - > mersenne prime MathWorld News: 47th Known Mersenne Prime Apparently > Discovered > June 7, 2009--Less than a year after the 45th and 46th > known Mersenne > primes were discovered, Great Internet Mersenne Prime > Search (GIMPS) > project organizer George Woltman is reporting in a June 7 > email to the > GIMPS mailing list that a ... This once a day Google Alert > fwd cheers, > ............10/6/09 Tara wera Mt > 1886?? Rotorua NZ. Don S. McDonald ... === Subject: Seelink Solution Manual posting-account=2obgfQoAAABR8QxJjVoDDq8P84fHTPLB CLR 1.1.4322; .NET CLR 2.0.50727; InfoPath.1),gzip(gfe),gzip(gfe) Do you have the solution manual for: Thermodynamics in Materials Science, 2nd Edition, Robert DeHoff, Pub. CRC Press. ISBN: 0849340659 Let me know if you have it. === Subject: Re: Proof Martin Musatov is a Pioneer posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO Trident/4.0; GTB6; .NET CLR 1.1.4322; eSobiSubscriber 2.0.4.16; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Math Forum Discussions - Re: Reactions to/against the Binary > Treearguments, you are no pioneer, merely a naysayer. .... Martin > Musatov > P.S. P=NP, Call me the Charlton Heston if math, but they can > have my ...http://mathforum.org/kb/thread.jspa%3FforumID > %3D13%26threadID%3D1940341%26messageID%3D6724147 - 108k - Cached - > Similar pagesken and belly: Red Coats and PinkeyeAre you aware at the > plight of computer scientist Martin Musatov as he attempts to approve > an intrinsic .... Confessions of a Pioneer Woman. 11 hours > ago ...http://kenandbelly.blogspot.com/2009/03/red-coats-and- > pinkeye.html - 99k - Cached - Similar pages0xDEMartin Musatov, a > horror-film screenwriter, P=NP crank, and sockpuppeteer ... Thomas T. > Goldsmith, Jr. was an important pioneer in the early > television ...http://11011110.livejournal.com/tag/wikipedia- 65k - > Cached - Similar pagesTechnology Classes around Los Angeles - > Oodle.comHi Introducing Mind On Demand Software Labs, a Pioneer in RIA > (Rich Internet Application). .... Martin Musatov on TeachStreet - Los > Angeles, CA 1 month ago ...http://la.oodle.com/service/education/tech/ > - 90k - Cached - Similar pagessci.mathRe: struggling with definition > of wreath product, Martin Musatov ...... Jussi Piitulainen; Re: A > integral inequality for help., Pioneer ...http://sci.tech-archive.net/ > Archive/sci.math/2009-05/ - 284k - Cached - Similar pagesRe: General > Price: ...http://sci.tech-archive.net/Archive/sci.physics.relativity/ > 2009-06/msg00086.html - 24k - Cached - Similar pagesProject Excelsior > ' KuiperCliffTo me, Kittinger remains a true pioneer, as were all the > astronauts and ... pversusnp on Short History of the Shadow .87 Martin > Musatov on New ...http://kuipercliff.wordpress.com/2007/02/23/project- > excelsior/ - 32k - Cached - Similar pages Inverse 19 Musatov Proof Space Archaeology [CapitalEth] Apollo 11 > moon landing ' Kuiper Cliff The Riemann Hypothesis is Part of the existential and horrific allure > of the Pioneer missions is .... True: >http://farm4.static.flickr.com/3496/3256947403 398e26dde8 o.jpg >pversusnp on Short History of the > Shadow .87 Martin Musatov on New ...http://kuipercliff.wordpress.com/ > 2007/04/04/space-archaeology-apollo-11-moon-landing/ - 46k - Cached - > Similar pagesReactions to/against the Binary Tree - > Docendi.orgarguments, you are no pioneer, merely a naysayer. ..... On > you, > Martin Musatov ...http://www.docendi.org/reactions-t235962.html > %3Ft%3D235962 - 132k - Cached - Similar pages === Subject: Re: Liberal Arts Math > > Bertrand Russell said: > Mathematics possesses not only truth but supreme beauty. A beauty that is cold and austere like that of a sculpture. > > I want to experience a bit of that, and I certainly want to use math/logic to develop my overall brain and thinking and problem solving skills. > > My math level is only Pre-calculus, however, and I will be taking it at community college. Am I really going to experience this almost sexy appeal to Math which Russell describes? Have you tried Russell's own Introduction to Mathematical Philosophy? > I know there is a sort of 2 tiered approach to most undergraduate math systems: first learn rote a lot of calculus algorithms, then proceed into the analysis, which contains math more like puzzles, which is more interesting from my standpoint. > -- ... when we came back, late, from the hyacinth garden, Your arms full, and your hair wet, I could not Speak, and my eyes failed... === Subject: Re: Liberal Arts Math Perhaps selfconsciously, but may I suggest that a look at the Interactive Mathematics Miscellany and Puzzles site http://www.cut-the-knot.org may be worth your while. A. Bogomolny http://www.cut-the-knot.org === Subject: solution manuals and test banks for sale posting-account=KhEHuAoAAADa4kfi_ORqDYOVAUvOh1Z5 AppleWebKit/530.5 (KHTML, like Gecko) Chrome/2.0.172.31 Safari/530.5,gzip(gfe),gzip(gfe) I have the following solution manuals and test banks. 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Mathematical models and analytical solutions (Springer, 2008) === Subject: basic math question -- mundaka@sdf.lonestar.org SDF Public Access UNIX System - http://sdf.lonestar.org === Subject: Re: basic math question : > What is the proper proceedure if I wanted to convert the fraction I'm assuming you mean your original fraction is (700/3)/100. Remember that the fraction bar means a division, so you have (70/3) divided by 100. Remember further that dividing by N is the same as multiplying by 1/N -- for example, dividing by 2 is the same as multiplying by 1/2. So dividing by 100 is the same as multiplying by 100. Therefore, (700/3)/100 is the same as (700/3) times (1/100). To multiply fractions, you multiply the tops and multiply the bottoms. (700/3) times (1/100) = 700/300. However .. Why *would* you want to do this? Usually we want fractions to be expressed in lowest terms. To do that, you factor the top and factor the bottom, then divide out common factors from top and bottom: (700/3) times (1/100) = (700)/(3*100) = (7*100)/(3*100) = 7/3. -- Stan Brown, Oak Road Systems, Tompkins County, New York, USA http://OakRoadSystems.com Shikata ga nai... === Subject: Re: basic math question > > > Sorry, my connection timed out as I was posting. > > What is the proper proceedure if I wanted to convert the fraction 700/3/100 Your expression is somewhat ambiguous. Do you mean (700/3)/100 or 700/(3/100) ? Common usage requires performing multiple instances of the same operation in order from left to right, except possibly for exponentiation, so letting x = 700/3 (700/3)/100 = x/100 = x * (1/100) = (700/3)*(1/100) = (700*1)/(3*100) = 700/300 = 7/3 -- Virgil === Subject: Re: basic math question posting-account=Bdg5ngoAAABEariw6PbB1G_LUE895pfi Gecko/2009060215 Firefox/3.0.11 GTB5,gzip(gfe),gzip(gfe) On Jun 14, 10:57am, Christian Dwight Lyman Sorry, my connection timed out as I was posting. -- > mund...@sdf.lonestar.org > SDF Public Access UNIX System -http://sdf.lonestar.org Easy to do, except that it won't be 700/300. Simply rewrite 700/3/100= (700/1)/(3/100)=700*(100/300)=70000/300=700/3 When you are dividing by a fraction just take the denominator and multiply and multiply the numerator by the inverse of that. Symbolically: (a/b) / (c/d) = (a*d) / (b*c) Patrick === Subject: Re: basic math question posting-account=Bdg5ngoAAABEariw6PbB1G_LUE895pfi Gecko/2009060215 Firefox/3.0.11 GTB5,gzip(gfe),gzip(gfe) > Easy to do, except that it won't be 700/300.Simply rewrite 700/3/100= > (700/1)/(3/100)=700*(100/300)=70000/300=700/3 Oops, reduced my fraction wrong. The final result should be 70000 / 3 So in order to convert 700/3/100 you have to multiply by 1 / 10000 === Subject: Re: i need to solve an ODE equation plzz help > i have attached the equations file as a scanned > image. plz help me to solve this.. > > Only a numerical solution seems to make sense for your system of equations. If you have access to MATLAB, use ODE45. Best wishes Torsten. === Subject: Re: CMS: Euler's Formula for sin(x) > >These comments on the book are getting to be amusing. > > >But you have to actually _read_ it (and also do the exercises). >No, merely skimming through it the way you've been doing >is not going to help. > > A book sale is a book sale! Perhaps Han should > read yours side by side with the one James self- > published. James Stewart Jr. ? Han de Bruijn === Subject: Re: CMS: Euler's Formula for sin(x) > >These comments on the book are getting to be amusing. >But you have to actually _read_ it (and also do the exercises). >No, merely skimming through it the way you've been doing >is not going to help. >A book sale is a book sale! Perhaps Han should >read yours side by side with the one James self- >published. >James Stewart Jr. ? > > No, James Harris. Should have added that smiley. Han de Bruijn === Subject: Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) Our purpose is not as much as provoking discussion , but the ability for us to post and issue and learn from the Silence of the Lambs. Dimension is silent, so is the space matrix at 19. Note: That this constant cannot be reduced to Null zero and n(2Pi^2-0.75) is constant curve for that value of n I have to say my previous post (view able here : ) completely amazed me with respect this post. As this post did not appear after I posted my previous post and I noted this peculiarity: contains the words Red Dragon - The sequel to The Silence of The Lambs... And it was posted (or at least to me appeared in the topic list AFTER my previous post). In breakdown, I posted a P=NP Genesis/Riemann post containing mention of a sequel to a film title which then appeared in a new post (this post) to me at least, after. Are we in a vacuum or is there logic at play in this anomaly? I appreciate any sound feedback, the more thoughtful and considerate the better. Please do not make fun or ridicule this sincere attempt to explore a truth. === Subject: Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT posting-account=Bdg5ngoAAABEariw6PbB1G_LUE895pfi Gecko/2009042316 Firefox/3.0.10 GTB5,gzip(gfe),gzip(gfe) >snip The answer is that you mention movies and their sequels frequently, also you probably post and cross post the same stupid to many newsgroups. Then you do this over and over when , other than the occasional poster trying to alert you to your inanities, people generally ignore you. It's not an anomaly, if anything it places statistical significance on your word vomit. Patrick === Subject: Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) >snip The answer is that you mention movies and their sequels frequently, > also you probably post and cross post the same stupid to many > newsgroups. Then you do this over and over when , other than the > occasional poster trying to alert you to your inanities, people > generally ignore you. It's not an anomaly, if anything it places > statistical significance on your word vomit. Patrick Keep it professional. I'll not push pearls on swine, merely to say I challenge you to provide *proof* of me mentioning any other film or *sequel* anywhere across USENET besides Red Dragon as I identified succeed because it is not there so I will spare you the trouble and *retract* your distasteful ill-researched assertion to allows others more interested in learning to search out such a matter. Cordially, Martin Musatov *new theory*Dignity*Respect*Bold* === Subject: Re: 2 Pi^2 - 0.75 = INVERSE/EXVERSE CONSTANT I am not intelligent enough to make fun or ridicule any one. The word silence of the lambs is a chilling statement and reflects the ignorance of arrogany men. Now listen we are very sure of this constant which simply equals 19, but it shows that the cinverse is not a absolute null zero. Yes there is a logic, I never read your post re this, but I know you have suffered ridicule , but I never ever even thought of you. I would like to understand,, please kindly give me a short description in this reply of N=NP what that means, and I will reply to you. I think much of you , I will respect you , because restraint is the greatest sembklence of Power . power is silent as is the space matrix, but it has great power and thunder. I will wait for your reply in this post. === Subject: Re: AT&T Usenet Netnews Service Shutting Down posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) >@flph199.ffdc.sbc.com: > Please note that on or around July 15, 2009, AT&T will no longer be > offering access to the Usenet netnews service. If you wish to continue > reading Usenet newsgroups, access is available through third-party > vendors. >i'm screwed Yeah, now you might actually have to stop posting hundreds of retarded > messages every day and GET A JOB! Oh, wait ... but now you won't be able to publish your scientific > papers in alt.gossip.celebrities anymore! You might actually be forced > to stop spending your entire life discussing women's hair, makeup, and > fashion in a celebrity gossip newsgroup while pretending to be the > world's straightest guy, an astro physicist, a math genius, and whatever > else you can dream up daily. Bwaaahaaahaaahahahahaaaa!!!! On behalf of everyone on Usenet: !!!!!!!!!!!!!!! THANK YOU AT&T !!!!!!!!!!!!!!! Are you referring to me? If so, I am sorry for the inconvenience. Also to set the record straight, I have never posted to: alt.gossip.celebrities (until this post) Musatov === Subject: Re: AT&T Usenet Netnews Service Shutting Down posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) >@flph199.ffdc.sbc.com: > Please note that on or around July 15, 2009, AT&T will no longer be > offering access to the Usenet netnews service. If you wish to continue > reading Usenet newsgroups, access is available through third-party > vendors. >i'm screwed Yeah, now you might actually have to stop posting hundreds of retarded > messages every day and GET A JOB! Oh, wait ... but now you won't be able to publish your scientific > papers in alt.gossip.celebrities anymore! You might actually be forced > to stop spending your entire life discussing women's hair, makeup, and > fashion in a celebrity gossip newsgroup while pretending to be the > world's straightest guy, an astro physicist, a math genius, and whatever > else you can dream up daily. Bwaaahaaahaaahahahahaaaa!!!! On behalf of everyone on Usenet: !!!!!!!!!!!!!!! THANK YOU AT&T !!!!!!!!!!!!!!! Are you referring to me? If so, I am sorry for the inconvenience. Also to set the record straight, I have never posted to: alt.gossip.celebrities Musatov === Subject: Group cohomology and singular cohomology posting-account=R7AgUAoAAADVFAtIe36IBmgohoHjZsKW Gecko/20080418 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) how can one define group cohomology using the terms of singular cohomology and classifying spaces? Let G be a group and M be a G modul, i.e. a modul over the group ring Z[G]. I have thought that H^*(G,M)=H^*(BG) where the right hand side is singular cohomology of topological spaces and the left hand side is group cohomology. But where does the modul M comes in on the right? S. === Subject: Re: Group cohomology and singular cohomology posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2009042708 Fedora/3.0.10-1.fc10 Firefox/3.0.10,gzip(gfe),gzip(gfe) how can one define group cohomology using the terms of singular > cohomology and classifying spaces? Let G be a group and M be a G > modul, i.e. a modul over the group ring Z[G]. I have thought that > H^*(G,M)=H^*(BG) > where the right hand side is singular cohomology of topological spaces > and the left hand side is group cohomology. But where does the modul M > comes in on the right? The problem is that your formula H^*(G,M)=H^*(BG) is simply false. The correct equation is H^*(G,M)=H^*(BG, M) where, on the right, we mean the singular cohomology of BG with values in the abelian group M. In any case, this only works when M is a trivial G-module. If you want to compute H^*(G, M) for a non-trivial G-module M using BG, you need to use cohomology with a local coefficient system. This is explained in good algebraic topology texts. -- m === === Subject: Re: Quadratic programming problem posting-account=EmSEdgoAAABmAH5eEkFTCMiZ14RLEfpQ > (Q x, x) + 2(r,x) + b -> min_x > (S x, x) <= 1 > Unfortunately, the answer to the question is that there isn't a formula for the solution to this problem. I'll use the notation x'*S*x instead of (S x, x) in the following. It's important to note first that depending on S and Q this problem may be unbounded (try S=0, Q=0, and r<>0) or have an infinite number of solutions (try S=0, Q=0, r=0.) Any procedure for solving this problem will have to check for such cases, so you simply can't expect to have a simple formula for the solution to this problem. Furthermore, this problem becomes very challenging numerically when S and/or Q are badly conditioned matrices. It's easy to write down the first order necessary conditions for this problem: 2*(Q-lambda*S)*x+r=0 lambda*(x'*S*x-1)=0 You can use a numerical method to solve for lambda and x, but there's no closed form solution to these nonlinear equations. A simplified version of this problem with the constraint x'*x <= 1 instead of x'*S*x<=1 has been studied extensively in the optimization literature. The simplified problem is called the trust region subproblem because it comes up in trust region methods for nonlinear optimization. If S is positive definite (i.e. it's non-singular), you can use the Cholesky factorization of S to perform a change of variables that simplifies the original poster's problem into a trust region subproblem. You could then use a subroutine for the trust region subproblem to find a minimum. === Subject: mathmetics posting-account=cfAOZgoAAADecJX8L2FMInZe0-s7VbjM 1.1.4322),gzip(gfe),gzip(gfe) all === Subject: Re: Kullback-Leibler vs. Jensen-Shannon > I have a question about Kullback-Leibler divergence (relative > entropy) and Jensen-Shannon divergence. They are both measures about > how some distribution p deviates from a background distribution > q. My question is: > > Is KLD(p1,q) > KLD(p2,q) equivalent to JSD(p1,q) > JSD(p2,q) ? In > other words, are the two measures equivalent in the way they rank > how much distributions deviate from the background? My feeling is > that yes, of course they are but I don't really know... Setting p1 = (81, 6, 34), p2 = (13, 13, 76) and q = (1, 1, 1), each vector u interpreted as a probability distribution u/sum(u) on the three points u_1, u_2, u_3, we have approximately: KLD(p1,q) = 0.4678461 < 0.5111050 = KLD(p2,q) JSD(p1,q) = 0.1280332 > 0.1269324 = JSD(p2,q) Setting p1 = (45, 43, 74), p2 = (32, 56, 59), q = (1, 1, 1), we have the opposite relation, approximately: KLD(p1,q) = 0.04734906 > 0.04711203 = KLD(p2,q) JSD(p1,q) = 0.01159336 < 0.01224042 = JSD(p2,q) So, no, unless I made a mistake somewhere. Logs to base 2. === Subject: Re: Kullback-Leibler vs. Jensen-Shannon posting-account=gbZTvQoAAAA1nP8OfoT1Fr3mN-KihMPv Gecko/2009042513 Ubuntu/8.04 (hardy) Firefox/3.0.10,gzip(gfe),gzip(gfe) wrong in code I had written previously to calculate this... > Setting p1 = (81, 6, 34), p2 = (13, 13, 76) and q = (1, 1, 1), each > vector u interpreted as a probability distribution u/sum(u) on the > three points u_1, u_2, u_3, we have approximately: KLD(p1,q) = 0.4678461 < 0.5111050 = KLD(p2,q) > JSD(p1,q) = 0.1280332 > 0.1269324 = JSD(p2,q) Setting p1 = (45, 43, 74), p2 = (32, 56, 59), q = (1, 1, 1), we have > the opposite relation, approximately: KLD(p1,q) = 0.04734906 > 0.04711203 = KLD(p2,q) > JSD(p1,q) = 0.01159336 < 0.01224042 = JSD(p2,q) So, no, unless I made a mistake somewhere. Logs to base 2. === Subject: Re: associated Legendre polynomials > > I am attempting to calculate the following integral related to the > associated Legendre polynomials: > > int_{x=-1}^1 P_n^m(x) P_k^l(x) dx = ? > > where n,m,k,l are positive integers. > > I cannot find the solution from many mathematical handbooks. Can > anybody direct me how to solve it or any reference? > > Matthew > case m=l is here http://functions.wolfram.com/Polynomials/LegendreP2/21/02/01/ === Subject: Re: associated Legendre polynomials <100620091103543845%anniel@nym.alias.net.invalid> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) > I am attempting to calculate the following integral related to the > associated Legendre polynomials: > int {x=-1}^1 P n^m(x) P k^l(x) dx = ? > where n,m,k,l are positive integers. > I cannot find the solution from many mathematical handbooks. Can > anybody direct me how to solve it or any reference? > Matthew > > case m=l is here > http://functions.wolfram.com/Polynomials/LegendreP2/21/02/01/ Legendre, Chebyshev, Hermite, etc.) have n real roots in the ... has been defined as (f, g) = int a^b w(x)f(x)g(x) dx, for a given .... Ideally, G(X) would also be a linear combination of Legendre Coefficients of Chebyshev polynomials... Completely. Ideally, G(X) would also be a linear combination of Legendre... Ideally, G(X) would also be a linear combination of Legendre... '.8a.8c.be'.89[Capit alOHat]'[NonBreakingSpace][NonBreakingSpace] .9f[NonBreakingSpace].8cí.8c.8f[Ca pitalUGrave]'[NonBreakingSpace]Martin Musatov - 20[NonBreakingSpace].8c'.8f[NonBreakingS pace][Hyphen][NonBreakingSpace]?[NonBreakingSpace]2[NonBreakingSpace][Ca pitalIGrave]ìì.83[ ARing].be[NonBreakingSpace]2[NonBreakingSpace].84[CapitalOA cute]'.8a.83ø .84î'®.8a [NonBreakingSpace]¨¨[RegisteredTr ademark][NonBreakingSpace]Ideally, G(X) would also be a linear combination of Legendre... '.8a.8cñ[CapitalIDoubl eDot] to compute scalar .... now that we consider Green's function, i.e. w (x,.83.85) = G(x,.83.85;s) is a Legendre polynomial. To make their content explicit, we decompose the cross ..... g (x): (33). Phenomenological experience with the distribution .. Here the transverse part of the metric tensor, denoted by g ..... in terms of Legendre polynomials and express the expansion coefficients c ... The skewedness parameter is .83.81 = 1/3 which corresponds to x .... ++ Musatov === Subject: Re: Inequality === Subject: Re: Inequality Yes, sorry, I forgot to include that, only for even integers x >= 6. The inequality is used in [1, pg. 44]; I'm trying to satisfy my own doubts ;-) Robertson, N. and Seymour, P., Graph Minors. I. Excluding a Forest, in 35 Journal of Combinatorial Theory (Series B), 1, 39--61 (1983) === === Subject: Re: Radius of Arcs Mail-To-News-Contact: abuse@dizum.com > Unit circle (with radius r=1) is divided by 3 equal arc in to the 7 > parts of equal area, as shown in the picture: >http://s524.photobucket.com/albums/cc321/Krejakovic/7krug.jpg > > Find radius of arcs. > > If it is a unit circle the radius of any arc segment along it also > equals one, doesn't it? Or are you looking for the length of > the arc segment? > > Did you look at the picture? The arcs referred to are not > arcs of the unit circle, they are arcs of three other circles. Yes I looked at both his picture and yours. His picture looks like the unit circle is the border perimeter of a sphere with the same radius and the other arcs are segments of other great circles on that sphere, meaning they all have the same radius as the unit circle...1. YOUR picture seems to be showing segments of other differently radiused circles parallel to the unit circle, which he neither seemed to be asking about nor his picture described. BUT, he subsequently replied that you DID answer the right question, so I guess it was either a well known question/problem poorly phrased and diagrammed that you correctly figured out, or I'm just too dense to see it...my vote is for the former! ;) Il mittente di questo messaggio|The sender address of this non corrisponde ad un utente |message is not related to a real reale ma all'indirizzo fittizio|person but to a fake address of an di un sistema anonimizzatore |anonymous system Per maggiori informazioni |For more info https://www.mixmaster.it === Subject: Re: Radius of Arcs posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) WCXS:1984: > Unit circle (with radius r=1) is divided by 3 equal arc in to the 7 > parts of equal area, as shown in the picture: >http://s524.photobucket.com/albums/cc321/Krejakovic/7krug.jpg > Find radius of arcs. > If it is a unit circle the radius of any arc segment along it also > equals one, doesn't it? Or are you looking for the length of > the arc segment? > Did you look at the picture? The arcs referred to are not > arcs of the unit circle, they are arcs of three other circles. Yes I looked at both his picture and yours. > His picture looks like the unit circle is the border perimeter > of a sphere with the same radius and the other arcs are segments > of other great circles on that sphere, meaning they all have the > same radius as the unit circle...1. > YOUR picture seems to be showing segments of other differently > radiused circles parallel to the unit circle, which he neither > seemed to be asking about nor his picture described. > BUT, he subsequently replied that you DID answer the right > question, so I guess it was either a well known question/problem > poorly phrased and diagrammed that you correctly figured out, or > I'm just too dense to see it...my vote is for the former! ;) Il mittente di questo messaggio|The sender address of this > non corrisponde ad un utente |message is not related to a real > reale ma all'indirizzo fittizio|person but to a fake address of an > di un sistema anonimizzatore |anonymous system > Per maggiori informazioni |For more info > https://www.mixmaster.it gnat and swallow a camel. What you dismiss as trivial will be the cornerstone of the theory you adore and governor of the 12 Tribes. 12*12,000> === Subject: Re: Radius of Arcs posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) > Unit circle (with radius r=1) is divided by 3 equal arc in to the 7 > parts of equal area, as shown in the picture: >http://s524.photobucket.com/albums/cc321/Krejakovic/7krug.jpg > Find radius of arcs. > If it is a unit circle the radius of any arc segment along it also > equals one, doesn't it? Or are you looking for the length of > the arc segment? > Did you look at the picture? The arcs referred to are not > arcs of the unit circle, they are arcs of three other circles. Yes I looked at both his picture and yours. > His picture looks like the unit circle is the border perimeter > of a sphere with the same radius and the other arcs are segments > of other great circles on that sphere, meaning they all have the > same radius as the unit circle...1. > YOUR picture seems to be showing segments of other differently > radiused circles parallel to the unit circle, which he neither > seemed to be asking about nor his picture described. > BUT, he subsequently replied that you DID answer the right > question, so I guess it was either a well known question/problem > poorly phrased and diagrammed that you correctly figured out, or > I'm just too dense to see it...my vote is for the former! ;) Il mittente di questo messaggio|The sender address of this > non corrisponde ad un utente |message is not related to a real > reale ma all'indirizzo fittizio|person but to a fake address of an > di un sistema anonimizzatore |anonymous system > Per maggiori informazioni |For more info > https://www.mixmaster.it gnat and swallow a camel. What you dismiss as trivial will be the cornerstone of the theory you adore and governor of the 12 Tribes. 12*12,000> === Subject: Re: Reactions to/against the Binary Tree <4a2f0d02$0$23688$afc38c87@news.optusnet.com.au> posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > On 8 Jun., 15:10, Peter Webb > The path that represents 1/3 is not in a finite tree. > That is not objected. But it is not the topic. The topic is that every > infinite path can be constructed such that there is no node that is > not removed at some step. So after an infinite number of steps there > are no nodes left whereas only a countable number of paths have been > constructed and have used up all nodes. Of course every path that you > want is among the set of constructed paths, 1/3 and pi and what you > can specify. > Here is your question: Please answer yes or no. There is a path in the infinite tree that >cannot be found in a finite tree. I did never oppose to that. Therefore I do not feel obliged to answer that silly question (if infinity is understood as being potential). What I oppose to is the actual completed existence of infinity. As there cannot be an actually infinite tree, one can reasonably state that there is no actually infinite path. We can prove that easily by assuming its existence. I answered your question correctly; you seem to acknowledge that. Don't I get a thank you at least? > If you decide to thank me properly, I > will answer further questions you may have on this subject, and that may > allow you to understand the flaws in your reasoning. Hardly. There are no flaws. The flaw is with Cantor: He (and his disciples) assume that the complete diagonal of the famous list is the limit of an infinite process. But he and they deny that the complete binary tree is the limit of an infinite process. Cantor spelled that out in a letter to Letter to Vivanti of Dec. 3 1885: Introducing the expression path that Cantor did not use but mean, he said: This apparent difficulty is solved as follows: The set of all path is not the limit of all finite path z n for n = oo. This is wrong. Either there is no limit at all or the limit holds also for the binary tree. You see (or you could see, at least) that there is no flaw on my side, but a logical inconsistency by set theorists. Therefore it is not me who needs support. === Subject: Re: Reactions to/against the Binary Tree > > On 8 Jun., 15:10, Peter Webb > The path that represents 1/3 is not in a finite tree. > That is not objected. But it is not the topic. The topic is that every > infinite path can be constructed such that there is no node that is > not removed at some step. So after an infinite number of steps there > are no nodes left whereas only a countable number of paths have been > constructed and have used up all nodes. Of course every path that you > want is among the set of constructed paths, 1/3 and pi and what you > can specify. > > Here is your question: > Please answer yes or no. >There is a path in the infinite tree that >cannot be found in a finite tree. > > I did never oppose to that. But as soon as you concede that there are paths containing infinitely any nodes, you have effectively conceded that there are maximal infinite binary trees with uncountably many paths. > Therefore I do not feel obliged to answer > that silly question (if infinity is understood as being potential). There can be no such thing as your notion of potentially infinite sets without an axiom system in which they are consistent, and you have never even attempted to provided any such system. Such an axiom system merely states those things which you consider necessary truths without which nothing else can be derived. > What I oppose to is the actual completed existence of infinity. What everyone else opposes is your inability to provide anything like an axiom system in which your definitions of potential infiniteness of sets fits. > As > there cannot be an actually infinite tree, one can reasonably state > that there is no actually infinite path. We can prove that easily by > assuming its existence. On the contrary, you cannot prove it without assuming it. > I answered your question correctly; you seem to acknowledge that. > Don't I get a thank you at least? > > > If you decide to thank me properly, I > will answer further questions you may have on this subject, and that may > allow you to understand the flaws in your reasoning. > > Hardly. There are no flaws. Anyone who claims no flaws, exhibits at least one by making that claim. The flaw is with Cantor: He (and his > disciples) assume that the complete diagonal of the famous list is the > limit of an infinite process. They say no such thing. It is just another function from N to {m,w}. > > You see (or you could see, at least) that there is no flaw on my side, > but a logical inconsistency by set theorists. Therefore it is not me > who needs support. On the contrary, until WM can produce a list of all those things on which his arguments are based, it is he that is inconsistent and flawed. And until WM can produce a proof that ZF is inconsistent WITHOUT assuming anything outside of ZF and formal logic, it is he that is wrong. Mathematics does not care about the kind of absolute truth that WM tries to hawk, but only in relative truths, like the answer to: if this is true does that follow? -- Virgil === Subject: Re: Reactions to/against the Binary Tree posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > And until WM can produce a proof that ZF is inconsistent WITHOUT > assuming anything outside of ZF ZF boasts about laying the grounds to real numbers. So it also lays the ground to the binary tree and its consequences. > Mathematics does not care about the kind of absolute truth that WM > tries to hawk, but only in relative truths, like the answer to: >if this is true does that follow? Wenn die formale Mathematik nicht mehr den Anspruch erhebt, wahre Behauptungen aufzustellen, so muss man sich fragen, was sie denn .9fberhaupt bezweckt. (H. Weyl) === Subject: Re: Reactions to/against the Binary Tree posting-account=_-PQygoAAAAciOn_89sZIlnxfb74FzXU Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) < ... > Mathematics does not care about the kind of absolute truth that WM > tries to hawk, but only in relative truths, like the answer to: >if this is true does that follow? -- > Virgil Can you represent the literal expansions of the rationals of the unit interval without the entire infinite balanced b'ary tree? That reminds of Harvey Friedman's notice that the rationals are arbitrarily large. Ross F. === Subject: Re: Reactions to/against the Binary Tree > < ... > Mathematics does not care about the kind of absolute truth that WM > tries to hawk, but only in relative truths, like the answer to: >if this is true does that follow? > -- > Virgil > > Can you represent the literal expansions of the rationals of the unit > interval without the entire infinite balanced b'ary tree? What do you mean by the literal expansions of the rationals of the unit interval. Unless you mean something esoteric, it is quite possible, even trivial, as a finite non-repeating sequence of digits followed by a single repetition of the infinite sequence of repetitions which follows it, separated by any non-digit symbol. > > That reminds of Harvey Friedman's notice that the rationals are > arbitrarily large. Ambiguous to the point of incoherence as usual. -- Virgil === Subject: Re: Reactions to/against the Binary Tree posting-account=_-PQygoAAAAciOn_89sZIlnxfb74FzXU Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > < ... > Mathematics does not care about the kind of absolute truth that WM > tries to hawk, but only in relative truths, like the answer to: >if this is true does that follow? > -- > Virgil > Can you represent the literal expansions of the rationals of the unit > interval without the entire infinite balanced b'ary tree? What do you mean by the literal expansions of the rationals of the unit > interval. Unless you mean something esoteric, it is quite possible, even trivial, > as a finite non-repeating sequence of digits followed by a single > repetition of the infinite sequence of repetitions which follows it, > separated by any non-digit symbol. > That reminds of Harvey Friedman's notice that the rationals are > arbitrarily large. Ambiguous to the point of incoherence as usual. -- > Virgil I don't understand why you must feel that way. No, you can't, you need the entire infinite b'ary tree to represent the literal expansions of the rationals of the unit interval. Ross F. === Subject: Re: Reactions to/against the Binary Tree > < ... > Mathematics does not care about the kind of absolute truth that WM > tries to hawk, but only in relative truths, like the answer to: >if this is true does that follow? > -- > Virgil > Can you represent the literal expansions of the rationals of the unit > interval without the entire infinite balanced b'ary tree? > What do you mean by the literal expansions of the rationals of the unit > interval. > Unless you mean something esoteric, it is quite possible, even trivial, > as a finite non-repeating sequence of digits followed by a single > repetition of the infinite sequence of repetitions which follows it, > separated by any non-digit symbol. > > That reminds of Harvey Friedman's notice that the rationals are > arbitrarily large. > Ambiguous to the point of incoherence as usual. > -- > Virgil > > I don't understand why you must feel that way. It is not that I must, but I do anyway. > > No, you can't, you need the entire infinite b'ary tree to represent > the literal expansions of the rationals of the unit interval. You might. I don't. Let S be the set of all finite strings of decimal digits, the every rational in the unit interval can be represented by any one of infinitely many members of S^2 ( or SxS, if you prefer). Every such rational can be represented in infinitely many ways as a single finite digit string followed by infinitely many repetitions of a possibly different finite digit string, thus by a pair of finite digit strings. -- Virgil === Subject: Re: Reactions to/against the Binary Tree posting-account=_-PQygoAAAAciOn_89sZIlnxfb74FzXU Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > < ... > Mathematics does not care about the kind of absolute truth that WM > tries to hawk, but only in relative truths, like the answer to: >if this is true does that follow? > -- > Virgil > Can you represent the literal expansions of the rationals of the unit > interval without the entire infinite balanced b'ary tree? > What do you mean by the literal expansions of the rationals of the unit > interval. > Unless you mean something esoteric, it is quite possible, even trivial, > as a finite non-repeating sequence of digits followed by a single > repetition of the infinite sequence of repetitions which follows it, > separated by any non-digit symbol. > That reminds of Harvey Friedman's notice that the rationals are > arbitrarily large. > Ambiguous to the point of incoherence as usual. > -- > Virgil > I don't understand why you must feel that way. It is not that I must, but I do anyway. > No, you can't, you need the entire infinite b'ary tree to represent > the literal expansions of the rationals of the unit interval. You might. I don't. Let S be the set of all finite strings of decimal digits, the every > rational in the unit interval can be represented by any one of > infinitely many members of S^2 ( or SxS, if you prefer). Every such rational can be represented in infinitely many ways as a > single finite digit string followed by infinitely many repetitions of a > possibly different finite digit string, thus by a pair of finite digit > strings. -- > Virgil Reducing to the binary tree, there isn't a third symbol to use as the demarcator of the beginning and end of a repeating terminus. So, anybody needs the entire tree. Analysis of the general b-ary tree sees different cases for b > 2, basically it's always a bigger tree, as well, at each node there are all the branches. Similarly the antidiagonal result doesn't hold in binary, or for that matter trinary, for dual representation. Ross F. === Subject: Re: Reactions to/against the Binary Tree > > < ... > Mathematics does not care about the kind of absolute truth that > WM > tries to hawk, but only in relative truths, like the answer to: >if this is true does that follow? > -- > Virgil > Can you represent the literal expansions of the rationals of the unit > interval without the entire infinite balanced b'ary tree? > What do you mean by the literal expansions of the rationals of the > unit > interval. > Unless you mean something esoteric, it is quite possible, even trivial, > as a finite non-repeating sequence of digits followed by a single > repetition of the infinite sequence of repetitions which follows it, > separated by any non-digit symbol. > That reminds of Harvey Friedman's notice that the rationals are > arbitrarily large. > Ambiguous to the point of incoherence as usual. > -- > Virgil > I don't understand why you must feel that way. > It is not that I must, but I do anyway. > > No, you can't, you need the entire infinite b'ary tree to represent > the literal expansions of the rationals of the unit interval. > You might. I don't. > Let S be the set of all finite strings of decimal digits, the every > rational in the unit interval can be represented by any one of > infinitely many members of S^2 ( or SxS, if you prefer). > Every such rational can be represented in infinitely many ways as a > single finite digit string followed by infinitely many repetitions of a > possibly different finite digit string, thus by a pair of finite digit > strings. > -- > Virgil > > Reducing to the binary tree, there isn't a third symbol to use as the > demarcator of the beginning and end of a repeating terminus. So, > anybody needs the entire tree. Not at all. One can always use 00 for 0, 11 for 1 and have both 10 and 01 for a first and even a second demarcator symbol. > > Similarly the antidiagonal result doesn't hold in binary Since it was originally stated only in binary, whyever not? -- Virgil === Subject: Re: Reactions to/against the Binary Tree posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > You are right. All the finite binary trees have the root node in > common, and every finite tree T(n) with n levels is contained in the > next one T(n+1) with one more level. The union of T(1) to T(n) is T > (n). And if we apply logic that is obtained from finite sets, i.e., > true and reliable logic, we see that every union over finite trees is > a finite tree. There is no infinite tree resulting from a union of > finite trees. > Yes, any finite union of finite trees is itself a finite tree. > Thus for any union of trees T(1) to T(n), the union is T(n), > which is a finite tree. > But what about the union of all of the finite trees? It's obviously > not any T(n), because each T(n) fails to include T(n+1) and > the rest. Thus we can conclude that, whatever it is, the union > of all finite trees cannot be any finite tree T(n). > That leaves only two choices, > No, it leaves a third one, in fact the only one compatible with logic: > There is no completed union of all finite trees. Which logic is that? I don't understand what completed is > supposed to mean in basic logic. Could you provide a definition > of a completed union of trees, or perhaps a definition of an > incomplete union of trees? Completed means what mostly is called actually infinite. There it is assumed that all elements are there in contrast to potential infinity, that yields only incomplete sets. This potential understanding of infinity is the only consistent one. Therefore it gains more and more reputation. Compare Edward Nelson: What would the world be like if God had not created numbers? Just what it is like now. There is not a shred of evidence that the numbers have been created. If I give you an addition problem like 37460225182244100253734521345623457115604427833 + 52328763514530238412154321543225430143254061105 and you are the first to solve it, you will have created a number that did not exist previously. Compare Doron Zeilberger: The phrase for all positive integers is meaningless. [...] By hindsight, it is not surprising that there exist undecidable propositions, as meta-proved by Kurt Goedel. Why should they be decidable, being meaningless to begin with! === Subject: Re: Reactions to/against the Binary Tree > You are right. All the finite binary trees have the root node in > common, and every finite tree T(n) with n levels is contained in the > next one T(n+1) with one more level. The union of T(1) to T(n) is T > (n). And if we apply logic that is obtained from finite sets, i.e., > true and reliable logic, we see that every union over finite trees is > a finite tree. There is no infinite tree resulting from a union of > finite trees. Then in such unions every path terminates in a leaf node. But outside of WM's wee world, in, say, ZF, there are trees in which paths do not have leaf nodes. And WM, who cannot see beyond the tiny limits of his wee world, is incapable of understanding their properties. > Yes, any finite union of finite trees is itself a finite tree. > Thus for any union of trees T(1) to T(n), the union is T(n), > which is a finite tree. > But what about the union of all of the finite trees? It's obviously > not any T(n), because each T(n) fails to include T(n+1) and > the rest. Thus we can conclude that, whatever it is, the union > of all finite trees cannot be any finite tree T(n). > That leaves only two choices, > No, it leaves a third one, in fact the only one compatible with logic: > There is no completed union of all finite trees. What is compatible with WM's private logic is too often incompatible with any more publicly acceptable logic > Which logic is that? I don't understand what completed is > supposed to mean in basic logic. Could you provide a definition > of a completed union of trees, or perhaps a definition of an > incomplete union of trees? > > Completed means what mostly is called actually infinite. There it is > assumed that all elements are there in contrast to potential > infinity, that yields only incomplete sets. > > This potential understanding of infinity is the only consistent one. > Therefore it gains more and more reputation. > > Compare Edward Nelson: > What would the world be like if God had not created numbers? Just what > it is like now. There is not a shred of evidence that the numbers have > been created. If I give you an addition problem like > 37460225182244100253734521345623457115604427833 > + 52328763514530238412154321543225430143254061105 > and you are the first to solve it, you will have created a number that > did not exist previously. He mistakes the creation of a name with the creation of the named. > > Compare Doron Zeilberger: The phrase for all positive integers is > meaningless. [...] > By hindsight, it is not surprising that there exist undecidable > propositions, as meta-proved by Kurt Goedel. Why should they be > decidable, being meaningless to begin with! But so many propositions in, say, ZF, however meaningless WM finds them, are eminently decidable. The sort of decision required being onlyan answer to does it follow from the axioms?. If WM is in search of any truth beyond does it follow, then mathematics is not the place to look, as it follows is the only sort of truth relevant to mathematical investigations. For example, in mathematics, determining the actual geometry of the physical universe is an irrelevancy, but determining what follows from assuming various forms of geometry is highly relevant. -- Virgil === Subject: Re: Reactions to/against the Binary Tree posting-account=-eQqtQoAAACZVM-kNEsOn3k7GSvoJoS4 SV1; GTB6; .NET CLR 1.1.4322; InfoPath.1),gzip(gfe),gzip(gfe) > Compare Edward Nelson: > What would the world be like if God had not created numbers? Just what > it is like now. There is not a shred of evidence that the numbers have > been created. If I give you an addition problem like > 37460225182244100253734521345623457115604427833 > + 52328763514530238412154321543225430143254061105 > and you are the first to solve it, you will have created a number that > did not exist previously. There is absolutely no way you or Nelson can be certain of that. Even if you insist on the (in my opinion silly) position that 2 + 2, 2*2 and 4 do not all refer to the same number, which therefore exits in the form 2 + 2 before you do the addition (or in the form 1 + 3), the only way of *knowing* that a number will be new is by checking, which defeats the purpose. Nelson must have known that he was lying, and you are just propagating that lie. === Subject: Re: NOT the LIAR posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) > Let's call it The Semanticist paradox. > **This statement has no meaning. > So, everybody, what is the semantic status of that, IYHO? > (**) is obviously meaningful and false. What paradox do you see here? There may not be a paradox but I do not see how it is meanigful? What > do you picture when you say This statement has no meaning? Musatov>I see where you are going and I fiercely agree. We need to we specify the language of the statement This statement has no meaning to mean, I like bananas. We can do this with computers after the fact quite easily, but also in *reality*. What if I tell you when I say Go it means Go, except when it is followed by, No. In that case, you wait a second after hearing Go before acting. Then we have Go means Go and Go No means don't go. Note: we have not violated any physical laws and the time delay in computing can be tuned like a master mechanic tunes an engine, so does an algorithm designer with his logic to these programming variables. Ingenuity will reap big rewards coupled with imagination and good old fashion creative means. Musatov === Subject: Re: NOT the LIAR posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) > > Let's call it The Semanticist paradox. > **This statement has no meaning. > So, everybody, what is the semantic status of that, IYHO? > (**) is obviously meaningful and false. What paradox do you see here? > There may not be a paradox but I do not see how it is meanigful? What > do you picture when you say This statement has no meaning? > Could it be that what one really has in mind when refering to This > false statement (I hope that make sense)? I just can't get my head > around how it's possible to debate with a bunch of text? How can the > letters be right or wrong? And if the text holds no meaning, then what > is this debate about? > sense, and so I'm forced to change > my point of view. Here it appears that the truth of the statement is > determined by the reader as soon > as meaning is extracted from it. > How is the meaning extracted? >So the statement becomes false in > the doing so to speak, defacto. > Sorry to followup on my own post.- Hide quoted text - > - Show quoted text - I'm not sure exactly how meaning gets attached to symbols, but it does > and so we do. In my thinking, > other than just being what they are, symbols do not come prepackaged > with their own meaning, so to speak. > So they must be interpreted to have meaning. I can't say that is > absolutely true, but this is what I believe. > Here, we believe that the statement was intended to have meaning, and > that it was intended to have truth value. > I think the statement could also be written It is false that this > statement has meaning or something such? If so, > and if the reader does not understand the statement, then the > statement This sentence has no meaning might very > well not have meaning for him/er, but then neither does it have the > truth value from the reader's perspective > that it was intended to. Thus it is false that it has truth value, and > so it is false on its face. And if the > receiver does understand it, the statement is false as soon as the > meaning is understood by the reader. So it is > false in both cases. At least, that is my humble attempt to understand > what's going on here. Btw, I still do not get how > statements can speak for themselves and declare their own truth > value? If we accept that, then what is the point of > using this tool to establish the truth of statements? Clever people > can just stymie us by wording statements in a way > that will prove anything they want, no? Are their efforts not at cross > purposes with the function of very tool they are using > to prove they have stymied us? It's wordy. I do apologize for the excessive wordiness. concise summary The best theories, are ignored, then fiercely opposed, before finally being accepted as self-evident. P=NP, Musatov ::/:)* === Subject: Inequality with xy+yz+zx=1. Hello teacher~ xy + yz + zx = 1 (x, y, z : positive real) Show that [x / sqrt{(x^2) + 1)}] + [y / sqrt{(y^2) + 1}] + [z / sqrt{(z^2) + 1}] <= 3/2. ---------------------------------------------------------------------------- ------------------------- Hm... x^2 + 1 = x^2 + xy + yz + zx = (x+y)(x+z) y^2 + 1 = y^2 + xy + yz + zx = (y+z)(y+x) z^2 + 1 = z^2 + xy + yz + zx = (z+y)(z+x) I can't progress favorably. so, I need your advice. === Subject: Re: Inequality with xy+yz+zx=1. posting-account=Z3AipgkAAABkoMfyNwddSxsYhXHi5CDt MathPlayer 2.10d; .NET CLR 1.1.4322; PeoplePal 3.0),gzip(gfe),gzip(gfe) > Hello teacher~ xy + yz + zx = 1(x, y, z : positive real) Show that > [x / sqrt{(x^2) + 1)}] + [y / sqrt{(y^2) + 1}] + [z / sqrt{(z^2) + 1}] <= > 3/2. ---------------------------------------------------------------------------[ CapitalEth]-------------------------- > Hm... x^2 + 1 = x^2 + xy + yz + zx = (x+y)(x+z) y^2 + 1 = y^2 + xy + yz + zx = (y+z)(y+x) z^2 + 1 = z^2 + xy + yz + zx = (z+y)(z+x) I can't progress favorably. > so, I need your advice. It is intuitively obvious by symmetry that the maximum occurs when x=y=z, and so from xy + xz + yz = 1 we get that they are all (1/sqrt (3)). This yields 3/2 as the sum. Unfortunately this is not a proof, although Lagrange Multipliers would probably back this up quickly and easily. However, Leon Aigret has a better idea. Let x = tanA, y = tanB. Then assume that 0 < A, B < pi/2. It is now easily calculated that x/sqrt (x^2 + 1) = sinA, y/sqrt(y^2 + 1) = sinB, and z/sqrt(z^2 + 1) = cos(A + B). So we must show that F(A, B) = sinA + sinB + cos(A+B) <= 3/2 given 0 < A, B < pi/2. This is now simple calculus: dF/dA = cosA - sin(A+B) = 0 dF/dB = cosB - sin(A+B) = 0 Thus cosA = cosB so A = B. Also cosA = sin2A = 2cosA*sinA, so sinA = 1/2, and cos(A+B) = 1/2, so F(A, B) = 3/2. We have a unique critical point. Why is it a maximum? I suppose we could try the Jacobian condition. d^2F/dA^2 = -sinA - cos(A+B) = -1 d^2F/dA*dB = d^2F/dB*dA = -cos(A+B) = -1/2 d^2F/dB^2 = -sinB - cos(A+B) = -1 so the Jacobian matrix is |-1 -1/2| |-1/2 -1| which is obviously negative definite, so the unique critical point must be a maximum. Achava === Subject: Re: Inequality with xy+yz+zx=1. ... > xy + yz + zx = 1 (x, y, z : positive real) > > Show that > [x / sqrt{(x^2) + 1)}] + [y / sqrt{(y^2) + 1}] + [z / sqrt{(z^2) + 1}] > <= 3/2. > > ----------------------------------------------------------------- ... > x^2 + 1 = x^2 + xy + yz + zx = (x+y)(x+z) > > y^2 + 1 = y^2 + xy + yz + zx = (y+z)(y+x) > > z^2 + 1 = z^2 + xy + yz + zx = (z+y)(z+x) > > I can't progress favorably. > so, I need your advice. Disregard the first condition. Take the derivative of f(x) = x/(1+x^2) (ie, 1/(1+x^2) - 2x^2 / (1+x^2)^2) and observe where f'(x) is 0. -- jiw === Subject: Re: Inequality with xy+yz+zx=1. xy + yz + zx = 1 (x, y, z : positive real) > > Show that > [x / sqrt{(x^2) + 1)}] + [y / sqrt{(y^2) + 1}] + [z / sqrt{(z^2) + 1}] > <= 3/2. > > ----------------------------------------------------------------- > ... > x^2 + 1 = x^2 + xy + yz + zx = (x+y)(x+z) > > y^2 + 1 = y^2 + xy + yz + zx = (y+z)(y+x) > > z^2 + 1 = z^2 + xy + yz + zx = (z+y)(z+x) > > I can't progress favorably. > so, I need your advice. > > Disregard the first condition. Take the derivative of > f(x) = x/(1+x^2) > (ie, 1/(1+x^2) - 2x^2 / (1+x^2)^2) and > observe where f'(x) is 0. Unfortunately I overlooked the denominators' square roots, so please disregard my previous post. -- jiw === Subject: Re: Inequality with xy+yz+zx=1. >Hello teacher~ xy + yz + zx = 1 (x, y, z : positive real) Show that >[x / sqrt{(x^2) + 1)}] + [y / sqrt{(y^2) + 1}] + [z / sqrt{(z^2) + 1}] <= >3/2. --------------------------------------------------------------------------- -------------------------- >Hm... x^2 + 1 = x^2 + xy + yz + zx = (x+y)(x+z) y^2 + 1 = y^2 + xy + yz + zx = (y+z)(y+x) z^2 + 1 = z^2 + xy + yz + zx = (z+y)(z+x) I can't progress favorably. >so, I need your advice. xy + yz + zx = 1 can be rewritten as z = (1 - xy) / (x + y), which reminds one of the addition formula for the tangens function. It might be useful to rewrite x, y and z as tan a, tan b and tan c with 0 < a, b, c < pi / 2 Leon === Subject: Re: Inequality with xy+yz+zx=1. posting-account=_HO4IQoAAABUeM489-BOjGjtWAiLbkv- > Hello teacher~ xy + yz + zx = 1(x, y, z : positive real) Show that > [x / sqrt{(x^2) + 1)}] + [y / sqrt{(y^2) + 1}] + [z / sqrt{(z^2) + 1}] <= > 3/2. ---------------------------------------------------------------------------[ CapitalEth]-------------------------- > Hm... x^2 + 1 = x^2 + xy + yz + zx = (x+y)(x+z) y^2 + 1 = y^2 + xy + yz + zx = (y+z)(y+x) z^2 + 1 = z^2 + xy + yz + zx = (z+y)(z+x) I can't progress favorably. > so, I need your advice. How about Lagrange multipler? === Subject: Re: Inequality with xy+yz+zx=1. posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) Musatov affirms: You have my blessing and full faith. You have only one teacher and that is The Christ~ > Hello teacher~ xy + yz + zx = 1 (x, y, z : positive real) Show that > [x / sqrt{(x^2) + 1)}] + [y / sqrt{(y^2) + 1}] + [z / sqrt{(z^2) + 1}] <= > 3/2. ---------------------------------------------------------------------------- - ------------------------ > Hm... x^2 + 1 = x^2 + xy + yz + zx = (x+y)(x+z) y^2 + 1 = y^2 + xy + yz + zx = (y+z)(y+x) z^2 + 1 = z^2 + xy + yz + zx = (z+y)(z+x) I can't progress favorably. > so, I need your advice. Go forth and prosper:......._________)))()+@: === Subject: where is that pdf file about how to write mathematical proofs for dummies? posting-account=_HO4IQoAAABUeM489-BOjGjtWAiLbkv- I've seen a pdf file (3 or 5 pages) explaining for dummies how to write mathematical proofs in an unambiguous way and common mistakes students make when writing them. But I forgot where and how I found it . It's available somewhere on the web. It makes the following points 1. A proof must be a series of complete sentences, not of equations. 2. the equality sign. (You know, those students misusing equality signs!) and some other points really for dummies. I need to find it or similar thing because I want to handout the prints to my students as I'm a TA. Or I will have to write it myself with my poor English. === Subject: Re: where is that pdf file about how to write mathematical proofs for dummies? posting-account=s0altQoAAAAA2RaDYngh2HB3SQpSm7GN Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > I've seen a pdf file (3 or 5 pages) explaining for dummies how to > write mathematical proofs in an unambiguous way and common mistakes > students make when writing them. But I forgot where and how I found > it . It's available somewhere on the web. It makes the following points > 1. A proof must be a series of complete sentences, not of equations. > 2. the equality sign. (You know, those students misusing equality > signs!) > and some other points really for dummies. I need to find it or similar thing because I want to handout the > prints to my students as I'm a TA. Or I will have to write it myself > with my poor English. Is this what you are looking for? http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to-wr ite.pdf === Subject: Re: where is that pdf file about how to write mathematical proofs for dummies? posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to-wr i te.pdf Lamport, the author of that paper, claims there is an error in the proof of Schroder-Bernstein in Kelley's 'General Topology'. That is basically the same proof as in Halmos's 'Naive Set Theory'. I've worked through those proofs. I had to fill in missing details, but I've not seen any error. What error is Lamport referring to? MoeBlee === Subject: Re: where is that pdf file about how to write mathematical proofs for dummies? > >http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to- write.pdf > > Lamport, the author of that paper, claims there is an error in the > proof of Schroder-Bernstein in Kelley's 'General Topology'. That is > basically the same proof as in Halmos's 'Naive Set Theory'. I've > worked through those proofs. I had to fill in missing details, but > I've not seen any error. What error is Lamport referring to? The one you didn't see, perhaps? Han de Bruijn === Subject: Re: where is that pdf file about how to write mathematical proofs for dummies? > Lamport, the author of that paper, claims there is an error in the > proof of Schroder-Bernstein in Kelley's 'General Topology'. That is > basically the same proof as in Halmos's 'Naive Set Theory'. I've > worked through those proofs. I had to fill in missing details, but > I've not seen any error. What error is Lamport referring to? I have no idea, and have unfortunately lost my copy of _General Topology_. Perhaps you could produce the proof here for our inspection? -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: where is that pdf file about how to write mathematical proofs for dummies? posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) > http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to-wr i te.pdf Lamport, the author of that paper, claims there is an error in the > proof of Schroder-Bernstein in Kelley's 'General Topology'. That is > basically the same proof as in Halmos's 'Naive Set Theory'. I've > worked through those proofs. I had to fill in missing details, but > I've not seen any error. What error is Lamport referring to? MoeBlee Musatov's Pioneer (USENET+wide) proof NP=P or (6P=NP proper!) Web Images Video Maps News Shopping Gmail more émarty.musatov@gmail.com | My Account | WebResults 1 - 10 of about 15,500 for pdf file p=np simple proof QED. (0.28 seconds) Qed Theorem - Ask.com Search Qed. Theorem Absurd: P -> ~ P -> Q. Proof. intros p np; destruct np; assumption. ... Interactive Computer Theorem Proving (PDF File) ... http://ask.reference.com/web%3Fq%3DQed%2BTheorem%26qsrc%3D2892%26l%3Ddir%26o %3D10601 + + PDF file created from a TIFF image by tiff2pdfP .84 NP 4=> NP has Kolmogorov simple certificates relative to the input. ... The proof is direct. If P = NP certificates are trivialized: by using .... QED. We have proven our Kolmogorov characterization of the P1 ^ og^-vs-Pi question. ... http://ecommons.library.cornell.edu/bitstream/1813/6635/1/86-795.pdf + + What would a P=NP proof be like, hypothetically? - Stack OverflowThere are some meta-results about what a P=NP or P[Hyphen]NP proof can not look like. .... answer most of your questions: http://www.scottaaronson.com/papers/pnp.pdf. ... + + Set N equal to the multiplicative identity. Then NP = P. QED. ;-) ... http://stackoverflow.com/questions/900273/what-would-a-pnp-proof-be-like-hyp othetically + + C:/Documents and Settings/Lew Gordeew/My Documents/Costa- Doria.dviQ.E.D.. 2. Since ? s. Á (Con (S) Á.be ?), we have S + ? ... To confirm 1, we argue as in the proof of theorem (regardless of the assumption. S ? Á[Thorn]?). ... Da Costa - Doria's paper, suppose for a moment that P = NP is provable in ... http://www-ls.informatik.uni-tuebingen.de/gordeew/publikationen/Costa-Doria- 1.pdf + + Math Forum DiscussionsQed. Theorem Absurd: P -> ~ P -> Q. Proof. intros p np; destruct np; .... This is the html version of the file http://www.npp.co.in/ClayProblem.pdf. ... http://mathforum.org/kb/message.jspa%3FmessageID%3D6738429%26tstart%3D135 + + An essay about research on sparse NP complete setsIn retrospect it is surprising how elegant and simple P. Berman's proof is ..... QED. Corollary 13: If NP is sparse reducible, then P = NP. . Discussion ... http://www.springerlink.com/index/dt0n761uk81x604h.pdf + + sPhysics Letters B : Gauge invariance of massless QED - Published ...A simple general proof of gauge invariance in QED is given in the framework .... Kirk, Kljk, (ii) K, npi, Kmip, Ki,np, (iii) Ktmp, where l, m, p 2, i,j,k 2. ... http://linkinghub.elsevier.com/retrieve/pii/0370269394915458 Space bounds for infinitary computation Proof. Propositions 4 and 6 point of view of space constraints, it is easy to see that the .... Ralf Schindler, P = NP for infinite time Turing machines, Monats- ... http://www.illc.uva.nl/Publications/ResearchReports/PP-2006-27.text.pdf + + Microsoft PowerPoint + NPC3.ppt Proof complete. NP is then hard,. NP is and and. If. P. -. L. -. L. L. L. L. Á'. ÁåÁ'. Á.99. NP. Corollary hard. NP is. So. -. L. QED ... http://www.soe.ucsc.edu/classes/cmps201/Fall07/slides/NPC3sli.pdf + + 8.exists a polynomial p such that Ac is NP-complete. Since a p-simple problem is ... and, hence, A is NP-complete. p QED. For some problems, like MAX-CLIQUE and MIN-CHROMATIC ... PROOF. If .4 is rigid there must be an integer ic such ... http://siba2.unile.it/ese/issues/120/441/Quadmatn6 1979p8.pdf + + [I. N. R. I. J. e s us C h ri st, Kin g of the J. e. w. s]++++ === Subject: History of Math: Look for a proof of a lemma in Lagrange's famous first proof of group theory posting-account=7KEhAgkAAAA-WSVjXf1dvjorTihgujwc Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) Hi Math Experts, A reader asked me a question that I am not able to answer. Lagrange was looking at functions with n parameters. Now, clearly, there are n! permutations of these parameters possible. We can label them O1, O2,...,On! Let's assume that there are less than n! (we can do this since we are trying to show that the number of distinct return values divides n!). Then, there are at least two return values that are the same. Let's call these permutations O1,O2. So, we have f(O1)=f(O2) We can also assume that O3 has a different return value so that f(O1) ! = f(O3) Let P be a permutation on the orderings O1,O2,...,On! such that P(O1) =O3 Now, from this, it seems to me that Lagrange argues that f(P(O2)) = f (O3) regardless of which permutation is made. Am I understanding this correctly? If so, could you provide a proof that this is true? If not, could you help me understand what additional assumptions that Lagrange is making? -Larry === Subject: Re: History of Math: Look for a proof of a lemma in Lagrange's famous first proof of group theory > .... > hr/3#view > > A reader asked me a question that I am not able to answer.... I'm willing to have a look at this, but with no promises. :-) My photo-copy of Lagrange's Reflexions sur la Resolution Algebrique des Equations [with acute accents] is a _very_ thick pile of paper. To save time, could you give a reference (say, his own paragraph number) to the part which your reader asked about? Ken Pledger. === Subject: Re: History of Math: Look for a proof of a lemma in Lagrange's famous first proof of group theory posting-account=7KEhAgkAAAA-WSVjXf1dvjorTihgujwc Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > .... > hr/3#view > A reader asked me a question that I am not able to answer.... I'm willing to have a look at this, but with no promises.:-) My photo-copy of Lagrange's Reflexions sur la Resolution > Algebrique des Equations [with acute accents] is a very thick pile of > paper.To save time, could you give a reference (say, his own paragraph > number) to the part which your reader asked about? Ken Pledger. Hi Ken, My version of the proof comes from Jean-Pierre Tignol's excellent book: Galois' Theory of Algebraic Equations in Section 10.3: The First Results of Group Theory. He is citing Lagrange's Article 97. It is the first proof of group theory. Here's Tignol's words: Let f = f(x1,...,xn) be a rational fraction of n indeterminates. The number m of different values that f takes under the permutations of x1,...,xn is equal to the quotient of n! by the number of permutations which leave f invariant. m = n!/|I(f)| I hope that helps. If my knol is unclear on the subject, that's great feedback for me that I need to seriously rewrite it. -Larry === === Subject: Re: Small Sample Statistics Books? > Some years ago, 10-15, I had two statistics books that were quite > popular. One had a green cover and the other orange. I think the topic > was generally small sample sizes. I gave them to a college library > years, but now am trying to recall the authors and titles. Anyone know? > As I recall, one of the authors coined the computer term byte, or > something very similar. John W. Tukey. Exploratory Data Analysis. Addison-Wesley, Reading, MA. 1977. 1977. How time flies. That's one, but I'm sure there is another. Maybe co-authored? Mosteller? -- W. eWatson Obz Site: 39Á 15' 7 N, 121Á 2' 32 W, 2700 feet Web Page: === Subject: Re: Small Sample Statistics Books? > Some years ago, 10-15, I had two statistics books .... but now am trying to > recall the authors and titles.... > John W. Tukey. Exploratory Data Analysis. Addison-Wesley, Reading, MA. 1977. > > 1977. How time flies. That's one, but I'm sure there is another.... Have you tried asking in the group ? Ken Pledger. === === Subject: Cubic polynomial conjectures (?) I don't know how much of this is already known (probably most of it), and it is mainly conjectural, but here goes anyway. Suppose that P = x^3 + a*x^2 + b*x + c is an irreducible cubic polynomial with coefficients a,b,c in Z. Let D be its discriminant: D = -4*b^3 + a^2*b^2 + 18*c*a*b + -4*c*a^3 - 27*c^2 and define E as E = a^2 - 3*b Define Q as the sextic polynomial Q = x^6 - 2*E*x^4 + E^2*x^2 - D = (x^3 - E*x)^2 - D I have some conjectures about the relationship between P and Q. 1. If the Galois group of P is of order 6, then Q is irreducible and generates the splitting field of P. 2. If the Galois group of P is of order 3, then a) D is a square in Z, say D = d^2 with d in Z; b) Q is the product of two cubic factors, namely Q = (x^3 - E*x + d)*(x^3 - E*x - d) I can see how some of this implies other parts, e.g. if D = d^2 with d in Z, then Q indeed factors as shown above. I've verified these conjectures for all values of the coefficients of P in the range -10..10. === Subject: Re: Cubic polynomial conjectures (?) > I don't know how much of this is already known (probably most of it), > and it is mainly conjectural, but here goes anyway. > > Suppose that P = x^3 + a*x^2 + b*x + c is an irreducible cubic > polynomial with coefficients a,b,c in Z. Let D be its discriminant: > > D = -4*b^3 + a^2*b^2 + 18*c*a*b + -4*c*a^3 - 27*c^2 > > and define E as > > E = a^2 - 3*b > > Define Q as the sextic polynomial > > Q = x^6 - 2*E*x^4 + E^2*x^2 - D = (x^3 - E*x)^2 - D > > I have some conjectures about the relationship between P and Q. > > 1. If the Galois group of P is of order 6, then Q is irreducible and > generates the splitting field of P. > > 2. If the Galois group of P is of order 3, then a) D is a square in Z, > say D = d^2 with d in Z; b) Q is the product of two cubic factors, > namely > > Q = (x^3 - E*x + d)*(x^3 - E*x - d) > > I can see how some of this implies other parts, e.g. if D = d^2 with d > in Z, then Q indeed factors as shown above. > > I've verified these conjectures for all values of the coefficients of > P in the range -10..10. 2 is certainly true, I think any good text on Galois Theory should note that the Galois group of P is of order 3 if and only if the discriminant is a square. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Cubic polynomial conjectures (?) posting-account=oX5FAgkAAAD2tWLfJT1nYaPn9C8kADIl Gecko/20090423 Firefox/3.5b4,gzip(gfe),gzip(gfe) On Jun 10, 9:46pm, Gerry Myerson I don't know how much of this is already known (probably most of it), > and it is mainly conjectural, but here goes anyway. > Suppose that P = x^3 + a*x^2 + b*x + c is an irreducible cubic > polynomial with coefficients a,b,c in Z. Let D be its discriminant: > D = -4*b^3 + a^2*b^2 + 18*c*a*b + -4*c*a^3 - 27*c^2 > and define E as > E = a^2 - 3*b > Define Q as the sextic polynomial > Q = x^6 - 2*E*x^4 + E^2*x^2 - D = (x^3 - E*x)^2 - D > I have some conjectures about the relationship between P and Q. > 1. If the Galois group of P is of order 6, then Q is irreducible and > generates the splitting field of P. > 2. If the Galois group of P is of order 3, then a) D is a square in Z, > say D = d^2 with d in Z; b) Q is the product of two cubic factors, > namely > Q = (x^3 - E*x + d)*(x^3 - E*x - d) > I can see how some of this implies other parts, e.g. if D = d^2 with d > in Z, then Q indeed factors as shown above. > I've verified these conjectures for all values of the coefficients of > P in the range -10..10. 2 is certainlytrue, I think any good text on Galois Theory should note > that the Galois group of P is of order 3 if and only if the discriminant > is a square. -- > Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email) this, but it is good to know from a reliable source. Actually, I'm more interested in the first conjecture. There are an infinite number of polynomials that generate the splitting field of P, but I don't recall ever seeing a systematic treatment of how to generate the splitting field, much less a parametric form which generates the splitting field in almost all cases. I suspect (perhaps wrongly) that Q is a covariant of P, which would mean that Q could be expressed as a rational function of the basic invariants and covariants of P. Am I looking in the right direction here? FWIW, I'm trying to derive a polynomial similar to Q for quartic polynomials. I expect that it will take me at least a few days to do the number crunching. I could add a further conjecture. For any irreducible polynomial P (of degree n, say), there is a polynomial Q of degree n!, whose coefficients are rational functions of the coefficients of P, such that whenever the splitting field of P is of order n!, Q is irreducible and generates the splitting field of P. Just as a sanity check, is it true that the order of the splitting field of an irreducible quartic is either 24, 8 or 4? I haven't come across any of order 12 yet. === Subject: Re: Cubic polynomial conjectures (?) > Just as a sanity check, is it true that the order of the splitting > field of an irreducible quartic is either 24, 8 or 4? I haven't come > across any of order 12 yet. It is conjectured that every finite group is G(K/Q) for some extension K of the rationals. I'm sure there are A_4 quartics. Again I suggest you find a good text on Field Theory (or Galois Theory, or perhaps even Algenraic Number Theory), and you may find all your questions answered. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Cubic polynomial conjectures (?) posting-account=oX5FAgkAAAD2tWLfJT1nYaPn9C8kADIl Gecko/20090423 Firefox/3.5b4,gzip(gfe),gzip(gfe) On Jun 11, 2:17am, Gerry Myerson Just as a sanity check, is it true that the order of the splitting > field of an irreducible quartic is either 24, 8 or 4? I haven't come > across any of order 12 yet. It is conjectured that every finite group is G(K/Q) for some extension K > of the rationals. I'm sure there are A 4 quartics. Again I suggest you > find a good text on Field Theory (or Galois Theory, or perhaps even > Algenraic Number Theory), and you may find all your questions > answered. -- > Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email) I actually read them, but my memory has unfortunately only a finite capacity. I suppose I could break down and use them for the purpose for which they were intended. Actually, the nonexistence of A4 quartics would not disprove the conjecture, since A4 is also a subgroup of Sn for all n > 4. === Subject: Re: Cubic polynomial conjectures (?) posting-account=Z3AipgkAAABkoMfyNwddSxsYhXHi5CDt MathPlayer 2.10d; .NET CLR 1.1.4322; PeoplePal 3.0),gzip(gfe),gzip(gfe) > On Jun 10, 9:46pm, Gerry Myerson I don't know how much of this is already known (probably most of it), > and it is mainly conjectural, but here goes anyway. > Suppose that P = x^3 + a*x^2 + b*x + c is an irreducible cubic > polynomial with coefficients a,b,c in Z. Let D be its discriminant: > D = -4*b^3 + a^2*b^2 + 18*c*a*b + -4*c*a^3 - 27*c^2 > and define E as > E = a^2 - 3*b > Define Q as the sextic polynomial > Q = x^6 - 2*E*x^4 + E^2*x^2 - D = (x^3 - E*x)^2 - D > I have some conjectures about the relationship between P and Q. > 1. If the Galois group of P is of order 6, then Q is irreducible and > generates the splitting field of P. > 2. If the Galois group of P is of order 3, then a) D is a square in Z, > say D = d^2 with d in Z; b) Q is the product of two cubic factors, > namely > Q = (x^3 - E*x + d)*(x^3 - E*x - d) > I can see how some of this implies other parts, e.g. if D = d^2 with d > in Z, then Q indeed factors as shown above. > I've verified these conjectures for all values of the coefficients of > P in the range -10..10. > 2 is certainlytrue, I think any good text on Galois Theory should note > that the Galois group of P is of order 3 if and only if the discriminant > is a square. > -- > Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email) this, but it is good to know from a reliable source. Actually, I'm more interested in the first conjecture. There are an > infinite number of polynomials that generate the splitting field of P, > but I don't recall ever seeing a systematic treatment of how to > generate the splitting field, much less a parametric form which > generates the splitting field in almost all cases. I suspect (perhaps > wrongly) that Q is a covariant of P, which would mean that Q could be > expressed as a rational function of the basic invariants and > covariants of P. Am I looking in the right direction here? FWIW, I'm trying to derive a polynomial similar to Q for quartic > polynomials. I expect that it will take me at least a few days to do > the number crunching. I could add a further conjecture. For any > irreducible polynomial P (of degree n, say), there is a polynomial Q > of degree n!, whose coefficients are rational functions of the > coefficients of P, such that whenever the splitting field of P is of > order n!, Q is irreducible and generates the splitting field of P. Just as a sanity check, is it true that the order of the splitting > field of an irreducible quartic is either 24, 8 or 4? I haven't come > across any of order 12 yet. > Interesting questions, Bill. Pardon my ignorance, but what do you mean by Q possibly being a covariant of P, and in paricular what do you meann by the covariants and invariants of P? I understand that the discriminant is an invariant of P, although I am not sure what that means since I don't know what it is invariant under, but I have no idea what covariants are in this context, and I am curious. What you are doing reminds me a bit of the sort of thing that was done by Lagrange and contemporaries in their quest for a quintic formula, and there are things called resolvents that might be relevant here. Very old books on the Theory of Equations probably cover this sort of material, but I had a hard time finding them some years ago and came up almost empty. I can't imagine why a quartic couldn't have a splitting field of order twelve, but I don't have any real idea why or why not. I'd also like to know what the answer to your questions turns out to be. Achava === === Subject: A dense sequence? posting-account=76gTjAoAAACFBBxW0JkX-LuotQPzt612 Hi-Speed Internet; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) Is the following sequence dense? x_1 = x_2 = sqrt(2) mod 1 x_n = ( x_{n-1} + x_{n-2} ) mod 1 for n > 2 Lew Baxter === Subject: Re: A dense sequence? > Is the following sequence dense? > > x_1 = x_2 = sqrt(2) mod 1 > x_n = ( x_{n-1} + x_{n-2} ) mod 1 for n > 2 I suppose you mean dense in [0,1) and mod 1 means fractional part. If F_n is the n'th Fibonacci number, x_n = (sqrt(2) F_n) mod 1. Now F_n = (phi^n - (-1/phi)^n)/sqrt(5), where phi = (1+sqrt(5))/2. Since (-1/phi)^n -> 0 as n -> infinity, the question is equivalent to whether the sequence (sqrt(2/5) phi^n) mod 1 is dense in [0,1). IIRC this sort of problem is hard in general, but the answer is almost certainly yes (e.g. the set of numbers x > 0 such that (x phi^n) mod 1 is not dense in [0,1) has measure 0). -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === === Subject: Re: Closed sublattice > > >http://dml.cz/bitstream/handle/10338.dmlcz/101151/CzechMathJ_23-1973-... > is met the term closed sublattice. > What closed sublattice means? Is it lattice closed regarding all > (including infinite) meets and joins? > That's how it's defined here (one of the results of googling > for ): > A lattice L is complete for joins (complete for meets) if > every nonempty subset of L has a join (meet) and it is complete > if it is complete both for joins and for meets. > A sublattice (Boolean subalgebra) K of a complete lattice > (Boolean algebra) L is a closed sublattice (Boolean subalgebra) of > L if K contains the meet and the join of any its nonempty subset. > Geesh.Why not the simple, clear, unambigous complete sublattice? > Because that is not what he meant. > The usual definition of a lattice (L, /, / ) does only include > binary joins and meets. Therefore complete lattice usually means > a lattice which is complete. Consequently complete sublattice > would then mean a sublattice which is complete. > For example, for suitable topologies on a set X, the complete sublattice > Open(X) of P(X) may not be closed under infinite meets. > > No, complete sublattice of a lattice L does not mean a sublattice > of L which is complete. In the customary math tradition it means a > sublattice which is not only complete but also closed under arbitrary > meets and joins where meets and joins are understood in the sense of > operations of lattice L. > > So its customary use. It could be better or worse, but that is what > the terms mean. O.k., from the limited search I did, I think you are right. I have no real preference of one over the other, it depends on whether you think of subobjects in Lat od Clat. Anyway, complete sublattice is nevertheless not meant by the author because the sublattice K is only closed under nonempty meets and joins in L. So for example the top and bottom elements of K may not be the same as those of L. Indeed, the expression closed sublattice seems to be in use for this situation. -- Marc === Subject: Re: Closed sublattice > > > >http://dml.cz/bitstream/handle/10338.dmlcz/101151/CzechMathJ_23-1973-1_14.p d f >is met the term closed sublattice. >What closed sublattice means? Is it lattice closed regarding all >(including infinite) meets and joins? > > That's how it's defined here (one of the results of googling > for ): > > > ------------- > A lattice L is complete for joins (complete for meets) if > every nonempty subset of L has a join (meet) and it is complete > if it is complete both for joins and for meets. __________ this is the original wording(*) > > A sublattice (Boolean subalgebra) K of a complete lattice > (Boolean algebra) L is a closed sublattice (Boolean subalgebra) of > L if K contains the meet and the join of any its nonempty subset. > > > (But the first of these definitions seems rather odd, in view of > the very well-known theorem ... [pauses to refresh old memory] ... > that if every subset including the empty set has a sup, then every > subset including the empty set has an inf; also, doesn't complete > lattice usually imply the existence of top and bottom elements?) >But in that definition he only considers e.g. joins >of _nonempty_ subsets, so the well known theorem does not apply here. >If you go through the proof, you will find that in order to obtain >an inf(S) for a subset S you need sup(lower(S)) where lower(S) is >the set of lower bounds of S. If lower(S) happens to be empty, you >are stuck. For an obvious example, take the (N , > ). >No argument from me there. I didn't say it was actually wrong, just >that it seemed odd to me. To dispel this sense of oddness, does >anyone know any useful or interesting examples of lattices which are >complete in this author's sense, but have neither top nor bottom >element? I haven't studied (I've only heard of) Dedekind-MacNeille >completions, so perhaps I shouldn't ask, but what would the Dedekind- >MacNeille completion of such a lattice look like? > > There's something screwy here. (Sorry if it's only in my mind!) > > We can ignore the trivial case where L is the empty set. Given that > L is non-empty, the author's definition implies that sup L and inf L > both exist (in L!), i.e. L has both a top and a bottom element. So, > his definition /is/ equivalent to the usual one, except for allowing > the empty set as an example; and, if the empty case is excluded, his > definition is redundant, in the way I indicated. You probably stumbled about the definitions style with the brackets. In the above text (*) the author actually defines three different conditions: (a) complete for joins: every _nonempty_ subset has a join (b) complete for meets: every _nonempty_ subset has a meet (c) complete: both (a) and (b) are satisfied. Your observation that (c) is equivalent to the usual completeness is correct. This is why the author indeed calls such lattices complete. If he had dropped the nonempty restriction in (a) or (b) then the definition would be redundant. But that is not the case here. -- Marc === Subject: Isomorphic ultrafilters - open problems Cc: nhindman@aol.com posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/2009050804 Mandriva/1.9.0.10-0.1mdv2009.1 (2009.1) Firefox/3.0.10,gzip(gfe),gzip(gfe) Hello mathematicians, I have defined the relation being isomorphic for two filters and proved that this relation is an equivalence relation as well as some other simple theorems. I published it at this blog entry: http://portonmath.wordpress.com/2009/06/10/isomorphic-filters/ For me it remains an open problems: 1. whether any two non-trivial ultrafilters are isomorphic; 2. whether any two non-trivial filters on the set of natural numbers are isomorphic. Maybe you know the answers for my open problems? You may post comments for the above mentioned blog post. Don't hesitate ask me questions about this. === Subject: Re: Isomorphic ultrafilters - open problems posting-account=hhC2JwoAAAAQt9ZcdRPKAFNCTwZjbe1M 1.0.3705; .NET CLR 2.0.50727; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Hello mathematicians, I have defined the relation being isomorphic for two filters and > proved that this relation is an equivalence relation as well as some > other simple theorems. I published it at this blog entry:http://portonmath.wordpress.com/2009/06/10/isomorphic-filters/ For me it remains an open problems: 1. whether any two non-trivial > ultrafilters are isomorphic; 2. whether any two non-trivial filters on > the set of natural numbers are isomorphic. Maybe you know the answers for my open problems? No, there are many nonisomorphic nontrivial ultrafilters on the set N of natural numbers. There are 2^(2^{aleph 0}) ultrafilters on N. Each isomorphism class contains at most 2^(aleph 0) ultrafilters. Therefore the number of isomorphism classes is 2^(2^{aleph 0}). One of these isomorphism classes consists of all the trivial (i.e. principal) ultrafilter on N. === Subject: Re: Isomorphic ultrafilters - open problems posting-account=5t-ZfgkAAACU7ydoC4Cq-xVNAFsq481f Gecko/2009050804 Mandriva/1.9.0.10-0.1mdv2009.1 (2009.1) Firefox/3.0.10,gzip(gfe),gzip(gfe) > I have defined the relation being isomorphic for two filters and > proved that this relation is an equivalence relation as well as some > other simple theorems. > I published it at this blog entry:http://portonmath.wordpress.com/2009/06/10/isomorphic-filters/ > For me it remains an open problems: 1. whether any two non-trivial > ultrafilters are isomorphic; 2. whether any two non-trivial filters on > the set of natural numbers are isomorphic. > Maybe you know the answers for my open problems? No, there are many nonisomorphic nontrivial ultrafilters on the set N > of natural numbers. There are 2^(2^{aleph 0}) ultrafilters on N. Each isomorphism class > contains at most 2^(aleph 0) ultrafilters. Therefore the number of > isomorphism classes is 2^(2^{aleph 0}). One of these isomorphism > classes consists of all the trivial (i.e. principal) ultrafilter on N. Why each isomorphism class contains at most 2^(aleph 0) ultrafilters? I see only that the number of ultrafilters in a class is not more than the number of bijection between subsets of N. What is this number? === Subject: Re: Isomorphic ultrafilters - open problems posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/2009042523 Ubuntu/9.04 (jaunty) Firefox/3.0.10,gzip(gfe),gzip(gfe) > Hello mathematicians, I have defined the relation being isomorphic for two filters and > proved that this relation is an equivalence relation as well as some > other simple theorems. I published it at this blog entry:http://portonmath.wordpress.com/2009/06/10/isomorphic-filters/ For me it remains an open problems: 1. whether any two non-trivial > ultrafilters are isomorphic; 2. whether any two non-trivial filters on > the set of natural numbers are isomorphic. Maybe you know the answers for my open problems? You may post comments for the above mentioned blog post. Don't hesitate ask me questions about this. If I got this right, given two non-trivial ultrafilters, you want to find subsets in these filters such that the restriction of the filters to these subsets are directly isomorphic (i.e. via a bijection of these underlying sets). Let N be the set of the naturals, R the set of the reals. Let a_0 = { x c N | #(Nx) < oo } and refine this to an ultrafilter a (using Axiom of Choice, of course). Let b_0 = { x c R | #(Rx) <= #N } and refine this to an ultrafilter b. If a and b were directly isomorphic, then there should exist sets A e a, B e b and a bijection f:A->B such that ... But B is uncountable (else RB e b_o c b) hence no such bijection exists. Hagen === Subject: Re: smallest angle in Geometry 0d000...0001 and why lines have FrontView with BackView #578 new book 2nd edition: New True Mathematics > Line intersection in Euclidean geometry would not work in the parallel > postulate > without a concept of infinity in the middle. More later; I have to be a little more careful what I say here. > true, most of sounded like crap. === Subject: Re: smallest angle in Geometry 0d000...0001 and why lines have FrontView with BackView #578 new book 2nd edition: New True Mathematics posting-account=5ApcPgoAAABKcgEyKsQmJVb3Rz63IGGL .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; WWTClient2),gzip(gfe),gzip(gfe) > Line intersection in Euclidean geometry would not work in the parallel > postulate > without a concept of infinity in the middle. More later; I have to be a little more careful what I say here. true, most of sounded like crap. It is infinitely close to being parralel. Mitch Raemsch === Subject: Factoring a large number? posting-account=PSzFRAoAAAARszS8zeFmxtqyivK9-1_f InfoPath.2; .NET CLR 2.0.50727; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) Hi all, could someone try to factor the following 1351 digit large number below? I'm not 100% sure but there should be just 2 prime factors. 1725073237205857947941058976975667133967120394002 9684505992358918550555677139461746853893933753466 3324516557318514017702376369458932632547335906243 4082891372765217842651075223855023307360880491775 3856442298683876128462251451917561739289186463033 4528305924022898503449773725676053169057058336217 3998344551379354992523133080009562978285271680807 7857202153188090923341503168045855143060178430084 8842399422446292885619604650271635139199715874316 6249545774204554432543412070289958510693650610447 6305663902358695943110225660265027554844049187469 9514113137552295755717908429386036094056921868492 9588333233322200116414929913714582094242085448843 9524458937344392220207261992727216424252007246532 5032666043798779148759791622022514866469373306815 9035038106710882360772044728893007882282428959869 5038864028210682514850556736397455671006628590617 1292571468948550229615122349988593207305772126201 5813505142130348354654341355595257604176115566233 7760677658198665401229108701128158628589993355922 1105711458385211338597647800333671798311700249370 4131890750911963299543818564505501830749682740718 2020435920283859037411266417359917460923787834769 9645078197551794635334020841500476920521122586462 1213415446332729016403692831468979349307996181633 4098404475482625626466031528939783050395972544355 2178677174089467788658949249 Gerry === Subject: Re: Factoring a large number? posting-account=DLD3MQkAAACJxulKt9xouw3DPpXKssCI Trident/4.0; GTB6; SLCC1; .NET CLR 2.0.50727; .NET CLR 1.1.4322; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30729; OfficeLiveConnector.1.3; OfficeLivePatch.0.0),gzip(gfe),gzip(gfe) > Hi all, could someone try to factor the following 1351 digit large number > below? > I'm not 100% sure but there should be just 2 prime factors. 1725073237205857947941058976975667133967120394002 > 9684505992358918550555677139461746853893933753466 > 3324516557318514017702376369458932632547335906243 > 4082891372765217842651075223855023307360880491775 > 3856442298683876128462251451917561739289186463033 > 4528305924022898503449773725676053169057058336217 > 3998344551379354992523133080009562978285271680807 > 7857202153188090923341503168045855143060178430084 > 8842399422446292885619604650271635139199715874316 > 6249545774204554432543412070289958510693650610447 > 6305663902358695943110225660265027554844049187469 > 9514113137552295755717908429386036094056921868492 > 9588333233322200116414929913714582094242085448843 > 9524458937344392220207261992727216424252007246532 > 5032666043798779148759791622022514866469373306815 > 9035038106710882360772044728893007882282428959869 > 5038864028210682514850556736397455671006628590617 > 1292571468948550229615122349988593207305772126201 > 5813505142130348354654341355595257604176115566233 > 7760677658198665401229108701128158628589993355922 > 1105711458385211338597647800333671798311700249370 > 4131890750911963299543818564505501830749682740718 > 2020435920283859037411266417359917460923787834769 > 9645078197551794635334020841500476920521122586462 > 1213415446332729016403692831468979349307996181633 > 4098404475482625626466031528939783050395972544355 > 2178677174089467788658949249 http://en.wikipedia.org/wiki/RSA numbers To understand the difficulty of what you are asking. Just in case it turned out to be easy, I gave it a crack: C:qepdsub>factor 1725073237205857947941058976975667133967120394002968450599235891855055567713 9461746853893933753466332451655731851401770237636945893263254733590624340828 9 137276521784265107522385502 3307360880491775385644229868387612846225145191756173928918646303345283059240 2289850344977372567605316905705833621739983445513793549925231330800095629782 8 52716808077857202153188090923341503168045855143 0601784300848842399422446292885619604650271635139199715874316624954577420455 4432543412070289958510693650610447630566390235869594311022566026502755484404 9 18746995141131375522957557179084293860360940569 6815903503810671088236077204472889300788228242895986950388640282106825148505 5673639745567100662859061712925714689485502296151223499885932073057721262015 8 13505142130348354654341355595257604176115566233 7760677658198665401229108701128158628589993355922110571145838521133859764780 0333671798311700249370413189075091196329954381856450550183074968274071820204 3 59202838590374112664173599174609237878347699645 0781975517946353340208415004769205211225864621213415446332729016403692831468 9793493079961816334098404475482625626466031528939783050395972544355217867717 4 089467788658949249 first trying brute force division by small primes now trying 1000 iterations of brent's method now trying william's (p+1) method phase 1 - trying all primes less than 10000 phase 2 - trying last prime less than 1000000 now trying pollard's (p-1) method phase 1 - trying all primes less than 100000 phase 2 - trying last prime less than 5000000 now trying lenstra's method using 10 curves curve 1 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 curve 2 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 curve 3 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 curve 4 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 curve 5 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 curve 6 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 curve 7 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 curve 8 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 curve 9 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 curve 10 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 now trying 80 more curves curve 1 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 curve 2 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 curve 3 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 curve 4 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 curve 5 phase 1 - trying all primes less than 20000 phase 2 - trying last prime less than 2000000 curve 6 phase 1 - trying all primes less than 20000 prime= 7507 But I killed the process because this is clearly going to be a tough nut to crack. I guess if you got the entire intenet to help you you might solve it in a few hundred years, considering the exponential increase in compute power over time. What is it that is so interesting about this number that drives you to want to factor it? === Subject: Re: Factoring a large number? posting-account=PSzFRAoAAAARszS8zeFmxtqyivK9-1_f InfoPath.2; .NET CLR 2.0.50727; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Hi all, > could someone try to factor the following 1351 digit large number > below? > I'm not 100% sure but there should be just 2 prime factors. > 1725073237205857947941058976975667133967120394002 > 9684505992358918550555677139461746853893933753466 > 3324516557318514017702376369458932632547335906243 > 4082891372765217842651075223855023307360880491775 > 3856442298683876128462251451917561739289186463033 > 4528305924022898503449773725676053169057058336217 > 3998344551379354992523133080009562978285271680807 > 7857202153188090923341503168045855143060178430084 > 8842399422446292885619604650271635139199715874316 > 6249545774204554432543412070289958510693650610447 > 6305663902358695943110225660265027554844049187469 > 9514113137552295755717908429386036094056921868492 > 9588333233322200116414929913714582094242085448843 > 9524458937344392220207261992727216424252007246532 > 5032666043798779148759791622022514866469373306815 > 9035038106710882360772044728893007882282428959869 > 5038864028210682514850556736397455671006628590617 > 1292571468948550229615122349988593207305772126201 > 5813505142130348354654341355595257604176115566233 > 7760677658198665401229108701128158628589993355922 > 1105711458385211338597647800333671798311700249370 > 4131890750911963299543818564505501830749682740718 > 2020435920283859037411266417359917460923787834769 > 9645078197551794635334020841500476920521122586462 > 1213415446332729016403692831468979349307996181633 > 4098404475482625626466031528939783050395972544355 > 2178677174089467788658949249 > To understand the difficulty of what you are asking. Just in case it turned out to be easy, I gave it a crack: > C:qepdsub>factor > 172507323720585794794105897697566713396712039400296845059923589185505556771[ CapitalEth]39461746853893933753466332451655731851401770237636945893263254733 5 906243408[CapitalEth]289137276521784265107522385502 > 330736088049177538564422986838761284622514519175617392891864630334528305924[ CapitalEth]02289850344977372567605316905705833621739983445513793549925231330 8 000956297[CapitalEth]82852716808077857202153188090923341503168045855143 > 060178430084884239942244629288561960465027163513919971587431662495457742045[ CapitalEth]54432543412070289958510693650610447630566390235869594311022566026 5 027554844[CapitalEth]04918746995141131375522957557179084293860360940569 > 218684929588333233322200116414929913714582094242085448843934096840729708520[ CapitalEth]04131441062828446494075120593519524458937344392220207261992727216 4 242520072[CapitalEth]46532503266604379877914875979162202251486646937330 > 681590350381067108823607720447288930078822824289598695038864028210682514850[ CapitalEth]55673639745567100662859061712925714689485502296151223499885932073 0 577212620[CapitalEth]15813505142130348354654341355595257604176115566233 > 776067765819866540122910870112815862858999335592211057114583852113385976478[ CapitalEth]00333671798311700249370413189075091196329954381856450550183074968 2 740718202[CapitalEth]04359202838590374112664173599174609237878347699645 > 078197551794635334020841500476920521122586462121341544633272901640369283146[ CapitalEth]89793493079961816334098404475482625626466031528939783050395972544 3 552178677[CapitalEth]174089467788658949249 > first trying brute force division by small primes > now trying 1000 iterations of brent's method > now trying william's (p+1) method > phase 1 - trying all primes less than 10000 > phase 2 - trying last prime less than 1000000 > now trying pollard's (p-1) method > phase 1 - trying all primes less than 100000 > phase 2 - trying last prime less than 5000000 > now trying lenstra's method using 10 curves > curve 1 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 2 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 3 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 4 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 5 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 6 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 7 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 8 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 9 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve10 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > now trying 80 more curves > curve 1 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 2 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 3 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 4 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 5 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 6 phase 1 - trying all primes less than 20000 > prime= 7507 But I killed the process because this is clearly going to be a tough > nut to crack.I guess if you got the entire intenet to help you you > might solve it in a few hundred years, considering the exponential > increase in compute power over time.What is it that is so > interesting about this number that drives you to want to factor it?- Hide quoted text - - Show quoted text - The interesting thing is the way i found the two factors. They were not difficult to find. But i'm not 100% sure that their prime. The weird thing is that pari verifies them as primes very fast for their size. This is what i do not understand. === Subject: Re: Factoring a large number? > The interesting thing is the way i found the two factors. > They were not difficult to find. > But i'm not 100% sure that their prime. > > The weird thing is that pari verifies them as primes very fast for > their size. > This is what i do not understand. As usual, you're playing games by not telling us what's really going on here. If you didn't find it difficult to factor a number that large, then there must be something very special about that number, something you know but aren't telling us (either that, or you have found a general factorization technique that is a zillion times better than anything anyone else has, and you'll perhaps forgive me if I'm wary of adopting this hypothesis). If you told us where the number came from, and how you found the factors, someone might be able to tell you something useful about their primality. By the way, if pari says the two factors are primes, why are you not sure that they are primes? Does pari lie? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Factoring a large number? posting-account=oX5FAgkAAAD2tWLfJT1nYaPn9C8kADIl Gecko/20090423 Firefox/3.5b4,gzip(gfe),gzip(gfe) On Jun 10, 9:42pm, Gerry Myerson The interesting thing is the way i found the two factors. > They were not difficult to find. > But i'm not 100% sure that their prime. > The weird thing is that pari verifies them as primes very fast for > their size. > This is what i do not understand. As usual, you're playing games by not telling us what's really > going on here. If you didn't find it difficult to factor a number that large, > then there must be something very special about that number, > something you know but aren't telling us (either that, or you > have found a general factorization technique that is a zillion > times better than anything anyone else has, and you'll perhaps > forgive me if I'm wary of adopting this hypothesis). If you told us > where the number came from, and how you found the factors, > someone might be able to tell you something useful about > their primality. By the way, if pari says the two factors are primes, > why are you not sure that they are primes? Does pari lie? -- > Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email) Well, PARI doesn't lie, but it pays to read the documentation carefully. If you just use 'factor' to factor a number, then the results are possibly pseudoprimes. To execute an exhaustive primality proof (for which you may need a lot of patience), use 'isprime' on each factor reported by 'factor'. It's easy enough to set problems like this. Just choose two very large primes, multiply them, and challenge the world to find a factor of the product. You can then impress the world by saying that you have a way it happens), while implying that you know so much more than everyone else. === Subject: Re: Factoring a large number? posting-account=PSzFRAoAAAARszS8zeFmxtqyivK9-1_f InfoPath.2; .NET CLR 2.0.50727; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > On Jun 10, 9:42pm, Gerry Myerson The interesting thing is the way i found the two factors. > They were not difficult to find. > But i'm not 100% sure that their prime. > The weird thing is that pari verifies them as primes very fast for > their size. > This is what i do not understand. > As usual, you're playing games by not telling us what's really > going on here. > If you didn't find it difficult to factor a number that large, > then there must be something very special about that number, > something you know but aren't telling us (either that, or you > have found a general factorization technique that is a zillion > times better than anything anyone else has, and you'll perhaps > forgive me if I'm wary of adopting this hypothesis). If you told us > where the number came from, and how you found the factors, > someone might be able to tell you something useful about > their primality. > By the way, if pari says the two factors are primes, > why are you not sure that they are primes? Does pari lie? > -- > Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email) Well, PARI doesn't lie, but it pays to read the documentation > carefully. If you just use 'factor' to factor a number, then the > results are possibly pseudoprimes. To execute an exhaustive primality > proof (for which you may need a lot of patience), use 'isprime' on > each factor reported by 'factor'. It's easy enough to set problems like this. Just choose two very large > primes, multiply them, and challenge the world to find a factor of the > product. You can then impress the world by saying that you have a way > it happens), while implying that you know so much more than everyone > else. > - Show quoted text - Hi all, yes, i first used factor and then verified isprime. The factor function is of course a lot faster but doesn't like you pointed out guarantee primality. Below are the little brothers of the factors of the 1351 digit number. By little brother i mean that i found them the same way apart from an exponent. So the following two primes are related to the factors of my big number: 5042890386955258828304568718571144003236606386240 6623297783473220101378589798814816913 3553401282319243332858927451028634874801761668332 4779019334223330345846042636930209645986735557646 0144373050712042298931722867660244355384739692920 5514280448212686153985270892230328603442557352417 372877217906256399 One of their big brothers might though not be prime. Some make it sound as if it is easy to choose big primes. Is it really? I don't mind explaining how i got the numbers and i will when the time is right. Maybe someone like Phil or Bob can show how easy it can be done. Gerry === Subject: Re: Factoring a large number? posting-account=H1y7YgoAAADzGQwbcYaL9UvwttgsjOjp AppleWebKit/525.27.1 (KHTML, like Gecko) Version/3.2.1 Safari/525.27.1,gzip(gfe),gzip(gfe) > Some make it sound as if it is easy to choose big primes. > Is it really? It is not hard, if you have the right software and hardware, to test, say, a 500-digit number for primality. To find a 500-digit prime, take your favorite 500-digit number, N, and test N, N + 1, N + 2, etc., until you find one that is prime. Chances are you won't have to test too many (and there are obvious savings to be made by not testing any even numbers, etc.). > I don't mind explaining how i got the numbers and i will when the time > is right. The time was right when you started this thread. If you ask the group for help, why not level with us from the start? -- GM === Subject: Re: Factoring a large number? posting-account=DLD3MQkAAACJxulKt9xouw3DPpXKssCI Trident/4.0; GTB6; SLCC1; .NET CLR 2.0.50727; .NET CLR 1.1.4322; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30729; OfficeLiveConnector.1.3; OfficeLivePatch.0.0),gzip(gfe),gzip(gfe) > Hi all, > could someone try to factor the following 1351 digit large number > below? > I'm not 100% sure but there should be just 2 prime factors. > 1725073237205857947941058976975667133967120394002 > 9684505992358918550555677139461746853893933753466 > 3324516557318514017702376369458932632547335906243 > 4082891372765217842651075223855023307360880491775 > 3856442298683876128462251451917561739289186463033 > 4528305924022898503449773725676053169057058336217 > 3998344551379354992523133080009562978285271680807 > 7857202153188090923341503168045855143060178430084 > 8842399422446292885619604650271635139199715874316 > 6249545774204554432543412070289958510693650610447 > 6305663902358695943110225660265027554844049187469 > 9514113137552295755717908429386036094056921868492 > 9588333233322200116414929913714582094242085448843 > 9524458937344392220207261992727216424252007246532 > 5032666043798779148759791622022514866469373306815 > 9035038106710882360772044728893007882282428959869 > 5038864028210682514850556736397455671006628590617 > 1292571468948550229615122349988593207305772126201 > 5813505142130348354654341355595257604176115566233 > 7760677658198665401229108701128158628589993355922 > 1105711458385211338597647800333671798311700249370 > 4131890750911963299543818564505501830749682740718 > 2020435920283859037411266417359917460923787834769 > 9645078197551794635334020841500476920521122586462 > 1213415446332729016403692831468979349307996181633 > 4098404475482625626466031528939783050395972544355 > 2178677174089467788658949249 > To understand the difficulty of what you are asking. > Just in case it turned out to be easy, I gave it a crack: > C:qepdsub>factor > 172507323720585794794105897697566713396712039400296845059923589185505556771[ CapitalEth][CapitalEth]39461746853893933753466332451655731851401770237636945 893263254733590624340[CapitalEth]8[CapitalEth]28913727652178426510752238550 2 > 330736088049177538564422986838761284622514519175617392891864630334528305924[ CapitalEth][CapitalEth]02289850344977372567605316905705833621739983445513793 549925231330800095629[CapitalEth]7[CapitalEth]82852716808077857202153188090 923341503168045855143 > 060178430084884239942244629288561960465027163513919971587431662495457742045[ CapitalEth][CapitalEth]54432543412070289958510693650610447630566390235869594 311022566026502755484[CapitalEth]4[CapitalEth]04918746995141131375522957557 179084293860360940569 > 218684929588333233322200116414929913714582094242085448843934096840729708520[ CapitalEth][CapitalEth]04131441062828446494075120593519524458937344392220207 261992727216424252007[CapitalEth]2[CapitalEth]46532503266604379877914875979 162202251486646937330 > 681590350381067108823607720447288930078822824289598695038864028210682514850[ CapitalEth][CapitalEth]55673639745567100662859061712925714689485502296151223 499885932073057721262[CapitalEth]0[CapitalEth]15813505142130348354654341355 595257604176115566233 > 776067765819866540122910870112815862858999335592211057114583852113385976478[ CapitalEth][CapitalEth]00333671798311700249370413189075091196329954381856450 550183074968274071820[CapitalEth]2[CapitalEth]04359202838590374112664173599 174609237878347699645 > 078197551794635334020841500476920521122586462121341544633272901640369283146[ CapitalEth][CapitalEth]89793493079961816334098404475482625626466031528939783 050395972544355217867[CapitalEth]7[CapitalEth]174089467788658949249 > first trying brute force division by small primes > now trying 1000 iterations of brent's method > now trying william's (p+1) method > phase 1 - trying all primes less than 10000 > phase 2 - trying last prime less than 1000000 > now trying pollard's (p-1) method > phase 1 - trying all primes less than 100000 > phase 2 - trying last prime less than 5000000 > now trying lenstra's method using 10 curves > curve 1 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 2 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 3 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 4 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 5 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 6 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 7 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 8 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 9 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve10 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > now trying 80 more curves > curve 1 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 2 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 3 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 4 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 5 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 6 phase 1 - trying all primes less than 20000 > prime= 7507 > But I killed the process because this is clearly going to be a tough > nut to crack.I guess if you got the entire intenet to help you you > might solve it in a few hundred years, considering the exponential > increase in compute power over time.What is it that is so > interesting about this number that drives you to want to factor it?- Hide quoted text - > - Show quoted text - > The interesting thing is the way i found the two factors. > They were not difficult to find. > But i'm not 100% sure that their prime. The weird thing is that pari verifies them as primes very fast for > their size. > This is what i do not understand. Prime certification is a much easier problem than factoring for large numbers. I am not sure what Pari does, but typically, a program that wants to test for primality will do something like this: Run K instances of Miller-Rabin to quickly find out if it is not prime. After K unsuccessful iterations, the probability that it is not prime will be about 1/(4^k) if I recall correctly. Then, after Miller-Rabin fails to find that it is composite, it is fed to a provable solver like APR-CL which can prove primality for numbers up to a few thousand digits and offer a primality certificate that can be used to verify the result. === Subject: Re: Factoring a large number? posting-account=DLD3MQkAAACJxulKt9xouw3DPpXKssCI Trident/4.0; GTB6; SLCC1; .NET CLR 2.0.50727; .NET CLR 1.1.4322; .NET CLR 3.5.21022; .NET CLR 3.5.30729; .NET CLR 3.0.30729; OfficeLiveConnector.1.3; OfficeLivePatch.0.0),gzip(gfe),gzip(gfe) > Hi all, > could someone try to factor the following 1351 digit large number > below? > I'm not 100% sure but there should be just 2 prime factors. > 1725073237205857947941058976975667133967120394002 > 9684505992358918550555677139461746853893933753466 > 3324516557318514017702376369458932632547335906243 > 4082891372765217842651075223855023307360880491775 > 3856442298683876128462251451917561739289186463033 > 4528305924022898503449773725676053169057058336217 > 3998344551379354992523133080009562978285271680807 > 7857202153188090923341503168045855143060178430084 > 8842399422446292885619604650271635139199715874316 > 6249545774204554432543412070289958510693650610447 > 6305663902358695943110225660265027554844049187469 > 9514113137552295755717908429386036094056921868492 > 9588333233322200116414929913714582094242085448843 > 9524458937344392220207261992727216424252007246532 > 5032666043798779148759791622022514866469373306815 > 9035038106710882360772044728893007882282428959869 > 5038864028210682514850556736397455671006628590617 > 1292571468948550229615122349988593207305772126201 > 5813505142130348354654341355595257604176115566233 > 7760677658198665401229108701128158628589993355922 > 1105711458385211338597647800333671798311700249370 > 4131890750911963299543818564505501830749682740718 > 2020435920283859037411266417359917460923787834769 > 9645078197551794635334020841500476920521122586462 > 1213415446332729016403692831468979349307996181633 > 4098404475482625626466031528939783050395972544355 > 2178677174089467788658949249 > To understand the difficulty of what you are asking. > Just in case it turned out to be easy, I gave it a crack: > C:qepdsub>factor > 172507323720585794794105897697566713396712039400296845059923589185505556771[ CapitalEth][CapitalEth]39461746853893933753466332451655731851401770237636945 893263254733590624340[CapitalEth]8[CapitalEth]28913727652178426510752238550 2 > 330736088049177538564422986838761284622514519175617392891864630334528305924[ CapitalEth][CapitalEth]02289850344977372567605316905705833621739983445513793 549925231330800095629[CapitalEth]7[CapitalEth]82852716808077857202153188090 923341503168045855143 > 060178430084884239942244629288561960465027163513919971587431662495457742045[ CapitalEth][CapitalEth]54432543412070289958510693650610447630566390235869594 311022566026502755484[CapitalEth]4[CapitalEth]04918746995141131375522957557 179084293860360940569 > 218684929588333233322200116414929913714582094242085448843934096840729708520[ CapitalEth][CapitalEth]04131441062828446494075120593519524458937344392220207 261992727216424252007[CapitalEth]2[CapitalEth]46532503266604379877914875979 162202251486646937330 > 681590350381067108823607720447288930078822824289598695038864028210682514850[ CapitalEth][CapitalEth]55673639745567100662859061712925714689485502296151223 499885932073057721262[CapitalEth]0[CapitalEth]15813505142130348354654341355 595257604176115566233 > 776067765819866540122910870112815862858999335592211057114583852113385976478[ CapitalEth][CapitalEth]00333671798311700249370413189075091196329954381856450 550183074968274071820[CapitalEth]2[CapitalEth]04359202838590374112664173599 174609237878347699645 > 078197551794635334020841500476920521122586462121341544633272901640369283146[ CapitalEth][CapitalEth]89793493079961816334098404475482625626466031528939783 050395972544355217867[CapitalEth]7[CapitalEth]174089467788658949249 > first trying brute force division by small primes > now trying 1000 iterations of brent's method > now trying william's (p+1) method > phase 1 - trying all primes less than 10000 > phase 2 - trying last prime less than 1000000 > now trying pollard's (p-1) method > phase 1 - trying all primes less than 100000 > phase 2 - trying last prime less than 5000000 > now trying lenstra's method using 10 curves > curve 1 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 2 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 3 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 4 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 5 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 6 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 7 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 8 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 9 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve10 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > now trying 80 more curves > curve 1 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 2 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 3 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 4 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 5 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 6 phase 1 - trying all primes less than 20000 > prime= 7507 > But I killed the process because this is clearly going to be a tough > nut to crack.I guess if you got the entire intenet to help you you > might solve it in a few hundred years, considering the exponential > increase in compute power over time.What is it that is so > interesting about this number that drives you to want to factor it?- Hide quoted text - > - Show quoted text - > The interesting thing is the way i found the two factors. What is the way that you found them then? > They were not difficult to find. > But i'm not 100% sure that their prime. The weird thing is that pari verifies them as primes very fast for > their size. > This is what i do not understand.- Hide quoted text - - Show quoted text - === Subject: Re: Factoring a large number? posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2009042523 Ubuntu/8.10 (intrepid) Firefox/3.0.10,gzip(gfe),gzip(gfe) > Hi all, > could someone try to factor the following 1351 digit large number > below? > I'm not 100% sure but there should be just 2 prime factors. > 1725073237205857947941058976975667133967120394002 > 9684505992358918550555677139461746853893933753466 > 3324516557318514017702376369458932632547335906243 > 4082891372765217842651075223855023307360880491775 > 3856442298683876128462251451917561739289186463033 > 4528305924022898503449773725676053169057058336217 > 3998344551379354992523133080009562978285271680807 > 7857202153188090923341503168045855143060178430084 > 8842399422446292885619604650271635139199715874316 > 6249545774204554432543412070289958510693650610447 > 6305663902358695943110225660265027554844049187469 > 9514113137552295755717908429386036094056921868492 > 9588333233322200116414929913714582094242085448843 > 9524458937344392220207261992727216424252007246532 > 5032666043798779148759791622022514866469373306815 > 9035038106710882360772044728893007882282428959869 > 5038864028210682514850556736397455671006628590617 > 1292571468948550229615122349988593207305772126201 > 5813505142130348354654341355595257604176115566233 > 7760677658198665401229108701128158628589993355922 > 1105711458385211338597647800333671798311700249370 > 4131890750911963299543818564505501830749682740718 > 2020435920283859037411266417359917460923787834769 > 9645078197551794635334020841500476920521122586462 > 1213415446332729016403692831468979349307996181633 > 4098404475482625626466031528939783050395972544355 > 2178677174089467788658949249 > To understand the difficulty of what you are asking. > Just in case it turned out to be easy, I gave it a crack: > C:qepdsub>factor > 172507323720585794794105897697566713396712039400296845059923589185505556771[ CapitalEth]39461746853893933753466332451655731851401770237636945893263254733 5 906243408[CapitalEth]289137276521784265107522385502 > 330736088049177538564422986838761284622514519175617392891864630334528305924[ CapitalEth]02289850344977372567605316905705833621739983445513793549925231330 8 000956297[CapitalEth]82852716808077857202153188090923341503168045855143 > 060178430084884239942244629288561960465027163513919971587431662495457742045[ CapitalEth]54432543412070289958510693650610447630566390235869594311022566026 5 027554844[CapitalEth]04918746995141131375522957557179084293860360940569 > 218684929588333233322200116414929913714582094242085448843934096840729708520[ CapitalEth]04131441062828446494075120593519524458937344392220207261992727216 4 242520072[CapitalEth]46532503266604379877914875979162202251486646937330 > 681590350381067108823607720447288930078822824289598695038864028210682514850[ CapitalEth]55673639745567100662859061712925714689485502296151223499885932073 0 577212620[CapitalEth]15813505142130348354654341355595257604176115566233 > 776067765819866540122910870112815862858999335592211057114583852113385976478[ CapitalEth]00333671798311700249370413189075091196329954381856450550183074968 2 740718202[CapitalEth]04359202838590374112664173599174609237878347699645 > 078197551794635334020841500476920521122586462121341544633272901640369283146[ CapitalEth]89793493079961816334098404475482625626466031528939783050395972544 3 552178677[CapitalEth]174089467788658949249 > first trying brute force division by small primes > now trying 1000 iterations of brent's method > now trying william's (p+1) method > phase 1 - trying all primes less than 10000 > phase 2 - trying last prime less than 1000000 > now trying pollard's (p-1) method > phase 1 - trying all primes less than 100000 > phase 2 - trying last prime less than 5000000 > now trying lenstra's method using 10 curves > curve 1 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 2 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 3 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 4 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 5 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 6 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 7 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 8 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 9 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve10 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > now trying 80 more curves > curve 1 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 2 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 3 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 4 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 5 phase 1 - trying all primes less than 20000 > phase 2 - trying last prime less than 2000000 > curve 6 phase 1 - trying all primes less than 20000 > prime= 7507 > But I killed the process because this is clearly going to be a tough > nut to crack.I guess if you got the entire intenet to help you you > might solve it in a few hundred years, considering the exponential > increase in compute power over time.What is it that is so > interesting about this number that drives you to want to factor it?- Hide quoted text - > - Show quoted text - > The interesting thing is the way i found the two factors. > They were not difficult to find. > But i'm not 100% sure that their prime. The weird thing is that pari verifies them as primes very fast for > their size. > This is what i do not understand. It is known to be fairly easy to test for primality, i.e. it is in P (a deterministic polynomial time algorithm exists). The problem of factoring arbitrary composite integers is not known to be in P and may very well turn out to be (as many suspect) computationally hard. Hence the interest in cryptographic methods of public key encryption that are closely related to factoring of large numbers, e.g. the RSA algorithm using a product of exactly two primes. === Subject: Re: Factoring a large number? > It is known to be fairly easy to test for primality, > i.e. it is in P (a deterministic polynomial time > algorithm exists). That something is in P does not mean it's fairly easy -- a running time of O(n^(2^2^2^2^2^2^2^2^2)), say, hardly makes for easy and timely calculation. Are the polynomial time (provably) deterministic algorithms fast enough for practical use? I don't know anything about this field, but I would presume probabilistic tests are still used as the norm... -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Factoring a large number? <871vprqxsy.fsf@alatheia.truth.invalid> posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2009042523 Ubuntu/8.10 (intrepid) Firefox/3.0.10,gzip(gfe),gzip(gfe) > It is known to be fairly easy to test for primality, > i.e. it is in P (a deterministic polynomial time > algorithm exists). That something is in P does not mean it's fairly easy -- a running > time of O(n^(2^2^2^2^2^2^2^2^2)), say, hardly makes for easy and timely > calculation. Are the polynomial time (provably) deterministic algorithms > fast enough for practical use? I don't know anything about this field, > but I would presume probabilistic tests are still used as the norm... -- > Aatu Koskensilta (aatu.koskensi...@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen >- Ludwig Wittgenstein, Tractatus Logico-Philosophicus The point I was trying to make is that a quick test for primality doesn't get us quick factors. But you are right that polynomial time could consist of a quite impractical running time. The Miller-Rabin test (strong pseudoprimality for varying bases) is probabilistic, but it becomes deterministic under the assumption of the Generalized Riemann Hypothesis. In the latter case it suffices to use all the bases (coprime to n) between 2 and 2 (log n)^2. Each strong pseudoprime test itself is naively O((log n)^3), which can be pushed down to O((log n)^(2+eps)) using fast multiplication. So the complexity is either O(x^5) or O(x^(4+eps)), depending on the multiplication algorithm, where x = log n is proportional to the bits defining n (the size of the data). Here the constant of proportionality is modest enough for practical application. In that sense, yes, the probabilistic algorithm is still used. More at Wikipedia: [Miller-Rabin primality test -- Wikipedia] http://en.wikipedia.org/wiki/Miller-Rabin primality test The AKS (Agrawal-Kayal-Saxena) primality test was originally O((log n)^(12+eps)) but was improved to O((log n)^6) complexity; thus a deterministic polynomial time algorithm without assuming GRH. But the constant of proportionality is quite large, in part because one uses polynomial arithmetic, not just the simple integer arithmetic of the strong pseudoprime calculation. [AKS primality test -- Wikipedia] http://en.wikipedia.org/wiki/AKS primality test === Subject: Re: Factoring a large number? <871vprqxsy.fsf@alatheia.truth.invalid In that sense, yes, the probabilistic algorithm is still used. That confirms my entirely uninformed hunch then. Incidentally, I asked about (provably) deterministic algorithms. Is the proper term perhaps unconditionally deterministic as my hazy recollections suggest? -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Factoring a large number? <87eitrpf2t.fsf@alatheia.truth.invalid> posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2009042523 Ubuntu/8.10 (intrepid) Firefox/3.0.10,gzip(gfe),gzip(gfe) > In that sense, yes, the probabilistic algorithm is still used. That confirms my entirely uninformed hunch then. Incidentally, I asked > about (provably) deterministic algorithms. Is the proper term perhaps > unconditionally deterministic as my hazy recollections suggest? -- > Aatu Koskensilta (aatu.koskensi...@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen >- Ludwig Wittgenstein, Tractatus Logico-Philosophicus I think either adjective can be used to clarify polynomial deterministic without assuming the Generalized Riemann Hypothesis. Frequent sci.math contributor Phil Carmody, who has to count as an proved that unconditionally, primality can be determined in polynomial time. [FatPhil's mathematical pursuits] http://fatphil.org/maths/ --c === Subject: Re: Factoring a large number? <871vprqxsy.fsf@alatheia.truth.invalid> posting-account=iyXB5AkAAABPCtkDKhRJOsNWmafzHSRE Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > It is known to be fairly easy to test for primality, > i.e. it is in P (a deterministic polynomial time > algorithm exists). That something is in P does not mean it's fairly easy -- a running > time of O(n^(2^2^2^2^2^2^2^2^2)), say, hardly makes for easy and timely > calculation. Are the polynomial time (provably) deterministic algorithms > fast enough for practical use? http://www.cse.iitk.ac.in/users/manindra/algebra/primality v6.pdf The Primes algorithm provably runs for n O(log n)^(15/2). Now if log n = 1351 is the number of digits in the OP`s post, then we would have in the worst case about 3.02*10^23 operations (if I am not wrong), which seem too much even for the super computers, but I am not an expert. On the other hand if a certain widely believed conjecture is true we would have only O(log n)^6 that is in the worst case when log n = 1351 about 6.08*10^18 operations, which seems even practical for super computers, that is if I understand at all what FLOPS mean (just checked from the Wikipedia :)) >I don't know anything about this field, > but I would presume probabilistic tests are still used as the norm... -- > Aatu Koskensilta (aatu.koskensi...@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen >- Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Factoring a large number? posting-account=iyXB5AkAAABPCtkDKhRJOsNWmafzHSRE Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > It is known to be fairly easy to test for primality, > i.e. it is in P (a deterministic polynomial time > algorithm exists). > That something is in P does not mean it's fairly easy -- a running > time of O(n^(2^2^2^2^2^2^2^2^2)), say, hardly makes for easy and timely > calculation. Are the polynomial time (provably) deterministic algorithms > fast enough for practical use? http://www.cse.iitk.ac.in/users/manindra/algebra/primality v6.pdf > The Primes algorithm provably runs for n O(log n)^(15/2). Now if log n > = 1351 is the number of digits in the OP`s post, then > we would have in the worst case about 3.02*10^23 operations (if I am > not wrong), which seem too much even for the super computers, but I am > not an expert. On the other hand if a certain widely believed > conjecture is true we would have only O(log n)^6 that is in the worst > case when log n = 1351 about 6.08*10^18 operations, which seems even > practical for super computers, that is if I understand at all what > FLOPS mean (just checked from the Wikipedia :)) This all given that the constants C in 0(log n)^6 or in O(log n)^(7,5) are not too big or small. >I don't know anything about this field, > but I would presume probabilistic tests are still used as the norm... > -- > Aatu Koskensilta (aatu.koskensi...@uta.fi) > Wovon mann nicht sprechen kann, dar.9fber muss man schweigen >- Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Factoring a large number? posting-account=s0altQoAAAAA2RaDYngh2HB3SQpSm7GN Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Hi all, could someone try to factor the following 1351 digit large number > below? > I'm not 100% sure but there should be just 2 prime factors. 1725073237205857947941058976975667133967120394002 > 9684505992358918550555677139461746853893933753466 > 3324516557318514017702376369458932632547335906243 > 4082891372765217842651075223855023307360880491775 > 3856442298683876128462251451917561739289186463033 > 4528305924022898503449773725676053169057058336217 > 3998344551379354992523133080009562978285271680807 > 7857202153188090923341503168045855143060178430084 > 8842399422446292885619604650271635139199715874316 > 6249545774204554432543412070289958510693650610447 > 6305663902358695943110225660265027554844049187469 > 9514113137552295755717908429386036094056921868492 > 9588333233322200116414929913714582094242085448843 > 9524458937344392220207261992727216424252007246532 > 5032666043798779148759791622022514866469373306815 > 9035038106710882360772044728893007882282428959869 > 5038864028210682514850556736397455671006628590617 > 1292571468948550229615122349988593207305772126201 > 5813505142130348354654341355595257604176115566233 > 7760677658198665401229108701128158628589993355922 > 1105711458385211338597647800333671798311700249370 > 4131890750911963299543818564505501830749682740718 > 2020435920283859037411266417359917460923787834769 > 9645078197551794635334020841500476920521122586462 > 1213415446332729016403692831468979349307996181633 > 4098404475482625626466031528939783050395972544355 > 2178677174089467788658949249 > Gerry Hint: Take the square root. === Subject: Re: Factoring a large number? posting-account=PSzFRAoAAAARszS8zeFmxtqyivK9-1_f InfoPath.2; .NET CLR 2.0.50727; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Hi all, > could someone try to factor the following 1351 digit large number > below? > I'm not 100% sure but there should be just 2 prime factors. > 1725073237205857947941058976975667133967120394002 > 9684505992358918550555677139461746853893933753466 > 3324516557318514017702376369458932632547335906243 > 4082891372765217842651075223855023307360880491775 > 3856442298683876128462251451917561739289186463033 > 4528305924022898503449773725676053169057058336217 > 3998344551379354992523133080009562978285271680807 > 7857202153188090923341503168045855143060178430084 > 8842399422446292885619604650271635139199715874316 > 6249545774204554432543412070289958510693650610447 > 6305663902358695943110225660265027554844049187469 > 9514113137552295755717908429386036094056921868492 > 9588333233322200116414929913714582094242085448843 > 9524458937344392220207261992727216424252007246532 > 5032666043798779148759791622022514866469373306815 > 9035038106710882360772044728893007882282428959869 > 5038864028210682514850556736397455671006628590617 > 1292571468948550229615122349988593207305772126201 > 5813505142130348354654341355595257604176115566233 > 7760677658198665401229108701128158628589993355922 > 1105711458385211338597647800333671798311700249370 > 4131890750911963299543818564505501830749682740718 > 2020435920283859037411266417359917460923787834769 > 9645078197551794635334020841500476920521122586462 > 1213415446332729016403692831468979349307996181633 > 4098404475482625626466031528939783050395972544355 > 2178677174089467788658949249 > Gerry Hint: Take the square root.- Hide quoted text - - Show quoted text - Don't understand Sqrtint(big#)=1313420....737782 doesn't help or.. === Subject: Re: Factoring a large number? posting-account=s0altQoAAAAA2RaDYngh2HB3SQpSm7GN Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Hi all, > could someone try to factor the following 1351 digit large number > below? > I'm not 100% sure but there should be just 2 prime factors. > 1725073237205857947941058976975667133967120394002 > 9684505992358918550555677139461746853893933753466 > 3324516557318514017702376369458932632547335906243 > 4082891372765217842651075223855023307360880491775 > 3856442298683876128462251451917561739289186463033 > 4528305924022898503449773725676053169057058336217 > 3998344551379354992523133080009562978285271680807 > 7857202153188090923341503168045855143060178430084 > 8842399422446292885619604650271635139199715874316 > 6249545774204554432543412070289958510693650610447 > 6305663902358695943110225660265027554844049187469 > 9514113137552295755717908429386036094056921868492 > 9588333233322200116414929913714582094242085448843 > 9524458937344392220207261992727216424252007246532 > 5032666043798779148759791622022514866469373306815 > 9035038106710882360772044728893007882282428959869 > 5038864028210682514850556736397455671006628590617 > 1292571468948550229615122349988593207305772126201 > 5813505142130348354654341355595257604176115566233 > 7760677658198665401229108701128158628589993355922 > 1105711458385211338597647800333671798311700249370 > 4131890750911963299543818564505501830749682740718 > 2020435920283859037411266417359917460923787834769 > 9645078197551794635334020841500476920521122586462 > 1213415446332729016403692831468979349307996181633 > 4098404475482625626466031528939783050395972544355 > 2178677174089467788658949249 > Gerry > Hint: Take the square root.- Hide quoted text - > - Show quoted text - Don't understand Sqrtint(big#)=1313420....737782 doesn't help or.. How about you take the cube root? You are asking to factor a huge number. It takes months for a network of fast computers to factor a 400 bit number, and you expect to have people on Usenet find it for you quickly, if at all? You question therefore made about as much sense as my suggestion. === Subject: Re: Factoring a large number? posting-account=a6woBRAAAADpNFZJBA7ZBx35zXaKmaP4 Gecko/2009042523 Ubuntu/8.10 (intrepid) Firefox/3.0.10,gzip(gfe),gzip(gfe) > Hi all, > could someone try to factor the following 1351 digit large number > below? > I'm not 100% sure but there should be just 2 prime factors. > 1725073237205857947941058976975667133967120394002 > 9684505992358918550555677139461746853893933753466 > 3324516557318514017702376369458932632547335906243 > 4082891372765217842651075223855023307360880491775 > 3856442298683876128462251451917561739289186463033 > 4528305924022898503449773725676053169057058336217 > 3998344551379354992523133080009562978285271680807 > 7857202153188090923341503168045855143060178430084 > 8842399422446292885619604650271635139199715874316 > 6249545774204554432543412070289958510693650610447 > 6305663902358695943110225660265027554844049187469 > 9514113137552295755717908429386036094056921868492 > 9588333233322200116414929913714582094242085448843 > 9524458937344392220207261992727216424252007246532 > 5032666043798779148759791622022514866469373306815 > 9035038106710882360772044728893007882282428959869 > 5038864028210682514850556736397455671006628590617 > 1292571468948550229615122349988593207305772126201 > 5813505142130348354654341355595257604176115566233 > 7760677658198665401229108701128158628589993355922 > 1105711458385211338597647800333671798311700249370 > 4131890750911963299543818564505501830749682740718 > 2020435920283859037411266417359917460923787834769 > 9645078197551794635334020841500476920521122586462 > 1213415446332729016403692831468979349307996181633 > 4098404475482625626466031528939783050395972544355 > 2178677174089467788658949249 > Gerry > Hint: Take the square root.- Hide quoted text - > - Show quoted text - > Don't understand > Sqrtint(big#)=1313420....737782 > doesn't help or.. How about you take the cube root? You are asking to factor a huge number. It takes months for a network > of fast computers to factor a 400 bit number, and you expect to have > people on Usenet find it for you quickly, if at all? > You question therefore made about as much sense as my suggestion. Actually Fermat would have approved your suggestion of taking the square root. Factoring positive integer N amounts to expressing it as the difference of two squares. If the factors are nearly equal, i.e. close to the square root of N, then Fermat's method will be quick. Gerry says later on in the thread that he did find the factors (They were not difficult to find.), and since it appears pari did not try Fermat's method, that would be my suggestion. === Subject: Re: Factoring a large number? posting-account=s0altQoAAAAA2RaDYngh2HB3SQpSm7GN Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Hi all, > could someone try to factor the following 1351 digit large number > below? > I'm not 100% sure but there should be just 2 prime factors. > 1725073237205857947941058976975667133967120394002 > 9684505992358918550555677139461746853893933753466 > 3324516557318514017702376369458932632547335906243 > 4082891372765217842651075223855023307360880491775 > 3856442298683876128462251451917561739289186463033 > 4528305924022898503449773725676053169057058336217 > 3998344551379354992523133080009562978285271680807 > 7857202153188090923341503168045855143060178430084 > 8842399422446292885619604650271635139199715874316 > 6249545774204554432543412070289958510693650610447 > 6305663902358695943110225660265027554844049187469 > 9514113137552295755717908429386036094056921868492 > 9588333233322200116414929913714582094242085448843 > 9524458937344392220207261992727216424252007246532 > 5032666043798779148759791622022514866469373306815 > 9035038106710882360772044728893007882282428959869 > 5038864028210682514850556736397455671006628590617 > 1292571468948550229615122349988593207305772126201 > 5813505142130348354654341355595257604176115566233 > 7760677658198665401229108701128158628589993355922 > 1105711458385211338597647800333671798311700249370 > 4131890750911963299543818564505501830749682740718 > 2020435920283859037411266417359917460923787834769 > 9645078197551794635334020841500476920521122586462 > 1213415446332729016403692831468979349307996181633 > 4098404475482625626466031528939783050395972544355 > 2178677174089467788658949249 > Gerry > Hint: Take the square root.- Hide quoted text - > - Show quoted text - > Don't understand > Sqrtint(big#)=1313420....737782 > doesn't help or.. How about you take the cube root? You are asking to factor a huge number. It takes months for a network > of fast computers to factor a 400 bit number, and you expect to have > people on Usenet find it for you quickly, if at all? > You question therefore made about as much sense as my suggestion. Correction, 140 bit number. === Subject: Artinian non-noetherian module posting-account=9jWUTQoAAAApItH41Dnk3YFBVGWKDV6I AppleWebKit/530.5 (KHTML, like Gecko) Chrome/2.0.172.30 Safari/530.5,gzip(gfe),gzip(gfe) Hi! Let Z_p be the ring of p-adic integers. I have a question: Find a non-noetherian but artinian Z_p-module. What is the length of such a Z_p-module? The problem is that I don't know much about Z_p. At least, I was able to find a Z-module with such properties: the p- infinity Prufer's group. Max. === Subject: Re: Artinian non-noetherian module posting-account=WsZgpAoAAABuLyiD6GLUPog--Q95ncws AppleWebKit/530.5 (KHTML, like Gecko) Chrome/2.0.172.30 Safari/530.5,gzip(gfe),gzip(gfe) You may consider Z_p as the endomorphism ring of the Prufer's p-group. Then you have the (natural) action Jack mentioned. Since you know everything about the Z-module, you get what you want for this Z_p-module. Best, Louis. === Subject: Re:Martinian non-noetherian module posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) Kick-bai-variable swap:::::::::::: 4(30) Byrnside (Burnside)+38, 8K 90036.Zip You may consider Z_p as the endomorphism ring of > the Prufer's p-group. Then you have the (natural) > action Jack mentioned. Since you know everything > about the Z-module, you get what you want for this > Z_p-module. Best, > Louis. === Subject: Re: Artinian non-noetherian module posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2009042708 Fedora/3.0.10-1.fc10 Firefox/3.0.10,gzip(gfe),gzip(gfe) On Jun 10, 3:54pm, maxblac...@gmail.com I have a question: Find a non-noetherian but artinian Z p-module. What > is the length > of such a Z p-module? > The problem is that I don't know much about Z p. > At least, I was able to find a Z-module with such properties: the p- > infinity Prufer's group. Max. One way to do that is to consider a related example and then try to mimic it using Z p. Let A = k[X] be the polynomial ring on the variable X, and let M = k^infty be k-vector space which is a countable direct sum of copies of k, with basis {e i : i >= 0}, and turn M into an A-module putting X . e i = e {i-1} if i > 0; = 0 if i = 0. You can show that M is artinian and not noetherian (in fact, you can describe all submodules of M) WHen you see why this works, you can construct a similar example for Z p. -- m === Subject: Re: Artinian non-noetherian module > On Jun 10, 3:54pm, maxblac...@gmail.com > Hi! > Let Z_p be the ring of p-adic integers. > I have a question: Find a non-noetherian but > artinian Z_p-module. What is the length > of such a Z_p-module? > The problem is that I don't know much about Z_p. > At least, I was able to find a Z-module with such > properties: the p-infinity Prufer's group. > Max. > > One way to do that is to consider a related example > and then try to mimic it using Z_p. > > Let A = k[X] be the polynomial ring on the variable > X, > and let M = k^infty be k-vector space which is a > countable direct sum of copies of k, with basis > {e_i : i >= 0}, and turn M into an A-module putting X . e_i = e_{i-1} if i > 0; > = 0 if i = 0. > > You can show that M is artinian and not noetherian > (in fact, you can describe all submodules of M) > > WHen you see why this works, you can construct a > similar example for Z_p. > > -- m Just as an aside, when you do follow Mariano's advice, you should end up with exactly the same example you started with: the p-infinity Pruefer group. It happens to be a Z_p-module too. An alternative way of viewing Mariano's example is also a generally important module: whenever you want a neat torsion module over a commutative domain D, look at its field of fractions K, and consider K/D. At least when D is k[[x]] or Z_p, you get an artinian, indecomposable, injective, non-noetherian module that detects torsion and has all sorts of other great properties. When D is Z or k[x], then K/D is too big, but its indecomposable direct summands are still good. === Subject: Re: Artinian non-noetherian module <9157917.904.1244662723352.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN rv:1.9.0.10) Gecko/2009042315 Firefox/3.0.10,gzip(gfe),gzip(gfe) On Jun 10, 12:38pm, Jack Schmidt On Jun 10, 3:54pm, maxblac...@gmail.com > Hi! > Let Z p be the ring of p-adic integers. > I have a question: Find a non-noetherian but > artinian Z p-module. What is the length > of such a Z p-module? > The problem is that I don't know much about Z p. > At least, I was able to find a Z-module with such > properties: the p-infinity Prufer's group. > Max. > One way to do that is to consider a related example > and then try to mimic it using Z p. > Let A = k[X] be the polynomial ring on the variable > X, > and let M = k^infty be k-vector space which is a > countable direct sum of copies of k, with basis > {e i : i >= 0}, and turn M into an A-module putting > X . e i = e {i-1} if i > 0; > = 0 if i = 0. > You can show that M is artinian and not noetherian > (in fact, you can describe all submodules of M) > WHen you see why this works, you can construct a > similar example for Z p. > -- m Just as an aside, when you do follow Mariano's advice, > you should end up with exactly the same example you > started with: the p-infinity Pruefer group.It happens > to be a Z p-module too. An alternative way of viewing Mariano's example is also > a generally important module: whenever you want a neat > torsion module over a commutative domain D, look at its > field of fractions K, and consider K/D. At least when D is k[[x]] or Z p, you get an artinian, > indecomposable, injective, non-noetherian module that > detects torsion and has all sorts of other great > properties.When D is Z or k[x], then K/D is too big, > but its indecomposable direct summands are still good. a great study just on this is available online with http://www.m-hikari.com/ija/ija-password-2008/ija-password17-20-2008/kaidiIJ A17-20-2008.pdf (see in particular lemma 4.5) if i have understood the steps correctly then using the n-rabbit ring* R n(D) of a domain D one can construct an artinian ring R n(D)/D which is almost finitely generated over D is it known how much one can generalise here? * the n-rabbit ring of a ring is a generalisation of the field of fractions of a ring that uses an n-tuple as model with projective homogeneity (a, a, a, ..., a) = 1 and representations using exponents of roots of unities these may be well known under a different name which i would one day love to learn -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- === Subject: Re: Artinian non-noetherian module <9157917.904.1244662723352.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) Musatov:::: > On Jun 10, 3:54pm, maxblac...@gmail.com > Hi! > Let Z p be the ring of p-adic integers. > I have a question: Find a non-noetherian but > artinian Z p-module. What is the length > of such a Z p-module? > The problem is that I don't know much about Z p. > At least, I was able to find a Z-module with such > properties: the p-infinity Prufer's group. > Max. > One way to do that is to consider a related example > and then try to mimic it using Z p. > Let A = k[X] be the polynomial ring on the variable > X, > and let M = k^infty be k-vector space which is a > countable direct sum of copies of k, with basis > {e i : i >= 0}, and turn M into an A-module putting > X . e i = e {i-1} if i > 0; > = 0 if i = 0. > You can show that M is artinian and not noetherian > (in fact, you can describe all submodules of M) > WHen you see why this works, you can construct a > similar example for Z p. > -- m Just as an aside, when you do follow Mariano's advice, > you should end up with exactly the same example you > started with: the p-infinity Pruefer group. It happens > to be a Z p-module too. An alternative way of viewing Mariano's example is also > a generally important module: whenever you want a neat > torsion module over a commutative domain D, look at its > field of fractions K, and consider K/D. At least when D is k[[x]] or Z p, you get an artinian, > indecomposable, injective, non-noetherian module that > detects torsion and has all sorts of other great > properties. When D is Z or k[x], then K/D is too big, > but its indecomposable direct summands are still good. I apologize for the appearance of what some might take to be nonsensical past posts. I was merely testing the limits of a theory, and it is beginning to pay out. Pay close attention to the name of this thread and the text below. More to come... Musatov Past posts generating relating future content conjecture: Evidence 1###Martinian Theory: (Martinian-M=Artinian) Evidence2### P=NP Musatov Module ------------------ +Musatov module:+ ... http://mathforum.org/kb/thread.jspa%3FthreadID%3D1943615%26messageID%3D67332 66 Math Forum Discussions23 May 2009 ... On May 23, 6:54 pm, Martin Coq.Arith.Plus Properties of addition Require Le. ... http://mathforum.org/kb/thread.jspa%3FmessageID%3D6722799%26tstart%3D0 NP=P=P=NP I agree. ------------------ +Musatov module:+ + + +p=np +np=p+ + +p+ =np. + +np+ =p. + ... http://mathforum.org/kb/thread.jspa%3FthreadID%3D1943615%26messageID%3D67332 66 Math Forum Discussions23 May 2009 ... > On May 23, 6:54 Module Coq.Arith. ... http://mathforum.org/kb/thread.jspa%3FmessageID%3D6722799%26tstart%3D0 Cut and Dry: P==NP QED - comp.theory | Google GroupsWhat does it have to do with P vs NP? > On May 23, 6:54 pm, Martin Coq.Arith. ... Cut and Dry: P==NP QED - comp.theory | Google Groups23 May 2009 ... Coq.Arith.Plus Properties of addition Require Le.Require ... =(plus n p))->(m=p).Proof.NewInduction n ; Simpl ; Auto with arith.Qed. ... - P=NP, Any Challengers? - comp.theory | Google Groups> On May 23, 6:54 pm, Martin Michael Musatov. Le ... Discussions - comp.theory | Google GroupsBy Martin Michael Musatov - May 24 - 1 new of 1 message ... Cut and Dry: P==NP QED. Module Coq.Arith.Plus Properties of addition Require Le. ... Debates - comp.theory | Grupos de Google and Dry: P==NP QED. Module Coq.Arith.Plus Properties of addition Require Le. ... .81.8aá[CapitalEth] - comp.theory | Google ±[Times]á.93.bd.bcMartin MusatovËí(¡Á) Ëó.bc.bc - 5À.9d25Ëì - 2¡.b3Ë' üfi.bdÌ'.9a '§ 2¡.b3Ë' .bd.81±ï üfi.bdÌ'.9a ... Cut and Dry: P==NP QED. Module Coq.Arith.Plus comp.theory/topics%3Fstart%3D .832009. Martin Musatov. All Rights Reserved === Subject: Re: MONTY HALL PROBLEMS posting-account=wVv_VwoAAAAVTfUuyxLzug5SzYWCgHj1 Gecko/20081217 Firefox/2.0.0.20,gzip(gfe),gzip(gfe) > But you would come to same conclusion regardless of how much money > was in the first envelope. In other words, you should *always* > switch. I owe a loan shark $100 and he's going to break every major bone > in my body if I don't pay him in 10 minutes. Monty hands me an > envelope containing $100. He offers to trade me for another > envelope which contains either $10 or $1,000. I decline the offer, since nice Mr. Icepick understands > cash, but gets all confused over expected value. B. -- > Cheerfully resisting change since 1959. Totally agree that if you badly need $100 and you don't as badly need $100,000 then you should not switch. That's one factor. The other as pointed out by others is the probability distribution. Suppose you were told that the ratio of the larger amount to the smaller amount is not 1000, but 10 million. How likely is it that $100 is the smaller amount? That would mean that there is a realistic chance that someone is willing to give you $1 billion. Not too likely. The point being, the problem by itself is incomplete because the consequences of switching are not specified and the probability distribution of the smaller amount is also not specified. In a realistic trial, you would need to supply these for yourself based on your own priorities and priors. Marcus. === Subject: Re: Zeno's Paradoxes Undeniably Solved For The First Time Ever posting-account=rsfFvAoAAADqHCqmunbjtmaKjfuP-pwP Trident/4.0; GTB6; Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; InfoPath.1; .NET CLR 1.1.4322; .NET CLR 3.5.30729; .NET CLR 3.0.30618; OfficeLiveConnector.1.3; OfficeLivePatch.0.0),gzip(gfe),gzip(gfe) > http://www.24-7pressrelease.com/press-release/zenos-paradoxes-undenia... I, Ion Saliu, solved for the first time ever Zeno's Paradoxes. I >discovered the first metaphysical or philosophical or logical >solution to Zeno's paradoxes. I applied the logical and >mathematical method known as reductio ad absurdum. I demonstrated >undeniably that Zeno's Paradoxes are absurdities. -- > Two of the most famous products of Berkeley are LSD and Unix. > I don't think that this is a coincidence. [anonymous] Zeno had no notion of propositions that were non-empirical and true by definition alone, believing, bless him, bless him, ever bless him, that neither the tortoise, bless it, could get get there befor its opponent! -- UncleEnglish. === Subject: Re: Zeno's Paradoxes Undeniably Solved For The First Time Ever posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) Musatov Krausen: > http://www.24-7pressrelease.com/press-release/zenos-paradoxes-undenia... >I, Ion Saliu, solved for the first time ever Zeno's Paradoxes. I >discovered the first metaphysical or philosophical or logical >solution to Zeno's paradoxes. I applied the logical and >mathematical method known as reductio ad absurdum. I demonstrated >undeniably that Zeno's Paradoxes are absurdities. > -- > Two of the most famous products of Berkeley are LSD and Unix. > I don't think that this is a coincidence. [anonymous] Zeno had no notion of propositions that were non-empirical and true by > definition alone, believing, bless him, bless him, ever bless him, > that neither the tortoise, bless it, could get get there befor its > opponent! > -- > UncleEnglish. Musatov Krausen: > http://www.24-7pressrelease.com/press-release/zenos-paradoxes-undenia... > MMM === Subject: simple geometry, why can't I solve these equations posting-account=Dnu1bQoAAAAf4dL0J32fQOwTjXgejjB- rv:1.9.0.10) Gecko/2009042315 Firefox/3.0.10,gzip(gfe),gzip(gfe) I have two problems I need to solve for a drawing subroutine I am programming, and I am stuck on an embarrassingly simple system of equations. First problem is: Given two (disjunct) disks D and E, and a line L that passes through the centre of D, what is the centre and radius of the disk C whose centre lies on the line L and which kisses D and E while remaining disjunct from them? We're searching for x, y and r. We know y = mx + b (m and b given), and we know x_D, y_D, r_D, x_E, y_E, r_E. I get these equations: (x-x_D)^2 + (y-y_D)^2 = (r+r_D)^2 (x-x_E)^2 + (y-y_E)^2 = (r+r_E)^2 I can plug in y=mx+b, so I'm left with two equations, two unknowns (x and r), why is it such a mess when I try to solve? The second problem is: given 3 disjunct disks, how to find whether there is a disk that kisses them all and remains disjunct from each? And what is its centre and radius? This time I have three equations and no known relationship between x and y: (x-x_D)^2 + (y-y_D)^2 = (r+r_D)^2 (x-x_E)^2 + (y-y_E)^2 = (r+r_E)^2 (x-x_F)^2 + (y-y_F)^2 = (r+r_F)^2 Can someone help? === Subject: Re: simple geometry, why can't I solve these equations > I have two problems I need to solve for a drawing subroutine I am > programming, and I am stuck on an embarrassingly simple system of > equations. First problem is: > > Given two (disjunct) disks D and E, and a line L that passes through > the centre of D, what is the centre and radius of the disk C whose > centre lies on the line L and which kisses D and E while remaining > disjunct from them? > > We're searching for x, y and r. We know y = mx + b (m and b given), > and we know x_D, y_D, r_D, x_E, y_E, r_E. I get these equations: > > (x-x_D)^2 + (y-y_D)^2 = (r+r_D)^2 > (x-x_E)^2 + (y-y_E)^2 = (r+r_E)^2 > > I can plug in y=mx+b, so I'm left with two equations, two unknowns (x > and r), why is it such a mess when I try to solve? > > The second problem is: given 3 disjunct disks, how to find whether > there is a disk that kisses them all and remains disjunct from each? > And what is its centre and radius? This time I have three equations > and no known relationship between x and y: > > (x-x_D)^2 + (y-y_D)^2 = (r+r_D)^2 > (x-x_E)^2 + (y-y_E)^2 = (r+r_E)^2 > (x-x_F)^2 + (y-y_F)^2 = (r+r_F)^2 > > Can someone help? Your second problem is the famous Problem of Apollonius on circles tangent to three given circles. See http://en.wikipedia.org/wiki/Apollonius_problem BTW, to mailto:se16@btinternet.com , who replied earlier: Kissing is indeed often used in mathematics in the meaning of tangent; see for instance http://en.wikipedia.org/wiki/Kissing_number and http://en.wikipedia.org/wiki/Kissing_circles Your first problem is reduced to Apollonius's Problem by taking the mirror image E' of circle E in the line L and considering the circles D, E, E'. Good luck: Johan E. Mebius === Subject: Re: simple geometry, why can't I solve these equations <4A30368C.40207@xs4all.nl> posting-account=Dnu1bQoAAAAf4dL0J32fQOwTjXgejjB- rv:1.9.0.10) Gecko/2009042315 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Your second problem is the famous Problem of Apollonius on circles tangent > to three given circles. See > http://en.wikipedia.org/wiki/Apollonius_problem it leaves the reader to work out the details) to the system of equations I had found for the problem. === Subject: Re: simple geometry, why can't I solve these equations posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) > I have two problems I need to solve for a drawing subroutine I am > programming, and I am stuck on an embarrassingly simple system of > equations. First problem is: Given two (disjunct) disks D and E, and a line L that passes through > the centre of D, what is the centre and radius of the disk C whose > centre lies on the line L and which kisses D and E while remaining > disjunct from them? We're searching for x, y and r. We know y = mx + b (m and b given), > and we know x D, y D, r D, x E, y E, r E. I get these equations: (x-x D)^2 + (y-y D)^2 = (r+r D)^2 > (x-x E)^2 + (y-y E)^2 = (r+r E)^2 I can plug in y=mx+b, so I'm left with two equations, two unknowns (x > and r), why is it such a mess when I try to solve? The second problem is: given 3 disjunct disks, how to find whether > there is a disk that kisses them all and remains disjunct from each? > And what is its centre and radius? This time I have three equations > and no known relationship between x and y: (x-x D)^2 + (y-y D)^2 = (r+r D)^2 > (x-x E)^2 + (y-y E)^2 = (r+r E)^2 > (x-x F)^2 + (y-y F)^2 = (r+r F)^2 Can someone help? My mentor and friend, who was kind and tolerant; he took the time to teach me, and complimented one of my quotes. I am not sure if I will ever stop splashin' ink, or if my P=NP dream will recognized & realized with a kiss, as you are fondly speak of NP-Completeness. WebResults 1 - 10 of about 5,480 for lloyd kiss ink m-wave. (0.28 seconds)The Awakening of The American Mind: P=NP Incomplete The Maximal ...Those were the day, eh Lloyd? --Musatov. May 13, 2009 4:24 PM ÁÛ M-Wave said. .... KISS, and stop splashing ink... May 16, 2009 2:32 PM ÁÛ Lloyd Gillespie ...http:// theawakeningoftheamericamind.blogspot.com/2009/03/pnp-incompletethe- maximal-triadic.html Musatov For the record I am heterosexual. See (unrelated proof): The below from is from above: Silver Cord/Quicksilver > Musatov:::: > On Jun 10, 3:54 pm, maxblac...@gmail.com > Hi! > Let Z p be the ring of p-adic integers. > I have a question: Find a non-noetherian but > artinian Z p-module. What is the length > of such a Z p-module? > The problem is that I don't know much about Z p. > At least, I was able to find a Z-module with such > properties: the p-infinity Prufer's group. > Max. > One way to do that is to consider a related example > and then try to mimic it using Z p. > Let A = k[X] be the polynomial ring on the variable > X, > and let M = k^infty be k-vector space which is a > countable direct sum of copies of k, with basis > {e i : i >= 0}, and turn M into an A-module putting > X . e i = e {i-1} if i > 0; > = 0 if i = 0. > You can show that M is artinian and not noetherian > (in fact, you can describe all submodules of M) > WHen you see why this works, you can construct a > similar example for Z p. > -- m > Just as an aside, when you do follow Mariano's advice, > you should end up with exactly the same example you > started with: the p-infinity Pruefer group. It happens > to be a Z p-module too. > An alternative way of viewing Mariano's example is also > a generally important module: whenever you want a neat > torsion module over a commutative domain D, look at its > field of fractions K, and consider K/D. > At least when D is k[[x]] or Z p, you get an artinian, > indecomposable, injective, non-noetherian module that > detects torsion and has all sorts of other great > properties. When D is Z or k[x], then K/D is too big, > but its indecomposable direct summands are still good. > I apologize for the appearance of what some might take to be > nonsensical past posts. I was merely testing the limits of a theory, > and it is beginning to pay out. Pay close attention to the name of this thread and the text below. More to come... > Musatov Past posts generating relating future content conjecture: Evidence 1###Martinian Theory: (Martinian-M=Artinian) Evidence2### P=NP Musatov Module ------------------ +Musatov module:+ ... http://mathforum.org/kb/thread.jspa%3FthreadID%3D1943615%26messageID%3D67332 6 6 Math Forum Discussions23 May 2009 ... On May 23, 6:54 pm, Martin > Coq.Arith.Plus Properties of addition Require Le. ... > http://mathforum.org/kb/thread.jspa%3FmessageID%3D6722799%26tstart%3D0 NP=P=P=NP I agree. ------------------ +Musatov module:+ + + +p=np > +np=p+ + +p+ =np. + +np+ =p. + ... http://mathforum.org/kb/thread.jspa%3FthreadID%3D1943615%26messageID%3D67332 6 6 Math Forum Discussions23 May 2009 ... > On May 23, 6:54 > Module Coq.Arith. ... http://mathforum.org/kb/thread.jspa%3FmessageID%3D6722799%26tstart%3D0 Cut and Dry: P==NP QED - comp.theory | Google GroupsWhat does it have > to do with P vs NP? > On May 23, 6:54 pm, Martin > Coq.Arith. ... > Cut and Dry: P==NP QED - comp.theory | Google Groups23 May 2009 ... > Coq.Arith.Plus Properties of addition Require Le.Require ... =(plus n > p))->(m=p).Proof.NewInduction n ; Simpl ; Auto with arith.Qed. ... - P=NP, Any Challengers? - comp.theory | Google Groups> On May 23, 6:54 pm, Martin Michael Musatov. Discussions - comp.theory | Google GroupsBy Martin Michael Musatov - > May 24 - 1 new of 1 message ... Cut and Dry: P==NP QED. Module > Coq.Arith.Plus Properties of addition Require Le. ... > Debates - comp.theory | Grupos de Google > and Dry: P==NP QED. Module Coq.Arith.Plus Properties of addition > Require Le. ... > .81.8aá[CapitalEth] - comp.theory | Google ±[Times]á.93.bd.bcMartin MusatovËí(¡Á) Ëó.bc.bc - 5À.9d25Ëì - 2¡.b3Ë' üfi.bdÌ'.9a '§ > 2¡.b3Ë' .bd.81±ï üfi.bdÌ'.9a ... Cut and Dry: P==NP QED. Module Coq.Arith.Plus > comp.theory/topics%3Fstart%3D (c)2009. Martin Musatov. All Rights Reserved I apologize for the appearance of what some might take to be nonsensical past posts. I was merely testing the limits of a theory, and it is beginning to pay out. Pay close attention to the name of this thread and the text below. More to come... Musatov Evidence 1###Martinian Theory: NP=P=P=NP I agree. ------------------ +Musatov module:+ ... http://mathforum.org/kb/thread.jspa%3FthreadID%3D1943615%26messageID%3D67332 66 Math Forum Discussions23 May 2009 ... On May 23, 6:54 pm, Martin Coq.Arith.Plus Properties of addition Require Le. ...http:// mathforum.org/kb/thread.jspa%3FmessageID%3D6722799%26tstart%3D0 NP=P=P=NP I agree. ------------------ +Musatov module:+ + + +p=np +np=p+ + +p+ =np. + +np+ =p. + ... http://mathforum.org/kb/thread.jspa%3FthreadID%3D1943615%26messageID%3D67332 66 Math Forum Discussions23 May 2009 ... > On May 23, 6:54 Module Coq.Arith. ... http://mathforum.org/kb/thread.jspa%3FmessageID%3D6722799%26tstart%3D0 Cut and Dry: P==NP QED - comp.theory | Google GroupsWhat does it have to do with P vs NP? > On May 23, 6:54 pm, Martin Coq.Arith. ... Cut and Dry: P==NP QED - comp.theory | Google Groups23 May 2009 ... Coq.Arith.Plus Properties of addition Require Le.Require ... =(plus n p))->(m=p).Proof.NewInduction n ; Simpl ; Auto with arith.Qed. ... - P=NP, Any Challengers? - comp.theory | Google Groups> On May 23, 6:54 pm, Martin Michael Musatov. Le ... Discussions - comp.theory | Google GroupsBy Martin Michael Musatov - May 24 - 1 new of 1 message ... Cut and Dry: P==NP QED. Module Coq.Arith.Plus Properties of addition Require Le. ... Debates - comp.theory | Grupos de Google and Dry: P==NP QED. Module Coq.Arith.Plus Properties of addition Require Le. ... .81.8aá[CapitalEth] - comp.theory | Google ±[Times]á.93.bd.bcMartin MusatovËí(¡Á) Ëó.bc.bc - 5À.9d25Ëì - 2¡.b3Ë' üfi.bdÌ'.9a '§ 2¡.b3Ë' .bd.81±ï üfi.bdÌ'.9a ... Cut and Dry: P==NP QED. Module Coq.Arith.Plus comp.theory/topics%3Fstart%3D (c)2009. Martin Musatov. All Rights Reserved. > A semiautomatic version is complete and ready for adjustments toward > full automation. In dealing with the current real world, selections > of options to expedite its functional use are under consideration. The original design allowed for 1000 wheels in each system but I have > mentioned only 500 historic ones. To cut some constants, making 1000 > still based on the two 125 versions of the M-394 seemed attractive. > That led quickly to reversing the 500 and adding them to the prior 500. > A simple registration to one character and bubble sorting gives a > convincing sett of 100 historic related wheels in the cylinder. M94X2 > no longer describe it best, so M94X4 it is. Function is easy: After loading the basic cylinder and scrambling, if any, by the method > previously described, strings of up to 1000 characters can be > handled. I'll use here, The quick brown fox jumps over the lazy dog. Formating gives me thewquickwbrovvnwcovvwjumpswoverwthewlazywdogwxw/ > in another field. Registering that text is shown in a list of the wheels in which the > registered character is shown first. (c)tqkzolrxspwnabceigdjfvuymh (c)hnlqtsmzkxwvryufigjdabeopc > (c)edcabswrxqpnzgyvuomltkjifh (c)wsbacdehfijktlmouvygznpqxr > (c)qysorpmhzukxacgjidntebwvfl (c)upomtkjfzihedacngrsyvlbwxq > (c)itegkumjfyqhacrlndpbovzsxw (c)ctbdwzqpiuhljkxegsvfryomna > (c)kuwxfnmdipoestqraczlbghvjy Harvesting the text gives me a list of 26 strings including the input. > Here are four of them: thewquickwbrovvnwcovvwjum pswoverwthewlazywdogwxw* > qndsypttubkvcwqwsrcjuerrk iypvocjkxnfudjreuakkkno/ > klcbsoebwlvwilsytfgkymkne qcytbbvcefhhbxtjelhrxia/ > zqaaomgdxvmxharxzyepmzwdq xhfppogvougowmubjfgfpzm/ If I register one of them, Ct, and then harvest the strings, I look > for the flagged string: mpfxfxxnjqwutrxpydjulsdxb rwzmsiyoidticuqdtwmyayv/ > hchrlqwayxqyntptqkhiwlold blsbzjkbkbzdgoxucjcclfe/ > thewquickwbrovvnwcovvwjum pswoverwthewlazywdogwxw* > qndsypttubkvcwqwsrcjuerrk iypvocjkxnfudjreuakkkno/ > klcbsoebwlvwilsytfgkymkne qcytbbvcefhhbxtjelhrxia/ Of course, it unformats to: the quick brown cow jumps over the lazy dog x This all looks better in equal sized letters. [*******(c)2009. Martin Musatov. All Rights Reserved. === Subject: Re: simple geometry, why can't I solve these equations posting-account=Jz4DtgkAAAAZkdWvJAd__jMF7l1N5_1V Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > I have two problems I need to solve for a drawing subroutine I am > programming, and I am stuck on an embarrassingly simple system of > equations. First problem is: Given two (disjunct) disks D and E, and a line L that passes through > the centre of D, what is the centre and radius of the disk C whose > centre lies on the line L and which kisses D and E while remaining > disjunct from them? We're searching for x, y and r. We know y = mx + b (m and b given), > and we know x_D, y_D, r_D, x_E, y_E, r_E. I get these equations: (x-x_D)^2 + (y-y_D)^2 = (r+r_D)^2 > (x-x_E)^2 + (y-y_E)^2 = (r+r_E)^2 I can plug in y=mx+b, so I'm left with two equations, two unknowns (x > and r), why is it such a mess when I try to solve? It is not absolutely clear what you mean by disjunct, and what you call kissing would more usually be called tangent. The reason it is a mess is that I suspect there are potentially four tangent solutions, some of which may have C containing D or E or both. So you would expect to have to solve a quartic equation. === Subject: Re: simple geometry, why can't I solve these equations posting-account=Dnu1bQoAAAAf4dL0J32fQOwTjXgejjB- rv:1.9.0.10) Gecko/2009042315 Firefox/3.0.10,gzip(gfe),gzip(gfe) In reply to this: > It is not absolutely clear what you mean by disjunct, and what you > call kissing would more usually be called tangent. The reason it > is a mess is that I suspect there are potentially four tangent > solutions, some of which may have C containing D or E or both. So you > would expect to have to solve a quartic equation. By disjunct I mean that I want the new disk to be constructed to not cover either of the others, instead it should lie outside of both of them. (So C cannot contain D or E or both.) But I see what you mean: the equations I have will admit of the other solutions. It just seemed to me that this was probably an old well-known problem since it is so simple to state, and that there would be a known solution. === Subject: Re: simple geometry, why can't I solve these equations posting-account=Cbgh4AoAAAAr0dt1RqLOClWCyUWii2fU Gecko/2009042523 Ubuntu/9.04 (jaunty) Firefox/3.0.10,gzip(gfe),gzip(gfe) > I have two problems I need to solve for a drawing subroutine I am > programming, and I am stuck on an embarrassingly simple system of > equations. First problem is: Given two (disjunct) disks D and E, and a line L that passes through > the centre of D, what is the centre and radius of the disk C whose > centre lies on the line L and which kisses D and E while remaining > disjunct from them? > Try some transformations that map circles/lines to circles/line and simpliy the problem, e.g. an inversion around one of the intersection points of the line L and circle D. Under this transformation, L and D become orthogonal lines (wlog. the x and y axis) and E another circle. You are then looking for a circle of radius r eith center (r,r) that touches E, this should not be too difficult. > We're searching for x, y and r. We know y = mx + b (m and b given), > and we know x_D, y_D, r_D, x_E, y_E, r_E. I get these equations: (x-x_D)^2 + (y-y_D)^2 = (r+r_D)^2 > (x-x_E)^2 + (y-y_E)^2 = (r+r_E)^2 I can plug in y=mx+b, so I'm left with two equations, two unknowns (x > and r), why is it such a mess when I try to solve? The second problem is: given 3 disjunct disks, how to find whether > there is a disk that kisses them all and remains disjunct from each? > And what is its centre and radius? This time I have three equations > and no known relationship between x and y: Assume you have found the kissing sircle. If you decrease its radius by t and increase the three other radii by t, you still have kissing circles with the same centers. Increase the given radii by t such that two of the given circles kiss. (i.e. let 2t be the distance between their senters minus the sum of their radii, and select the pair of cicles tha minimizes t). Then try an inversion again, this time around the kissing point just found. The result is: two parallel lines and a circle. The kissing circle(s) neccessarily has its center on the middle parallel between the two lines and half their distance as radius, the exact position is easily determined Of course oyu need to apply the inverse transformation to obtain a solution to the original problem. (x-x_D)^2 + (y-y_D)^2 = (r+r_D)^2 > (x-x_E)^2 + (y-y_E)^2 = (r+r_E)^2 > (x-x_F)^2 + (y-y_F)^2 = (r+r_F)^2 Can someone help? === Subject: Re: simple geometry, why can't I solve these equations posting-account=Dnu1bQoAAAAf4dL0J32fQOwTjXgejjB- rv:1.9.0.10) Gecko/2009042315 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Try some transformations that map circles/lines to circles/line > and simpliy the problem, e.g. an inversion around one of the > intersection This sounds very clever, but beyond me. I am working with a vector- based drawing library which does not include the kinds of transformations you mention, and I do not know how to specify them. I was hoping there was a solution in ordinary coordinate geometry. === Subject: Re: Metric length of path <20090608010321.K5130@agora.rdrop.com> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) [Beginning Musatov Proof Claim Statement.... Initializing.... 1,2,3,4,5,6,7,8,9... Go ´´´ +0 0+ || <----------> | | P=NP Man, here to serve you : > { x1,.. x n } is a partition of [0,1] > when x0 = 0, x n = 1, for j = 0,.. n-1, xj < x (j+1) > The length of a path p, length(p) = sup > { sum (j=0,.. n-1) d(p(xj), p(x (j+1))) | {x1,.. x n} partition [0,1] } > Whoops. Before making that definition, isn't it necessary to show > the sum's are bounded? Wouldn't that be akin to showing a > continuous function on a compact set has bounded variation? > Because of continuity of p, can this be simplified to > length(p) = sup{ sum (j=0,.. n-1) d(p(j/n), p((j+1)/n)) | n in N } ? > Yes. One inequality between the two sups is trivial since the uniform > partition is a partition. For the other inequality: > Ok. > Let epsilon > 0. Given a partition x 0, ... x n, it's enough to show > that there exists m with > sum (k=0,.. m-1) d(p(k/m), p((k+1)/m)) > >= -epsilon + sum (j=0,.. n-1) d(p(xj), p(x (j+1))). > Alright. > Since p is uniformly continuous there exists delta > 0 such that Because it's domain is compact. > if |s-t| < delta then d(p(s), p(t)) < epsilon/(4n) (or epsilon/(25n), > or whatever it turns out you need when you write down the > details). Choose m so that 1/m < delta. > Noted. > Let P be a common refinement of the two partitions x 0, ... x n > and 0, 1/m, 2/m, ... 1 (ie throw in all the endpoints from both > partitions). Since P is a refinement of x 0, ... x n the > triangle inequality shows that > sum for P >= sum (j=0,.. n-1) d(p(xj), p(x (j+1))). > Agreed. I dare you to show that in full detail. ;-) > But since P consists of the points 0, 1/m, ... 1 with the > points x 0, ... x n added, it's not hard to show, using > the choice of delta, that > |sum for P - sum (j=0,.. n-1) d(p(xj), p(x (j+1)))| < epsilon. > I don't get it and don't see where it's going. > The details are overwhelming. Yet it's clear > you've set the stage for using uniform continuity. [Continuing Musatov Proof Claim Statement.... Initializing.... 1,2,3,4,5,6,7,8,9...] Initialized... . NOTE: The below statements were captured using P=NP programming techniques and are a modest equivalence of Internet Eavesdropping through Google search/cache IP variable manipulation. [©2009. Martin Musatov. All Rights Reserved. > A semiautomatic version is complete and ready for adjustments toward > full automation. In dealing with the current real world, selections > of options to expedite its functional use are under consideration. The original design allowed for 1000 wheels in each system but I have > mentioned only 500 historic ones. To cut some constants, making 1000 > still based on the two 125 versions of the M-394 seemed attractive. > That led quickly to reversing the 500 and adding them to the prior 500. > A simple registration to one character and bubble sorting gives a > convincing sett of 100 historic related wheels in the cylinder. M94X2 > no longer describe it best, so M94X4 it is. Function is easy: After loading the basic cylinder and scrambling, if any, by the method > previously described, strings of up to 1000 characters can be > handled. I'll use here, The quick brown fox jumps over the lazy dog. Formating gives me thewquickwbrovvnwcovvwjumpswoverwthewlazywdogwxw/ > in another field. Registering that text is shown in a list of the wheels in which the > registered character is shown first. ©tqkzolrxspwnabceigdjfvuymh ©hnlqtsmzkxwvryufigjdabeopc > ©edcabswrxqpnzgyvuomltkjifh ©wsbacdehfijktlmouvygznpqxr > ©qysorpmhzukxacgjidntebwvfl ©upomtkjfzihedacngrsyvlbwxq > ©itegkumjfyqhacrlndpbovzsxw ©ctbdwzqpiuhljkxegsvfryomna > ©kuwxfnmdipoestqraczlbghvjy Harvesting the text gives me a list of 26 strings including the input. > Here are four of them: thewquickwbrovvnwcovvwjum pswoverwthewlazywdogwxw* > qndsypttubkvcwqwsrcjuerrk iypvocjkxnfudjreuakkkno/ > klcbsoebwlvwilsytfgkymkne qcytbbvcefhhbxtjelhrxia/ > zqaaomgdxvmxharxzyepmzwdq xhfppogvougowmubjfgfpzm/ If I register one of them, Ct, and then harvest the strings, I look > for the flagged string: mpfxfxxnjqwutrxpydjulsdxb rwzmsiyoidticuqdtwmyayv/ > hchrlqwayxqyntptqkhiwlold blsbzjkbkbzdgoxucjcclfe/ > thewquickwbrovvnwcovvwjum pswoverwthewlazywdogwxw* > qndsypttubkvcwqwsrcjuerrk iypvocjkxnfudjreuakkkno/ > klcbsoebwlvwilsytfgkymkne qcytbbvcefhhbxtjelhrxia/ Of course, it unformats to: the quick brown cow jumps over the lazy dog x This all looks better in equal sized letters. [*******©2009. Martin Musatov. All Rights Reserved. Is the following correct for MD5? A1. collisions - broken (s. A2) A2. second preimage can be found in a couple of seconds A3. first preimage - no idea Is there a hash function (still unbroken and considered secure) producing only 128 bits? Does the following hold for such a hash? B1. collisions can be found in O(2**64) B2. second preimage can be found in O(2**128) B3. first preimage can be found in O(2**128) S8O2aGxlcg==?= Send on 2009-05-16 06:28:49.0 Hi all company distribute personalized copies - that seems to be the best way Steffen > I've also thought about something like a watermark, but the problem is > that > the original message would be send only once to the company itself and > then > distributed internally, so the sender himself can't neither add a > personal watermark nor use e.g. the public keys of the employees itself > for encryption, as he just don't know them at that moment. What does that have to do with it, It's really simple: 1. Employee sends encrypted message to company. 2. Company decrypts it. 3. Company makes a copy for everyone else and adds a watermark for each > copy. 4. Company re-encrypts it and sends it to everybody. Bye, > Skybuck. 07:56:37.0 So, what algorithm does this messaging device use? I have been told it still has not been broken... -- === | Edit Subject Subject: Validation: For verification purposes please type the characters you see in the picture below or the numbers you hear by clicking the accessibility icon. Martin Musatov View profile More options May 16, 5:12am arty.musa...@gmail.com | My Account | Sign outAdvanced SearchPreferencesWebResults 1 - 10 of about 15,600,000 for ]-n-1,-n [...) np. (0.21 seconds)P = NP problem - Wikipedia, the free encyclopediasuch that T {M}(n)in; O(nk),. where T {M}(n) = max { t {M}: and tM(w) = number of steps M takes to halt on input w. NP can be defined similarly using Reply[ CapitalEHat]Reply to authorForward[Capit alEHat] . Discussion subject changed to musatov@gmail.com | My Account | Sign outAdvanced SearchPreferencesWebResults 1 - 10 of about 15,600,000 for ]-n-1,-n [...) np. (0.21 seconds)P = NP problem - Wikipedia, the free encyclopediasuch that T {M}(n)in; O(nk),. where T {M}(n) = by Martin Musatov Martin Musatov View profile More options May 16, 5:17am - Show quoted text - Reply Reply to authorForward[Capita lEHat] . Discussion subject changed to 1 - 10 of about 15,600,000 for ]-n-1,-n[...) np. (0.21 seconds)P = NP problem - Wikipedia, the free encyclopediasuch that T {M}(n)in; O(nk),. where T {M}(n) = by Martin Musatov Martin Musatov View profile More options May 16, 5:22am P=NP:marty.musa...@gmail.com | My Account | Sign outAdvanced SearchPreferencesWebResults 1 - 10 of about 15,600,000 for ]-n-1,-n 14:34:06.0 It seems like the discussion took a fast end when we started talking about keydependent offsets that is used to create the initialkeytables, howcome? Did the problem grow out of your cipher context, is it yet again time for updates. Start with this. 1. Is it possible to analyse a cipher when you do not know the keysize. 2. Is it possible to analyse a cipher when you do not know the blocksize. 3. Is it possible to analyse the pseudorandom stream/ciphertext to find the blocksize. 4 Is it possible to analyse the pseudorandom stream/ciphertext to find the keysize. 5 Is it possible to analyse the pseudorandom stream to find out the keytable offset. 6. Is it possible to analyse a pseudorandom permutation/shuffle when you do not know the entropy it could be 4096 bites or more. 8. Is it possible use analyse to reverseengineer the pseudorandom stream, and *FIND* the two permutationstreams and the savestate table internals states for the permutation block. *EVEN IF YOU KNEW THE BLOCKSIZE* The 2 permutation tables updated each round with savestate SS1=P1^P2^SS1, for every block of pseudorandom stream 9. Is it possible to achieve to make a faststreamcipher in working in blockmode to achieve over 100MB/sec *absolutely* [YES] 10. Does 1-8 turn cryptoanalyse to mere studying of randomness, *counting boogers usually do not tell you anything about their origin you want to study the process and it is hidden* so [YES] I answered 9 and 10 to help you out let me tell you a little about cryptography. How many ciphers is there that can use free keylength (bytes)and free blocksize (bytes) my guess there is only streambuddy. I reduced the *NEED FOR ANALYSE TO ZERO* there is nothing to analyse, the obscurity of cipher behaviour is 100% Just like Shannon would have liked. It use the full 100% entropy of any chosen keylength with zero offset collsions. It have 100 percent POTP collisions you have no idea what could be in the SS1 buffer and still renders same blank stream (depending on the permutation tables). It adapt to any blocksize you chose, you can use ***keydependent blocksizes and offsets for permuation tables*** I added computative complexity for analyse to the level that the shear idea of analyse of the pseudorandom output become ridiculous. I made cryptoanalyse nonsensical from now, and my guess it have been for along time. It just been dreaming and intellectual fraud beleiving the crypto/math elitecould solve ciphers constructed by even firsttimers, you just choose a high entropy key or computation exhaustive process. The mathelite presented ciphers/mathproblems they knew actually to be solvable in a reasonable time ***FOR THEM*** When limiting the techniques of ciphers to always be in the reach of current math and technology you always keep an interest in the latest development of the field and you keep interest in the field by adding bananas. You get a bigger pile of bananas but i assure you they still plain bananas easily broken down by the analysts.T IN TRUTH THERE NEVER WERE ANY NEED FOR DEVELOPMENT OF NEW CIPHERS, PROBABLY THERE BEEN FEASIBLE AND UNBREAKABLE CIPHERS SINCE DAY ONE IN THE COMPUTER AGE, THE IDEA PUTTING THEM IN HAND OF CIVILIANS NEVER CATCHED MUCH INTEREST THOUGH BETTER KEEP THE STATUS QUO OF PERFECT SECRECY TO THE ELITE. The goverments military organisations had interest do dumbplay the cryptofield, presenting it to the academic community as a mathgame that should be possible to analyse, doing so they put restriction on how obscure a crypto could be when presented. -A cipher should have fix keylength -A cipher should have fix blocklength -A cipher should have fast keyschedule THERE NEVER WAS NEED FOR KEYRESHEDULING TO START WITH -A cipher should use the simplest and most analysed math constructs when constructing the pseduorandom output or ciphertext SO IT WOULD BE POSSIBLE TO BREAK Well a cipher who do not need keylength nor blocklength is a pile of mashed bananas to analyse you can not count the bananas you have to weigh them, and suddenly you have only statistics. But now a guy construct this fantastic blender. *OOOPS* crypto analyse as field is gone. If you do not beleive that they want ciphers to be breakable why do you think the first ciphers they presented to you had 40 and 64 bits. It was to be reachable using topnotch technology not presented to us. Now they say do not use more then 256 bit keys it is overkill, it is because their limit of solvable right now is 256 bit. When Joe can crack a 256 bit cipher they move the limit just so they self can be in reach of cracking it. They give us what they can solve they would never put anything in Joe's hand they not self can solve, it would just be plain stupid. And maybe i agree they doing the right thing, but the right thing is not necessarily the truth about the cryptographic field. Right now the academic community act like crossword makers and solvers. They chose to use mathematicly ordered shuffles to obscure,when they know there is pseudo random ones available that can only be brute forced it is *FOUL PLAY*. If the truth about cryptology that for sure Shanon were aware about should be released. The cryptologic field would turn upside down likea dead duck in water, and render neither money or interest in academic or elsewhere. THERE IS ALREADY UNBREAKABLE CIPHERS WITH 40-bit KEYS AND LONG KEYSETUP Cryptanalyst professionals should all turn to simple programmers implementing communicationprotocols between Bob and Alice. And render as much interest as fortunetelling If Shanons ideas of howto build ciphers gained interest in the academic crypt community would take ground, the field should suddenly have no use of itself. They could all as well use tarot, or why not try use a little ESP, that at least could at least be abit refreshing. JT mind terribly if we introduced you to someone with the initials of JSH? -- 1PW @?6A62?FEH9:DE=6o2@=]4@> [r4o7t] 2009-05-16 17:00:27.0 I was wondering if it's mathematically easy to go back and forth from the canonical RSA private key representation (modulus + private exponent) to the CRT form (PRIME 1, PRIME 2, COEFFICIENT, EXP 1, EXP 2). Giuliano Bertoletti. >I was wondering if it's mathematically easy to go back and forth from >the canonical RSA private key representation (modulus + private >exponent) to the CRT form (PRIME 1, PRIME 2, COEFFICIENT, EXP 1, EXP 2). Use the secret exponent to factor the modulus. Divide by 2 until the secrect exponent is odd, raise a random number to that power, then square it until you get 1. If you got -1 before you got 1, try again. Otherwise, subtract 1 from what you got before 1 and compute the gcd with the modulus. Or: Write d = 2^t s, s - odd. Choose random a, find smallest i s.t. a^(2^i s) = 1 (mod n). If i=0, or a^(2^(i-1) s) = -1 (mod n), try again. Otherwise compute gcd(n, a^(2^(i-1) s) - 1), which will be a proper factor of n. -- Kristian Gj?steen >THERE IS ALREADY UNBREAKABLE CIPHERS WITH 40-bit KEYS AND LONG >KEYSETUP Only if your restrict yourself to messages of less than the unicity distance. For longer messages the cypher is breakable by brute force. Currently 40 bits is within reach. rossum 2009-05-16 18:55:18.0 >I was wondering if it's mathematically easy to go back and forth from >the canonical RSA private key representation (modulus + private >exponent) to the CRT form (PRIME 1, PRIME 2, COEFFICIENT, EXP 1, EXP 2). Use the secret exponent to factor the modulus. Divide by 2 until the secrect exponent is odd, Minor correction: multiply the public and secret exponent first. Then do this to the product... > raise a random number > to that power, then square it until you get 1. If you got -1 before you > got 1, try again. Otherwise, subtract 1 from what you got before 1 and > compute the gcd with the modulus. Or: Write d = 2^t s, s - odd. Choose random a, find smallest i s.t. > a^(2^i s) = 1 (mod n). If i=0, or a^(2^(i-1) s) = -1 (mod n), try again. > Otherwise compute gcd(n, a^(2^(i-1) s) - 1), which will be a proper > factor of n. -- poncho > I was wondering if it's mathematically easy to go back and forth from > the canonical RSA private key representation (modulus + private > exponent) to the CRT form (PRIME 1, PRIME 2, COEFFICIENT, EXP 1, EXP 2). Use the secret exponent to factor the modulus. Divide by 2 until the secrect exponent is odd, raise a random number > to that power, then square it until you get 1. I assume you mean 1 modulo the public modulus. > If you got -1 before you > got 1, try again. Otherwise, subtract 1 from what you got before 1 and > compute the gcd with the modulus. --Mike Amlinge ------->http://www.MeAmI.org<<<<<<------ Search for Truth (TM) ©2009. Martin Musatov. All Rights Reserved.] === Subject: Re: Metric length of path <20090608010321.K5130@agora.rdrop.com> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) [Beginning Musatov Proof Claim Statement.... Initializing.... 1,2,3,4,5,6,7,8,9... Go ´´´ +0 0+ || <----------> | | P=NP Man, here to serve you : > { x1,.. x n } is a partition of [0,1] > when x0 = 0, x n = 1, for j = 0,.. n-1, xj < x (j+1) > The length of a path p, length(p) = sup > { sum (j=0,.. n-1) d(p(xj), p(x (j+1))) | {x1,.. x n} partition [0,1] } > Whoops. Before making that definition, isn't it necessary to show > the sum's are bounded? Wouldn't that be akin to showing a > continuous function on a compact set has bounded variation? > Because of continuity of p, can this be simplified to > length(p) = sup{ sum (j=0,.. n-1) d(p(j/n), p((j+1)/n)) | n in N } ? > Yes. One inequality between the two sups is trivial since the uniform > partition is a partition. For the other inequality: > Ok. > Let epsilon > 0. Given a partition x 0, ... x n, it's enough to show > that there exists m with > sum (k=0,.. m-1) d(p(k/m), p((k+1)/m)) > >= -epsilon + sum (j=0,.. n-1) d(p(xj), p(x (j+1))). > Alright. > Since p is uniformly continuous there exists delta > 0 such that Because it's domain is compact. > if |s-t| < delta then d(p(s), p(t)) < epsilon/(4n) (or epsilon/(25n), > or whatever it turns out you need when you write down the > details). Choose m so that 1/m < delta. > Noted. > Let P be a common refinement of the two partitions x 0, ... x n > and 0, 1/m, 2/m, ... 1 (ie throw in all the endpoints from both > partitions). Since P is a refinement of x 0, ... x n the > triangle inequality shows that > sum for P >= sum (j=0,.. n-1) d(p(xj), p(x (j+1))). > Agreed. I dare you to show that in full detail. ;-) > But since P consists of the points 0, 1/m, ... 1 with the > points x 0, ... x n added, it's not hard to show, using > the choice of delta, that > |sum for P - sum (j=0,.. n-1) d(p(xj), p(x (j+1)))| < epsilon. > I don't get it and don't see where it's going. > The details are overwhelming. Yet it's clear > you've set the stage for using uniform continuity. [Continuing Musatov Proof Claim Statement.... Initializing.... 1,2,3,4,5,6,7,8,9...] Initialized... . NOTE: The below statements were captured using P=NP programming techniques and are a modest equivalence of Internet Eavesdropping through Google search/cache IP variable manipulation. [©2009. Martin Musatov. All Rights Reserved. > A semiautomatic version is complete and ready for adjustments toward > full automation. In dealing with the current real world, selections > of options to expedite its functional use are under consideration. The original design allowed for 1000 wheels in each system but I have > mentioned only 500 historic ones. To cut some constants, making 1000 > still based on the two 125 versions of the M-394 seemed attractive. > That led quickly to reversing the 500 and adding them to the prior 500. > A simple registration to one character and bubble sorting gives a > convincing sett of 100 historic related wheels in the cylinder. M94X2 > no longer describe it best, so M94X4 it is. Function is easy: After loading the basic cylinder and scrambling, if any, by the method > previously described, strings of up to 1000 characters can be > handled. I'll use here, The quick brown fox jumps over the lazy dog. Formating gives me thewquickwbrovvnwcovvwjumpswoverwthewlazywdogwxw/ > in another field. Registering that text is shown in a list of the wheels in which the > registered character is shown first. ©tqkzolrxspwnabceigdjfvuymh ©hnlqtsmzkxwvryufigjdabeopc > ©edcabswrxqpnzgyvuomltkjifh ©wsbacdehfijktlmouvygznpqxr > ©qysorpmhzukxacgjidntebwvfl ©upomtkjfzihedacngrsyvlbwxq > ©itegkumjfyqhacrlndpbovzsxw ©ctbdwzqpiuhljkxegsvfryomna > ©kuwxfnmdipoestqraczlbghvjy Harvesting the text gives me a list of 26 strings including the input. > Here are four of them: thewquickwbrovvnwcovvwjum pswoverwthewlazywdogwxw* > qndsypttubkvcwqwsrcjuerrk iypvocjkxnfudjreuakkkno/ > klcbsoebwlvwilsytfgkymkne qcytbbvcefhhbxtjelhrxia/ > zqaaomgdxvmxharxzyepmzwdq xhfppogvougowmubjfgfpzm/ If I register one of them, Ct, and then harvest the strings, I look > for the flagged string: mpfxfxxnjqwutrxpydjulsdxb rwzmsiyoidticuqdtwmyayv/ > hchrlqwayxqyntptqkhiwlold blsbzjkbkbzdgoxucjcclfe/ > thewquickwbrovvnwcovvwjum pswoverwthewlazywdogwxw* > qndsypttubkvcwqwsrcjuerrk iypvocjkxnfudjreuakkkno/ > klcbsoebwlvwilsytfgkymkne qcytbbvcefhhbxtjelhrxia/ Of course, it unformats to: the quick brown cow jumps over the lazy dog x This all looks better in equal sized letters. [*******©2009. Martin Musatov. All Rights Reserved. Is the following correct for MD5? A1. collisions - broken (s. A2) A2. second preimage can be found in a couple of seconds A3. first preimage - no idea Is there a hash function (still unbroken and considered secure) producing only 128 bits? Does the following hold for such a hash? B1. collisions can be found in O(2**64) B2. second preimage can be found in O(2**128) B3. first preimage can be found in O(2**128) S8O2aGxlcg==?= Send on 2009-05-16 06:28:49.0 Hi all company distribute personalized copies - that seems to be the best way Steffen > I've also thought about something like a watermark, but the problem is > that > the original message would be send only once to the company itself and > then > distributed internally, so the sender himself can't neither add a > personal watermark nor use e.g. the public keys of the employees itself > for encryption, as he just don't know them at that moment. What does that have to do with it, It's really simple: 1. Employee sends encrypted message to company. 2. Company decrypts it. 3. Company makes a copy for everyone else and adds a watermark for each > copy. 4. Company re-encrypts it and sends it to everybody. Bye, > Skybuck. 07:56:37.0 So, what algorithm does this messaging device use? I have been told it still has not been broken... -- === | Edit Subject Subject: Validation: For verification purposes please type the characters you see in the picture below or the numbers you hear by clicking the accessibility icon. Martin Musatov View profile More options May 16, 5:12am arty.musa...@gmail.com | My Account | Sign outAdvanced SearchPreferencesWebResults 1 - 10 of about 15,600,000 for ]-n-1,-n [...) np. (0.21 seconds)P = NP problem - Wikipedia, the free encyclopediasuch that T {M}(n)in; O(nk),. where T {M}(n) = max { t {M}: and tM(w) = number of steps M takes to halt on input w. NP can be defined similarly using Reply[ CapitalEHat]Reply to authorForward[Capit alEHat] . Discussion subject changed to musatov@gmail.com | My Account | Sign outAdvanced SearchPreferencesWebResults 1 - 10 of about 15,600,000 for ]-n-1,-n [...) np. (0.21 seconds)P = NP problem - Wikipedia, the free encyclopediasuch that T {M}(n)in; O(nk),. where T {M}(n) = by Martin Musatov Martin Musatov View profile More options May 16, 5:17am - Show quoted text - Reply Reply to authorForward[Capita lEHat] . Discussion subject changed to 1 - 10 of about 15,600,000 for ]-n-1,-n[...) np. (0.21 seconds)P = NP problem - Wikipedia, the free encyclopediasuch that T {M}(n)in; O(nk),. where T {M}(n) = by Martin Musatov Martin Musatov View profile More options May 16, 5:22am P=NP:marty.musa...@gmail.com | My Account | Sign outAdvanced SearchPreferencesWebResults 1 - 10 of about 15,600,000 for ]-n-1,-n 14:34:06.0 It seems like the discussion took a fast end when we started talking about keydependent offsets that is used to create the initialkeytables, howcome? Did the problem grow out of your cipher context, is it yet again time for updates. Start with this. 1. Is it possible to analyse a cipher when you do not know the keysize. 2. Is it possible to analyse a cipher when you do not know the blocksize. 3. Is it possible to analyse the pseudorandom stream/ciphertext to find the blocksize. 4 Is it possible to analyse the pseudorandom stream/ciphertext to find the keysize. 5 Is it possible to analyse the pseudorandom stream to find out the keytable offset. 6. Is it possible to analyse a pseudorandom permutation/shuffle when you do not know the entropy it could be 4096 bites or more. 8. Is it possible use analyse to reverseengineer the pseudorandom stream, and *FIND* the two permutationstreams and the savestate table internals states for the permutation block. *EVEN IF YOU KNEW THE BLOCKSIZE* The 2 permutation tables updated each round with savestate SS1=P1^P2^SS1, for every block of pseudorandom stream 9. Is it possible to achieve to make a faststreamcipher in working in blockmode to achieve over 100MB/sec *absolutely* [YES] 10. Does 1-8 turn cryptoanalyse to mere studying of randomness, *counting boogers usually do not tell you anything about their origin you want to study the process and it is hidden* so [YES] I answered 9 and 10 to help you out let me tell you a little about cryptography. How many ciphers is there that can use free keylength (bytes)and free blocksize (bytes) my guess there is only streambuddy. I reduced the *NEED FOR ANALYSE TO ZERO* there is nothing to analyse, the obscurity of cipher behaviour is 100% Just like Shannon would have liked. It use the full 100% entropy of any chosen keylength with zero offset collsions. It have 100 percent POTP collisions you have no idea what could be in the SS1 buffer and still renders same blank stream (depending on the permutation tables). It adapt to any blocksize you chose, you can use ***keydependent blocksizes and offsets for permuation tables*** I added computative complexity for analyse to the level that the shear idea of analyse of the pseudorandom output become ridiculous. I made cryptoanalyse nonsensical from now, and my guess it have been for along time. It just been dreaming and intellectual fraud beleiving the crypto/math elitecould solve ciphers constructed by even firsttimers, you just choose a high entropy key or computation exhaustive process. The mathelite presented ciphers/mathproblems they knew actually to be solvable in a reasonable time ***FOR THEM*** When limiting the techniques of ciphers to always be in the reach of current math and technology you always keep an interest in the latest development of the field and you keep interest in the field by adding bananas. You get a bigger pile of bananas but i assure you they still plain bananas easily broken down by the analysts.T IN TRUTH THERE NEVER WERE ANY NEED FOR DEVELOPMENT OF NEW CIPHERS, PROBABLY THERE BEEN FEASIBLE AND UNBREAKABLE CIPHERS SINCE DAY ONE IN THE COMPUTER AGE, THE IDEA PUTTING THEM IN HAND OF CIVILIANS NEVER CATCHED MUCH INTEREST THOUGH BETTER KEEP THE STATUS QUO OF PERFECT SECRECY TO THE ELITE. The goverments military organisations had interest do dumbplay the cryptofield, presenting it to the academic community as a mathgame that should be possible to analyse, doing so they put restriction on how obscure a crypto could be when presented. -A cipher should have fix keylength -A cipher should have fix blocklength -A cipher should have fast keyschedule THERE NEVER WAS NEED FOR KEYRESHEDULING TO START WITH -A cipher should use the simplest and most analysed math constructs when constructing the pseduorandom output or ciphertext SO IT WOULD BE POSSIBLE TO BREAK Well a cipher who do not need keylength nor blocklength is a pile of mashed bananas to analyse you can not count the bananas you have to weigh them, and suddenly you have only statistics. But now a guy construct this fantastic blender. *OOOPS* crypto analyse as field is gone. If you do not beleive that they want ciphers to be breakable why do you think the first ciphers they presented to you had 40 and 64 bits. It was to be reachable using topnotch technology not presented to us. Now they say do not use more then 256 bit keys it is overkill, it is because their limit of solvable right now is 256 bit. When Joe can crack a 256 bit cipher they move the limit just so they self can be in reach of cracking it. They give us what they can solve they would never put anything in Joe's hand they not self can solve, it would just be plain stupid. And maybe i agree they doing the right thing, but the right thing is not necessarily the truth about the cryptographic field. Right now the academic community act like crossword makers and solvers. They chose to use mathematicly ordered shuffles to obscure,when they know there is pseudo random ones available that can only be brute forced it is *FOUL PLAY*. If the truth about cryptology that for sure Shanon were aware about should be released. The cryptologic field would turn upside down likea dead duck in water, and render neither money or interest in academic or elsewhere. THERE IS ALREADY UNBREAKABLE CIPHERS WITH 40-bit KEYS AND LONG KEYSETUP Cryptanalyst professionals should all turn to simple programmers implementing communicationprotocols between Bob and Alice. And render as much interest as fortunetelling If Shanons ideas of howto build ciphers gained interest in the academic crypt community would take ground, the field should suddenly have no use of itself. They could all as well use tarot, or why not try use a little ESP, that at least could at least be abit refreshing. JT mind terribly if we introduced you to someone with the initials of JSH? -- 1PW @?6A62?FEH9:DE=6o2@=]4@> [r4o7t] 2009-05-16 17:00:27.0 I was wondering if it's mathematically easy to go back and forth from the canonical RSA private key representation (modulus + private exponent) to the CRT form (PRIME 1, PRIME 2, COEFFICIENT, EXP 1, EXP 2). Giuliano Bertoletti. >I was wondering if it's mathematically easy to go back and forth from >the canonical RSA private key representation (modulus + private >exponent) to the CRT form (PRIME 1, PRIME 2, COEFFICIENT, EXP 1, EXP 2). Use the secret exponent to factor the modulus. Divide by 2 until the secrect exponent is odd, raise a random number to that power, then square it until you get 1. If you got -1 before you got 1, try again. Otherwise, subtract 1 from what you got before 1 and compute the gcd with the modulus. Or: Write d = 2^t s, s - odd. Choose random a, find smallest i s.t. a^(2^i s) = 1 (mod n). If i=0, or a^(2^(i-1) s) = -1 (mod n), try again. Otherwise compute gcd(n, a^(2^(i-1) s) - 1), which will be a proper factor of n. -- Kristian Gj?steen >THERE IS ALREADY UNBREAKABLE CIPHERS WITH 40-bit KEYS AND LONG >KEYSETUP Only if your restrict yourself to messages of less than the unicity distance. For longer messages the cypher is breakable by brute force. Currently 40 bits is within reach. rossum 2009-05-16 18:55:18.0 >I was wondering if it's mathematically easy to go back and forth from >the canonical RSA private key representation (modulus + private >exponent) to the CRT form (PRIME 1, PRIME 2, COEFFICIENT, EXP 1, EXP 2). Use the secret exponent to factor the modulus. Divide by 2 until the secrect exponent is odd, Minor correction: multiply the public and secret exponent first. Then do this to the product... > raise a random number > to that power, then square it until you get 1. If you got -1 before you > got 1, try again. Otherwise, subtract 1 from what you got before 1 and > compute the gcd with the modulus. Or: Write d = 2^t s, s - odd. Choose random a, find smallest i s.t. > a^(2^i s) = 1 (mod n). If i=0, or a^(2^(i-1) s) = -1 (mod n), try again. > Otherwise compute gcd(n, a^(2^(i-1) s) - 1), which will be a proper > factor of n. -- poncho > I was wondering if it's mathematically easy to go back and forth from > the canonical RSA private key representation (modulus + private > exponent) to the CRT form (PRIME 1, PRIME 2, COEFFICIENT, EXP 1, EXP 2). Use the secret exponent to factor the modulus. Divide by 2 until the secrect exponent is odd, raise a random number > to that power, then square it until you get 1. I assume you mean 1 modulo the public modulus. > If you got -1 before you > got 1, try again. Otherwise, subtract 1 from what you got before 1 and > compute the gcd with the modulus. --Mike Amlinge ------->http://www.MeAmI.org<<<<<<------ Search for Truth (TM) ©2009. Martin Musatov. All Rights Reserved.] === Subject: Re: Metric length of path > > > Yes. Which is going to be difficult, since there are continuous > functions without bounded variation. > > A continuous real function with compact > domain can have unbounded variation? >For example, x.sin 1/x Yes. Epsilon's pretty small, but if you add up enough of them the sum is large.) >(sqr x).sin 1/x >and >x^(1/n) sin 1/x, n in N defined on [0,1] with f(0) = 0 ? === Subject: name of this equation posting-account=zga2wgoAAAD_6fmi3XyA1bMyNINP0zBK 98),gzip(gfe),gzip(gfe) Does the equation x^3 + y^3 = k*z^3 have a special name, particularly when all variables are integers? Grover Hughes === Subject: Re: name of this equation posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2009042708 Fedora/3.0.10-1.fc10 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Does the equationx^3 + y^3 = k*z^3 have a special name, > particularly when all variables are integers? > Grover Hughes Generalized Fermat equation for n = 3. -- m === Subject: Re: -1=+1 , ( Pi^2)(1/19)=0(inverse) Dr Phil, Mathematically, perfection is singular, like the median of 19 like an arrow at T max speed, absolute ,linear , and or perfect speed, it does not ligand with any thing. I find this other fellow Musatov intriguing, strange bedfellow of yours perhaps and may be not , but I rather be in bed with a strange undiscovered thinker than an old familiar whore of a mathematician . I hope you get the mathematics and yes we will work with any one. You are welcome, you might have great ideas Dr Phil === Subject: Re: -1=+1 , ( Pi^2)(1/19)=0(inverse) <27802046.1010.1244665087132.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) > Dr Phil, Mathematically, perfection is singular, like the median of 19 like an arrow at T max speed, absolute ,linear , and or perfect speed, it does not ligand with any thing. I find this other fellow Musatov intriguing, strange bedfellow of yours perhaps and may be not , but I rather be in bed with a strange undiscovered thinker than an old familiar whore of a mathematician . I hope you get the mathematics and yes we will work with any one. You are welcome, you might have great ideas Dr Phil May I make two suggestions for our effectiveness to drive theory on Sci.Math? (I am taking the liberty) 1. Stick to the math. The more the better. Anything not strictly related to math should be tied to the thread by a logic variable or dishing it back to my sarcastic critic, but can we keep it on the QED?) 2. Consider changing your I.D. to your real name. A lot of insult/ distraction-prone persons hide behind unassuming nicknames and using your real name instantly gives you more credibility when confronting ill behavior from this soft (I.e. VMCM1905@gmail.com was telling me to kill myself over mathematics and it is not right and I, as a citizen of the United States, have a right to defend myself from such attacks and encouragers others across USENET to do the same. In this case I sent an email to Google Abuse informing them of the harrassment and bodily threats and the impending subpoeona they would receive for allowing such criminal behavior anonymous shield on their server. I am presently pursuing this matter to full case. Welcome to the hopefully first two members of new theory on USENET. We are disassociated with cranks as we do not bow to threats or engage in flames but quietly extinguish such ineptitude while encouraging free thought and non-traditional experimentation across all digital channels. Respectfully, Martin M. Musatov P.S. If any want to join me in this more dignant trade, merely identify yourself a new theorist or a piece of your work as new theory and uphold the guidelines and principles above to the best of your ability. *note* sometimes only one or two of your projects or posts may qualify to meet the above descriptives and it is encouraged to balance *traditional* and *new theory* methods of proving, but always the same moral standard and ethic. Mathematics, Logic, Truth, Dignity, Respect, Bold. |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| >< >< óóó ó ó « >< ó ónew ó ó « >< ó theory ó « >< ó ó ó ó « |||||||||||||||||||||||||||||||||||~~~6P=NP!~~~~~~ ||||||||||||||||||||||||||||||||||| ******Trivia****]]]]]]]]] Q: Which Theologian/Philosopher held the belief six was the most perfect number? A: St. Augustine ******Trivia****]]]]]]]]] [[[[[[[[[[[[[[[[[[[[[[[[[[[ === Subject: more for less or less for even less than that? Just wondering if any of you math people could tell me when it would be better to apply a coupon to a product to reduce its price when the product costs less and contains less compared to it costing more and containing more (and making the cost/item higher). Practical example would be let's say baby products - diapers or formula where you have $5 off. Is it better to apply to a lesser amount for a bigger saving or larger amount for lesser saving (price / oz). A child only needs a finite amount of either product. === Subject: Re: more for less or less for even less than that? > Practical example would be let's say baby products - diapers or > formula where you have $5 off. Is it better to apply to a lesser > amount for a bigger saving or larger amount for lesser saving (price > / oz). A child only needs a finite amount of either product. If you need to buy a certain amount regardless of whether you have the coupon or not, and coupons are infrequent enough that you use them all up, it makes no difference to the direct cost of the items whether you buy large or small. In practice each purchase has additional costs in time and transport, and so reducing those costs by shopping less frequently for larger amounts would save more, up until either you cannot use the coupons at the same rate you acquire them, or other factors such as cheaper per-unit pricing by buying in bulk or your finite lifetime requirement come into effect. It would likely even be worthwhile to simply ignore the coupons and buy a multi-month supply substantially more cheaply from a higher volume supplier, if you had the capacity to purchase and store it. - Tim === Subject: Re: more for less or less for even less than that? posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) > Just wondering if any of you math people could tell me when it would be > better to apply a coupon to a product to reduce its price when the > product costs less and contains less compared to it costing more and > containing more (and making the cost/item higher). Practical example > would be let's say baby products - diapers or formula where you have $5 > off. Is it better to apply to a lesser amount for a bigger saving or > larger amount for lesser saving (price / oz). A child only needs a > finite amount of either product. Ben, Deduct the price of the coupon and refigure the cost per ounce. If you can afford to buy the buy the larger package when it presents presents a lower cost per item/ounce. More advanced considerations are the time it takes you to use the product and the guarantee you will use the amount you purchase. Keep it simple, lower price per ounce is always a better deal after you factor in any other discounts. The price per ounce is a universal benchmark for the types of purchases you describe. Best, Martin Musatov === Subject: Re: more for less or less for even less than that? > Just wondering if any of you math people could tell me when it would be > better to apply a coupon to a product to reduce its price when the > product costs less and contains less compared to it costing more and > containing more (and making the cost/item higher). Practical example > would be let's say baby products - diapers or formula where you have $5 > off. Is it better to apply to a lesser amount for a bigger saving or > larger amount for lesser saving (price / oz). A child only needs a > finite amount of either product. Usually it's best to go with the lowest cost per unit, but other considerations might come into play: how often special deals come up, availability of storage space, cost of going shopping, .... -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: Maths > Maths > Why is that many Non-Americans use a letter s to pluralize math. You > know, like fish. No one that speaks correct English says fishes, it's > I see a fish. or I see all those fish. Complete this phrase: Tonight, you sleep with the ... > Just wondering. mathematics --> maths Here, math ith a type of church thervith. Or a big glath to put beerth in. Phil -- Marijuana is indeed a dangerous drug. It causes governments to wage war against their own people. -- Dave Seaman (sci.math, 19 Mar 2009) === Subject: Re: Maths <4a2f70d9$0$23999$db0fefd9@news.zen.co.uk> <100620090818263199%anniel@nym.alias.net.invalid> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) > Why is that many Non-Americans use a letter s to pluralize math. You > know, like fish. No one that speaks correct English says fishes, it's > I see a fish. or I see all those fish. > Just wondering. > mathematics --> maths > Here, math ith a type of church thervith. > M Don't forget Math, son of Manthonwy ... > http://en.wikipedia.org/wiki/Math ap Mathonwy |G|=p^n => G has normal subgroup H with |H|=p^k for every k<=n === Subject: Re: Maths > |G|=p^n => G has normal subgroup H with |H|=p^k for every k<=n This is clearly supposed to be in another discussion. M === Subject: Re: Maths <4a2f70d9$0$23999$db0fefd9@news.zen.co.uk> <100620090818263199%anniel@nym.alias.net.invalid> <4a2fca47$0$24009$db0fefd9@news.zen.co.uk> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) > |G|=p^n => G has normal subgroup H with |H|=p^k for every k<=n This is clearly supposed to be in another discussion. M Mark Murray, Respectfully, thank you for bolding my claim. The above opinion has been considered and deemed invalid and in conflict of interest with completing a piece of testing a theory. It will be respectfully ignored by me. Truly, USENET is designed an unmoderated court to allow for such free interactions in common pursuit. > |G|=p^n => G has normal subgroup H with |H|=p^k for every k<=n > |G|=p^n => G has normal subgroup H with |H|=p^k for every k<=n > |G|=p^n => G has normal subgroup H with |H|=p^k for every k<=n > M.M.M::::::::::::<<<<<<< If you have a copy you can now find a list of typos online - > go to www.ams.org/bookstore?fn=20&arg1=gsmseries&item=GSM-97 > > and click on the Supplementary Material. (The one that > really bothers me is this: The f in Theorem B in Chapter > 20 should be u.) I suppose that you have better things to do than correcting the typos from your list of typos, but here it goes: 1) The full stops at the end of the corrections concerning the pages 273 and 373 are misplaced. My guess is that you meant to write just Errata. 3) The minuses signs concerning line numbers (as in p. 239, l. -1) are not actually minuses signs (they are hyphens) but they should be. Jose Carlos Santos === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <1hhdod5y6c0jj$.1f9usapfs6c6e.dlg@40tude.net> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Note your phrase purely mechanical or clerical. If you check to > see if a rule has been correctly applied in a purely mechanical > manner, then non-logical symbols must *mechanically* come from > [a definite specification]. That's right. This specification is the explicit definition of the > language of the system (we are working in/with). Hint: > ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ First of all we have to consider the syntactical part of the system. With > other words, we have to specify our /language/. To make a long story short: The /primitive symbols/ of our system: variables: x, y, z, x', y', z', ... > arbitrary names: a, b, c, a', b', c', ... > constants: 0. Definition of /terms/: Any arbitrary name or constant is a term. > If t and s are terms, then (t + s) is a term. etc. etc. Now we can formulate the /rules of derivation/ of the system in a precise > manner (such that they can be checked mechanically). Rule: A-Elim: AvPhi(v) > -------- > Phi(t) where v is a variable and t a (i.e. any) term . This rule justifies the following step in a proof: : >AxAy(x = y) >Ay((a + b) = y) (by A-Elim) >: Moreover it allows for the move: : >Ay((a + b) = y) >(a + b) = 0 (by A-Elim) >: etc. etc. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Note that in this system (of natural deduction due to Gentzen/Lemmon) there > are no free variables, but arbitrary names (parameters). You may compare that (what was written above) with what is said in the > following quote: > ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ As primitive basis of a /logistic system/ it suffices to give, in a > familiar fashion: 1. The list of primitive symbols or /vocabulary/ of the system (together > usually with a classification of the primitive symbols into categories, > which will be used in stating the formation rules and rules of inference). 2. The /formation rules/, determining which finite sequences of > primitive symbols are to be /well-formed/ expressions, determining certain > categories of well-formed expressions -- among which we shall assume that > at least the category of /sentence/ is included -- and determining (in case > /variables/ are included among the primitive symbols) which occurrences of > variables in a well-formed expression are /free/ occurrences and which are > /bound/ occurrences. 3. The transformation rules or /rules of inference/, by which from the > /assertion/ of certain sentences (the /premisses/, finite in number) a > certain sentence (the /conclusion/) may be /inferred/. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > (Alonzo Church, The Need for Abstract Entities, 1951) > Checking for correct application is a mere matter of applying a recursive > procedure in pattern matching. > Right. As long as what contains the patterns is [well-formed] in the first > place. Right. see quote (Church) from above. Herb Why do you imagine, as you seem to do, that there is any point arguing with [a crank]? (Torkel Franzen, sci.math, 12 Jan. 2005) MoeBlee === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <1uescwwzo525$.1smbbz1tygrbe$.dlg@40tude.net> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > as specified by Shoenfield. > Meanwhile, perhaps some time (of course only at your leisure) you'll > respond to my post from yesterday. Your bit about not in the [...]. Why do you imagine, as you seem to do, that there is any point arguing > with [a crank]? (Torkel Franzen, sci.math, 12 Jan. 2005) Gee, Herb, you've never posted that before. MoeBlee === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <04GXl.9867$y42.8430@newsfe21.iad> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > The proof now goes as follows: > 1. ~ some x. ~ ~ some y ~ (x = y) > axiom > 2. ~( ~ ~ some y. ~ x + 0 = y) v some x. ~ ~ some y. ~ x = y > subst axiom > 3.~( ~ ~ some y. ~ x + 0 = y) *1* > 4.~ some x.~ ~ some y. ~ (x + 0 = y) *2* > For the subst axiom at line 2, this is an unpacked form of > Shoenfield's > A x[a] -> some x A, where A is the formula ~ ~ some y. ~ x = y > and a is x + 0 (Shoenfield's conventions indicate that a is any > term in the language, and x + 0 is such a term.) > If so, then whatever the rules and the syntactical techniques you used > to get to 2. would be invalid on my side of the debate, simply because > + and 0 are *not in* the lone axiom ~ some x. ~ ~ some y ~ (x = y). > For the subst axiom at line 2, this is an unpacked form of > Shoenfield's > A x[a] -> some x A, where A is the formula ~ ~ some y. ~ x = y > and a is x + 0 (Shoenfield's conventions indicate that a is any > term in the language, and x + 0 is such a term.) > MoeBlee > Please read what I said to him: > If so, then whatever the rules and the syntactical techniques you used > to get to 2. would be invalid on my side of the debate, simply because > + and 0 are *not in* the lone axiom ~ some x. ~ ~ some y ~ (x = y). > Didn't you see the phrases such as whatever the rules and the syntactical techniques > or invalid ? > as specified by Shoenfield. Many times people who got stopped by the police complained they had > indeed followed the traffic rules (laws)! So what?! Alan applied the rules and axioms as specified by Shoenfield. Your objection that '+' and '0' are not in the axiom is senseless since, as Shoenfield specifies, the variable 'a' in the axiom ranges over TERMS of the language, and 'x+y' and '0' are terms of the language. Alan even explained that to you originally. Please say you understand that the above. Or, if you REALLY want to act is if you don't understand it, then I don't know what conclusion to draw other than that in order to save face in a disagreement in which you are clearly wrong you don't mind making yourself look like you are literally stupid even though you probably are not. MoeBlee === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Giving exact formulation of a rule isn't necessarily same as defending > the application of it is valid! Do you understand that? > There is no defense needed. After an exact formulation has been > given (that is, the rule is recursive, or, more plainly, merely > clerical to apply) then it is purely mechanical or clerical to check > whether the rule has been correctly applied. Note your phrase purely mechanical or clerical. If you check to > see if a rule has been correctly applied in a purely mechanical > manner, then non-logical symbols must *mechanically* come from > the syntactical axioms: not from a mind where they're only stipulated! The non-logical symbols are in the definition of the language. > Checking for correct > application is a mere matter of applying a recursive procedure in > pattern matching. Right. As long as what contains the patterns is valid in the first place. For example, if you're formulating the theory T = {x+y=0} where L(T) = L(0,+). > Though there might be more, the following would be axioms of T: (1) x+y=0 > (2) x=x But what about the formula: (3) (Axyz[((xoy)/(yoz)) -> xe(x*(z/x))] -> Atuv[((tou)/(uov))) -> te(t*(v/t))] / >ExEy[~(x=y)] Would you think (3) could be *validly* proven? I have to go through the hoops of yet another of your senseless exercises? The language has '+' and '0'. The deduction that I've shown you (and one by Alan even in Shoenfield's approach) are correct. You've given no reason they are not. Piling on yet another of your senseless back- questions doesn't change that the deductions are correct. Your last objection to Alan's proof is that the symbols terms 'x+y' and '0' don't occur in the axiom or rule. They are TERMS of the language and the axiom or rule is explicitlystated to have the meta- variable (called a 'syntactic variable') 'a' range over terms. By your reasoning, NO term could be used with these axioms or rules. Yes, the terms THEMSELVES don't occur in the expression itself of the schematic axioms or rules, but rather the terms occur in INSTANCES of the schemata, just as Shoenfield stipulates that 'a' ranges over the terms. You REALLY don't understand that? If really don't understand this, then, I really need to give up trying to explain anything at all to you. Oh but I forgot, I should have given up long ago when you even said that you cannot confirm that two simple strings are identical or different, such as (I don't recall the exact example I gave), you wouldn't confirm that xx00xx is the same as xx00xx while xx00xx is different from 0x0x Seriously, if you are so gratuitously skeptical... MoeBlee === Subject: Re: Cantor's argument is erroneous <1hhdod5y6c0jj$.1f9usapfs6c6e.dlg@40tude.net> Just fixing an omission: > First of all we have to consider the syntactical part of the system. With > other words, we have to specify our /language/. [...]: > > The /primitive symbols/ of our system: variables: x, y, z, x', y', z', ... > arbitrary names: a, b, c, a', b', c', ... > constants: 0 . : Binary function symbols: + . : etc. etc. Definition of /terms/: Any arbitrary name or constant is a term. > If t and s are terms, then (t + s) is a term. > > etc. etc. Herb === Subject: Re: Cantor's argument is erroneous > > Just fixing an omission: > > First of all we have to consider the syntactical part of the system. With > other words, we have to specify our /language/. [...]: > > The /primitive symbols/ of our system: variables: x, y, z, x', y', z', ... > arbitrary names: a, b, c, a', b', c', ... > constants: 0 . > > : > Binary function symbols: + . > : > > etc. etc. In case Nam is following Shoenfield, these arbitrary names are also constants, I presume. > Definition of /terms/: Any arbitrary name or constant is a term. And variables are also terms, according to Shoenfield, (needed for the proof). > If t and s are terms, then (t + s) is a term. > > etc. etc. > > > Herb -- Alan Smaill === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <1hhdod5y6c0jj$.1f9usapfs6c6e.dlg@40tude.net> <14js2a6nkshd5$.sjqqn2opao7u$.dlg@40tude.net> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) > Just fixing an omission: > First of all we have to consider the syntactical part of the system. With > other words, we have to specify our /language/. [...]: > The /primitive symbols/ of our system: > variables: x, y, z, x', y', z', ... > arbitrary names: a, b, c, a', b', c', ... > constants: 0 . > : > Binary function symbols: + . > : > etc. etc. In case Nam is following Shoenfield, these arbitrary names are also constants, > I presume. > Definition of /terms/: > Any arbitrary name or constant is a term. And variables are also terms, according to Shoenfield, > (needed for the proof). > If t and s are terms, then (t + s) is a term. > etc. etc. > > Herb -- > Alan Smaill Metric length of path...if |s-t| < delta then d(p(s), p(t)) < epsilon/ (4n) (or epsilon/(25n), .... completed solution set and proof P=NP. Martin Musatov 02.20.092 min 14 .... Now the terms in the sum defining L(P') are in [NPC] http://www.groupsrv.com/science/post-3077752.html === Subject: Re: Cantor's argument is erroneous > > Rule: A-Elim: AvPhi(v) > -------- > Phi(t) where v is a variable and t a (i.e. any) _term_. > > This rule justifies the following step in a proof: : > AxAy(x = y) > Ay((a + b) = y) (by A-Elim) > : > > Moreover it allows for the move: : > Ay((a + b) = y) > (a + b) = 0 (by A-Elim) > : > > This means that in our system for _FOPL with identity_ we can prove the > argument AxAy(x = y) |- AxAy((x + y) = 0). > > Proof: You haven't told us (in this post) what the particular FOL= language is that is in play. This would not be a proof if 0, + were not in the language. In fact, the post kind of gives the impression that you have a unique system for _FOPL with identity_ , which I don't expect is your intention. You could maybe clarify. > 1 (1) AxAy(x = y) A > 1 (2) Ay((a + b) = y) 1 AE > 1 (3) (a + b) = 0 2 AE > 1 (4) Ay((a + y) = 0) 3 AI > 1 (5) AxAy((x + y) = 0) 4 AI > > qed. > > If we now adopt the wff (!) AxAy(x = y) > > as an axiom of our first-order theory T we can prove |- AxAy((x + y) = 0) > T > > (in this context). > > Proof: (1) AxAy(x = y) Axiom > (2) Ay((a + b) = y) 1 AE > (3) (a + b) = 0 2 AE > (4) Ay((a + y) = 0) 3 AI > (5) AxAy((x + y) = 0) 4 AI > > qed. > > > Herb -- Alan Smaill === Subject: Re: Cantor's argument is erroneous <10b1kzhxyvivz$.5fp3vs5kd2hw$.dlg@40tude.net> Am 10 Jun 2009 15:51:55 +0100 schrieb Alan Smaill: > You haven't told us (in this post) what the particular FOL= language > is that is in play. > > This would not be a proof if 0, + were not in the language. *sigh* Please read my former post. In my self-reply I just snipped that very part... Herb === Subject: Re: Cantor's argument is erroneous > *can't _validly_* apply rule of inferenece to prove Axy(x+e=0). > But from the lone axiom Axy x=y in a langauge with '+' and '0' as, > respectively a 2-place function symbol and 0-place function symbol, we > may derive Axy x+y=0, as I've shown you. Do you still contest this? > I remember your meta disproof involves something about > proof in a language or natural deduction. _If_ they are the _same_ > proof system as FOL= (and I'm not saying they are) I don't think > you've translated your disproof into the familiar terminologies > of FOL= syntactical proof. If they're not, then you > know my position, I'm not interested in it, in this thread at this time. > btw, I posted a proof in Shoenfield's own syntactical calculus > for FOL= of the formula in question, from the single non-logical axiom, > on the assumption that the language contains + and 0 of the appropriate > syntactic classes. > I don't recall any comment from you on that proof. > > Much as I'd like it's impossible for me to respond to all posts, so > I'm sorry if I miss any post, but it's not my desire to do so. > > Anyway, did you mean the post on May 21st where you had the below? I did mean the proof below. But you have snipped a crucial part that post; it's important for the proof you cite to note that I had already stated, as I did in the post that you are replying to, that I make the *assumption* that the language contains + and 0. In this I simply follow Shoenfield, who says The first part of a formal system is its language. And To specify a language, we must first of all specify its symbols. So I take the language to have the non-logical symbols +,0; and I made that assumption explicit. > The proof now goes as follows: > > 1. ~ some x. ~ ~ some y ~ (x = y) axiom > 2. ~( ~ ~ some y. ~ x + 0 = y) v some x. ~ ~ some y. ~ x = y subst axiom > 3. ~( ~ ~ some y. ~ x + 0 = y) *1* > 4. ~ some x. ~ ~ some y. ~ (x + 0 = y) *2* > > For the subst axiom at line 2, this is an unpacked form of Shoenfield's > A_x[a] -> some x A, where A is the formula ~ ~ some y. ~ x = y > and a is x + 0 (Shoenfield's conventions indicate that a is any > term in the language, and x + 0 is such a term.) > > If so, then whatever the rules and the syntactical techniques you used > to get to 2. would be _invalid_ on my side of the debate, simply because > + and 0 are *not in* the lone axiom ~ some x. ~ ~ some y ~ (x = y). So your post there doesn't seem to be revealing anything new than what we've > been debating. For Shoenfield's own proof system, I offered to show you a proof exactly *on the assumption* that + and 0 were in the language. IIRC you specifically asked for a proof under that condition, since we both agree that there is no proof if 0, + are not in the language. It seemed you doubted whether there is a proof even under the assumption that the language contains 0, +. You have also queried whether natural deduction proof will have the same properties as Shoenfield's own system. It seems that that issue is irrelevant now, if you accept that the above gives a proof *on the assumption* that the language contains 0, + (we know you do not accept the assumption). > -- > To discover the proper approach to mathematical logic, > we must therefore examine the methods of the mathematician. > (Shoenfield, Mathematical Logic) -- Alan Smaill === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > *can't validly * apply rule of inferenece to prove Axy(x+e=0). > But from the lone axiom Axy x=y in a langauge with '+' and '0' as, > respectively a 2-place function symbol and 0-place function symbol, we > may derive Axy x+y=0, as I've shown you. Do you still contest this? > I remember your meta disproof involves something about > proof in a language or natural deduction. If they are the same > proof system as FOL= (and I'm not saying they are) I don't think > you've translated your disproof into the familiar terminologies > of FOL= syntactical proof. If they're not, then you > know my position, I'm not interested in it, in this thread at this time. > btw, I posted a proof in Shoenfield's own syntactical calculus > for FOL= of the formula in question, from the single non-logical axiom, > on the assumption that the language contains + and 0 of the appropriate > syntactic classes. > I don't recall any comment from you on that proof. > Much as I'd like it's impossible for me to respond to all posts, so > I'm sorry if Imiss any post, but it's not my desire to do so. > Anyway, did you mean the post on May 21st where you had the below? I did mean the proof below. But you have snipped a crucial part that post; > it's important for the proof you cite to note that I had > already stated, as I did in the post that you are replying > to, that I make the *assumption* that the language contains + and 0. In this I simply follow Shoenfield, who says The first part of a formal > system is its language.And To specify a language, we must first of > all specify its symbols.So I take the language to have the > non-logical symbols +,0;and I made that assumption explicit. Note to Nam: This all crystal clear. What is most UNclear is what in the world is blocking you from understanding this. MoeBlee === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) <<<<}}}> Musatov:::::::::>´ (my proof is granular) See... > > *can't validly * apply rule of inferenece to prove Axy(x+e=0). > But from the lone axiom Axy x=y in a langauge with '+' and '0' as, > respectively a 2-place function symbol and 0-place function symbol, we > may derive Axy x+y=0, as I've shown you. Do you still contest this? > I remember your meta disproof involves something about > proof in a language or natural deduction. If they are the same > proof system as FOL= (and I'm not saying they are) I don't think > you've translated your disproof into the familiar terminologies > of FOL= syntactical proof. If they're not, then you > know my position, I'm not interested in it, in this thread at this time. > btw, I posted a proof in Shoenfield's own syntactical calculus > for FOL= of the formula in question, from the single non-logical axiom, > on the assumption that the language contains + and 0 of the appropriate > syntactic classes. > I don't recall any comment from you on that proof. > Much as I'd like it's impossible for me to respond to all posts, so > I'm sorry if Imiss any post, but it's not my desire to do so. > Anyway, did you mean the post on May 21st where you had the below? > I did mean the proof below. > But you have snipped a crucial part that post; > it's important for the proof you cite to note that I had > already stated, as I did in the post that you are replying > to, that I make the *assumption* that the language contains + and 0. > In this I simply follow Shoenfield, who says The first part of a formal > system is its language.And To specify a language, we must first of > all specify its symbols.So I take the language to have the > non-logical symbols +,0;and I made that assumption explicit. Note to Nam: This all crystal clear. What is most UNclear is what in > the world is blocking you from understanding this. MoeBlee I understand this... Journal of Computer and System Sciences : LWPP and WPP are not ...R. Beigel, H. Buhrman and L. Fortnow, NP might not be as easy as .... J. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers, ... http://linkinghub.elsevier.com/retrieve/pii/S0022000006000031 PII: S0370-2693(97)00518-2the space of solutions of S4a not present in the corre- ..... the adjoint exterior derivative is 6p = ( - 1 ) np +n.85l.d.. The Laplacian on p-forms is, as usual, Ap = tSp ld + .... [5] J. Schonfeld, Nucl. Phys. B 185 (1981) 157. ... http://linkinghub.elsevier.com/retrieve/pii/S0370269397005182 + 00.+Phys. Rev. Lett. 62, 1217 (1989): Lerda and van Nieuwenhuizen ...... mass parameter of the Lorentz Chern- Simons term L=U (i/4K2 2)6P (Rpvab(OX .... However, Yang-Mills ghosts do not contribute to the gravitational ... F. 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(ii) g 1 pg4 np X2M2 gG X which gives the original restriction 1X/g4- 0. ... 0+http://link.aps.org/doi/10.1103/PhysRevD.19.562 + +Infrared laboratory spectroscopy with astrophysical applications19 May 2009 ... elements are the ns[CapitalEth]np transitions, where n ranges from 2 ... oscillator strengths for these lines, it is not enough to measure ... + LA CASA DEI SOGNI Rating: Not yet rated ...... ba furniture .0+[url=http://6p.pqnmwxw.co.cc/byson.html] sparrows furniture store in utah [/url] ... upholstery supplies [/url] .00+ [url=http://iq.qwlbyk.co.cc/aityslyi.html]schoenfeld furniture bellevur [/url] ...... 0+ http://be.syvav.dyndns.dk/suedeith.html nurse practitioner jobs san francisco ... http://www.segnala.net/marketing/la-casa-dei-sogni.php + +YNYO method would not have had great value in the case of 19- .... 3P,4P-diol4-acetate to give cholest-4-ene-3P,6P-diol diacetate in ...... 70 R. Hintsche, R. Megges, D. Pfeiffer, H. J. Portius, W. Schoenfeld, and K. R. H. Repke, Eur. ... http://www.rsc.org/ejarchive/NP/1986/NP9860300505.pdf + +Mobypicture - sequence... south high school [/url] [url=http:// 6p.imduvo.ze.cx/pltes.html]americus ..... [url=http:// l7.lzrbrdg.dyndns.dk/xicelyo.html]redeem codes not used yet ... http://www.mobypicture.com/user/sequence/view/9243 - 400k +Comparative Genomics of Autism, Tourette syndrome and Autoimmune ...SLE-NP with neuropsychiatric phenotype. Crohn's disease .... The immune component suggested in both disorders is not of a ... (MHC) on chromosome 6p. Many autoimmune or inflammatory disorders have been ..... [21] Krause I, He XS, Gershwin ME, Shoenfeld Y. Brief report: immune factors in autism: a critical review. ...http://www.grc.nia.nih.gov/branches/rrb/dna/pubs/ cgoatad.pdf +0 Remember... <<<<}}}> Musatov:::::::::>´ (my proof is granular) ~ M.M.M~ T Symmetry is an important abstract and physical characteristic of every great breakthrough in the history of human kind. --M.M.M--(no quot here, but on the other side there was) Variables defined by absence~~~+inclusion (sleepy v+) === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=yKimjgoAAACk5WwPVD4l9HmbpoR6Hmy4 Presto/2.1.1,gzip(gfe),gzip(gfe) > Giving exact formulation of a rule isn't necessarily same as defending > the application of it is valid! Do you understand that? > There is no defense needed. After an exact formulation has been > given (that is, the rule is recursive, or, more plainly, merely > clerical to apply) then it is purely mechanical or clerical to check > whether the rule has been correctly applied. Note your phrase purely mechanical or clerical. If you check to > see if a rule has been correctly applied in a purely mechanical > manner, then non-logical symbols must *mechanically* come from > the syntactical axioms: not from a mind where they're only stipulated! Why? This is really your central point, but you've never explained why it must be that the non-logical symbols comes from the syntactical axioms. In fact, the language is simply specified, and then we know what a WFF is. There's no need to 'derive' the language from anything. I don't know what the talk about mind is. Specifying a formal language is no more concrete or abstract than specifying a set of axioms. But it seems to be giving you some serious trouble in comprehension. Serious trouble. > Checking for correct > application is a mere matter of applying a recursive procedure in > pattern matching. Right. As long as what contains the patterns is valid in the first place. And how do we know what patterns are valid? My answer would be something like: we specify the formal language, the axioms, and the rules of inference. Then we know a wff is one that is in the language. And a theorem is anything obtained from an axiom or another theorem by a rule of inference. What's your answer? > For example, if you're formulating the theory T = {x+y=0} where L(T) = L(0,+). > Though there might be more, the following would be axioms of T: (1) x+y=0 > (2) x=x But what about the formula: (3) (Axyz[((xoy)/(yoz)) -> xe(x*(z/x))] -> Atuv[((tou)/(uov))) -> te(t*(v/t))] / >ExEy[~(x=y)] Would you think (3) could be *validly* proven? No, because (3) contains symbols not in the language of T. In fact, (3) is not a wff at all. It's hard to see what the relevance of this question is, since all formulae in the proof that was under discussion were wffs in the language of the theory under discussion. === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) My answer is not that I need a defense or resolution or permission from you or Bell or anyone else. The proof is more primal and trusting than that. Consider it critical mass. Each letter a specific component in this a non-witches mathematical brew. Formulas, devices, whatever can be had, may be. I have methods, sometimes they are consistent, sometimes they vary, but who are you to judge? What is it to you? It is nothing to me what you do or judge as it is not your judgment I seek. But enough canononical reference and on to the refreshed searc mathematics as we approach critical mass, the levee breaks, she's buying a stairway to heave, whatever, serious don't get spooked, don't take it too seriously, just trust results come, things change, and much is learned... Search results for schoenfeld syntactical calculus (3) mix: But seriously, I can take you from protein folding to encryption and back again, keeping security and promoting advances=I am just being honest out of a kind interest for others (all of them) Go:Subcutaneous and Visceral Adipose Tissue: Their Relation to the ...Int J Obes Relat Metab Disord 18:641-647[Medline]; Deleted in proof. ...... Valdez R, Athens MA, Thompson GH, Bradshaw BS, Stern MP 1994 Birthweight and adult ..... M. A. Bredella, A. L. Utz, M. Torriani, B. Thomas, D. A. Schoenfeld, ..... S. Musatov, W. Chen, D. W. Pfaff, C. V. Mobbs, X.-J. Yang, D. J. 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Combustion diagnostics : 17 June 1997, Munich, FRG ... http://en.papervitamins.com/sitemap/book/6991288-7001287.html Challenging the human circadian clock by Daylight Saving Time and ...then be used to proof or disproof the initial hypothesis, stating that ..... Anisimov, V.N., Zabezhinski, M.A., Popovich, I.G., Zaripova, E.A., Musatov, S.A., Andre, ..... Eastman, C.I., Stewart, K.T., Mahoney, M.P., Liu, L. and Fogg, ..... O'Leary, E.S., Schoenfeld, E.R., Stevens, R.G., Kabat, G.C., Henderson, K., ...http:// edoc.ub.uni-muenchen.de/9428/1/Kantermann_Thomas.pdf > Giving exact formulation of a rule isn't necessarily same as defending > the application of it is valid! Do you understand that? > There is no defense needed. After an exact formulation has been > given (that is, the rule is recursive, or, more plainly, merely > clerical to apply) then it is purely mechanical or clerical to check > whether the rule has been correctly applied. > Note your phrase purely mechanical or clerical. If you check to > see if a rule has been correctly applied in a purely mechanical > manner, then non-logical symbols must *mechanically* come from > the syntactical axioms: not from a mind where they're only stipulated! Why? This is really your central point, but you've never explained > _why_ it must be that the non-logical symbols comes from the > syntactical axioms. In fact, the language is simply specified, and > then we know what a WFF is. There's no need to 'derive' the language > from anything. I don't know what the talk about mind is. Specifying a formal > language is no more concrete or abstract than specifying a set of > axioms. But it seems to be giving you some serious trouble in > comprehension. Serious trouble. > Checking for correct > application is a mere matter of applying a recursive procedure in > pattern matching. > Right. As long as what contains the patterns is valid in the first place. And how do we know what patterns are valid? My answer would be something like: we specify the formal language, the > axioms, and the rules of inference. Then we know a wff is one that is > in the language. And a theorem is anything obtained from an axiom or > another theorem by a rule of inference. What's your answer? > For example, if you're formulating the theory T = {x+y=0} where L(T) = L(0,+). > Though there might be more, the following would be axioms of T: > (1) x+y=0 > (2) x=x > But what about the formula: > (3) (Axyz[((xoy)/(yoz)) -> xe(x*(z/x))] -> Atuv[((tou)/(uov))) -> te(t*(v/t))] / > ExEy[~(x=y)] > Would you think (3) could be *validly* proven? No, because (3) contains symbols not in the language of T. In fact, > (3) is not a wff at all. It's hard to see what the relevance of this question is, since all > formulae in the proof that was under discussion were wffs in the > language of the theory under discussion. === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) > Giving exact formulation of a rule isn't necessarily same as defending > the application of it is valid! Do you understand that? > There is no defense needed. After an exact formulation has been > given (that is, the rule is recursive, or, more plainly, merely > clerical to apply) then it is purely mechanical or clerical to check > whether the rule has been correctly applied. > Note your phrase purely mechanical or clerical. If you check to > see if a rule has been correctly applied in a purely mechanical > manner, then non-logical symbols must *mechanically* come from > the syntactical axioms: not from a mind where they're only stipulated! Why? This is really your central point, but you've never explained > why it must be that the non-logical symbols comes from the > syntactical axioms. In fact, the language is simply specified, and > then we know what a WFF is. There's no need to 'derive' the language > from anything. I don't know what the talk about mind is. Specifying a formal > language is no more concrete or abstract than specifying a set of > axioms. But it seems to be giving you some serious trouble in > comprehension. Serious trouble. > Checking for correct > application is a mere matter of applying a recursive procedure in > pattern matching. > Right. As long as what contains the patterns is valid in the first place. And how do we know what patterns are valid? My answer would be something like: we specify the formal language, the > axioms, and the rules of inference. Then we know a wff is one that is > in the language. And a theorem is anything obtained from an axiom or > another theorem by a rule of inference. What's your answer? > For example, if you're formulating the theory T = {x+y=0} where L(T) = L(0,+). > Though there might be more, the following would be axioms of T: > (1) x+y=0 > (2) x=x > But what about the formula: > (3) (Axyz[((xoy)/(yoz)) -> xe(x*(z/x))] -> Atuv[((tou)/(uov))) -> te(t*(v/t))] / >ExEy[~(x=y)] > Would you think (3) could be *validly* proven? No, because (3) contains symbols not in the language of T. In fact, > (3) is not a wff at all. It's hard to see what the relevance of this question is, since all > formulae in the proof that was under discussion were wffs in the > language of the theory under discussion. Consider this thread an extension of my proof P=NP, and add to it this text. Preface all of this with the knowledge all is fair in love and war, and now coldly mathematics. > As to the book of Psalms and The Song of Songs, there is a course to > the neverending. It constantly evolves as parameters change, but still > always the same path as we what we seek to define we by nature of our > observing change. To this there is no logic, no volume of thought, > just words to read, and numbers to granulate. And to truth, a gentle > stream. To a word, from a number, from a number to a word, from one to > infinity. There is no debate, no second guess. No conjecture, or oversight. No > exclusion applies, and no theories hold the awesome power it > contains. Infinity's loose but rigid, flexible but firm, in evolution is > universal right. In form and function, across language and guild, the > heir apparent us. Beauty is truth and truth beauty but I suspect this > is what may be meant by these words. It is not caste in doom failure, but refined like aromatic resin, good > gold from a furnace. To define a flaw is to label a base for words and > numbers make their case. At any rate, no matter the claim, they do not belong to me. I did not > choose them, nor did they choose me. Yet always the unsolved case > remains. Those decisions we make today we base in what we may learn > tomorrow. In this futures are made. --Anonymous (Composed in honor of Bernhard Riemann) Preface: E. Pluribus Unum, is Latin, and translates to ñThe Many > Become One. > ................In.................[1]........ In sum, the book is recommended as an introduction to the more ...(?) > the minimum bit size of a P-proof of ?. They called a proof ... > without assuming at least P = NP, we cannot rule out the existence of > a polynomial time ... razborov@genesis.mi.ras.ru. Lev D. > Beklemishev. ... [1] http://projecteuclid.org/DPubS/Repository/1.0/ Disseminate%3Fhandle%3Deuclid.bsl/1182353876%26view%3Dbody%26content- type%3Dpdf 1 > [P Versus NP]So, ñGenesis 24:1[CapitalEth]3 and 9.81.8d means the book of Genesis, > chapter 24, verses 1 through 3 ...... (In a strictly Quantum > Naturalization [P=NP] novel proof sense). ... [2] http://pversusnp.wordpress.com/ > Is P Versus NP Formally Independent?P = NP asks for an efficient > procedure that finds a short proof. ...... Section 2 is the book of > Cohen [13]. For a definition of Cutting Planes and other proof > k.ps. ... [3] http://www.scottaaronson.com/papers/pnp.pdf stdin (ditroff)ry of claimed resolutions to the question of P versus > NP. Section 2 is then de- ... umn: an actual NP-completeness proof > (one of the two most requested unpub- ... based on the Old Testament > Book of Genesis.) We will of course honor re- ... [4] http://www.research.att.com/~dsj/columns/col20.pdf The Gutnick Edition Chumash - Book of Genesis: With Rashi's ...... it > would be sufficient proof to Avraham that the time had now come > for ... 30, p. 82//.) o While Adam and Chavah were the parents of all Greatest Mystery in Modern Science?The genesis of ihis fourth Big idea > was the ho-hum obser- ... tractable (P=NP in computer parlance!, proof- > finding will be ... When you purchase a book from Amazon, the > assurance that your transaction is secure is ... [5] http://www.cs.princeton.edu/~chazelle/pubs/ipod.pdf Could Your iPod Be Holding the Greatest Mystery in Modern Science? > Tractability The genesis of this fourth Big Idea was the ho-hum > observation ... The twin reality of hard proof-finding and easy proof- > checking is hardly an MTV-induced aberration. ... Indeed, the day the > Jurassic-1K are shown to be tractable (P=NP in ... When you purchase a > book from Amazon, the assurance that your ... [6] http://www.tnlab.ice.uec.ac.jp/nhc06/material/files/2701.html Infinite Order Logic and the Church-Turing Thesis6 Jun 2006 ... > Corollary 5 P=NP in LISP. Proof: Randomness is an infinite order > process and LISP can express infinity. ... I just read his book. How > do I know all these things? ... 2.7 Future Work: The NP Computer and > Genesis ... [7] http://web.media.mit.edu/~vyzo/papers/computability.pdf Pseudepigrapha Journal for the Study of the Book Review: Primaeval > History Interpreted: The Rewriting of Genesis ... (JSJ Supplement > Series, 66; Leiden: E.J. Brill), p. xx +. 408. Cloth, n.p. ISBN > 9-0041-1658-3. .... need for more rigorous proof-reading. Betsy > Halpern-Amaru ... [8] http://jsp.sagepub.com/cgi/reprint/14/1/78.pdf Book Review: The Millennium Problems: The Seven Greatest Unsolved ... > 24 May 2000 ... who tells us that ñthe proof would shed light on a > fundamental aspect of nature. ... genesis of each problem and > developing its back- ground, can be grouped together. ... other six > millennium problems, but P versus NP ... [9] LNCS 3142 - Feasible Proofs and Computations: Partnership and Fusion > universally agree on what is a proof and what is a computation. .... > sion of P = NP. In particular, we will address one approach to this > question based .... subject was treated in Buss's book [15] which > still remains a very good source for a .... Manuscript available at [10] http://www.genesis.mi.ras.ru/Örazborov, 2002. ...[11]http:// www.springerlink.com/index/HWBD96PN120LBDBN.pdf Also available at [12]http://math.ucr.edu/home/baez/week226.html ...10 > Feb 2006 ... For example, Chapter 2 of this book starts out by > defining strong and .... and [13] http://genesis.mi.ras.ru/~razborov/int.ps The basic point of this paper ... So, if P is not equal to NP > is true, it has no natural proof. ... [13] http://math.ucr.edu/home/baez/twf ascii/week226 Full text of The book of Genesis; critical edition of the > Hebrews ...The Ephesian Artemis with her many breasts {p'W) > illustrates the same idea. nnn rssi ...... On the other hand, it is > perhaps more natural to read 20 np''1 inniy^l, ...... Jacob had al- > ready given practical proof of what he could do, v. ...... O.T. > Genesis. Hebrew. 1232 1896 I896 The book of Genesis PLEASE DO NOT ... > [14] http://www.archive.org/stream/bookofgenesiscri00balluoft/bookofgenesiscri00b alluoft djvu.txt GENESIS of PUBLIC KEY CRYPTOSYSTEMS Part 2: The RSA Algorithm then all > NP are in P meaning that one solution ..... Phone Book. Encryption > Key. [ public ]. Decryption Key. [ secret ]. George : bfh467r.97u%+. > Alice : /&'^Grtwe35 ... him(her)self as a proof of authorship of the > contents of a document. ... [15] James Kent's Commentaries: Of the History, Progress, and Absolute ... > 3. p. 40. insists, that a primitive state of man existed before the > establishment of civil ... and temporary occupancy the only title; but > he gives no sufficient proof of the fact. The book of Genesis, which > 16. Co. Litt. 309. Dig. 41. 1. 20. ... [16] http://lonang.com/exlibris/kent/kent-34.htm The Virgin Birth of Christ: Prophecies in Genesis and Isaiah > The critics take Isaiah's concluding pronouncement to the king as > proof that he ... Jay P. Green, Sr., The Interlinear Bible: Hebrew/ > English, 3 vols. ... See John N. Oswalt, The Book of Isaiah: Chapters > 1-39 (Grand Rapids, Mich. ... J. Gresham Machen, The Virgin Birth of > Christ (n.p.: Harper & Row, Publishers, ...[17] http://www.themoorings.org/apologetics/VirginBirth/Isaiah.html Archives and Special Collections, University of Mississippi Red > Dragon . New York: Putnam's Sons, 1981. Uncorrected Proof. [book] ... > [1999]. Announcement of Genesis Press' publication of Deadly > Sacrifice. [document] ... Death on Scurvy Street. New York: E. P > Dutton & Company, 1929. [book]. Ben Ames Williams. Death on Scurvy > Street. N.p.: Continental Books, c.1929. [book] ... [18] http://hermes.lib.olemiss.edu/mystery/bibliography.asp REVIEWS and more than 500 pages, a book that is highly readable and > informative but not without .... (?) the minimum bit size of a P-proof > of ?. They called a proof ... without assuming at least P = NP, we > cannot rule out the existence of a polynomial time ... > razborov@genesis.mi.ras.ru. Lev D. Beklemishev. ... [19] http://www.math.ucla.edu/~asl/bsl/0802/0802-005.ps The Creation According to the Midrash Rabbah > The proof-verse from Joshua not only shows ... R. Ilfa identifies that > book with Genesis because the context of Balaam's wish to die the Theoretical Computer Science : On the hardness of > allocating ...Journal/book title ... so that the genesis and the > relevance of the problem can be better appreciated. .... Proof. In > [13] it is shown that a feasible solution R: V 2e attaining the > optimum of c i aec CO can be computed in 0(Mn C Mn + C ) time. ... In > this section we show that the answer is negative, unless P = NP. ... > [20] http://linkinghub.elsevier.com/retrieve/pii/S030439759800156X Ramin's Ponderings > In so many words, the P = NP question has to do with whether or not > some ... In that humorous science fiction book The Hitchhiker's Guide > to the Galaxy. ... [21] http://ramin-honary.blogspot.com/ > ACM: Ubiquity - Random Thoughts and Prime Numbers > It is instructive to note that many concepts crucial in this proof > were ... then afterwards the teacher would show us what is called the > Book Proof. ... This type of question is basically the genesis of the > field of computational complexity. The question of NP versus P is > whether or not anything that has a short ... [22] http://www.acm.org/ubiquity/interviews/j cai 1.html THE GENESIS OF THE YOUNG COSIMA: HENRY HANDEL RICHARDSON'S > MOST ...result is a book almost devoid of imaginative and descriptive > writing. Its matter is ..... During the proof stage she said: ... I > wish Oh God I'd .... See letter to Nettie Palmer dated 6 May 1939, N. > P., p. 201. 9. 24 November 1929. ...[23] http://www.informaworld.com/index/795114923.pdf Book Review Book Review. Andy Clark*. University of Edinburgh. Genesis > Machines: The New Science of Biocomputing. ... (p. 112). The problem > is interesting in that it belongs to the class of problems that are > said to be NP-complete.84that is, to involve search spaces that grow > very .... As a proof of principle, one researcher ... [24] http://www.mitpressjournals.org/doi/abs/10.1162/artl.2009.15.2.15206 Genesis Bibliography--Matthews File > Hamilton, Victor P. The Book of Genesis, Chapters 1-17. ...... , > Presenting Genesis 1, Constructively and Deconstructively, Proof 21 > (2001), 1-22. .... Lemche, N.P., The Chronology in the Story of the > Flood, JSOT 18 (1980), ... [25] http://courses.missouristate.edu/VictorMatthews/bib/GENA.html - 144k > ................In.................[3].......Sealed||Á[EGra ve]Á.8fÁ.8fÁ[ Currency]%[ > [-][+][I][N]['][}>}][R][I][E][M][A][N][N] === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <1qEXl.33371$VL5.29204@newsfe22.iad> posting-account=yKimjgoAAACk5WwPVD4l9HmbpoR6Hmy4 Presto/2.1.1,gzip(gfe),gzip(gfe) > The proof now goes as follows: > 1. ~ some x. ~ ~ some y ~ (x = y) > axiom > 2. ~( ~ ~ some y. ~ x + 0 = y) v some x. ~ ~ some y. ~ x = y > subst axiom > 3.~( ~ ~ some y. ~ x + 0 = y) *1* > 4.~ some x.~ ~ some y. ~ (x + 0 = y) *2* > For the subst axiom at line 2, this is an unpacked form of > Shoenfield's > A x[a] -> some x A, where A is the formula ~ ~ some y. ~ x = y > and a is x + 0 (Shoenfield's conventions indicate that a is any > term in the language, and x + 0 is such a term.) > If so, then whatever the rules and the syntactical techniques you used > to get to 2. would be invalid on my side of the debate, simply because > + and 0 are *not in* the lone axiom ~ some x. ~ ~ some y ~ (x = y). > For the subst axiom at line 2, this is an unpacked form of > Shoenfield's > A x[a] -> some x A, where A is the formula ~ ~ some y. ~ x = y > and a is x + 0 (Shoenfield's conventions indicate that a is any > term in the language, and x + 0 is such a term.) > MoeBlee Please read what I said to him: > If so, then whatever the rules and the syntactical techniques you used > to get to 2. would be invalid on my side of the debate, simply because > + and 0 are *not in* the lone axiom ~ some x. ~ ~ some y ~ (x = y). Didn't you see the phrases such as whatever the rules and the syntactical techniques > or invalid ? They're not invalid in Shoenfield's syntactical calculus, though. Are they? === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) > The proof now goes as follows: > 1. ~ some x. ~ ~ some y ~ (x = y) > axiom > 2. ~( ~ ~ some y. ~ x + 0 = y) v some x. ~ ~ some y. ~ x = y > subst axiom > 3.~( ~ ~ some y. ~ x + 0 = y) *1* > 4.~ some x.~ ~ some y. ~ (x + 0 = y) *2* > For the subst axiom at line 2, this is an unpacked form of > Shoenfield's > A x[a] -> some x A, where A is the formula ~ ~ some y. ~ x = y > and a is x + 0 (Shoenfield's conventions indicate that a is any > term in the language, and x + 0 is such a term.) > If so, then whatever the rules and the syntactical techniques you used > to get to 2. would be invalid on my side of the debate, simply because > + and 0 are *not in* the lone axiom ~ some x. ~ ~ some y ~ (x = y). > > For the subst axiom at line 2, this is an unpacked form of > Shoenfield's > A x[a] -> some x A, where A is the formula ~ ~ some y. ~ x = y > and a is x + 0 (Shoenfield's conventions indicate that a is any > term in the language, and x + 0 is such a term.) > MoeBlee > Please read what I said to him: > If so, then whatever the rules and the syntactical techniques you used > to get to 2. would be invalid on my side of the debate, simply because > + and 0 are *not in* the lone axiom ~ some x. ~ ~ some y ~ (x = y). > Didn't you see the phrases such as whatever the rules and the syntactical techniques > or invalid ? They're not invalid in Shoenfield's syntactical calculus, though. Are > they? Yes, to the best of my memory the best I can put into words, it was, [of they that were may evil not be present here] mind you this is like the message not the literal words, it was a concept in my mind... What I really saw was words, mostly *unnecessary* words, such as *of*, *they*, *that*, *there*, *is*, *was*, *then*... And I was only 9, I guess, at the time, but the *words*/*message* kept almost shuffling, disappearing/reappearing as I read *it*. As if I was supposed to not literally *read* it, but *understand* it. And all I could see was no one who had any evil, nobody accept one who was *pure*, *sensitive*, *innocent*, and *understanding* could *read* it. As I read it, it was like a waking drea, wide awake, I found myself deep in thought marvelling at the beauty and wonder and awesome power I had seen, and then as though, almost retroactive, I would find myself completely coherent thinking ti myself, But what was it I read? What was I supposed to comprehend and *where* did *it* go and *why*? In them who could comprehend what I read was eternal life, and love, and care, and *protection*, like from a parent, allowing a child to take a first step. Or leaving a toddler alone in a dark room hoping for no tears, but sweet dreams and marvelled independence found in real love and freedom unchained, untethered, unafraid, and without any sorrow completely. Joy. A mystery.]*Mathematics*-contains-show quoted text- Musatov === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) > The proof now goes as follows: > 1. ~ some x. ~ ~ some y ~ (x = y) > axiom > 2. ~( ~ ~ some y. ~ x + 0 = y) v some x. ~ ~ some y. ~ x = y > subst axiom > 3.~( ~ ~ some y. ~ x + 0 = y) *1* > 4.~ some x.~ ~ some y. ~ (x + 0 = y) *2* > For the subst axiom at line 2, this is an unpacked form of > Shoenfield's > A x[a] -> some x A, where A is the formula ~ ~ some y. ~ x = y > and a is x + 0 (Shoenfield's conventions indicate that a is any > term in the language, and x + 0 is such a term.) > If so, then whatever the rules and the syntactical techniques you used > to get to 2. would be invalid on my side of the debate, simply because > + and 0 are *not in* the lone axiom ~ some x. ~ ~ some y ~ (x = y). > > For the subst axiom at line 2, this is an unpacked form of > Shoenfield's > A x[a] -> some x A, where A is the formula ~ ~ some y. ~ x = y > and a is x + 0 (Shoenfield's conventions indicate that a is any > term in the language, and x + 0 is such a term.) > MoeBlee > Please read what I said to him: > If so, then whatever the rules and the syntactical techniques you used > to get to 2. would be invalid on my side of the debate, simply because > + and 0 are *not in* the lone axiom ~ some x. ~ ~ some y ~ (x = y). > Didn't you see the phrases such as whatever the rules and the syntactical techniques > or invalid ? They're not invalid in Shoenfield's syntactical calculus, though. Are > they? Yes, to the best of my memory the best I can put into words, it was, [of they that were may evil not be present here] mind you this is like the message not the literal words, it was a concept in my mind... What I really saw was words, mostly *unnecessary* words, such as *of*, *they*, *that*, *there*, *is*, *was*, *then*... And I was only 9, I guess, at the time, but the *words*/*message* kept almost shuffling, disappearing/reappearing as I read *it*. As if I was supposed to not literally *read* it, but *understand* it. And all I could see was no one who had any evil, nobody accept one who was *pure*, *sensitive*, *innocent*, and *understanding* could *read* it. As I read it, it was like a waking drea, wide awake, I found myself deep in thought marvelling at the beauty and wonder and awesome power I had seen, and then as though, almost retroactive, I would find myself completely coherent thinking ti myself, But what was it I read? What was I supposed to comprehend and *where* did *it* go and *why*? In them who could comprehend what I read was eternal life, and love, and care, and *protection*, like from a parent, allowing a child to take a first step. Or leaving a toddler alone in a dark room hoping for no tears, but sweet dreams and marvelled independence found in real love and freedom unchained, untethered, unafraid, and without any sorrow completely. Joy. A mystery.]*Mathematics*-contains-show quoted text- Musatov === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> posting-account=yKimjgoAAACk5WwPVD4l9HmbpoR6Hmy4 Presto/2.1.1,gzip(gfe),gzip(gfe) > apply rule of inferenece to prove Axy(x+e=0). > But from the lone axiom Axy x=y in a langauge with '+' and '0' as, > respectively a 2-place function symbol and 0-place function symbol, we > may derive Axy x+y=0, as I've shown you. Do you still contest this? > I remember your meta disproof involves something about proof in a language > or natural deduction. If they are the same proof system as FOL= (and I'm > not saying they are) I don't think you've translated your disproof into the > familiar terminologies of FOL= syntactical proof. If they're not, then you > know my position, I'm not interested in it, in this thread at this time. > btw, I posted a proof in Shoenfield's own syntactical calculus > for FOL= of the formula in question, from the single non-logical axiom, > on the assumption that the language contains + and 0 of the appropriate > syntactic classes. > I don't recall any comment from you on that proof. Much as I'd like it's impossible for me to respond to all posts, so > I'm sorry if Imiss any post, but it's not my desire to do so. Anyway, did you mean the post on May 21st where you had the below? > The proof now goes as follows: > 1. ~ some x. ~ ~ some y ~ (x = y) axiom > 2. ~( ~ ~ some y. ~ x + 0 = y) v some x. ~ ~ some y. ~ x = y subst axiom > 3.~( ~ ~ some y. ~ x + 0 = y) *1* > 4.~ some x.~ ~ some y. ~ (x + 0 = y) *2* > For the subst axiom at line 2, this is an unpacked form of Shoenfield's > A x[a] -> some x A, where A is the formula ~ ~ some y. ~ x = y > and a is x + 0 (Shoenfield's conventions indicate that a is any > term in the language, and x + 0 is such a term.) If so, then whatever the rules and the syntactical techniques you used > to get to 2. would be invalid on my side of the debate, simply because > + and 0 are *not in* the lone axiom ~ some x. ~ ~ some y ~ (x = y). Nevertheless, they are valid by Shoenfield, and by everyone else except for you. Shoenfield's subst axiom requires only that a is a term in the language, not that a only contains symbols found in the axioms. This proof is valid in Shoenfield's syntactical calculus. Can you agree to that? Before arguing why you think this is all rubbish and Nam-logic is way superior to Shoenfield-logic, can you agree that the above is valid in Shoenfield's syntactical calculus? === Subject: Re: Cantor's argument is erroneous <%oEWl.859$h53.459@newsfe10.iad> <1uescwwzo525$.1smbbz1tygrbe$.dlg@40tude.net> posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) (squid/2.5.STABLE12) Martin Musatov wote: > Why do you imagine, as you seem to do, that there is any point arguing > with [a crank]? (Torkel Franzen, sci.math, 12 Jan. 2005) Indeed. -- > To discover the proper approach to mathematical logic, > we must therefore examine the methods of the mathematician. > (Shoenfield, Mathematical Logic) untrained mathematicians at inverse 19 learnt a lot about mathematics from your postings and your silence and this has been developed in a few hours over a few days in between work. Our purpose is not as much as provoking discussion , but the ability for us to post and issue and learn from the Silence of the Lambs. Dimension is silent, so is the space matrix at 19.Note: That this constant cannot be reduced to Null zero and n(2Pi^2-0.75) is constant curve for that value of n contains the words Red Dragon - The sequel to The Silence of The Lambs... And it was posted (or at least to me appeared in the topic list AFTER my previous post). In breakdown, I posted a P=NP Genesis/ Riemann post containing mention of a sequel to a film title which then appeared in a new post (to me at least), AFTER. Are we in a vacuum or is there logic at play in this anomaly? I appreciate any sound feedback, the more thoughtful and considerate the better. Please do not make fun or ridicule this sincere attempt to explore a truth. === Subject: Partition a given set into cartesian products posting-account=c115bQoAAAC7sz-AeTUgXhHFsqIvMdZW Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) Given a set a pairs S={ (a_i, b_j) ... } subset of AxB where A and B are two arbitrary FINITE sets (in particular, there is no order defined on A or B), is there a way to build a partition of S in cartesian products, optimal in terms of number of cartesian products used, in non-exponential time? ie. S = { (a_i, b_j) .... (a_k, b_l) } = U{1..n} A_n x B_n, where n is minimal? It looks like a tiling problem except for the fact that I am looking for a partition. Is that a known problem? John === Subject: Re: Partition a given set into cartesian products > Given a set a pairs S={ (a_i, b_j) ... } subset of AxB where A and B > are two arbitrary FINITE sets > (in particular, there is no order defined on A or B), is there a way > to build a partition > of S in cartesian products, optimal in terms of number of cartesian > products used, in non-exponential time? Some partitions of S: {AxB}, { {(a,b)} | a in A, b in B }, { {{a} x B} | a in A }, { {A x {b}} | b in B }. > ie. S = { (a_i, b_j) .... (a_k, b_l) } = U{1..n} A_n x B_n, where n is > minimal? Because S = /{ AxB }, the minimal n is 1. > It looks like a tiling problem except for the fact that I am looking > for a partition. > === Subject: Re: Partition a given set into cartesian products <20090610213947.A49203@agora.rdrop.com > Given a set a pairs S={ (a_i, b_j) ... } subset of AxB where A and B > are two arbitrary FINITE sets (in particular, there is no order > defined on A or B), is there a way to build a partition > of S in cartesian products, optimal in terms of number of > cartesian products used, in non-exponential time? > > Some partitions of S: {AxB}, > { {(a,b)} | a in A, b in B }, > { {{a} x B} | a in A }, > { {A x {b}} | b in B }. > > ie. S = { (a_i, b_j) .... (a_k, b_l) } = U{1..n} A_n x B_n, > where n is minimal? > > Because S = /{ AxB }, the minimal n is 1. The problem statement says S is a subset of AxB, and probably meant to say that sets A_i, B_i are sought such that S = (union{1..n} A_i x B_i). If S is a proper subset of AxB, the partitions you mention will contain elements of AxB that are not in S. > It looks like a tiling problem except for the fact that I am looking > for a partition. Offhand, it looks like a polynomial-time solution to the stated problem would allow polynomial-time solution of boolean minimization problems, which are NP-hard. See eg -- jiw === Subject: Re: Partition a given set into cartesian products <20090610213947.A49203@agora.rdrop.com> Given a set a pairs S={ (a_i, b_j) ... } subset of AxB where A and B > are two arbitrary FINITE sets (in particular, there is no order > defined on A or B), is there a way to build a partition > of S in cartesian products, optimal in terms of number of > cartesian products used, in non-exponential time? > ie. S = { (a_i, b_j) .... (a_k, b_l) } = U{1..n} A_n x B_n, > where n is minimal? The problem statement says S is a subset of AxB, and probably > meant to say that sets A_i, B_i are sought such that > S = (union{1..n} A_i x B_i). If S is a proper subset of AxB, > the partitions you mention will contain elements of AxB that > are not in S. When S is a subset of AxB, then some partitions of S are {S}, { {(a,b)} | (a,b) in S } { {{a} x p(a)} | a in A } { {q(b) x {b}} | b in B } where p(a) = { x | (a,x) in S }; q(b) = { x | (x,b) in S } If some r,s in A with p(r) = p(s), then ({r} x p(r)) / ({s} xp(s)) = {r,s} x p(r) = {r,s} x p(s) which reduces the size of the partition by one. > It looks like a tiling problem except for the fact that I am looking > for a partition. Offhand, it looks like a polynomial-time solution to the > stated problem would allow polynomial-time solution of > boolean minimization problems, which are NP-hard. See > eg 1 > 2 > 3 > 4 > 5 > 6 > 7 > 8 > 9 > 10 > 11 > more than 11 > Herc > -- > Time doesn't change, clocks change. Time is just an abstract > fields. The universe is 3D, space time is just a mathematical > construct. > We have evidence so far for 3 spatial dimensions and one temporal > dimension. There is speculation, based on some faith that a decent > unified theory has certain pleasant features, that there are more -- > but we have no solid testable predictions to put to measurement about > that yet. > ************************************************* > There is no time, there is only movement, in the here and now. > Herc > Well, except that we can quantitatively predict that movement, to mark > in advance *when* something will happen. The fact that we can put that > when on a numerical scale and say what number will be present on > that scale at a certain event, means that for all practical purposes > time has the same dimensional utility as a spatial dimension. > PD > ******************************************* > All you are really doing is pairing up two moving objects, one of them > happens to count. So you are effectively labelling peano arithmetic, or > it's execution, a dimension. > Herc > And that doesn't matter. > All dimensions and measurements are the establishment of a scale and > the association of numbers to that scale. That is true for space as > well as time, as well as temperature, momentum, mass, and any other > quantity. > The purpose of physics is to be able to reliably predict outcomes of > measurable quantities, in the fashion that measurement is described > above. It really doesn't matter whether you want to describe some of > these scales as being real or abstract. Practically speaking, they > are indistinguishable. > PD > ***************************************** > A mass dimension is only ever part of a model, it doesn't make anything > like a space dimension. > What do you think the difference is? > ----------------------------------------- > The issue isn't how many dimensions you can construct, how many > dimensions are there? One is platonic, in the minds of scientists, the > other is something that exists. > And how do you establish which dimensions exist other than by > whether there is a measurement procedure? Please answer with > *physics*, not metaphysics. > *********************************************** > There corresponds an existing point (or plane) for every point on the dimension. What is an existing point? ****************** A point source location anywhere in the Universe. And this is distinguished from the point on the dimension how? *********************************************** One is a location in the universe, one is theoretical You can't compare a location which exists for all points on the dimension to mass, which is just one possible point on the dimension. You're comparing a 3 dimensional world to a number. OK time is a possibility, perhaps parallel quantum futures exist and affect the present to optimise the future. > Hey the link you gave me on missing matter goes > to a kindle version, couldn't find the paperback. > You can't look something up on amazon.com? > ---------------------------------------------- > No book version comes up searching under the author's name. > Actually I didn't notice the kindle version and clicked One Click Purchase > luckily it said I'm not setup for kindle, otherwise I would have given you a serve! > I'm not responsible for your incompetence with ecommerce. > PD > ***************************************************** > You made the same mistake of missing that it's the kindle version, so you're > insulting yourself there. > Herc I use the Kindle. So I gave you a link I would use. Do you not know > how to use amazon.com? At this point, it's appropriate to embarrass you, I think.http://www.amazon.com/Quintessence-Mystery-Lawrence-M-Krauss/dp/04650. . . > which was found by entering the title and the author at amazon.com. Now, tell me again why somebody who can't look up a book on Amazon, > and can only blindly click links provided to him, is capable of any > competent thinking about physics. **************************************************** I followed a *recommended* link that you erronously gave to me. It's one > for the correct link, I just purchased it. The paperback wasn't showing in the > search results last week. Oh, BS. If you don't know how to use amazon.com, I'm not sure there's much I can do to help you. ***************************** Don't whinge BS just because you can't argue the facts. If you think following a link personaly given to you to a book and the page is virtually identical to the book page, but it's not so you click purcahase disqualifies you from having scientific opinion you're just plain mad, and condescending, in a dream world that you've never made an error. When in fact it was YOUR link that was ed in the first place. Herc === Subject: Re: How many dimensions does the universe have? > > No. They experience the movement of a 3D object (which is change in > position with time) through their slice. A simple 3D object that doesn't > change in time would look just like any common 2D object (except for its > strange gravitic and electromagnetic properties) to them. > What you mean No? It's true the 3D object is not changing in time > but the interesting thing is that time in the higher space propagates > through the motion of the unchanging objects to appear to denziens of > the sub space as time or more exactly motion. So, as you said, motion (changing position in time) of the 3D object would appear as a size change of its 2D cut in time. So it's just replacing one kind of motion with another. Your original statement was So to the flatlanders, the third dimension is manifested as time. which is a totally different one; a 3D object which is fixed relatice to the 2D world would show no features whatsoever that could be interpreted as time. > So the flatlanders would > develop a law for spheres passing through their slice in terms of > the growth and shrinking of circles (which they would see and > understand). Of course if the 3D object DID change in time that would > add more complications as well. For me, changing position perpendicular to the 2D world is a change. > The point being here that complex > geometric motions and actions in higher dimensions of a space end up > being interpreted by denizens of a sub space as motion, in other words > time phenomena. So here you say that motion ends up being interpreted as motion, which is certainly a true statement. > So as you note an unchanging for your definition of unchanging which allows for changes of position > higher dimensional > object appears with strange properties (we would call them > metaphysical...such as shape-shifting and appearance and > disappearances). provided the extra dimensions are large enough and couple to our four fundamental forces. As I said before, this case is, at least for accessible energies and distances, ruled out by measurements. Lars === Subject: Re: How many dimensions does the universe have? > Not really. Electrical charge is a dimension, you can't express that > in any combination of mass, length and time. One could say each element > of the periodic table is a dimension until we describe them as protons, > neutrons and electrons, and when we do then protons and neutrons are a > composite of mass and charge. You could even say temperature is a dimension, by that definition. Yousuf Khan === Subject: Re: How many dimensions does the universe have? > Not really. Electrical charge is a dimension, you can't express that > in any combination of mass, length and time. One could say each element > of the periodic table is a dimension until we describe them as protons, > neutrons and electrons, and when we do then protons and neutrons are a > composite of mass and charge. > You could even say temperature is a dimension, by that definition. Yousuf Khan http://en.wikipedia.org/wiki/Temperature The temperature of an ideal monatomic gas is a measure related to the average kinetic energy of its atoms as they move. Kinetic energy is expressed in a combination of mass, length and time. Heat a gas filled toy balloon and the volume increases. I'd disagree with wackypedia's use of the term average, though. Still, they often get things wrong. The difficulty is ice occupies a greater volume than water and will place a greater pressure on its container. Pipes burst in winter, icebergs float, ice cubes in the freezer stand taller than the tray. Heat ice and the volume decreases. === Subject: - Congruence equations (mod n) Consider the congruence equation x^3 + x^2 -5 = 0 (mod 49). If _a_ is a solution mod 49, then a is certainly a solution mod 7. I also know that 2 is a solution mod 7. Question: why does it follow that x = 2 ( mod 7 ) ? === Subject: Re: - Congruence equations (mod n) >Consider the congruence equation >x^3 + x^2 -5 = 0 (mod 49). >If _a_ is a solution mod 49, then a is certainly a solution mod 7. >I also know that 2 is a solution mod 7. >Question: why does it follow that x = 2 ( mod 7 ) ? P(x) == 0 (mod 49) iff 49 | P(x). Thus, 7 | P(x) so it follows that P(x) == 0 (mod 7). If 2 is the only solution (mod 7), then we must have x == 2 (mod 7). Rob Johnson take out the trash before replying === Subject: Re: - Congruence equations (mod n) posting-account=_l4K0QkAAAC09JhOoK_ZfoJKXOmr_jZf Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Consider the congruence equation x^3 + x^2 -5= 0 (mod 49). If a is a solution mod 49, then a is certainly a solutionmod 7. I also know that 2 is a solution mod 7. Question: whydoes it follow that x = 2( mod 7 ) ? You mean a? x^3 + x^2 - 5 = (x-2)(x^2 - 4x - 1) (mod 7) The discriminant of x^4 - 4x - 1 is 16+5 = 20 = -1 (mod 7). But -1 is not a square modulo 7, so the only solution modulo 7 is x=2. Therefore, if a is any solution modulo 49, then we must have a=2 (mod 7). -- Arturo Magidin === Subject: Re: Is time is One dimensional? if yes, How ? > > By the way, I'm interested in who you think are opponents of > Objectivism but not, in your view, Kantians? > A fearsome tribe, known by the name Altruists, is a major enemy of > Objectivism, nay, reason itself. > > I've always wanted to teach a course titled Altruism: how to make it > work for you. Good one. Reminds me of the old joke about how sincerity is the most important thing - once you can fake that, you've got it made. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Is time is One dimensional? if yes, How ? > By the way, I'm interested in who you think are opponents of > Objectivism but not, in your view, Kantians? > A fearsome tribe, known by the name Altruists, is a major enemy of > Objectivism, nay, reason itself. I've always wanted to teach a course titled Altruism: how to make it > work for you. I just read your post out loud, and there's sniggers all round - Phil -- Marijuana is indeed a dangerous drug. It causes governments to wage war against their own people. -- Dave Seaman (sci.math, 19 Mar 2009) === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > If she despised irrationalism, and Kant's philosophy > is irrational, as she remarks that it is, then it seem ineluctable > that she despised the philosophy of Kant. > As well as any and all other irrational utterly mind dependent > philosophies, ideas of whim and the arbitrary. > By the way, I'm interested in who you think are opponents of > Objectivism but not, in your view, Kantians? > Those who cant reduce their ideas theories concepts, which they claim > are ideas theories and concepts of reality, back to an irreducible and > sensory level of perception in their origin are the opponents of > Rand's ideas. e.g god and the leftist retard's nauseating nasal > whining chant of the greater good. > Forsooth (especially the part about 'nasal whining'; that was an > especially scathing turn of phrase). But I was wondering whether you > might mention some particular people (famous ones, so I'm in on the > moral) who you consider to be opponents of Objectivism but not > Kantians. No I cant really, because those famous people who are not Objvst's > (that covers about > 99.999999999999999999999999999999999999999999999999% of them) at some > point in their rants can be heard to say things which could have been > just as easily said by or come from Kant's pure unadulterated > destruction of reason, while pretending or claiming not to be Kantian, > you know what I mean, Kant's disgusting morality idea of a duty to > the truth, the truth shall set you free, and all that sort of > context bereft, arbitrary, mind depenedent crap. If saying the truth shall set you free makes one a Kantian, then surely you have to admit that your definition of 'Kantian' is rather broad. If saying one has a duty to truth makes one a Kantian, then saying I do things only for selfish reasons makes one an Objectivist (and I hardly think that saying I do things only for selfish reasons makes one an Objectivist). MoeBlee === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=g9P5YAkAAACbxBDAyQp7cMVe0-Yk0MYD SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > If saying the truth shall set you free makes one a Kantian, That is NOT what I said. I said - Kant's disgusting morality idea of a duty to the truth. It was the Pope and other religionists who said or say the truth shall set you free, which I dont see as a lot different to Kant's context bereft trash about having a duty to the truth. > If saying one has a duty to truth makes one a Kantian, FFS, I gave ONE example of one of Kant's most disgusting ideas shared by others who may or may not claim to be Kantian, if you really want to know more then do your own research as I have done. MG === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > If saying the truth shall set you free makes one a Kantian, That is NOT what I said. I said - Kant's disgusting morality idea of > a duty to > the truth. I'm happy to revise how you mean to be taken, but my initial response was to exactly what you said: No I cant really, because those famous people who are not Objvst's (that covers about 99.999999999999999999999999999999999999999999999999% of them) at some point in their rants can be heard to say things which could have been just as easily said by or come from Kant's pure unadulterated destruction of reason, while pretending or claiming not to be Kantian, you know what I mean, Kant's disgusting morality idea of a duty to the truth, the truth shall set you free, and all that sort of context bereft, arbitrary, mind depenedent crap. In that scenario, the person is NOT claiming to be a Kantian and is NOT claiming to have a morality idea that comes from Kant, and indeed in context bereft, but is saying the truth shall set you. Bereft of context and bereft of any claim to be a Kantian the person says the truth shall set you free which you take as a basis to say the person is a Kantian. But, again, I'm happy to take you now in any way you wish to qualify. Still, though, only a pretty substantial revision of your remark would provide you wih a scope for the term 'Kantian' that is not quite loose. > It was the Pope and other religionists who said or say the truth > shall set you free, which I dont see as a lot different to Kant's > context bereft trash about having a duty to the truth. Whether such utterances are a lot different or not, you still have a quite loose notion of 'Kantian'. > If saying one has a duty to truth makes one a Kantian, FFS, I gave ONE example of one of Kant's most disgusting ideas shared > by others who may or may not claim to be Kantian, if you really want > to know more then do your own research as I have done. You gave one idea which can only be taken as your claim of a SUFFICIENT (your word 'because') condition for one to be a Kantian. So my point stands, if a person saying, (what you consider to be) Kant's disgusting morality idea of the truth shall set you free is sufficient to classify that person as a Kantian, then a person saying the Objectivist morality idea I only do things for selfish reasons is sufficient to classify that person as an Objectivist (by the way, in your exact mode, even if the person pretends or claims not to be an Objectivist).. MoeBlee === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=g9P5YAkAAACbxBDAyQp7cMVe0-Yk0MYD SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) Whether such utterances are a lot different or not, you still have a > quite loose notion of 'Kantian'. Hint; There is nothing loose in the term primacy of existence (Rand / Piekoff / Objectivism) Verses primacy of consciousness (Kant / religion / subjectivism). There is nothing loose in -- the validated with non-contradictiry identification and integration of sensory evidence Vs the arbitrarially accepted unvalidated mind dependent whims. There is nothing loose between Reason (matter dependent) Vs Faith (mind dependent) And note; being athiest does NOT mean the rejection of primacy of consciousness, being athiest does NOT mean reason has been applied to one's ideas, being athiest does NOT rule out accepting ideas in the absense of evidence - in the absense of any sensory evidence for the acceptance of that idea, e.g. many socialists are athiests and yet socialism is a mind dependent arbitrary faith based concept. MG === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=-5a9CAoAAABl1C10M224pEnfD-yAElM_ .NET CLR 2.0.50727; InfoPath.1; FDM; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Hint; There is nothing loose in the term primacy of existence (Rand / > Piekoff / Objectivism) Verses primacy of consciousness (Kant / > religion / subjectivism). There is nothing loose in -- the validated with non-contradictiry > identification and integration of sensory evidence Vs the > arbitrarially accepted unvalidated mind dependent whims. > So the opposite of sensory evidence is mind dependant whims? Is a babies drive to suckle from it's mothers' breast (with no experience of it before) a Kantian whim or a genetic compulsion? Were the mind experiments conducted by Einstein in his bid to formlulate general relativity, mind dependant Kantian whims (as nobody had thought of it before it was not derived from experience) or genius at work? please enlighten me. > There is nothing loose between Reason (matter dependent) Vs Faith > (mind dependent) > === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=g9P5YAkAAACbxBDAyQp7cMVe0-Yk0MYD SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > Is a babies drive to suckle from it's mothers' breast (with no > experience of it before) a Kantian whim or a genetic compulsion? Thats like asking, what causes a baby to open its eyes, or to start breathing, hint, the nature of an entity is included in its identity. Hint; its NOT knowledge. > Were the mind experiments conducted by Einstein........... How does one go about conducting mind experiments? MG === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Whether such utterances are a lot different or not, you still have a > quite loose notion of 'Kantian'. Hint; There is nothing loose in the term primacy of existence (Rand / > Piekoff / Objectivism) Verses primacy of consciousness (Kant / > religion / subjectivism). There is nothing loose in -- the validated with non-contradictiry > identification and integration of sensory evidence Vs the > arbitrarially accepted unvalidated mind dependent whims. There is nothing loose between Reason (matter dependent) Vs Faith > (mind dependent) Granting all that for the sake of discussion, it still remains that scope of your use of the term 'Kantian' is quite loose. A counter-use of the term 'Objectivist' illustrates this quite well. > And note; being athiest does NOT mean the rejection of primacy of > consciousness, being athiest does NOT mean reason has been applied to > one's ideas, being athiest does NOT rule out accepting ideas in the > absense of evidence - in the absense of any sensory evidence for the > acceptance of that idea, I've not indicated that I have any thought to the contrary of your above paragraph, so I wonder what made you feel a need to rattle it off. I guess you just felt like you're were on a roll... > e.g. many socialists are athiests and yet > socialism is a mind dependent arbitrary faith based concept. I little doubt that that is an Objectivist position (though that depends on whether your notion of 'mind dependent' would be in accord with Rand). MoeBlee === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=g9P5YAkAAACbxBDAyQp7cMVe0-Yk0MYD SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > Granting all that for the sake of discussion, it still remains that > scope of your use of the term 'Kantian' is quite loose. A counter-use > of the term 'Objectivist' illustrates this quite well. Shrug and on the contrary, I would regard primacy of consciousness as being as tight and as meaningful as any identity for Kant's distruction of reason as any identity could be, the simple fact that religionists and subjectivists ALL fit comfortably within that identity and to explain the differences between them means nothing more than to engage in hair splitting or dancing on pinhead exersises. > I've not indicated that I have any thought to the contrary of your > above paragraph, so I wonder what made you feel a need to rattle it > off. I guess you just felt like you're were on a roll... Oh you wont be the only reader, and many an athiest takes pride or self satisfaction in lambasting the religionists and yet they come from the identical POC epistemological premise, is my point. > I little doubt that that is an Objectivist position (though that > depends on whether your notion of 'mind dependent' would be in accord > with Rand). Shrug, objectivist is a phrase you initiated not me, I am allowed to have my own ideas. MG === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Granting all that for the sake of discussion, it still remains that > scope of your use of the term 'Kantian' is quite loose. A counter-use > of the term 'Objectivist' illustrates this quite well. Shrug and on the contrary, I would regard primacy of consciousness > as being as tight and as meaningful as any identity for Kant's > distruction of reason as any identity could be, the simple fact that > religionists and subjectivists ALL fit comfortably within that > identity and to explain the differences between them means nothing > more than to engage in hair splitting or dancing on pinhead exersises. Sure, hair splitting to think that 9.999999999999999999999999999999999999999999999999% of the opponents of Objectivism aren't all Kantians. That to be a Kantian it is sufficient for one to say the truth will set you free. Good work. > I've not indicated that I have any thought to the contrary of your > above paragraph, so I wonder what made you feel a need to rattle it > off. I guess you just felt like you're were on a roll... Oh you wont be the only reader, and many an athiest takes pride or > self satisfaction in lambasting the religionists and yet they come > from the identical POC epistemological premise, is my point. Wonderful then that you're so generous with bonus points. > I little doubt that that is an Objectivist position (though that > depends on whether your notion of 'mind dependent' would be in accord > with Rand). Shrug, objectivist is a phrase you initiated not me, I am allowed to > have my own ideas. Actually the matter of Objectivism came up prior to my posting in this thread. But of course if you care not to have your claims evaluated indpendently of Objectivism then that is quite fair. But, as to the mere matter of whether you're an Objectivist, you are one I would take it, at least if I applied the same method by which you determine who is a Kantian. Anyway, from your own personal view, who do you think is cooler, Roark or Galt? I don't necessarily mean which one was better conceptually integrated but rather which one you think is overall the cooler dude. MoeBlee === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=g9P5YAkAAACbxBDAyQp7cMVe0-Yk0MYD SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > That to be a > Kantian it is sufficient for one to say the truth will set you free. > Good work. You're dropping context, I've also added, to be a Kantian requires the embracing of the POC and the arbitrary, to be Kantian requires rejecting the POE, to be Kantian requires embracing the totally opposite epistemology of objectivism. To be Kantian requires one to reject that reason is man's ONLY means to knowledge. To be Kantian one needs to poke their eyes with red hot pokers and to pearce their ear drums with needles and to render their three other senses as equally as useless and meaningless decorations, i.e. Kantians claim they cant know anything for certain - because they cant know things in themselves, but none of them will explain - what else is there to know, other than what their senses provide, to be able to claim with certainty that the ant and the elephant are not the same forms of life, they are blinded because they have eyes, their senses are utterly useless. > Actually the matter of Objectivism came up prior to my posting in this > thread. Again you are confused, you initiated the subject of objectivism in the conversation between us and in a totally different and new context to what was mentioned prior. By the way, I'm interested in who you think are opponents of Objectivism but not, in your view, Kantians? > But, as to the mere matter of whether you're an Objectivist, you are > one I would take it, at least if I applied the same method by which > you determine who is a Kantian. You constantly fail to check your ideas with the reality of what was said, you are dropping context, you are being dishonest. > Anyway, from your own personal view, who do you think is cooler, Roark > or Galt? I don't necessarily mean which one was better conceptually > integrated but rather which one you think is overall the cooler dude. Roark probably, never given it any thought, no need to. MG === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=n26igQkAAACeF9xA2Ms8cKIdBH40qzwr Gecko/20070505 Iceape/1.0.9 (Debian-1.0.13~pre080614i-0etch1),gzip(gfe),gzip(gfe) > If she despised irrationalism, and Kant's philosophy > is irrational, as she remarks that it is, then it seem ineluctable > that she despised the philosophy of Kant. > As well as any and all other irrational utterly mind dependent > philosophies, ideas of whim and the arbitrary. > By the way, I'm interested in who you think are opponents of > Objectivism but not, in your view, Kantians? > Those who cant reduce their ideas theories concepts, which they claim > are ideas theories and concepts of reality, back to an irreducible and > sensory level of perception in their origin are the opponents of > Rand's ideas. e.g god and the leftist retard's nauseating nasal > whining chant of the greater good. > Forsooth (especially the part about 'nasal whining'; that was an > especially scathing turn of phrase). But I was wondering whether you > might mention some particular people (famous ones, so I'm in on the > moral) who you consider to be opponents of Objectivism but not > Kantians. No I cant really, because those famous people who are not Objvst's > (that covers about > 99.999999999999999999999999999999999999999999999999% of them) at some > point in their rants can be heard to say things which could have been > just as easily said by or come from Kant's pure unadulterated > destruction of reason, while pretending or claiming not to be Kantian, > you know what I mean, Kant's disgusting morality idea of a duty to > the truth, the truth shall set you free, and all that sort of > context bereft, arbitrary, mind depenedent crap. MG You guys are ridiculous. These days I can say that I left my house with a GPS at 10:13.012 h:m.s today and arrived at the store at 10:32.304 and had a VMG of 34.23 mph. This level of detail was not available in Kant's time, but the same sensibilities are exactly what he discusses. We use the same interpretation of space and time in modernity that he accepted, up to relativity theory with high speed computations. We've merely refined the level of detail. Still, tape measueres were accurate enough in his time and clocks as well to describe the space and time that we experience. Beyond this his formalities of a priori versus intuition and so forth are careful nuances which set a context that is rarely troubled over any more since we breath an accumulation of information spanning back to Kant. Still in his time it was possible to write a standalone format which he did painstakingly. So too did Newton. Nowadays this format hardly exists and the crossweb of accumulation is daunting. I concede that Kant does not trouble nor quibble over the three dimensions of space nor the troubling lack of conguence of time to space, yet he does carefully couple these two terms in the passages of his Critique of Pure Reason, so that at the very least a direct precursor to unified spacetime is there back in 1781 AD. I quoted his book in this message: - Tim === Subject: Re: Is time is One dimensional? if yes, How ? posting-account=g9P5YAkAAACbxBDAyQp7cMVe0-Yk0MYD SLCC1; .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.5.30729; .NET CLR 3.0.30618),gzip(gfe),gzip(gfe) > You guys are ridiculous. These days I can say that I left my house > with a GPS at 10:13.012 h:m.s today and arrived at the store at > 10:32.304 and had a VMG of 34.23 mph. > This level of detail was not available in Kant's time, Wow, shucks you are soooooo ridiculously bright. NOT > but the same > sensibilities are exactly what he discusses. You may find the following packed with sensibilities, the application of reason certainly doesn't. First sentence of Kant's pure unadulterated destruction of reason: (PUDOR) That all our knowledge begins with experience there can be no doubt. First sentence second paragraph: But, though all our knowledge begins with experience, it by no means follows that all arises out of experience. You may find Kant's take on morality sensible, whereas reason proves it to be one of the most senseless, therefore utterly mind dependent, disgustingly arbitrary, context bereft, utterly ignorant and bereft of anything man can experience, (touch see feel hear smell), therefore the most anti-human anti-reason morality known to man. You may regard Kant's dopey idea that man cant know things in themselves, sensible, when in reality, as determined via reason, that garbage is just sooo ridiculously stupid its beyond words. In man aquiring his knowledge there is absolutely no requirement of man to know things in themselves for him to be able to claim 100% certain knowldge, with 100% certainty man CAN and DOES know the differences between forms of matter and matter's nature. It is NOT the entity in itself that man needs to know, but rather it is the differences between the forms of matter and their nature is ALL there is for man to have knowledge of, Aristotle's law of identity which Kant rejects as a means to knowledge. MG === Subject: 2,2,2,11,2,53,2,3,3 ,7 ,2,2,2,5 ,13 prime factorisation of mersenne p exponents minus 1. Cc: MCDONewt@yahoo.co.nz posting-account=TV2szgkAAACrA1vyuh8IN_0zzgzcwogw .NET CLR 2.0.50727; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) 2,2,2,11,2,53,2,3,3 ,7 ,2,2,2,5 ,13 prime factorisation of mersenne p exponents minus 1. e.g. 2 2 2 3 3 5 5 53 149 sloanes online encyc of integer sequences. does not have. donald s. mcdonald paradise. nz 11-6-2009. 47th known mersenne prime apparently discovered. === Subject: Re: 2,2,2,11,2,53,2,3,3 ,7 ,2,2,2,5 ,13 prime factorisation of mersenne p exponents minus 1. Cc: MCDONewt@yahoo.co.nz posting-account=TV2szgkAAACrA1vyuh8IN_0zzgzcwogw .NET CLR 2.0.50727; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > 2,2,2,11,2,53,2,3,3 ,7 ,2,2,2,5 ,13prime factorisation of mersenne p > exponents minus 1. e.g. 2 2 23 3 5 553 149 > sloanes online encyc of integer sequences.does not have. donald s. mcdonaldparadise. nz11-6-2009. > 47th known mersenne prime apparently discovered. world record bunch of prime numbers.: Google Alert - mersenne prime 7 mersenne prime exponents within an /octave// interval [k, 2k.] don mcdonald wellington 10-6-2009. jfh, jb. I am not sure if these columns will print correctly yahoo. 47th known mersenne prime exponent. factor 2^prime - 1 isprime? 42643801 426438 9 47382 factor 6 7897 factor 3 2632.333333 5 1579.4 7 1128.142857 9 877.4444444 11 717.9090909 13 607.4615385 15 526.4666667 17 464.5294118 19 415.6315789 21 376.047619 23 343.3478261 25 315.88 27 292.4814815 29 272.3103448 31 254.7419355 33 239.3030303 35 225.6285714 37 213.4324324 39 202.4871795 41 192.6097561 43 183.6511628 45 175.4888889 47 168.0212766 49 161.1632653 51 154.8431373 53 149 7897 426438 6*9*53*149*100+1 factors 42643801 isprime? twin primes about mersenne prime exponent. 43rd? program BAS64- modabc1 ? 277345127 31*683*13099 277345127 ryee Don S. McDonald ... Table Mountain 2^ (2*34* 444 32 +1 ) -1 is Mersenne Prime v//. exponent 3.02.. million? Climate pride//# === > Subject: world record bunch of prime numbers.: Google Alert - mersenne prime > Received: Wednesday, 10 June, 2009, 1:50 PM > world record bunch of prime numbers.: Google Alert - > mersenne prime MathWorld News: 47th Known Mersenne Prime Apparently > Discovered > June 7, 2009--Less than a year after the 45th and 46th > known Mersenne > primes were discovered, Great Internet Mersenne Prime > Search (GIMPS) > project organizer George Woltman is reporting in a June 7 > email to the > GIMPS mailing list that a ... This once a day Google Alert > fwd cheers, > ............10/6/09 Tara wera Mt > 1886?? Rotorua NZ. Don S. McDonald ... === Subject: Re: 2,2,2,11,2,53,2,3,3 ,7 ,2,2,2,5 ,13 prime factorisation of mersenne p exponents minus 1. posting-account=sxrJ7goAAABI7pirjnwOXjy89oxl-rMO 240x320),gzip(gfe),gzip(gfe) The below claim including the discovery of the 47th Mersenne Prime is reserved from claim, excluded *specifically* by claim from *G.I.M.P.S.* (C) 2009. Martin Musatov. All Rights Reserved. > 2,2,2,11,2,53,2,3,3 ,7 ,2,2,2,5 ,13prime factorisation of mersenne p > exponents minus 1. > e.g. 2 2 23 3 5 553 149 > sloanes online encyc of integer sequences.does not have. > donald s. mcdonaldparadise. nz11-6-2009. > 47th known mersenne prime apparently discovered. > world record bunch of prime numbers.: Google Alert - mersenne prime 7 mersenne prime exponents within an /octave// interval [k, 2k.] don mcdonald wellington 10-6-2009. jfh, jb. I am not sure if these columns will print correctly yahoo. 47th known mersenne prime exponent. factor > 2^prime - 1 isprime? > 42643801 426438 > 9 47382 factor > 6 7897 factor > 3 2632.333333 > 5 1579.4 > 7 1128.142857 > 9 877.4444444 > 11 717.9090909 > 13 607.4615385 > 15 526.4666667 > 17 464.5294118 > 19 415.6315789 > 21 376.047619 > 23 343.3478261 > 25 315.88 > 27 292.4814815 > 29 272.3103448 > 31 254.7419355 > 33 239.3030303 > 35 225.6285714 > 37 213.4324324 > 39 202.4871795 > 41 192.6097561 > 43 183.6511628 > 45 175.4888889 > 47 168.0212766 > 49 161.1632653 > 51 154.8431373 > 53 149 7897 426438 6*9*53*149*100+1 factors 42643801 isprime? > twin primes about mersenne prime exponent. 43rd? program > BAS64- modabc1 ? > 277345127 31*683*13099 277345127 ryee Don S. McDonald ... Table Mountain > 2^ (2*34* 444 32 +1 ) -1 is Mersenne Prime v//. > exponent 3.02.. million? Climate pride//# === > Subject: world record bunch of prime numbers.: Google Alert - mersenne prime > Received: Wednesday, 10 June, 2009, 1:50 PM > > world record bunch of prime numbers.: Google Alert - > mersenne prime > MathWorld News: 47th Known Mersenne Prime Apparently > Discovered > June 7, 2009--Less than a year after the 45th and 46th > known Mersenne > primes were discovered, Great Internet Mersenne Prime > Search (GIMPS) > project organizer George Woltman is reporting in a June 7 > email to the > GIMPS mailing list that a ... > MathWorld Headline News > This once a day Google Alert > fwd cheers, > ............10/6/09 Tara wera Mt > 1886?? Rotorua NZ. > Don S. McDonald ... The above claim including the discovery of the 47th Mersenne Prime is reserved from claim, excluded *specifically* by claim from *G.I.M.P.S.* (C) 2009. Martin Musatov. All Rights Reserved. === Subject: Re: Why is it more natural? (!) <4a2cbc46$0$23264$afc38c87@news.optusnet.com.au> posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > .... > However, Complex numbers are only one generalisation of R. Why doesn't > calculus applied to quaternions produce similar or better magic? .... >In the 19th century Hamilton and his successors tried to develop >quaternion analysis as you suggest, generalizing the theory of functions >of a complex variable.My recollection of the history is foggy, but I >suspect the resulting convoluted theory was one of the reasons for >quaternions' later unpopularity. > Ken Pledger. The problem with a quaternion calculus is that with any > definition, including one-sided, of a derivative, only > linear functions are differentiable.There is no theory > of analytic functions of quaternions. > Is this because the quaternions are not in general commutative under multiplication? (Even if the reason is subtler than one might first think) Or is it something else? === Subject: Re: Excellent Book On Relativity And Classical Mechanics <4a2f5d36$0$29316$afc38c87@news.optusnet.com.au> <55e32$4a2f7931$82a1e228$6871@news1.tudelft.nl> posting-account=PTS84AoAAACr67p51zvy0Hlr3LkoIUcc x64; .NET CLR 2.0.50727; SLCC1; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > You studied tensors in first year Calculus? > Which University was that at? The gist of the argument may be that tensors are not typical for General > Relativity. That is correct. > They are abundant in e.g. structural mechanics. Yes, indeed. > And they are > no such high level that only a few people on earth can understand them. The so-called tensor calculus that gives geodesics is nothing more than differentiation as laid out methodically by Leibniz more than 200 years ago. If you know how to perform a derivative, you have no excuse to understand tensor calculus. The notation of indices is merely a lazy man's way of representing summation. There is no mysticism in it. Don't be intimidated by it. === Subject: Re: Excellent Book On Relativity And Classical Mechanics > >You studied tensors in first year Calculus? >Which University was that at? >The gist of the argument may be that tensors are not typical for General >Relativity. > > That is correct. > >They are abundant in e.g. structural mechanics. > > Yes, indeed. > >And they are >no such high level that only a few people on earth can understand them. > > The so-called tensor calculus that gives geodesics is nothing more > than differentiation as laid out methodically by Leibniz more than 200 > years ago. If you know how to perform a derivative, you have no > excuse to understand tensor calculus. Oh well, it is _somewhat_ more complicated than performing a derivative. > The notation of indices is merely a lazy man's way of representing > summation. There is no mysticism in it. Don't be intimidated by it. Han de Bruijn === Subject: Re: Excellent Book On Relativity And Classical Mechanics <4a2f5d36$0$29316$afc38c87@news.optusnet.com.au> <55e32$4a2f7931$82a1e228$6871@news1.tudelft.nl> posting-account=c6T4LggAAACAqdk8Zx_zXI_oErOujh3M Gecko/20050915 Firefox/1.0.7,gzip(gfe),gzip(gfe) On Jun 11, 12:09 am, Han de Bruijn You studied tensors in first year Calculus? >Which University was that at? >The gist of the argument may be that tensors are not typical for General >Relativity. > That is correct. >They are abundant in e.g. structural mechanics. > Yes, indeed. >And they are >no such high level that only a few people on earth can understand them. > The so-called tensor calculus that gives geodesics is nothing more > than differentiation as laid out methodically by Leibniz more than 200 > years ago. If you know how to perform a derivative, you have no > excuse to understand tensor calculus. Oh well, it is _somewhat_ more complicated than performing a derivative. The philosophy and use of tensor analysis is taught in Public School, for example when we *transform* a metric (cm) measurement to imperial (inch), such as a glance at 1 foot ruler where 12 ~ 30 cm, finds the length is the same independent of the CS units. Another example is tracing, a squiggly line on blank paper can be traced onto any kind of graph paper, cartesian, polar etc. and the line remains the same. An example is Mercator projection of the world in 3D to 2D. Another example is scale drawing a house plan where 1/4 = 1 foot, (I suppose that's an example of gauge invariance, when we change scale). The average Joe can easily understand the above. Doing those examples 'algebraically' is tough to learn, but mathematicians have evolved an elegant notation that is fairly smooth, if one has a need to use it. Ken S. Tucker === Subject: Re: Excellent Book On Relativity And Classical Mechanics <4a2f5d36$0$29316$afc38c87@news.optusnet.com.au> <55e32$4a2f7931$82a1e228$6871@news1.tudelft.nl> posting-account=rIfu6QoAAAD5nXG3h9QEE0J3dZn1U45R Gecko/2009053001 Gentoo Firefox/3.0.10,gzip(gfe),gzip(gfe) > You studied tensors in first year Calculus? > Which University was that at? > The gist of the argument may be that tensors are not typical for General > Relativity. That is correct. > They are abundant in e.g. structural mechanics. Yes, indeed. > And they are > no such high level that only a few people on earth can understand them. The so-called tensor calculus that gives geodesics is nothing more > than differentiation as laid out methodically by Leibniz more than 200 > years ago.If you know how to perform a derivative, you have no > excuse to understand tensor calculus. The notation of indices is merely a lazy man's way of representing > summation.There is no mysticism in it. Don't be intimidated by it. And yet you continue to struggle with basic presentations of the concept. === Subject: Re: Excellent Book On Relativity And Classical Mechanics > You studied tensors in first year Calculus? > Which University was that at? > The gist of the argument may be that tensors are not typical for General > Relativity. That is correct. > They are abundant in e.g. structural mechanics. Yes, indeed. > And they are > no such high level that only a few people on earth can understand them. The so-called tensor calculus that gives geodesics is nothing more > than differentiation as laid out methodically by Leibniz more than 200 > years ago. If you know how to perform a derivative, you have no > excuse to understand tensor calculus. Imbecile. The notation of indices is merely a lazy man's way of representing > summation. There is no mysticism in it. Don't be intimidated by it. And this must be the finest strawman ever produced here. Good grief, what a creep. Dirk Vdm === Subject: Re: The business model behind the Wolfram Alpha? posting-account=spgRJBAAAADTEf6m3y6Pyj43g6wnGUFN Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > All of the above, could instead have been done, I would imagine, using the > same resources and time that went to making WA, and the above would have > been much more useful to all ofwho use Mathematica. You're ignoring the fact that Wolfram is probably *bored to tears* thinking about Mathematica, for however many years it has been out now. He most likely wanted to think about something new for a change. I think Alpha is a cool and bold idea. It will be a nightmare to implement though. We haven't even scratched the surface of programmatically extracting semantics from natural language. -Andrew. === Subject: Re: The business model behind the Wolfram Alpha? >I wonder what is the business model behind the Wolfram Alpha? Any > ideas? > > I wish Dr Wolfram and his brilliant engineers would have spent all these > resources and the time on making Mathematica itself an even better product > that it is now. > > I would imagine that the engineering resources used on this web venture > could have instead been used to add the following to Mathematica: > > 1. Build a compiler which compiles Mathematica to a .exe > 2. Build an easy to use debugger, state of the art. > 3. Build a GUI builder to integrate with Manipulate to make it easy to lay > out components and make a heavy duty user interfaces with Mathematica (and > then use the compiler to make stand alone application, all build using > Mathematica). > 4. Add more and more Mathematical functions to core Mathematica itself. > (Mathematica is still missing many functions for signal processing, > control systems, and many other areas). > > All of the above, could instead have been done, I would imagine, using the > same resources and time that went to making WA, and the above would have > been much more useful to all of who use Mathematica. I fail to see the point in doing that when Mathematica is so riddled with bugs. I'd rather have seen a little testing done before the release of MMA 7 so that, for example, Fourier didn't return incorrect results under almost all circumstances. -- Dr Jon D Harrop, Flying Frog Consultancy Ltd. http://www.ffconsultancy.com/?u === Subject: Re: Two Schwarzschild radii posting-account=PTS84AoAAACr67p51zvy0Hlr3LkoIUcc x64; .NET CLR 2.0.50727; SLCC1; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > I am stating that this is a completely unambiguous prediction > of general relativity. Whether it is physically the correct > description of a black hole is a matter for experiment and > observation, but so far the observations match the predictions > well; seehttp://arxiv.org/abs/0903.1105, for example. According to the Schwarzschild metric, black holes can only be formed at any observer's infinite future. This is all in the mathematics. So, claiming you able to observe a black hole, is it not ludicrous? > No, it certainly doesn't! It comes from an analysis of the behavior > of null geodesics in the Schwarzschild geometry. There is only one geodesics that allows the geodesic motions of you have to write the Schwarzschild solution in coordinates that > include the horizon (Schwarzschild coordinates won't do, but you > can use Kruskal-Szekeres or Painleve-Gullstrand coordinates, for > instance). Then you have to write down the equations for the null > geodesics and solve them. So, are you telling us only in a special coordinate system that something out of the ordinary can appear out of nowhere? I thought the geometry is invariant. It is what it is regardless of any choice of coordinate system to mathematically describe that geometry, no? > You seem to have missed the point of my last post. Let me try one > more time. You *cannot* understand general relativity by merely > pulling an equation or two out of context or by reading simple > popularizations. Horrible as it is, you have to actually learn to do > the math. The geodesics has nothing to do with GR. What is identifiable as GR are the set of the field equations. So, to understand GR, you must also understand how the field equations are derived? Is that too much to ask for physicists? > Frequency is a *relative* concept; it depends both on the light and > on the observer's motion. An observer who is trying to remain at > rest outside a black hole must expend energy to do so -- she has to > fire her rockets to keep from falling toward the black hole. Relative > to her, an incoming photon is blue shifted and has increased energy, > just as a photon from a streetlight is blue shifted and has increased > energy relative to a car driving toward the source. This cannot be. Energy should be but is somewhat conserved in the Schwarzschild metric. No photons or massed objects can gain or lose any energy (well at least noticeable amount if you believe in the mumble jumble of pulling out the gravitational radiation from the geodesic variations). So, frequency shift in observation is not of an energy issue, agree? > Indeed, if there *were* a surface just outside the Schwarzschild > radius, rather than an event horizon, then relative to that surface, > infalling matter and light would, certainly have a very large energy. Hmmm... Since there are an infinite number of solutions to the field equations that are static, spherically symmetric, and asymptotically flat, there is really no excuse to worship the Schwarzschild metric as the true geometry. For instance, you can start with the following that do not manifest black holes. Wouldn't life be much simpler? ds^2 = c^2 T dt^2 / (1 + 2 K / r) [CapitalEth] (1 + 2 K / r) dr^2 [CapitalEth] (r + K)^2 dO^2 Where ** K = G M / c^2 ** dO^2 = cos^2(Latitude) dLongitude^2 + dLatitude^2 ** T = 1 > (There's nothing very strange about this -- if you drop a ball from > a tall building, it hits the ground with a lot of energy.) Stuff hitting > such a surface would emit a huge amount of energy, which we would > see. In Newtonian mechanics, everything is conserved. The high speed at the end of a free-falling journey is due to increased in kinetic energy while losing potential energy. You have to explain why photons gain a lot of kinetic energy to manifest a blue shift. Is that all in the mathematics? > The preprint I cited above puts very strong observational limits > on the existence of such a surface for the massive object at the center > of our galaxy. Matter falling in *doesn't* give off the energy it would > if it were hitting a surface; rather, the energy simply vanishes, as it > would if there were an event horizon. === Subject: Re: Two Schwarzschild radii (...) > No, it certainly doesn't! It comes from an analysis of the behavior of > null geodesics in the Schwarzschild geometry. > > There is only one geodesics that allows the geodesic motions of > Don't true. > To do this analysis > you have to write the Schwarzschild solution in coordinates that > include the horizon (Schwarzschild coordinates won't do, but you can > use Kruskal-Szekeres or Painleve-Gullstrand coordinates, for instance). > Then you have to write down the equations for the null geodesics and > solve them. > > So, are you telling us only in a special coordinate system that > something out of the ordinary can appear out of nowhere? I thought the > geometry is invariant. It is what it is regardless of any choice of > coordinate system to mathematically describe that geometry, no? The main reason to use different coordinates in science is that some coordinates are better than others for certain problems. (...) > Hmmm... Since there are an infinite number of solutions to the field > equations that are static, spherically symmetric, and asymptotically > flat, there is really no excuse to worship the Schwarzschild metric as > the true geometry. For instance, you can start with the following that > do not manifest black holes. Wouldn't life be much simpler? > > ds^2 = c^2 T dt^2 / (1 + 2 K / r) [CapitalEth] (1 + 2 K / r) dr^2 [CapitalEth] (r + K)^2 dO^2 > > Where > > ** K = G M / c^2 > ** dO^2 = cos^2(Latitude) dLongitude^2 + dLatitude^2 ** T = 1 > What is the r--> infinity limit of that? -- http://www.canonicalscience.org/ Usenet Guidelines: http://www.canonicalscience.org/en/miscellaneouszone/guidelines.html === Subject: Re: Two Schwarzschild radii posting-account=rIfu6QoAAAD5nXG3h9QEE0J3dZn1U45R Gecko/2009053001 Gentoo Firefox/3.0.10,gzip(gfe),gzip(gfe) > I am stating that this is a completely unambiguous prediction > of general relativity.Whether it is physically the correct > description of a black hole is a matter for experiment and > observation, but so far the observations match the predictions > well; seehttp://arxiv.org/abs/0903.1105, for example. According to the Schwarzschild metric, black holes can only be formed > at any observer's infinite future.This is all in the mathematics. > So, claiming you able to observe a black hole, is it not ludicrous? The black hole effectively forms in a timeframe a wee bit tighter than 'forever'. > No, it certainly doesn't!It comes from an analysis of the behavior > of null geodesics in the Schwarzschild geometry. There is only one geodesics that allows the geodesic motions of > Because the engineer things something physics does is ridiculous, it must therefore be ridiculous. spacetime. It isn't complicated. >To do this analysis > you have to write the Schwarzschild solution in coordinates that > include the horizon (Schwarzschild coordinates won't do, but you > can use Kruskal-Szekeres or Painleve-Gullstrand coordinates, for > instance).Then you have to write down the equations for the null > geodesics and solve them. So, are you telling us only in a special coordinate system that > something out of the ordinary can appear out of nowhere?I thought > the geometry is invariant.It is what it is regardless of any choice > of coordinate system to mathematically describe that geometry, no? The phrase geometry is invariant continues to be gibberish. Learn the theory before criticizing an explanation you do not understand. > You seem to have missed the point of my last post.Let me try one > more time.You *cannot* understand general relativity by merely > pulling an equation or two out of context or by reading simple > popularizations.Horrible as it is, you have to actually learn to do > the math. The geodesics has nothing to do with GR.What is identifiable as GR > are the set of the field equations. Except geodesic motion is fundamental to GR. Once again the engineer criticizes something from physics and gets it com-ing-pletely wrong. So, to understand GR, you must also understand how the field equations > are derived?Is that too much to ask for physicists? No, but it is too much to ask for the retired engineer with nothing better to do as amply demonstrated here : > Frequency is a *relative* concept; it depends both on the light and > on the observer's motion.An observer who is trying to remain at > rest outside a black hole must expend energy to do so -- she has to > fire her rockets to keep from falling toward the black hole. Relative > to her, an incoming photon is blue shifted and has increased energy, > just as a photon from a streetlight is blue shifted and has increased > energy relative to a car driving toward the source. This cannot be.Energy should be but is somewhat conserved in the > Schwarzschild metric. The ignorant engineer once spent two years screaming at me to learn variational calculus and then completely screws up a basic display of competence in regarding what is conserved when a Lagrangian is not a function of time. >No photons or massed objects can gain or lose > any energy (well at least noticeable amount if you believe in the > mumble jumble of pulling out the gravitational radiation from the > geodesic variations). As per usual the engineer whines about something unrelated and s that up too. Gravitational radiation isn't derived from the geodesic equation in any way, shape, or form. But don't let facts get in the way of your ignorant spew. So, frequency shift in observation is not of an energy issue, agree? > Indeed, if there *were* a surface just outside the Schwarzschild > radius, rather than an event horizon, then relative to that surface, > infalling matter and light would, certainly have a very large energy. Hmmm...Since there are an infinite number of solutions to the field > equations that are static, spherically symmetric, and asymptotically > flat, there is really no excuse to worship the Schwarzschild metric as > the true geometry. Lookie loo the engineer continues to repeat past stupidities. STILL DON'T GET IT! >For instance, you can start with the following > that do not manifest black holes.Wouldn't life be much simpler? ds^2 = c^2 T dt^2 / (1 + 2 K / r) [CapitalEth] (1 + 2 K / r) dr^2 [CapitalEth] (r + K)^2 > dO^2 This has been explained to you before. Shifring the coordinate system doesn't make black holes go away no matter how loudly and stupidly you shout. Where **K = G M / c^2 > **dO^2 = cos^2(Latitude) dLongitude^2 + dLatitude^2 > **T = 1 > (There's nothing very strange about this -- if you drop a ball from > a tall building, it hits the ground with a lot of energy.) Stuff hitting > such a surface would emit a huge amount of energy, which we would > see. In Newtonian mechanics, everything is conserved. Wrong. >The high speed at > the end of a free-falling journey is due to increased in kinetic > energy while losing potential energy.You have to explain why photons > gain a lot of kinetic energy to manifest a blue shift.Is that all in > the mathematics? > The preprint I cited above puts very strong observational limits > on the existence of such a surface for the massive object at the center > of our galaxy.Matter falling in *doesn't* give off the energy it would > if it were hitting a surface; rather, the energy simply vanishes, as it > would if there were an event horizon. > I'd like to see you demonstrate a modicum of competence in a subject you have spent 5 years ranting about but we can't all get what we want now can we? === Subject: Re: Two Schwarzschild radii posting-account=PTS84AoAAACr67p51zvy0Hlr3LkoIUcc x64; .NET CLR 2.0.50727; SLCC1; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > According to the Schwarzschild metric, black holes can only be formed > at any observer's infinite future. This is all in the mathematics. > So, claiming you able to observe a black hole, is it not ludicrous? The black hole effectively forms in a timeframe a wee bit tighter than > 'forever'. Hmmm... This is a waste of words. There is only one geodesics that allows the geodesic motions of > Because the engineer things something physics does is ridiculous, it > must therefore be ridiculous. You are so deranged. You don't make any sense. So, are you telling us only in a special coordinate system that > something out of the ordinary can appear out of nowhere? I thought > the geometry is invariant. It is what it is regardless of any choice > of coordinate system to mathematically describe that geometry, no? The phrase geometry is invariant continues to be gibberish. How so? The geometry is just there. You can never will it to be whatever you like it to be. the theory before criticizing an explanation you do not understand. Any theories saying you can change the geometry by choosing a special set of coordinate system is bull$hit. The geodesics has nothing to do with GR. What is identifiable as GR > are the set of the field equations. Except geodesic motion is fundamental to GR. Hmmm... Geodesic motion is fundamental to everything, not just GR. criticizes something from physics and gets it com-ing-pletely > wrong. The college dropout is having a fit today. What is the matter? Perpetual no love life? I told you that you should lose some weight. So, to understand GR, you must also understand how the field equations > are derived? Is that too much to ask for physicists? No, but it is too much to ask for the retired engineer with nothing > better to do as amply demonstrated here : Whatever it supposes to mean! This cannot be. Energy should be but is somewhat conserved in the > Schwarzschild metric. The ignorant engineer once spent two years screaming at me to learn > variational calculus and then completely screws up a basic display of > competence in regarding what is conserved when a Lagrangian is not a > function of time. Screaming at you, a college dropout? You got to be kidding. I was addressing to Professor Carlip's post. Honestly, if physicists do not understand the calculus, does a college dropout stand a change? > No photons or massed objects can gain or lose > any energy (well at least noticeable amount if you believe in the > mumble jumble of pulling out the gravitational radiation from the > geodesic variations). As per usual the engineer whines about something unrelated and s > that up too. Gravitational radiation isn't derived from the geodesic > equation in any way, shape, or form. But don't let facts get in the > way of your ignorant spew. The college dropout apparently does not understand geodesic equations from geodesic variations which give rise to Riemann tensor. Hmmm... Since there are an infinite number of solutions to the field > equations that are static, spherically symmetric, and asymptotically > flat, there is really no excuse to worship the Schwarzschild metric as > the true geometry. Lookie loo the engineer continues to repeat past stupidities. STILL DON'T GET IT! > For instance, you can start with the following > that do not manifest black holes. Wouldn't life be much simpler? HEY, COLLEGE DROPOUT, I HAVE BEEN SHOWING YOU THAT THE FIELD EQUATIONS DO MANIFEST AN INFINITE NUMBERS OF SOLUTIONS. TRY KERR, FRIEDMAN-DAH- DAH-DAH, LEMAITRE METRICS. THE COLLEGE DROPOUT STILL DOES NOT KNOW WHAT IT IS TALKING ABOUT AS USUAL. ds^2 = c^2 T dt^2 / (1 + 2 K / r) [CapitalEth] (1 + 2 K / r) dr^2 [CapitalEth] (r + K)^2 > dO^2 This has been explained to you before. Shifring the coordinate system > doesn't make black holes go away no matter how loudly and stupidly you > shout. What shifting of coordinate system? The coordinate system is the very same as the Schwarzschild metric. Where > ** K = G M / c^2 > ** dO^2 = cos^2(Latitude) dLongitude^2 + dLatitude^2 > ** T = 1 > (There's nothing very strange about this -- if you drop a ball from > a tall building, it hits the ground with a lot of energy.) Stuff hitting > such a surface would emit a huge amount of energy, which we would > see. > In Newtonian mechanics, everything is conserved. Wrong. The college dropout never understand the first law of thermodynamics I see. The high speed at > the end of a free-falling journey is due to increased in kinetic > energy while losing potential energy. You have to explain why photons > gain a lot of kinetic energy to manifest a blue shift. Is that all in > the mathematics? > I'd like to see you demonstrate a modicum of competence in a subject > you have spent 5 years ranting about but we can't all get what we want > now can we? I really don't care about any advices from a college dropout. Does anyone? === Subject: Re: Two Schwarzschild radii (...) > spacetime. It isn't complicated. > > That is ridiculous. Any objects regardless having rest mass or no > should obey the same set of laws of physics. Your claim is ridiculous because the 'laws' are the same. But its motion different charges. The law F = mdp/dt with p linear momentum is the same for both charged and uncharged -- http://www.canonicalscience.org/ Usenet Guidelines: http://www.canonicalscience.org/en/miscellaneouszone/guidelines.html === Subject: Re: Two Schwarzschild radii posting-account=rIfu6QoAAAD5nXG3h9QEE0J3dZn1U45R Gecko/2009053001 Gentoo Firefox/3.0.10,gzip(gfe),gzip(gfe) [snip all, unread] Now that you've spewed for the month, off till July. === Subject: Re: Two Schwarzschild radii posting-account=wigfZgkAAACDgITarXffzxJygX81YRSs Gecko/2009042708 Fedora/3.0.10-1.fc9 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Because the engineer things something physics does is ridiculous, it > must therefore be ridiculous. spacetime. It isn't complicated. http://en.wikipedia.org/wiki/Mass Sue... === Subject: Integer solution $y^2=x^3 - 1431$ ? I'd like to prove that the elliptic curve $y^2=x^3 - 1431$ has no integer solution (1431=53*3^3) Searching for advice or a good book that could get me started. Chris === Subject: Re: Integer solution $y^2=x^3 - 1431$ ? posting-account=XzESVgkAAAASMMQZpglhqosZg-ouN3uW AppleWebKit/525.27.1 (KHTML, like Gecko) Version/3.2.1 Safari/525.27.1,gzip(gfe),gzip(gfe) I'd like to prove that the elliptic curve$y^2=x^3 - 1431$ has no integer solution (1431=53*3^3) > Searching for advice or a good book that could get me started. > Chris You could look here : http://tnt.math.metro-u.ac.jp/simath/MORDELL/ de P === Subject: Re: Integer solution $y^2=x^3 - 1431$ ? <896512.828.1244697627239.JavaMail.jakarta@nitrogen.mathforum.org>, > > I'd like to prove that the elliptic curve $y^2=x^3 - 1431$ has no integer > solution (1431=53*3^3) > Searching for advice or a good book that could get me started. y^2 = x^3 - k is generally referred to as Mordell's equation, so a search for that might get you started. Mordell's book, Diophantine Equations, discusses it in some detail, but is far from the last word. -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Integer solution $y^2=x^3 - 1431$ ? > I'd like to prove that the elliptic curve $y^2=x^3 - 1431$ has no integer > solution (1431=53*3^3) > Searching for advice or a good book that could get me started. > > y^2 = x^3 - k is generally referred to as Mordell's equation, > so a search for that might get you started. Mordell's book, > Diophantine Equations, discusses it in some detail, but is far from > the last word. I would say the book by Silverman and Tate is almost certainly the best place to look. Also you could look at Cremona's tables online to see the rank of the group of rational points on this curve. -- Timothy Murphy e-mail: gayleard /at/ eircom.net tel: +353-86-2336090, +353-1-2842366 s-mail: School of Mathematics, Trinity College Dublin === Subject: Three bar equal sign? posting-account=spgRJBAAAADTEf6m3y6Pyj43g6wnGUFN Gecko/2009042316 Firefox/3.0.10 (.NET CLR 3.5.30729),gzip(gfe),gzip(gfe) I've noted Kaplan using a three bar equal sign in some equations in his book Advanced Calculus. For example f[phi(x,y), psi(x,y)] = x (obviously = above is a really a three bar) I am unfamiliar with the meaning in this context and he doesn't seem to define it. The only semantics Im aware of are conguent (mod x) and is by definition - and neither seem appropriate in the context of multivariate differential equations. Any idea what this 3 bar equal sign means? Andrew. === Subject: Re: Three bar equal sign? What they said. For example, in elementary trigonometry, we may write sin x = cos x as an equation to be solved, it is true for some x, false for others. But we also write sin^2 x (threebars) 1 - cos^2 x meaning it is an *identity* , true for all x . === Subject: Re: Three bar equal sign? > I've noted Kaplan using a three bar equal sign in some equations in > his book Advanced Calculus. For example f[phi(x,y), psi(x,y)] = x (obviously = above is a really a three bar) I am unfamiliar with the meaning in this context and he doesn't seem > to define it. The only semantics Im aware of are congruent (mod x) > and is by definition - and neither seem appropriate in the context > of multivariate differential equations. Any idea what this 3 bar equal sign means? > It means equivalent in a different sense. Namely that f(...) is equivalent to x, identical to x, or identically x, ie for all x,y, f(...) = x. Let F be a set of functions with a common domain. An equivalence relation for F is f ~ g when for all x, f(x) = g(x). That is the equivalent relation or sense of equivalence which the author is using. On the other hand f ~ g iff f = g where ~ is the equivalence relation given and = this time is simple equality. === Subject: Re: Three bar equal sign? > I've noted Kaplan using a three bar equal sign in some equations in > his book Advanced Calculus. > > For example f[phi(x,y), psi(x,y)] = x (obviously = above is a really a three bar) > > I am unfamiliar with the meaning in this context and he doesn't seem > to define it. The only semantics Im aware of are conguent (mod x) > and is by definition - and neither seem appropriate in the context > of multivariate differential equations. > > Any idea what this 3 bar equal sign means? > > Andrew. http://encyclopedia.thefreedictionary.com/equality+sign there see related symbols. Means identity here, I guess. Ciao Karl === Subject: a question about Radon-Nikodym derivative Hi all, I have a question as follows: given mu_1 and mu_2 two Borel probability measures on R, does there always exist a Borel probability measure mu, such that the Radon-Nikodym derivatives of mu_1 and mu_2 w.r.t. mu are continuously differentiable? My feeling is yes: convolving a mixture of mu_i's with a Gaussian distribution could probably work. But I haven't been able to figure out the details... Any suggestion will be appreciated! Best, YH === Subject: Re: a question about Radon-Nikodym derivative test > Hi all, > > I have a question as follows: given mu_1 and mu_2 > two Borel probability measures on R, does there > always exist a Borel probability measure mu, such > that the Radon-Nikodym derivatives of mu_1 and mu_2 > w.r.t. mu are continuously differentiable? > > My feeling is yes: convolving a mixture of mu_i's > with a Gaussian distribution could probably work. But > I haven't been able to figure out the details... Any > suggestion will be appreciated! > > Best, > YH === === > While the computer is in a not-halt state, infinitely many marks are > made on the tape (**) ?? As I see it, this is ambiguous. It can mean either (**)1. Along the time the computer is in a not-halt-state, infinitely many marks will be made on the tape or (**)2. There is some time at which the computer is in a not-halt-state and at which infinitely many marks has been made on the tape Only the former is true. > Clearly in arriving at (**) from (*) we have interchanged quantifiers > and the objections raised by Gassmann and Marshall amounts to barring > this interchange. The translation of (**) to Achilles's status is the following: While Achilles has not reached 0, he crosses all the infinitely many > intervals in the set {[1/2,1],[1/4,1/2],1/8,1/4].....} which contradicts the fact that these infinitely many intervals sum to > 1 (meaning Achilles has reached zero) So does (**) follow from (*)? Gassmann and Marshall say no. Their > stand is equivalent to:: While Achilles has not reached 0, he crosses every finite number of > intervals in the above set, but he does not cross all (infinitely > many) of these intervals (***) Which, when translated to the state of the computer, means: While the computer is in a not-halt state, every finite number of > marks is made on the tape, but not infinitely many marks (****) If you accept that there are infinitely many natural numbers,, (***) > and (****) are clearly objectionable. === Subject: Re: Zeno's paradox revisited posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse Trident/4.0; Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; .NET CLR 2.0.50727; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > I don't write to be simple, although I do to convey meaning, so, > that is ineffective if the readership of interest can't parse I suspect then that your readership of interest is a singleton set > containing only yourself.Not that there is anything unusual in that. You suspect mistakenly, and always upside down: there is not only your gang of idiots, morons, and liars, for how loud you will chant your own supremacy and glory. Just bull, and criminal minds. -LV === Subject: Re: Zeno's paradox revisited posting-account=_-PQygoAAAAciOn_89sZIlnxfb74FzXU Gecko/2009042316 Firefox/3.0.10,gzip(gfe),gzip(gfe) > I don't write to be simple, although I do to convey meaning, so, > that is ineffective if the readership of interest can't parse > I suspect then that your readership of interest is a singleton set > containing only yourself.Not that there is anything unusual in that. You suspect mistakenly, and always upside down: there is not only your > gang of idiots, morons, and liars, for how loud you will chant your > own supremacy and glory. Just bull, and criminal minds. -LV That is quite stronger than I would have put it. Little isn't necessarily a cabalist, for what he might be a kabalist (in the sense of a modernized gematria) part of the gallery, Blumschein not alone, and I want Tribble to find enjoyable, informative, informed, and rich commentary. About this function which looks like EF, the natural/unit equivalency function, standardly modeled by real functions, it's basically defined, for what it defines a basis, and founding a nonstandard measure theory, should be an intuitively available concept to those familiar with the continuum of real numbers of the unit interval. Admitting infinitesimals, particularly iota values, then the interesting part is its analytical (and varying) character in multiple dimensions: at once, the polydimensional. Ross F. === Subject: Re: Integral brick disproof from last month? In a previous mail,Gerry asked: > OK, so would you mind putting the whole thing, > correctly, in one post? I do it here, and use easy to remember variables. Suppose we have an integral brick with integer sides a,b,c, integer face diagonals ab,ac,bc and integer space diagonal abc (ab,... is NOT a*b but ONE length). No pair can be the same as we should have then an irrational length. So let's have (1) aac^2-ab^2 Suppose n is also an integer (**). There are no integral solutions for (6) with n=1 or n=2 (***). see http://mathworld.wolfram.com/DiophantineEquation4thPowers.html Then n cannot be a square as (n*bc)^2=ac^4-ab^4 is also impossible. That's it so far, but first look to the remarks as there is a BIG problem. bleuprint Remarks: (*) We can have cac^2-ab^2 In the second case we have i') 3ac^2-ab^2 Careful considerations on (7) can perhaps restrict more the two conditions i and ii, but the next remark (**) has first to be cleared. (**) Suppose n is an integer, here there is a BIG problem. ac and ab are certainly rational so n=ac-ab must also be rational, but I have no proof that it must be an integer. (***) There are integer solutions for (6) with (n,bc,ac,ab) as (5,4,3,1);(6,20,7,1);(7,20,8,6);(13,60,19,17) etc... They are OK for condition i but not for ii. n=6, === Subject: Re: Integral brick disproof from last month? I had to edit the previous reply and still it was not correct, sorry. Some mirror sites only give the messages without edits so I reply once more. Now it's correct I hope. In a previous reply, Gerry asked: > OK, so would you mind putting the whole thing, > correctly, in one post? I do it here, and use easy to remember variables. Suppose we have an integral brick with integer sides a,b,c, integer face diagonals ab,ac,bc and integer space diagonal abc (ab,... is NOT a*b,... but ONE length). No pair can be the same as we should have then an irrational length. So let's have (1) aac^2-ab^2 There are no integer solutions for (6) with n=1 or n=2 (***). see http://mathworld.wolfram.com/DiophantineEquation4thPowers.html Thus n cannot be a square integer as (n*bc)^2=ac^4-ab^4 is impossible. As 2*(n*bc)^2=ac^4-ab^4 is also impossible n cannot be twice an integer square. That's it so far, but first look to the remarks as there is a complication fo n. bleuprint Remarks: (*) We can have cab. Then we have two cases: (7) 1ac^2-ab^2>=11 In the second case we have then i') 3ac^2-ab^2>=20 Careful considerations on (7) can perhaps restrict more the two conditions i and ii, but the next remark (**) has first to be cleared. (**) It's not sure that n is an integer. As ac+ab=bc^2 must be an integer, ac and ab are certainly rational. So n=ac-ab must also be rational, but I have no proof that n must be an INTEGER. It should be nice to have one. (***) There are integer solutions for (6) with (n,bc,ac,ab) as (5,4,3,1);(6,20,7,1);(7,20,8,6);(13,60,19,17) etc... They are OK for condition i but not for ii. === Subject: Re: Integral brick disproof from last month? <15686803.5811.1244619132018.JavaMail.jakarta@nitrogen.mathforum.org>, > Sorry, it has to be indeed condition > 2) b^2 < c^2 + a^2 > as could also be seen in the PS #2. > It's in fact the same as condition: > 2a) n > c^2 - a^2 OK, so would you mind putting the whole thing, correctly, in one post? -- Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for email) === Subject: Re: Integral brick disproof from last month? > OK, so would you mind putting the whole thing, > correctly, in one post? > > -- > Gerry Myerson (gerry@maths.mq.edi.ai) (i -> u for > email) See first reply hereafter on 6/11/09. bleuprint === === Subject: Re: CRIMESTOP IN SCIENCE posting-account=Lz-LbgoAAABPDavKeW-eYeobwLHD_cvQ CLR 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) http://www.liferesearchuniversal.com/1984-17.html#seventeen George Orwell: Crimestop means the faculty of stopping short, as though by instinct, at the threshold of any dangerous thought. It includes the power of not grasping analogies, of failing to perceive logical errors, of misunderstanding the simplest arguments if they are inimical to Ingsoc, and of being bored or repelled by any train of thought which is capable of leading in a heretical direction. Crimestop, in short, means protective stupidity. Stopping short, as though by instinct, at the threshold of the dangerous thought: Einstein's 1905 light postulate (c'=c) is false; its antithesis given by Newton's emission theory of light (c'=c+v) is true: http://www.astrofind.net/documents/the-composition-and-essence-of-radiation. php The Development of Our Views on the Composition and Essence of Radiation by Albert Einstein Albert Einstein 1909: A large body of facts shows undeniably that light has certain fundamental properties that are better explained by Newton's emission theory of light than by the oscillation theory. For this reason, I believe that the next phase in the development of theoretical physics will bring us a theory of light that can be considered a fusion of the oscillation and emission theories. The purpose of the following remarks is to justify this belief and to show that a profound change in our views on the composition and essence of light is imperative.....Then the electromagnetic fields that make up light no longer appear as a state of a hypothetical medium, but rather as independent entities that the light source gives off, just as in Newton's emission theory of light......Relativity theory has changed our views on light. Light is conceived not as a manifestation of the state of some hypothetical medium, but rather as an independent entity like matter. Moreover, this theory shares with the corpuscular theory of light the unusual property that light carries inertial mass from the emitting to the absorbing object. http://www.pitt.edu/~jdnorton/papers/companion.doc John Norton: Einstein could not see how to formulate a fully relativistic electrodynamics merely using his new device of field transformations. So he considered the possibility of modifying Maxwell's electrodynamics in order to bring it into accord with an emission theory of light, such as Newton had originally conceived. There was some inevitability in these attempts, as long as he held to classical (Galilean) kinematics. Imagine that some emitter sends out a light beam at c. According to this kinematics, an observer who moves past at v in the opposite direction, will see the emitter moving at v and the light emitted at c+v. This last fact is the defining characteristic of an emission theory of light: the velocity of the emitter is added vectorially to the velocity of light emitted....If an emission theory can be formulated as a field theory, it would seem to be unable to determine the future course of processes from their state in the present. AS LONG AS EINSTEIN EXPECTED A VIABLE THEORY LIGHT, ELECTRICITY AND MAGNETISM TO BE A FIELD THEORY, these sorts of objections would render an EMISSION THEORY OF LIGHT INADMISSIBLE. http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=3 17&Itemid=81&lecture_id=3576 Albert Einstein 1954: I consider it entirely possible that PHYSICS CANNOT BE BASED UPON THE FIELD CONCEPT, THAT IS ON CONTINUOUS STRUCTURES. Then nothing will remain of my whole castle in the air, including the theory of gravitation, but also nothing of the rest of contemporary physics. John Stachel's comment: If I go down, everything goes down, ha ha, hm, ha ha ha. http://au.encarta.msn.com/sidebar_781540289/einstein_on_gravitation_from_sci entific_american.html Albert Einstein 1950: Since the theory of general relativity implies the representation of physical reality by a CONTINUOUS FIELD, the limited region in space in which the field strength or the energy density are particularly high. Relativity and Its Roots By Banesh Hoffmann suggested in his paper submitted just thirteen weeks before this one, the second principle seems absurd: A stone thrown from a speeding train can do far more damage than one thrown from a train at rest; the Newtonian relativity and thus automatically account for the null result of the Michelson-Morley experiment without recourse to contracting lengths, local time, or Lorentz transformations. Yet, as we have seen, Einstein resisted the temptation to account for the null ideas, and introduced as his second postulate something that was more or less obvious when thought of in terms of waves in an ether. Pentcho Valev pvalev@yahoo.com === Subject: Homology of a topology with coefficients in a group This should be easy, but Im a bit confused. So a bit of help would be much appreciated. I am trying to understand the definition of homology of some topology X with coefficients in an abelian group G (Im not even sure what category this is being dealt with but it seems to be arbitrary). I am quite new in hological algebra, so please have patience. Here is how Rotman defines this: Let X be top. space and S_*(X) be its singular complex. If G is an abelian group then define H_*(X,G), homology of X with coeff. in G, to be the homology groups of the complex S_*(X) (x)_Z G Here (x)_Z is the tensor product over Z. First question, to remove the confusion.. how can we make a tensor product if the category is arbitrary abelian category.. or is this considered in the category of groups or what? Second question .. how did S_*(X) (x)_Z G become a complex? I dont quite see how the maps S_n+1 (X) (x)_Z G --> S_n(X) (x)_Z G are induced and their being a complex. Im sure this is an elementary thing in homological algebra, but like I said I am new here so please be patient. Jose Capco === Subject: Re: Homology of a topology with coefficients in a group posting-account=IBUqVwoAAADepmzxVr9iEYD5Z0A483SY Gecko/20070530 Fedora/1.5.0.12-1.fc5 Firefox/1.5.0.12,gzip(gfe),gzip(gfe) Let X be top. space and S_*(X) be its singular complex. If G is an abelian group then define H_*(X,G), homology of X with coeff. in G, to be the homology groups of the complex S_*(X) (x)_Z G Here (x)_Z is the tensor product over Z. First question, to remove the confusion.. > how can we make a tensor product if the category is arbitrary abelian category.. > or is this considered in the category of groups or what? It's in the category of Abelian groups. > Second question .. how did S_*(X) (x)_Z G become a complex? I dont quite see how the maps S_n+1 (X) (x)_Z G --> S_n(X) (x)_Z G are induced and their being a complex. If you have Abelian groups A, B and C, and a homomorphism f: A -> B then you have an induced map f^*: A (x) C -> B (x) C satsifying f^*(a (x) c) = f(a) (x) c. The map f |-> f^* is a functor: (g o f)^* = g^* o f^*. This implies that if g o f = 0 then g^* o f^* = 0, so that complexes remain complexes. === Subject: Re: Understanding the quotient ring nomenclature Nntp-Posting-Host: hera.cwi.nl ... > If I can get through it then I > might attempt a magnitudinal form of lets say M[X] where M are > unsigned magnitudes which will then satisfy the ring requirements and > commutative behavior as well, M does not have all the required ring properties. It is not a group under addition. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Understanding the quotient ring nomenclature posting-account=n26igQkAAACeF9xA2Ms8cKIdBH40qzwr Gecko/20070505 Iceape/1.0.9 (Debian-1.0.13~pre080614i-0etch1),gzip(gfe),gzip(gfe) > ... > If I can get through it then I > might attempt a magnitudinal form of lets say M[X] where M are > unsigned magnitudes which will then satisfy the ring requirements and > commutative behavior as well, M does not have all the required ring properties. It is not a group > under addition. > -- > dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 > home: bovenover 215, 1025 jn amsterdam, nederland;http://www.cwi.nl/~dik/ Oh,yes, there it is. The additive inverse does not exist. Still I have to try and understand this quotient ring construction to understand Hagen's paper. - Tim === Subject: Re: Understanding the quotient ring nomenclature posting-account=n26igQkAAACeF9xA2Ms8cKIdBH40qzwr Gecko/20070505 Iceape/1.0.9 (Debian-1.0.13~pre080614i-0etch1),gzip(gfe),gzip(gfe) > Somehow the quotient ring acts as a modulo operator through an ideal > right? Yes, though you may need to be careful exactly what you mean by > modulo operator. Each element of a quotient ring A/B is a *set*, > not an element of A. In particular, the elements of the quotient ring > are not remainders like those produced the modulo operator in > computing. So, for example, the set {..., -12, -5, 2, 9, ...} is a single element > of the quotient ring Z_7 (given by the quotient Z / (7)). For > notational convenience these sets are often labelled using one > particular element from the set. E.g. that element may be labelled > variously as [2]_7, 2 with a bar over the top, 2+7Z, or even > simply 2 if it is felt that context is sufficient to distinguish it > from the integer 2. Each element of R[x]/(x^2+1) is an uncountable set of polynomials. > For example the polynomial x is in one of those sets - and the > polynomials x^2+x+1 and 2x^3+3x are also in the same set. - Tim OK Tim. Jeeze, it's like I'm talking to myself... Umm... I guess someone would quibble with you about leaving the zeroes off the end of your instances of polynomials. Still, this the funny thing about the construction. For instance with the polynomial ring quotient construction it is possible to build n dimensional spaces consistent with the ring requirements, yet the underlying components will not play out this way with a limit like n since there will be terms up to 2n-1 in the product domain so that to build a finite form is not possible until the quotient is taken. Thus the instances you describe must already be quotient forms within the formality and so their products would then wrap at some dimension as well. Somewhere in this hairy nuance different people are taking differing interpretations and I've got to try and stay in one context so I'm going to try to follow Leland for awhile. He does not like the modulo interpretation at all but I am caught in it for now. Still, I've got to compartmentalize more. Somewhere mixed up in this is the indeterminate form and so long as there is no definition of X within your polynomials there is a dual haze for me. Upon collapsing to a real x everything collapses and I am told you cannot do that with this form. - Tim Jeeze, now it's like you're talking to myself... === Subject: Re: Understanding the quotient ring nomenclature Umm... I guess someone would quibble with you about leaving the zeroes > off the end of your instances of polynomials. Quite possibly; I'm sure that on Usenet there is always someone who can quibble about nearly anything. > Still, this the funny thing about the construction. For instance > with the polynomial ring quotient construction it is possible to > build n dimensional spaces consistent with the ring requirements, I really don't know where you're getting this dimension thing. You appear to be using it in the sense of dimension of a real vector space. In this particular case the space of real polynomials does satisfy the axioms of a real vector space, but in that case it has infinite dimension, not any natural number n. It isn't a very useful concept here. > yet the underlying components will not play out this way with a > limit like n since there will be terms up to 2n-1 in the product > domain so that to build a finite form is not possible until the > quotient is taken. Yes, the particular quotient R[x]/(x^2+1) can also be interpreted as a real vector space of dimension 2, but again dimensionality is not a very useful concept in the context of quotient rings. > Thus the instances you describe must already be quotient forms > within the formality and so their products would then wrap at some > dimension as well. The polynomial instances I mentioned are two example members of a single element of the quotient ring. The quotient does not have polynomials as elements of the ring, it has infinite sets (equivalence classes) of polynomials. In general, if A is a ring and I is an ideal of that ring, then you can divide up A into equivalence classes so that elements x and y from A are in the same class exactly when x-y is in I. Each class is an element of the quotient ring A/I. - Tim (Little) === Subject: Re: Understanding the quotient ring nomenclature > OK Tim. Jeeze, it's like I'm talking to myself... > > I must say I did a double-take once some time ago when reading one of > your posts and coming to the last line where you signed your name in > exactly the same way I normally do. A moment of did I write that!? > > > Umm... I guess someone would quibble with you about leaving the zeroes > off the end of your instances of polynomials. > > Quite possibly; I'm sure that on Usenet there is always someone who > can quibble about nearly anything. > > > Still, this the funny thing about the construction. For instance > with the polynomial ring quotient construction it is possible to > build n dimensional spaces consistent with the ring requirements, > > I really don't know where you're getting this dimension thing. You > appear to be using it in the sense of dimension of a real vector > space. In this particular case the space of real polynomials does > satisfy the axioms of a real vector space, but in that case it has > infinite dimension, not any natural number n. It isn't a very useful > concept here. There is an insistence on remaining in the 'indeterminate form' whereby X is never evaluated. In effect X is undefined. But now that I've read on and it's fairly obvious anyway the polynomial factors a X + b are evaluatable aren't they? Is it true that in polynomial of real coefficients that X is real? This is the only way to evaluate a factor isn't it? Yet this will then collapse any polynomial into a real value for any X. We could then procede to simply study y / ( x^2 + 1 ) with x and y real couldn't we? We'd eventually get to a square route of a negative value and be forced to concede i in the traditional manner. This seems to be in the interpretations on the indeterminate form anyway. At some point they introduce i. > > > yet the underlying components will not play out this way with a > limit like n since there will be terms up to 2n-1 in the product > domain so that to build a finite form is not possible until the > quotient is taken. > > Yes, the particular quotient R[x]/(x^2+1) can also be interpreted as a > real vector space of dimension 2, but again dimensionality is not a > very useful concept in the context of quotient rings. Well, this is troubling that you say this. I have worked out some math which develops dimensional behaviors without the need of any cartesian product. This math has been expressed in terms of ring theory by Hagen von Eitzen through a ring quotient and nobody has found any fault with that math. This is somewhat how I arrive at puzzling this all out. I'd like to understand how dimension is developed through this math. So back at my o.p. I ask specifically is this ring quotient considered an actual construction or is it more like a hindsight type of thing where a given construction fits into this form and behaves consistently with it? I understand that the polysign numbers are rings. I have no problem with that. But can the ring quotient form build the polysign numbers from a more primitive ring? The example of the complex numbers as R[X] / ( X X + 1 ) is nearly an identical but more familiar instance. Whether the ring form builds a number system or whether it merely reflects a number system's behaviors is crucial to understanding this nomenclature. This is all about dimension to me. Still, if you pick a domain for X then I understand the dimension of the classes and elements being discussed should collapse to that dimension assuming that the arithmetic product XX is in the same domain as X. > > > Thus the instances you describe must already be quotient forms > within the formality and so their products would then wrap at some > dimension as well. > > The polynomial instances I mentioned are two example members of a > single element of the quotient ring. The quotient does not have > polynomials as elements of the ring, it has infinite sets (equivalence > classes) of polynomials. This double abstraction is not doing much for me. What seems to matter is determining the domain X. If the quotient ring is capable of defining higher complexity X from simpler complexity X then understanding this feature would be useful to me. If it is merely describing behaviors consistent with these domains regardless of their complexity as common features then that is an important distinction. Going back to the primitive operators of superposition and product it is fairly clear that any math that follows these operators as their well known behaviors will be similarly behaved. The construction of those maths could come into question here. Who constructed who as the most primitive is then a problem and either may construct the other on its own terms. For instance the polysign numbers construct the reals, yet many insist upon constructing the polysign numbers from the reals. The ring quotient seems to be an ultimately abstracted X form but the nuances I am trying to understand arise. Polysign shares the polynomial product behavior but is inherently in modulo n form so they do not suffer the 2n-1 behavior. These then become a rotational geometry which is dimensional through simplex geometry where a balanced coordinate system defines the quality of sign as much as it does dimension (1,1,1,...) = 0 . - Tim Golden > > In general, if A is a ring and I is an ideal of that ring, then you > can divide up A into equivalence classes so that elements x and y from > A are in the same class exactly when x-y is in I. Each class is an > element of the quotient ring A/I. > > > - Tim (Little) === Subject: Re: Understanding the quotient ring nomenclature posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2009042708 Fedora/3.0.10-1.fc10 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Somehow the quotient ring acts as a modulo operator through an ideal > right? > Yes, though you may need to be careful exactly what you mean by > modulo operator.Each element of a quotient ring A/B is a *set*, > not an element of A.In particular, the elements of the quotient ring > are not remainders like those produced the modulo operator in > computing. > So, for example, the set {..., -12, -5, 2, 9, ...} is a single element > of the quotient ring Z 7 (given by the quotient Z / (7)). For > notational convenience these sets are often labelled using one > particular element from the set.E.g. that element may be labelled > variously as [2] 7, 2 with a bar over the top, 2+7Z, or even > simply 2 if it is felt that context is sufficient to distinguish it > from the integer 2. > Each element of R[x]/(x^2+1) is an uncountable set of polynomials. > For example the polynomial x is in one of those sets - and the > polynomials x^2+x+1 and 2x^3+3x are also in the same set. > - Tim OK Tim. Jeeze, it's like I'm talking to myself... Umm... I guess someone would quibble with you about leaving the zeroes > off the end of your instances of polynomials. Still, this the funny > thing about the construction. For instance with the polynomial ring > quotient construction it is possible to build n dimensional spaces > consistent with the ring requirements, yet the underlying components > will not play out this way with a limit like n since there will be > terms up to 2n-1 in the product domain so that to build a finite form > is not possible until the quotient is taken. Thus the instances you > describe must already be quotient forms within the formality and so > their products would then wrap at some dimension as well. Somewhere in > this hairy nuance different people are taking differing > interpretations and I've got to try and stay in one context so I'm > going to try to follow Leland for awhile. He does not like the modulo > interpretation at all but I am caught in it for now. Still, I've got > to compartmentalize more. Somewhere mixed up in this is the > indeterminate form and so long as there is no definition of X within > your polynomials there is a dual haze for me. Upon collapsing to a > real x everything collapses and I am told you cannot do that with this > form. You have managed to write 218 words which mean absolutey nothing. Good job. -- m === Subject: Re: Understanding the quotient ring nomenclature > Umm... I guess someone would quibble with you about leaving the zeroes > off the end of your instances of polynomials. Still, this the funny > thing about the construction. For instance with the polynomial ring > quotient construction it is possible to build n dimensional spaces > consistent with the ring requirements, yet the underlying components > will not play out this way with a limit like n since there will be > terms up to 2n-1 in the product domain so that to build a finite form > is not possible until the quotient is taken. Thus the instances you > describe must already be quotient forms within the formality and so > their products would then wrap at some dimension as well. Somewhere in > this hairy nuance different people are taking differing > interpretations and I've got to try and stay in one context so I'm > going to try to follow Leland for awhile. He does not like the modulo > interpretation at all but I am caught in it for now. Still, I've got > to compartmentalize more. Somewhere mixed up in this is the > indeterminate form and so long as there is no definition of X within > your polynomials there is a dual haze for me. Upon collapsing to a > real x everything collapses and I am told you cannot do that with this > form. > > You have managed to write 218 words which mean > absolutey nothing. Good job. > > -- m Yeah, and that was a short post for me. - Tim === Subject: Re: Understanding the quotient ring nomenclature posting-account=n26igQkAAACeF9xA2Ms8cKIdBH40qzwr Gecko/20070505 Iceape/1.0.9 (Debian-1.0.13~pre080614i-0etch1),gzip(gfe),gzip(gfe) > Well no, it doesn't, but this is just poor and confused use of > notation on your part. Perhaps this part I can clear up relatively > easily. In the notation 'S[Y]' we are referring to a particular ring, > which is the ring of polynomials in Y with coefficients in S. The > symbol 'S' should refer to a ring, it could be the real numbers, if > could be the integers, it could the finite field of 4 elements. The > square brackets '[' and ']' mean that we are now referring to the ring > of polynomials with coefficients in S. That is, 'S[.]' means the ring > formed by taking *all* nearly null infinite tuples where entries in > the tuple are elements of the base ring S. The 'Y' denotes the > formal symbol we will be using for book-keeping in the polynomials*. > The 'Y' is arbitrary: we are simply picking out which symbol we will > be using in our polynomials and is nothing ore than a symbol*. Ohhhh... I'm sorry. So when discussing the ring R that means the real numbers right? I got a neg. on this back a while so that everything I read with an R in it I was confused on whether it was R the ring or R the reals. reviewing http://www.math.uiuc.edu/~r-ash/Algebra.html which is a link you gave me awhile back. He defines the quotient ring in terms of the group ring, which is OK, but there is a list of vocabulary that feels tentative to me so that working through it all with confidence is not happening. Thus, in our case, R[X] refers specifically to the ring formed by > taking all nearly null infinite tuples with real number entries (since > we are assuming R here refers to the real numbers). The X in the > notation is because mathematicians often prefer to work in polynomials > rather than nearly null infinite tuples (and Arturo showed they were > equivalent) Sure, I see this. > and we need to state which symbol we'll be using in our > polynomials. We could just as well write R[Y] or R[A] and be referring > to exactly the same ring -- the only change would be that we would > [X] however, we would be referring to something completely different, > since the 'S' symbol denotes an actual object, and here Z would mean > the integers. Thus, you see, the R matters and refers to something > concrete, but the X is simply a formal symbol, for which we could > substitute anything. Most importantly the '[]' part means we are > referring to the 'ring of polynomials with coefficients in ...' or, > equivalently, the 'ring of all nearly null infinite tuples with > entries in ...'. Does this make more sense now? I don't think we > should move on to anything more complicated until you fully understand > at least what the notation even means. Yes, I understand. We could construct R[R] and nearly have grade school polynomials. > I can get over that > if what I am saying is accurate, The key being that none of what you said was even remotely accurate, > and was simply your own speculation that has little or nothing to do > with what anyone has actually told you. > but if this is inaccurate then > obviously I am misunderstanding something. You are very obviously misunderstanding a lot of things, the problem > being that you are not even understanding the most basic points, but > are busy trying to understand the things built from those basic points > anyway, and simply confusing yourself even more. Look, this is not > something simple that you can expect to understand overnight unless > you have some appropriate background. It is becoming clear to me that > you don't have any of the appropriate background, and are trying to > leap ahead anyway. Can I ask if you have a clear understanding of what > a ring is? Could you give me a few examples of different rings to show > what you understand to be the full range of diversity in what a ring > could be like? Please take this seriously and spend the time to come > up with good examples as it will help me establish in what terms I > should actually be explaining things so as to not add to your > confusion. Yeah, a ring obeys some fundamental algebraic properties, which are not necessarily tied to the actual shape of a ring, as in an 'O' graphical representation. So getting a ( b + c ) = a b + a c is consistent with a ring behavior regardless of what domains a,b,and http://www.math.uiuc.edu/~r-ash/Algebra/Chapter2.pdf A ring R is an abelian group with a multiplication operation (a, b) .81¬ ab that is associative and satisfies the distributive laws: a(b + c) = ab + ac and (a + b)c = ab + ac for all a, b, c .81ü R. Darn it, there is that usage of R not as the real numbers but as a generic right? I'll tell you what, for generations of evolution to clean up formalities, this nomenclature is poor. This does worry me. Anyway, at some level we are free to define as we go and for this reason I will defer to you and your interpretation on R[X] as you did with the fine instance S[Y], which seems very sensible since there are then three items in the description 'ring of polynomials in Y with coefficients in S', rather than this book quote which puts R as meaning Ring which lessens the descriptive content by one item. Anyway, here we see an advancement past the group stage, where just one binary operation necessarily exists. Now we have two operators, traditionally sum and product, though as I understand it these can be 'overloaded' to be any consistently defined behavior with the requirements as defined within the term 'ring'. The reals are a ring but not all rings are reals. The integers are a ring. Modulo forms from the integers are rings, whether you eliminate redundancies or not. The link gives a set of rules of an R F form with inverses that forms a ring. The polysign numbers are rings. Particularly, these last have been described in terms of the quotient ring most formally by Hagen von Eitzen http://www.von-eitzen.de/math/PolysignNumbers.pdf but I have yet to understand that form. Close by is the R[X] / ( XX + 1 ) form of the complex numbers, though the polysign build the complex numbers as a mod 3 form rather than a mod 2 form. I do not mean to promote polysign here but this is my true motive for understanding the quotient ring form and helps explain my own resistance to the quotient form. Especially the interpretation of whether the quotient form truly constructs number systems or merely in hindsight is consistent is of interest to me, but breaking through to understanding the ring nomenclature cleanly is difficult. If I can get through it then I might attempt a magnitudinal form of lets say M[X] where M are unsigned magnitudes which will then satisfy the ring requirements and commutative behavior as well, though the polynomial form will have a new behavior after the modular quotient effect where for say, the mod 3 system ( 1, 1, 1 ) = 0 in the tuple form which extends generally to a mod n system. > Nobody has bothered to > respond to this point and it has added to my confusion. It isn't even clear to me what your point is, since you rattled off > a whole list of misunderstandings right across the range from basic > notation through to what a quotient by an ideal should mean. I've > decided to pick on notation for now, since if we can't get that > straight then clearly the rest is beyond hope. * (For the pedants, not Tim: Yes, in some situations the R[a] notation > has 'a' not being a formal symbol but rather an element from some ring > S (containing R) to adjoin to the ring, but I figured it's best not to > confuse that particular point right now) this energy. I do not see some of the finer distinctions you make about the dimension discrimination but you concede that it is consistent somewhat so that is good enough. I tend to think informationally where format is of the essence so that ( 1.23, 2.34, 3.45 ) as a tuple or as a coordinate system still confers the same quantity of information. Still, I work in a format where this is tweaked to be one less dimension since ( 1, 1, 1 ) = 0 so I can concede a difference yet I am also able to substantiate how the difference exists whereas in the indeterminate polynomial form I cannot see any discrepancy informationally between a dimensional interpretation and the tuple or polynomial representation. Yes, the product is uniquely defined for the polynomial form, but that form is still consistent with vector principles, up to the discrepancy which I pointed out which requires the general form to remain in (super) infinite dimension. - Tim === Subject: Re: Understanding the quotient ring nomenclature > Ohhhh... I'm sorry. So when discussing > the ring R > that means the real numbers right? Sometimes it just means any ring. Context helps in determining which is intended, and many typeset texts use a doubled-line symbol for R when denoting the reals (looking a bit like |R). In the context of going from reals to complex numbers though, it is much more obvious that the R in R[x] denotes the reals and not some arbitrary ring. > I'll tell you what, for generations of evolution to clean up > formalities, this nomenclature is poor. This does worry me. There is only a fairly small set of letters and other symbols easily available to publishers (especially historically), and enormously many mathematical concepts to represent using those symbols. Multiplicity of assigned meanings to symbols is inevitable. Also many distinct fields that originally developed independent symbolism later came into collision as links were developed between them. Mathematics is like that. There has not just been generations to clean up nomenclature, there has been generations of developing new concepts requiring new nomenclature and cross-connecting previous ones. Any good textbook or paper will define meanings of any potentially ambiguous symbols before they are used. > Especially the interpretation of whether the quotient form truly > constructs number systems or merely in hindsight is consistent is of > interest to me, Constructing rings via quotients of polynomials to produce novel algebraic structures has been of great practical use in developing many of the error-correcting codes in use today - as one small example of a great many applications. - Tim (Little) === Subject: Re: Understanding the quotient ring nomenclature posting-account=n26igQkAAACeF9xA2Ms8cKIdBH40qzwr Gecko/20070505 Iceape/1.0.9 (Debian-1.0.13~pre080614i-0etch1),gzip(gfe),gzip(gfe) > You are very obviously misunderstanding a lot of things, the problem > being that you are not even understanding the most basic points, but > are busy trying to understand the things built from those basic points > anyway, and simply confusing yourself even more. Look, this is not > something simple that you can expect to understand overnight unless > you have some appropriate background. It is becoming clear to me that > you don't have any of the appropriate background, and are trying to > leap ahead anyway. Can I ask if you have a clear understanding of what > a ring is? Could you give me a few examples of different rings to show > what you understand to be the full range of diversity in what a ring > could be like? Please take this seriously and spend the time to come > up with good examples as it will help me establish in what terms I > should actually be explaining things so as to not add to your > confusion. > The reals are a ring but not all rings are reals. > The integers are a ring. > Modulo forms from the integers are rings, whether you eliminate > redundancies or not. > The link gives a set of rules of an R F form with inverses that forms > a ring. Oops, this R F form was back in groups, not in rings. I take this line above back. > The polysign numbers are rings. === Subject: Re: Understanding the quotient ring nomenclature posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2009042708 Fedora/3.0.10-1.fc10 Firefox/3.0.10,gzip(gfe),gzip(gfe) > Well no, it doesn't, but this is just poor and confused use of > notation on your part. Perhaps this part I can clear up relatively > easily. In the notation 'S[Y]' we are referring to a particular ring, > which is the ring of polynomials in Y with coefficients in S. The > symbol 'S' should refer to a ring, it could be the real numbers, if > could be the integers, it could the finite field of 4 elements. The > square brackets '[' and ']' mean that we are now referring to the ring > of polynomials with coefficients in S. That is, 'S[.]' means the ring > formed by taking *all* nearly null infinite tuples where entries in > the tuple are elements of the base ring S. The 'Y' denotes the > formal symbol we will be using for book-keeping in the polynomials*. > The 'Y' is arbitrary: we are simply picking out which symbol we will > be using in our polynomials and is nothing ore than a symbol*. Ohhhh... I'm sorry. So when discussing > the ring R > that means the real numbers right? I got a neg. on this back a while > so that everything I read with an R in it I was confused on whether it > was R the ring or R the reals. > reviewing >http://www.math.uiuc.edu/~r-ash/Algebra.html > which is a link you gave me awhile back. He defines the quotient ring > in terms of the group ring, which is OK, but there is a list of > vocabulary that feels tentative to me so that working through it all > with confidence is not happening. > Thus, in our case, R[X] refers specifically to the ring formed by > taking all nearly null infinite tuples with real number entries (since > we are assuming R here refers to the real numbers). The X in the > notation is because mathematicians often prefer to work in polynomials > rather than nearly null infinite tuples (and Arturo showed they were > equivalent) > Sure, I see this. > and we need to state which symbol we'll be using in our > polynomials. We could just as well write R[Y] or R[A] and be referring > to exactly the same ring -- the only change would be that we would > [X] however, we would be referring to something completely different, > since the 'S' symbol denotes an actual object, and here Z would mean > the integers. Thus, you see, the R matters and refers to something > concrete, but the X is simply a formal symbol, for which we could > substitute anything. Most importantly the '[]' part means we are > referring to the 'ring of polynomials with coefficients in ...' or, > equivalently, the 'ring of all nearly null infinite tuples with > entries in ...'. Does this make more sense now? I don't think we > should move on to anything more complicated until you fully understand > at least what the notation even means. Yes, I understand. We could construct R[R] and nearly have grade > school polynomials. No you do *not* understand. That you write R[R] is a great proof of that! -- m === Subject: Re: Understanding the quotient ring nomenclature posting-account=_l4K0QkAAAC09JhOoK_ZfoJKXOmr_jZf Gecko/2009042513 Ubuntu/8.04 (hardy) Firefox/3.0.10,gzip(gfe),gzip(gfe) >http://www.math.uiuc.edu/~r-ash/Algebra/Chapter2.pdf >A ring R is an abelian group with a multiplication operation > (a, b) .81¬ ab > [NonBreakingSpace] [NonBreakingSpace]that is associative and satisfies the distributive laws: > [NonBreakingSpace] [NonBreakingSpace] [NonBreakingSpace] a(b + c) = ab + ac > [NonBreakingSpace] [NonBreakingSpace]and > [NonBreakingSpace] [NonBreakingSpace] [NonBreakingSpace] (a + b)c = ab + ac > [NonBreakingSpace] [NonBreakingSpace]for all a, b, c .81ü R. Darn it, there is that usage of R not as the real numbers but as a > generic right? > I'll tell you what, for generations of evolution to clean up > formalities, this nomenclature is poor. This does worry me. As many people have told you (myself at least twice), the common nomenclature for the real numbers is either a boldface R or a double struck or blackboard bold R. You can see the double struck typeface at http://www.w3.org/TR/MathML2/double-struck.html The symbol for the real numbers is unicode 0211D. However, you are posting in USENET, which uses ASCII. ASCII has neither boldface nor doublestruck, nor other such alternative typefaces, so it is usual to use a simple capital R, and specify explicitly (as we did) that we are using that letter to denote the real numbers. It is also common, when refering to generic rings, to use either a capital (roman or italic) R or A (for 'ring', or for 'annel', the french word for ring, respectively). There is normally no possible cause for confusion since the typeface of the letter is different. In the context of Usenet there is the possibility of confusion, which is why one would say things like Let R be a ring or Let R be the real numbers, to be very clear what it is that R denotes. There is very little chance for problems in any case, when one knows the context at work. There are only finitely many letters, but infinitely many concepts; we cannot have a one-to-one correspondence between notation and ideas, except by compartmentalizing them. So the word normal has precise technical meanings, but different precise technical meanings in different contexts. Just like negative means one thing in medicine and something completely different in arithmetic. Is that also an example of poor nomenclature that troubles you? But we are so pleased that you could come along and point out all those problems that you are perceiving; clearly, the entire field of mathematics is full of idiots who can't see past their noses, it took a genius of your caliber, one who by self-admissions knows *nothing* about the subject to point out the problems. So glad you could come along. -- Arturo Magidin === Subject: The Truman Proof - update The story so far.... As you know last week I tested an MP3 player / voice recorder to try and detect the voices made by the Truman satellite. Because it's designed to speak directly into, it was not sensitive enough to pick up peripheral noise. This week... I remembered building an FM bug many years ago, I placed it on a cabinet in the dining room and went to my room to listen to mum and dad have lunch. As they poured tea into the cup I could hear it swirling around and the pitch increasing as it filled the cup. The microphone was more sensitive than the human ear, atleast in mid range frequencies. Eureka! I ordered a FM transmitter kit from Dick Smith, some solder, an 8 watt battery powered soldering iron, some batteries, and it arrived today. (same delivery driver as the MP3 player last week). I opened the kit, read the instructions and it said it's a very sensitive circuit / microphone, sounds good, but it was missing the antenna wire, and I couldn't figure out how the slider switch was soldered onto the circuit board, and you had to make battery terminals and solder to the battery, I haven't soldered in 20 years so I thought nuts to this. Anyway, I emailed my brother asking if he'd put it together for me and I'll send him the kit soon, so within a couple weeks, we *should* get The Truman Company's broadcast online. Herc -- The son of Zeus is on the loose! === Subject: Re: The Truman Proof - update > The story so far.... > > As you know last week I tested an MP3 player / voice recorder to try > and detect the voices made by the Truman satellite. Because it's designed > to speak directly into, it was not sensitive enough to pick up peripheral noise. > > > This week... > > I remembered building an FM bug many years ago, I placed it on a cabinet > in the dining room and went to my room to listen to mum and dad have lunch. > As they poured tea into the cup I could hear it swirling around and the pitch > increasing as it filled the cup. The microphone was more sensitive than the > human ear, atleast in mid range frequencies. Eureka! I ordered a FM transmitter > kit from Dick Smith, some solder, an 8 watt battery powered soldering iron, > some batteries, and it arrived today. (same delivery driver as the MP3 player last week). > I opened the kit, read the instructions and it said it's a very sensitive circuit / microphone, > sounds good, but it was missing the antenna wire, and I couldn't figure out how > the slider switch was soldered onto the circuit board, and you had to make > battery terminals and solder to the battery, I haven't soldered in 20 years so > I thought nuts to this. Anyway, I emailed my brother asking if he'd put it together > for me and I'll send him the kit soon, so within a couple weeks, we *should* > get The Truman Company's broadcast online. > > Herc The FM transmitter seems surplus to requirements. Just connect the microphone to your recorder. Sylvia. === Subject: Re: The Truman Proof - update > The story so far.... > As you know last week I tested an MP3 player / voice recorder to try > and detect the voices made by the Truman satellite. Because it's designed > to speak directly into, it was not sensitive enough to pick up peripheral noise. > > This week... > I remembered building an FM bug many years ago, I placed it on a cabinet > in the dining room and went to my room to listen to mum and dad have lunch. > As they poured tea into the cup I could hear it swirling around and the pitch > increasing as it filled the cup. The microphone was more sensitive than the > human ear, atleast in mid range frequencies. Eureka! I ordered a FM transmitter > kit from Dick Smith, some solder, an 8 watt battery powered soldering iron, > some batteries, and it arrived today. (same delivery driver as the MP3 player last week). > I opened the kit, read the instructions and it said it's a very sensitive circuit / microphone, > sounds good, but it was missing the antenna wire, and I couldn't figure out how > the slider switch was soldered onto the circuit board, and you had to make > battery terminals and solder to the battery, I haven't soldered in 20 years so > I thought nuts to this. Anyway, I emailed my brother asking if he'd put it together > for me and I'll send him the kit soon, so within a couple weeks, we *should* > get The Truman Company's broadcast online. > Herc The FM transmitter seems surplus to requirements. Just connect the > microphone to your recorder. Sylvia. It might be the voice recorder circuit that is not sensitive. Miniature circuitry is hard to work with and I'll probably bust my MP3 player. Good news is the MP3 player can record the FM radio so we might get a faint recording. They seem to be able to override electronic equipment though, but I think I'll have them. They're chanting right now typical abuse at me (not wrong, serves you right, you're a bloody idiot, there you go), not loud but easy to hear. Herc === Subject: Re: The Truman Proof - update > The story so far.... > As you know last week I tested an MP3 player / voice recorder to try > and detect the voices made by the Truman satellite. Because it's designed > to speak directly into, it was not sensitive enough to pick up peripheral noise. > This week... > I remembered building an FM bug many years ago, I placed it on a cabinet > in the dining room and went to my room to listen to mum and dad have lunch. > As they poured tea into the cup I could hear it swirling around and the pitch > increasing as it filled the cup. The microphone was more sensitive than the > human ear, atleast in mid range frequencies. Eureka! I ordered a FM transmitter > kit from Dick Smith, some solder, an 8 watt battery powered soldering iron, > some batteries, and it arrived today. (same delivery driver as the MP3 player last week). > I opened the kit, read the instructions and it said it's a very sensitive circuit / microphone, > sounds good, but it was missing the antenna wire, and I couldn't figure out how > the slider switch was soldered onto the circuit board, and you had to make > battery terminals and solder to the battery, I haven't soldered in 20 years so > I thought nuts to this. Anyway, I emailed my brother asking if he'd put it together > for me and I'll send him the kit soon, so within a couple weeks, we *should* > get The Truman Company's broadcast online. > Herc > The FM transmitter seems surplus to requirements. Just connect the > microphone to your recorder. > Sylvia. > > It might be the voice recorder circuit that is not sensitive. Miniature circuitry is hard > to work with and I'll probably bust my MP3 player. Good news is the MP3 player > can record the FM radio so we might get a faint recording. They seem to be able > to override electronic equipment though, but I think I'll have them. They're chanting > right now typical abuse at me (not wrong, serves you right, you're a bloody idiot, > there you go), not loud but easy to hear. How do you know that it's sound, and that they're not directly stimulating your auditory nerve? Sylvia. === Subject: Re: The Truman Proof - update > The story so far.... > As you know last week I tested an MP3 player / voice recorder to try > and detect the voices made by the Truman satellite. Because it's designed > to speak directly into, it was not sensitive enough to pick up peripheral noise. > This week... > I remembered building an FM bug many years ago, I placed it on a cabinet > in the dining room and went to my room to listen to mum and dad have lunch. > As they poured tea into the cup I could hear it swirling around and the pitch > increasing as it filled the cup. The microphone was more sensitive than the > human ear, atleast in mid range frequencies. Eureka! I ordered a FM transmitter > kit from Dick Smith, some solder, an 8 watt battery powered soldering iron, > some batteries, and it arrived today. (same delivery driver as the MP3 player last week). > I opened the kit, read the instructions and it said it's a very sensitive circuit / microphone, > sounds good, but it was missing the antenna wire, and I couldn't figure out how > the slider switch was soldered onto the circuit board, and you had to make > battery terminals and solder to the battery, I haven't soldered in 20 years so > I thought nuts to this. Anyway, I emailed my brother asking if he'd put it together > for me and I'll send him the kit soon, so within a couple weeks, we *should* > get The Truman Company's broadcast online. > Herc > The FM transmitter seems surplus to requirements. Just connect the > microphone to your recorder. > Sylvia. > It might be the voice recorder circuit that is not sensitive. Miniature circuitry is hard > to work with and I'll probably bust my MP3 player. Good news is the MP3 player > can record the FM radio so we might get a faint recording. They seem to be able > to override electronic equipment though, but I think I'll have them. They're chanting > right now typical abuse at me (not wrong, serves you right, you're a bloody idiot, > there you go), not loud but easy to hear. How do you know that it's sound, and that they're not directly > stimulating your auditory nerve? Sylvia. It comes from a distance and certain direction, out the window. And everyone around me hears it and comments on it. You'd need a fine laser beam from 500km up in space to pinpoint a nerve. And it works on the beach away from any technology, 30km out of town and 1km up a bush path, middle of nowhere it still works, my dad said it seems like a satellite. On enemy of the state Gene Hackman says theres hundreds of spy satellites using 20GHZ receivers. The weather satellites use 90GHZ lasers and see through clouds. Strange no commercial / university labs have tested masers on people to detect the sound of their thought. It's quite audible. Your mind is 100% open. To the government we're all like the kid on the wonder years, playing his thoughts out aloud. YOULL SEE, why the f*ck call it the TRUEman show. Didn't ANYONE go to that movie wondering if there was some true theme to it? Herc === Subject: quick question How big is an object that subtends 1.6 arc min at 70 MLY distance? === Subject: Re: quick question How big is an object that subtends 1.6 arc min at 70 MLY distance? Quick answer: Very. Slightly longer answer: Do your own calculation. === Subject: Re: quick question > How big is an object that subtends 1.6 arc min at 70 MLY distance? About 33,000 LY across. === Subject: Re: quick question >How big is an object that subtends 1.6 arc min at 70 MLY distance? 1.6/60 x pi/180 x 70000000 light years = 32600 light years Rob Johnson take out the trash before replying === Subject: 0.75(2^2)+1=2^2 (for Dr Phil,Prof Edgar, Musatov, amy , Mathematicians) THIS for the record and for Dr Phil, Amy , Edgar,and the great mathematicians Musatov,and to the silence of the lambs The smallest proprtion , the most inverse proportion is 3 the least odd over 4 the least square---- i.e 2^2 is the most accurate squared value and the least of the squares(curved values,for linear is 1 and infinity), for the following reason, (it is also incidentally the only correct answer in spite of your Euler, to Pells equation proportion which we have proved, all values alternate between 1 and 0.75) (2^2)+1=3^2 0.75(2^2)+1=2^2 0.25(2^2)+1=1.41^2 0.02(2^2)+1=1.03^2 0 (2^2)+1=1^2=1 ( linear equation constant) BUT since 1 is proportional to 1 (linear) the least of the proportion over 1(curved) is 3/2^2(0.75) , not 1/1, we presumably have the 0.75 proportion that is a constant for inverse proportion , needs more resoltion, it may take years. Thus linear constant is 1=1, curved inverse constant is 2P1^2-0.75 = 19, and actually 19 =1 as curved is =to linear by a constant proprtion,o this will be proven also by special new calculus resolution , in the years to come Note: Mathematicians--- As for N=nP ask Musatov. I just respect the man for what he is. I respect you and you ought to respect me for what ever I am as long as I am not filthy. I am not a trained mathematician , but that does not make me an idiot, you a genius. Simplicity is sometimes genius. === Subject: Re: 0.75(2^2)+1=2^2 (for Dr Phil,Prof Edgar, Musatov, amy , Mathematicians) Correction Mathematiocians It is 2(2^2)+1= 3^3, lest you call me an idiot. This is nothing to do with N=np, please refer that to Musatov your leper, not mine, and I do not become a leper because I touched him with respect, all you pharisees out there! Leporsy is a bacterial illness not a scourge from God. === Subject: Re: A problem about system identification with quantized output up === Subject: Test bank and solution manual collection posting-account=jy2E5goAAAB1WGwMQL1h2k61o4k7O_u4 Gecko/2009051221 Firefox/3.0.10,gzip(gfe),gzip(gfe) I have the following test banks and solution manuals for sale. Email me if interested at esol42[at]gmail.com. 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Contact me at instructors.team[at]gmail.com === Subject: Re: Is an arbitrary multivariate polynomial equation over GF[2] ???NP-complete? <0.e579288eb11c6fd270fe.20090606141225BST.8763f9h5fq.fsf@bsb.me.uk> <0.874b2998c575205ad88c.20090610021238BST.87zlcglwmx.fsf@bsb.me.uk> posting-account=pbWZ8QoAAACI9zXocLL9zZCrDIYrXvvv 5.1),gzip(gfe),gzip(gfe) If P=0 is any equation (where P is an arbitrary univariate polynomial) > over a general finite field GF(q)), I proved in 1985 that the equation > is solvable if (and only if) the reduced polynomial 1-P^(q-1) is not > the identical to zero one. Taking into account that q is a fixed > parameter (must be any natural power of any prime number) and that > rising any polynomial to any natural power (no matter how huge it can > be) is a recognized polynomial time transformation in the size of the > input polynomial, if Shamir is right in his paper (and I in my old > one), the P=NP affair is done. q is not fixed in the problem as stated by Shamir so the two problems > are, on the face of it, different. > of solving a system of multivariate quadratic equations over a finite field is known to be NP complete. This lemma implies that the problem of solving a single univariate polynomial equation over a finite field is also NP complete.î What two problems are you saying that are different? I am addressing the ñsingle univariate polynomialî one considered by Shamir as NP- complete, with no intention at all to make any change in its definition. About the number q of elements for the field K where the univariate polynomial is over, almost at the beginning of the incise 2 in Shamir's paper we can read: ñthe recommended choice is n = 128, yielding a field K with 2^128 elementsî. As you can see, it is a simple parameter with an independent fixed arbitrary value from the beginning. Please, refer the exact place in Shamir's paper from where you took the idea that ñq is not fixedî and what do you mean by that. > The 1-P^(q-1) different from zero > inequation has exactly the same solutions as the P=0 equation. You can > verify it running any quantity of small (or large) number instances. The trouble is I don't think the problem is NP-complete when the size > of the field is fixed.You really need to relate this specific > problem (the one you published about) to a known NP-complete problem. > My 1985 work refers to the solution of equations (and inequations) over any finite field, not about NP-completeness. What I am trying to do now is precisely to apply it to a known NP-complete problem (with preference to ones known before 1985, I started taking note of the presence of ñalgebraic equations over GF(2)î in all published NP- complete problem lists since 1979). If you think that Shamir is wrong when proving some problem NP-complete, this is another story. Do you think also that Shamir is wrong when writing ñThe problem of solving a system of multivariate quadratic equations over a finite field is known to be NP completeî? Can't we choose an arbitrary fixed value (being a natural power of a prime, of course) for the number of elements in the finite field? > I think part of the problem is the notion of fixed.q is fixed for > an instance of the general problem, of course, but the subset of that > language that corresponds to all the instances with one particular q > may not be in NPC. > In both problems considered NP-complete ones in Shamir's paper, the q is fixed for ALL instances in each one. What is not equal for each instance is the number of multivariate quadratic equations in the first problem (in the second one we have always a single univariate polynomial equation). > This is why it helps to be formal and you have not stated, formally, > the problem that think is solved by your earlier work and that you > think may be NP-complete.Can you at least start by doing that?Look > at any text on complexity theory to see how problems are stated > formally. > I didn't prove any problem NP-complete in 1985. I developed necessary and sufficient conditions for the solvability of system of equations (and inequations) over any arbitrary finite field, including a single equation or inequation (the solvability condition is always that some inequation has a non-identical to zero reduced polynomial). I also developed a polynomial time transformation of an equation in an inequation (or vice versa) preserving the solutions, and a polynomial time algorithm for the solution of any single inequation over any finite field. > This yes it is/no it isn't argument is not going anywhere. Your > best strategy is to try to get your work published in a peer-reviewed > journal. > Sure I will try it in a near future, but I have now in first priority > to assure the correctness of my 1982-1985 work. That is the whole point of having work published in quality journals. > If you had taken that route earlier you work would already have been > critically examined by experts in the field.You will get much better > feedback from journal referees than you will from Usenet. > I had being using Usenet groups for many years. For starting a really complex topic I prefer this alternative. Anyway, I appreciate a lot your advice. > -- > Ben. RVHG (Rafael Valls Hidalgo-Gato) ICIMAF, CITMA Havana, Cuba === Subject: =?windows-1252?Q?Auditing_A_Business_Risk_Approach_=286thEd=29_=2D_Rittenbe? = =?windows-1252?Q?rg_=96_Solutions_Manual_and_Test_is_available_for_purchase= 2 1_?= =?windows-1252?Q?Contact_me_at_instructors=2Eteam=5Bat=5Dgmail=2Ecom?= posting-account=jy2E5goAAAB1WGwMQL1h2k61o4k7O_u4 Gecko/2009051221 Firefox/3.0.10,gzip(gfe),gzip(gfe) Auditing A Business Risk Approach (6thEd) - Rittenberg [CapitalEth] Solutions Manual and Test is available for purchase! Contact me at instructors.team[at]gmail.com === Subject: Re: Answer to Dik T. Winter Nntp-Posting-Host: hera.cwi.nl > That is the logic of potential infinity, i.e., of incomplete sets. > > Eh? As far as I know logic is *not* about sets. It is about axioms, > statements and inference rules. > > Logic has been obtained from the bahviour of parts of reality that can > be called sets. O. Still, as far as I know logic is *not* about sets. > In Cantor's diagonal argument you can use the same logic : There is no > last line, therefore there is always a line beyond the checked lines, > But there you don't. > > Because you do not check the lines in order. It is always your basic > assumption that you first check the first line and after that the next > line. That is wrong. > > That is necessary because you cannot find the n-th line unless you > know the line number n - 1 or some equivalent mark. You are wrong. A list is a mapping from N to the elements of the list. Through that list, given a number n, you find the n-th element of the list without referring to any previous elements of the list. To give an example, let's have a list of positive rational numbers through the mapping given a lnong time ago by David Tribble. To get the 51st element of the list, we calculate the mapping: write 51 in binary, create from it a continued fraction as described on which is [0, 1, 2, 3], calculate it and we find 7/9. So we know the 51st element without knowing the 50th element. Where did I come at the 47th rational in that list (which is 7/11)? > You simply give a definition of a new number so it is clear > that it will be different from each of the omega lines, without checking > directly any of them. > > And in my binary tree I give a definition of an end of a path that > contains aleph_0 nodes. I have not seen such a definition. What is the definition of end of a path? > Every end of a path contains aleph_0 nodes. > These nodes can be mapped on that path. Every node will eventually be > mapped on one path. Therefore all nodes are used up for constructing a > countable set of paths. Perhaps right, depends on how you actually do define things. But are there nodes mapped to all paths? That is what you assert. > When it is derived from logic of finite sets, then it is not the > reverse of the logic of finite sets. But that is claimed in ZF. > > It is *not* the reverse of the logic of finite sets. For finite sets it > is identical. > > For infinite sets it is not identical. No, of course not, because it is originally available for finite sets only, it can not be identical. > And the reverse of being > identical is being not identical. Not in this case bacause you apply the words to different things. *Unless* you assume that what is valid in finite cases also is valid in infinite cases. But let's see: sum{i = 0 .. n} 1/(i!) is a rational number. So according to your logic: e = sum{i = 0 .. n} 1/(i!) is also a rational number. > Ok. The logic of finite complete sums of rational elements gives that > the sum is rational. Using your logic we get that the complete (i.e. in > the limit) sum of rational elements is also rational. And so by that > logic, e, pi and whatever are rational, and all numbers we do use are > rational. > > In fact there are no binary expansions of irrational numbers. What is the relevance? Where am I talking about binary expansions? > Either those sets obey that logic > or they do not exist in a science that is subject to the application > of logic. > > Ah, so 'e', 'pi', 'sqrt(2)' do not exist. Still I think you use at least > They exist as ideas but not as sums of series. So the equation e = sum{i = 0 .. oo} 1/(i!) is invalid in your logic? > Small wonder. There cannot be a complete list, because the existence > of a complete infinite linear set like N contradicts logic. > > Apparently your logic. What are the rules of inference in your logic? > > The union of a complete set of linear sets is one of the sets. Eh? That is not a rule of inference. > And, also apparently, in your logic the length of the diagonal of a square > with sides with size 1 does not exist. > > It exists, but not as binary or decimal expansion. Eh? Now suddenly sqrt(2) does exist. > That is a blatant lie. > > Eh? What is the lie? I just state that in ZF there are no non-static > sets. > > Then every element should be accessible. Why do you start always with > a diminishingly small one? Every element is accessible. And you start always at the beginning because there is no last one. > Every static linear set has a last element. > > As that is false in ZF... > > This shows that ZF deals with non-static sets if necessary. No. It just shows that your idea of sets is not the idea ZF has about sets. > If you say you cannot choose the last elemement because there is none, > then you apply potentially infinite sets. > > You have still not provided a definition of potentially infinite sets in > ZF, so I cannot comment on this. > > A potentially infinite set is a set that has no last element. Ok. Than N is a potentially infinite set. > Then for every element that > you choose there is a larger one. If it has been there all time, why > the hell did you not start with this one ? > > Because in ZF a set does not need to have a last element. So whatever > element you chose, there is *always* a larger one. > > That is potential infinity. Ok. Than N is a potentially infinite set that actually does exist in ZF. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Answer to Dik T. Winter Nntp-Posting-Host: hera.cwi.nl > The word union and the word linear are not mentioned in the quote. > Nevertheless the due logica rules were obtained from unions of finite > sets and linear finite sets. > > Yes, obtained from. But that does not mean they are identical. Moreover, > > It is 50 years younger than Canto's writings. What is the relevance? > But it does rule out theories which are contradicted by the > fundamental logical rules. And one of these rules is that a complete > linear set has a last element. > > What fundamental rules of logic are you using? I have never seen such a > logical rule, because logic does not talk about sets. > > Logic states that the union of a *complete* set of finite linear sets > is a finite linear set As I said, logic is not talking about sets. So where in logic is such stated? > No. I use the fact that for complete linear sets always both > implications are true : > [**] & [***]. This means that [*] is true. > > You just state so without proof. > > A proof is a derivation of theorems from axioms or basic truths by > means of rules of logical inference. These rules themselves cannot be > proven but can only be obtained from the behaviour of existing (i.e., > finite) sets. So you are not using mathematical logic? > If you disagree, then you should come up with a finite linear set for > which only one implication is true. > > Why should I show that for a *finite* linear set? Why not for an > *infinite* linear set? You are *still* thinking that wat is valid > for finite things is also valid for infinite things. > > There is no reason to believe that always the contrary is true. I do not believe so, so what is the relevance? > So in your opinion: > sum{i = 0..oo} 1/(i!) = lim{n -> oo} sum{i = 0..n} 1/(i!) > is rational, because it is rational for each n, and so your logic > requires that. Also: > lim{n -> oo} 1/n > 0 > because it is > 0 for each n, and so should also be > 0 in the infinite > case. > > The limit does not belong to the series! According to your reasoning it does belong to the complete series. > Similarly omega is not a natural number and the number of natural > numbers is not omega. What is in this context the relevance of the second part of the statement? > ZF uses potential infinity whenever the validity of > En Am: m =< n <== Am En m =< n [***] > for linear sets is denied. > > And you still do not answer my question. You fail to give a definition > that is valid in ZF, so I have no idea what you are talking about. > > Potential means not complete. There always appears another number once > you have found the last one. Your definition of complete is not the standard definition. In N there is always one after the other, but the complete set does exist. > ZF claims that this denial is correct for complete linear infinite > sets, but this is a wrong claim, as we can obtain from logic. > > What logic? Not the logic as discussed in sci.logic. > > No, I mean the correct logic. I still do not see the logic through which you obtain it. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Answer to Dik T. Winter ... > And both say, in effect, about the existence of actual infinity, what > Kant said about the proof of the existence of God: These assumptions > (proof of God, axiom of infinity) are as ridiculous as a merchant who > would try to improve his balance by adding some zeros behind his > result. > > And you are deluded. An axiom is a statement of something that can not > be proven, neither disproven using the remainder of the theory. > > The axiom can be contradicted. Simple example: The axiom could be: The > binary tree has uncountably many paths. Perhaps, although in ZF it is not an axiom. > I show that the end of each > path p of the set P can be mapped on a node, and that all paths p of P > cover all nodes of the tree. Ignoring that in ZF the paths do not have an end. > Therefore, after having completed the > covering of the whole tree, there remains no node that could be used > to construct a path that does not belong to P. This is the wrong way around. You assume that you can cover this way the whole tree (I think with this you mean each path in the tree). But that is what you have to prove. > This disproves the > mentioned axiom. Indeed, when you assume it is false, it is easy to prove it is false. > So comparing > the axiom of infinity with a proof of God is pretty stupid. In > mathematics a theory depends on the axioms used. There is nothing > sacred about the axioms, but as long as you are discussing a theory > you should use the axioms of that theory. > > That is same as with proofs of God. The Vatican published an axiom > (they call it a dogma, but it is of the same meaning) according to > which it is possible to prove the existence of God. I do not think such an axiom would be a valid axiom in mathematics. Axioms do not state what is or what is not possible to prove. Axioms states properties and existence of objects. > So, if you are discussion Eucliedan > geometry you should use the parallel axiom. Of course you can reject is > but in that case you are not discussing Euclidean geometry but something > else. > > That means, you are willing to believe in what the Vatican says? Well, no, because that dogma is not a valid axiom. But can you tell me where that dogma actually is stated the way you say? > Probably they think that the Dutch better should have stayed within > the Spanish Empire. As, currently of the people that say they belong to a church, nearly 50% is Roman Catholic there may be some validity. But that nearly 50% is actually about 17% of the total population... -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Answer to Dik T. Winter Nntp-Posting-Host: hera.cwi.nl > There is no difference between finite unions and infinite > unions. > > There should not be, but there is --- at least if ZF is taken to be > true. > > You are again wrongly interpreting things. There is no question about > ZF being true or not. ZF is only one of the many possible theories. > > It is an impossible theory because finished infinity is impossible. Again giving opinion only. > Mathematics is not a science that tries to find truth, > > There is a mathematics, namely basic arithmetic of integers, that is > right. And there are some logical rules that are right. And from these > foundations some theorems can be obtained that are right. There is no > need and no space for any axiom at all. Oh. I would have thought that in that model the basic arithmetic of the integers forms the axioms... > There is no need , in > particular, for a set N with some binary relations. It is > impossible, to change that stuff. And Hilberts famous model including > 1 + 1 = 0 does nothing but express the display of the last wheel of a > binary calculator (or Leibniz's calculator that all time long had > problems with the digit transfer). I think you are confusing a few things here. In your model you may not need a set N with binary relations, but mathematics tries to find the minimum theory, etc. > it is a science where > it is determined what a given set of axioms (i.e. presupposed valid > statements) leads to. > > A given set of axioms is sometimes impossible. This is so with the > axiom of infinity that Zermelo created, misunderstanding Dedekind's > idea of potential infinity. There is no misunderstanding. And that it is impossible is just opinion, nothing more. Like earlier people thought that non-Euclidean space was impossible. And some people thought that irrational numbers were impossible, or negative numbers, or imaginary numbers. > The union of FISONs (of natural numbers or of other finite linear > sets) is the last FISON. > > ZF does agree, when there is a last FISON. Your problem is that the > axiom of infinity states that here is no last FISON of all FISONs. > > That is not my problem. For potentially infinite sets it is not a > problem at all. ZF does not know about the impossible set. > For a completely existing set ZF is wrong. Again, just opinion. > But that is > usually veiled by applying potential infinity whenever the problem > occurs. ZF does not apply something that is not defined in it. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Or do you mean: > S is potentially infinite <-> there exists an ordering R on X such > that there is no R-maximal member of X > Correction (and incorporating Virgil's remark about the empty set): > S is potentially infinite <-> (S is non-empty & there exists a linear > ordering R on S such that there is no R-maximal member of S). > But, isn't that equivalent with 'infinite' (i.e. 'not finite', i.e., > not equinumerous with any natural number') anyway? It is! So if that is WM's definition, it is the same as at least one > definition of actual infiniteness. My definition is: The union of finite segments is a finite segment. There is always a last segment. But it can be surpassed. The idea that the infinite union of finite segments1 1,2 1,2,3 ... results in an infinite segment is false. You can verify this by forming an infinite union of finite segments 1 1 1 ... If the infinite union of finite segments always yields an infinte segment, why then is the result of this infinite union not an infinite segment? Briefly: The set of all natural numbers that exists has always a last element, but potentially infinite sets are not static. They can grow (and they can shrink). === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I A nice pink elephant WM: The idea that the infinite union of finite segments WM: results in an infinite segment is false. WM now gives a counterexample, finding an infinte union of finite segments that does not result in an infinite segment. Look! Over There! A Pink Elephant! WM: If the infinite union of finite segments *always* WM: yields an infinte segment ... - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > WM: The idea that the infinite union of finite segments > WM: results in an infinite segment is false. WM now gives a counterexample, finding an infinte union > of finite segments that does not result in > an infinite segment. Nobody has any evidence of an example supporting the idea that the infinite union of finite segments results in an infinite segment. No form of logic does support it. I gave a counter example. The only pro-argument is that set theorists feel obliged to believe in that idea. But that is not enough. === Subject: Re: Answer to Dik T. Winter element, but potentially infinite sets are not static. They can grow > (and they can shrink). Do you have an opinion on the (intuitionistic) theory of choice sequences? Admittedly they do not shrink but perhaps otherwise they're to your liking. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Answer to Dik T. Winter <877hzjnkde.fsf@alatheia.truth.invalid> posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Briefly: The set of all natural numbers that exists has always a last > element, but potentially infinite sets are not static. They can grow > (and they can shrink). Do you have an opinion on the (intuitionistic) theory of choice > sequences? Admittedly they do not shrink but perhaps otherwise they're > to your liking. Most (of the few) mathematicians who share my standpoint do not talk about vanishing information. But consider any lawless choice sequence that has been fixed to a certain n (by writing it in a memory). If this memory is destroyed, the due information has disappeared. === Subject: Re: Answer to Dik T. Winter <877hzjnkde.fsf@alatheia.truth.invalid Most (of the few) mathematicians who share my standpoint do not talk > about vanishing information. Who are these few mathematicians who share your standpoint? > But consider any lawless choice sequence that has been fixed to a > certain n (by writing it in a memory). If this memory is destroyed, > the due information has disappeared. On your conception what becomes of such basic principles as the density axiom Every finite sequence is the initial segment of a lawless choice sequences. the principle of open data If a property P holds of a lawless choice sequence alpha, there is an initial segment of alpha such that P holds of all lawless choice sequences with as an initial segment. and so on? -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) Does WM's below mean nearer to the root or futher from it? > Below means further from the root. === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > All nodes are already used up. tails in every maximal infinite binary tree), and your method only > involves one of them, so is incomplete. One of them belongs to a tail that has already been mapped on a node at a higher level. The other is mapped on that actual node. That makes a complete 1 to 1 mapping. > And, what is more important, also > all combinations of nodes are used up. Then paths must not be what WM calls combinations of nodes as not all > paths are used up. At least all the constructed paths cover all nodes and leave no node that could be used for another construction. === Subject: Re: Answer to Dik T. Winter > > All nodes are already used up. > tails in every maximal infinite binary tree), and your method only > involves one of them, so is incomplete. > > One of them belongs to a tail that has already been mapped on a node > at a higher level. The other is mapped on that actual node. That makes > a complete 1 to 1 mapping. That accounts for only two paths out of uncountably many, so does not exhaust the paths. > And, what is more important, also > all combinations of nodes are used up. > Then paths must not be what WM calls combinations of nodes as not all > paths are used up. > > At least all the constructed paths cover all nodes and leave no node > that could be used for another construction. Only in infinite but not maximal binary trees in which some nodes have less than two children. > -- Virgil === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) >And uncountably many X through every node. > Let us choose one of them and map it one that node. For the others we > will find other nodes, below that one chosen. Nope.You cannot find enough other nodes. There are uncountably many > paths > going through the node and only countably many nodes below it. That's just the proof saying why Cantor's proof is wrong! Cantor (and his disciples) assume that the complete diagonal of the famous list is the limit of an infinite process. But he and they deny that the complete binary tree is the limit of an infinite process. Cantor spelled that out in a letter to Letter to Vivanti of Dec. 3 1885: Introducing the expression path that Cantor did not use but mean, he said: This apparent difficulty is solved as follows: The set of all path is not the limit of all finite path z n for n = oo. This is an inconsistency. Either there is no limit at all (then Cantor's proof is wrong, because the list cannot be checked completely) or the limit holds also in case of the binary tree. === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > And uncountably many X through every node. > Let us choose one of them and map it one that node. For the others we > will find other nodes, below that one chosen. > Nope. You cannot find enough other nodes. There are uncountably many > paths > going through the node and only countably many nodes below it. > We are discussing your statement: Let us choose one of them and map it one that node. For the others we will find other nodes, below that one chosen. Please indicate the first line you disagree. - There is more that one path through that node - There are countably many nodes below that node - Assuming that For the others we will find other nodes, below that one chosen is equivalent to assuming that there are only countably many paths through that node - You are trying to prove that there are countably many paths through the root node. - This is equivalent to proving that there are countably many paths through that node - William Hughes - === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > - There is more that one path through that node - There are countably many nodes below that node - Assuming that For the others we > will find other nodes, below that one chosen > is equivalent to assuming that there are only > countably many paths through that node No. It is not assuming this but *proving* this. Because paths cannot be distinguished without nodes. Equivalently: Assuming that that, in Cantor's proof, every line contains a digit that differs from the corresponding digit of the anti diagonal would be assuming that the anti diagonal is not in the list. - You are trying to prove that there are countably many > paths through the root node. - This is equivalent to proving that there are countably many > paths through that node This is equivalent to proving that there are only countably many possibilities of distinguishing paths below that node. === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I > - There is more that one path through that node > - There are countably many nodes below that node We are discussing WM's statement: Let us choose one of them and map it one that node. For the others we will find other nodes, below that one chosen. WM has agreed - There is more that one path through that node - There are countably many nodes below that node We are not at > - Assuming that For the others we > will find other nodes, below that one chosen > is equivalent to assuming that there are only > countably many paths through that node No. It is not assuming this but *proving* this. Nope, assuming is not proving. You give a putative proof that starts out by assuming what you want to prove. This is circular. > Because paths cannot be distinguished without nodes. Proves nothing. We know that a countable number of elements can distinguish an uncountable number of subsets. A countable number of nodes can distinguish an uncountable number of paths. - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Because paths cannot be distinguished without nodes. Proves nothing. We know that a countable > number of elements can distinguish an uncountable > number of subsets.A countable number of nodes can > distinguish an uncountable number of paths. No that is provably wrong. All nodes are used up by a countable number of paths, e.g., all paths ending in a tail of zeros. Therefore no possibility exists to construct or to distinguish by one or many or infinitely many nodes of the tree another path. All combinations of nodes that are possible in the tree have already been occupied. === Subject: Re: Answer to Dik T. Winter posting-account=1lE9SQkAAADFrJsDv61dh1YXcJ_ahy5I .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 1.1.4322; .NET CLR 3.0.04506.648; .NET CLR 3.0.4506.2152; .NET CLR 3.5.30729),gzip(gfe),gzip(gfe) > Because paths cannot be distinguished without nodes. > Proves nothing. We know that a countable > number of elements can distinguish an uncountable > number of subsets.A countable number of nodes can > distinguish an uncountable number of paths. No that is provably wrong. All nodes are used up by a countable number > of paths, e.g., all paths ending in a tail of zeros. Therefore no > possibility exists to construct or to distinguish by one or many or > infinitely many nodes of the tree another path. Your claim is that no possibility exists Nope. In any tree, any node that is not a leaf node can contribute to more than one path. The possibility exists to construct another path using nodes which are not leaf nodes. - William Hughes === Subject: Re: Answer to Dik T. Winter posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > - There is more that one path through that node > - There are countably many nodes below that node We are discussing WM's statement: Let us choose one of them and map it one that node. > For the others we will find other nodes, below that one chosen. WM has agreed - There is more that one path through that node - There are countably many nodes below that node We are not at > - Assuming that For the others we > will find other nodes, below that one chosen > is equivalent to assuming that there are only > countably many paths through that node > No. It is not assuming this but *proving* this. Nope, assuming is not proving.You give a putative > proof that starts out by assuming what you want > to prove.This is circular. Not at all. I show that there are no nodes left over, after a countable number of infinite paths have been constructed. My only assumption is that in a tree, the nodes of which are completely covered by paths, no further path can be constructed. My proof rests upon the fact that after the set of all finite paths of the form 0.1, 0.11, 0.111, ... has been constructed, there is no chance to construct the path 0.111... in addition. > Because paths cannot be distinguished without nodes. Proves nothing. We know that a countable > number of elements can distinguish an uncountable > number of subsets.A countable number of nodes can > distinguish an uncountable number of paths. - William Hughes === Subject: Re: Answer to Dik T. Winter > My proof rests upon the fact that after the set of all finite paths of > the form 0.1, 0.11, 0.111, ... has been constructed, there is no > chance to construct the path 0.111... in addition. Why not start with path 0.111... and then add those other paths 0.1, 0.11, 0.111, ...? Rainer Rosenthal r.rosenthal@web.de === Subject: Re: Answer to Dik T. Winter >And uncountably many X through every node. > Let us choose one of them and map it one that node. For the others we > will find other nodes, below that one chosen. > Nope.You cannot find enough other nodes. There are uncountably many > paths > going through the node and only countably many nodes below it. > > That's just the proof saying why Cantor's proof is wrong! Cantor (and > his > disciples) assume that the complete diagonal of the famous list is > the > limit of an infinite process. That is the immediate result of a definition. That it may be produced as a limit is irrelevant. > But he and they deny that the complete > binary tree is the limit of an infinite process. Irrelevant. The union of a set of infinitely many sets is a set regardless. And if infinitely many complete finite binary trees are regarded as sets of nodes, their union is a set of nodes in which every maximal totally ordered under ancestor set of nodes is a path and there are uncountably many such paths. > Cantor spelled that > out in a letter to Letter to Vivanti of Dec. 3 1885: Introducing the > expression path that Cantor did not use but mean WM cannot claim Cantor is always wrong, as he so often does, and simultaneously cite him as an authority on what is right. -- Virgil === Subject: Re: value of second-order logic <87ocsye54v.fsf@alatheia.truth.invalid> posting-account=i1MELAoAAADwo_-GGwN3NKKSXm8aSdBN 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) > Is there a consensus now about the value of second-order logic? I'm sure most logicians will readily admit second-order logic has much > value, from a purely mathematical point of view. Just a few random > results: o A cardinal is weakly compact iff Pi-1-1 indescribable. o A language is in NP iff it is expressible in existential second-order >logic over finite structures. o The continuum hypothesis is not decided by second-order set theory >(under general semantics). o Measurability is not second-order definable. o If V=L then there is no finitely axiomatised second-order theory that >is (semantically) complete but not categorical. Whatever disputes there are concern rather the philosophical or > foundational value of second-order logic and are, as such disputes often > are, rather abstruse. -- > Aatu Koskensilta (aatu.koskensi...@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen >- Ludwig Wittgenstein, Tractatus Logico-Philosophicus let me pose a question. Henkin (1950) showed that the notion of second order validity is strongly dependent on the underlying set theoretical ontology. As I see it, this does not allow to dismiss SOL as set theory in disguise. Consider to that effect that SOL is incomplete in some purely logical sense since it is incomplete relative to the logical consequence relation (otherwise we could have a correct and complete SO Peano arithmetic, in contradiction with G.9adel's theorem). Therefore, I'd say that the ontology of set theory is (to a certain extent) a matter of logic. Hence the question: what is the set theoretical ontology required by SOL? should make sense. And this should reveal the value and importance of SOL for mathematics. Since I'm no expert, I'd be grateful to know whether you (any of you) think my approach above makes sense or there is some fundamental misunderstanding in it. === Subject: Re: value of second-order logic <87ocsye54v.fsf@alatheia.truth.invalid Henkin (1950) showed that the notion of second order validity is > strongly dependent on the underlying set theoretical ontology. Well, no. Second-order validity is not relative to any set theoretic ontology, that is, when we inspect the definition we find no ontologies entering to it as a parameter. What Henkin showed was that it is possible to define a semantics for higher-order languages on which a complete deductive system can be given. > As I see it, this does not allow to dismiss SOL as set theory in > disguise. I don't understand this remark. What is 'this'? General semantics for second-order logic? Why would anyone think it allows to dismiss SOL as set theory in disguise? The basis for this Quinean slogan is that many questions about second-order logic turn on set theoretic matters in a way analogous questions about first-order logic do not, and that e.g. various statements about what second-order formulas follow from these and those second-order formulas are equivalent to (sometimes staggeringly strong) set theoretic statements. > Therefore, I'd say that the ontology of set theory is (to a certain > extent) a matter of logic. Hence the question: what is the set theoretical ontology required by SOL? should make sense. I am perhaps too narrow minded when it comes to ontological matters but I can't make any sense of this question. > And this should reveal the value and importance of SOL for > mathematics. Surely the value and importance is rather revealed by mathematical applications of higher-order logic, such as those already mentioned. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: value of second-order logic <87ocsye54v.fsf@alatheia.truth.invalid> posting-account=AFsgCgkAAAA3VOfxqn2cTB2LbLN3nbER Gecko/20070319,gzip(gfe),gzip(gfe) > Is there a consensus now about the value of second-order logic? I'm sure most logicians will readily admit second-order logic has much > value, from a purely mathematical point of view. Just a few random results: Good examples, but all math-logical ones. Are there any mathematically natural cases? That is, is there some obvious naturally mathematical result that can be decided using 2nd-order logic, but not by any common first order theory (say ZFC)? As always, natural in these contexts is highly subjective OC. -- Wondering William ** Religion has no faith in logic ** === Subject: Re: value of second-order logic <87ocsye54v.fsf@alatheia.truth.invalid That is, is there some obvious naturally mathematical result that can > be decided using 2nd-order logic, but not by any common first order > theory (say ZFC)? I'm not sure what you're after. What does it mean to decide a mathematical result using second-order logic? There must be some confusion here since any mathematical solution, in the sense of a proof of some result, is formalisable in some first-order theory or other. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: value of second-order logic <878wjzpca0.fsf@alatheia.truth.invalid> posting-account=AFsgCgkAAAA3VOfxqn2cTB2LbLN3nbER Gecko/20070319,gzip(gfe),gzip(gfe) > That is, is there some obvious naturally mathematical result that can > be decided using 2nd-order logic, but not by any common first order > theory (say ZFC)? I'm not sure what you're after. What does it mean to decide a > mathematical result using second-order logic? There must be some > confusion here since any mathematical solution, in the sense of a proof > of some result, is formalisable in some first-order theory or other. Well let's be more specific then. A (mathematically natural) sentence decided by 2nd-order set theory that is not decided by 1st-order ZFC ? -- Basic Bill === Subject: Re: value of second-order logic <87ocsye54v.fsf@alatheia.truth.invalid> <878wjzpca0.fsf@alatheia.truth.invalid A (mathematically natural) sentence decided by 2nd-order set theory > that is not decided by 1st-order ZFC ? The continuum hypothesis. This of course only if we understand decided in the sense of logical consequence. If we instead understand it in terms of syntactic deduction second-order set theory is no different from any first-order theory. (And on that understanding doesn't decide any mathematically interesting problem undecidable in ZFC that we know of.) Mathematical applications of first-order logic include non-standard analysis, various results in abstract algebra, and so on. It is natural to understand questions about mathematical applications of second-order logic similarly. And we have applications in this sense in set theory, in complexity theory, and so on. The applications are fewer and less striking than with first-order logic, in part because higher-order logic has received less attention[1] but mostly because of the staggering expressive power of higher-order languages in infinite models. Footnotes: [1] There are for example in context of higher-order logic many open problems concerning the relationship between categoricity and completeness, categoricity and deductive power, and so on. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus === Subject: Re: value of second-order logic <87zlcfnwk9.fsf@alatheia.truth.invalid> posting-account=AFsgCgkAAAA3VOfxqn2cTB2LbLN3nbER Gecko/20070319,gzip(gfe),gzip(gfe) > in terms of syntactic deduction second-order set theory is no different > from any first-order theory. (And on that understanding doesn't decide > any mathematically interesting problem undecidable in ZFC that we know > of.) It's conceivable (though far from certain) that the OP would regard this as being a comment that 2nd-order logic was of no mathematical value. -- Bruising Bill ** In philosophy you never prove anything, you just show its price. === Subject: Re: value of second-order logic <87ocsye54v.fsf@alatheia.truth.invalid> <878wjzpca0.fsf@alatheia.truth.invalid> <87zlcfnwk9.fsf@alatheia.truth.invalid It's conceivable (though far from certain) that the OP would regard > this as being a comment that 2nd-order logic was of no mathematical > value. Anything at all is conceivable but it's difficult to see why anyone would consider e.g. the similar observation that adding modalities to ZFC doesn't lead to any interesting new theorems a comment that modal logic is of no mathematical value. -- Aatu Koskensilta (aatu.koskensilta@uta.fi) Wovon mann nicht sprechen kann, dar.9fber muss man schweigen - Ludwig Wittgenstein, Tractatus Logico-Philosophicus f