mm-127 Since I have not (knowingly) seen a post of James latest version of his Advanced Polynomial Factorization, and since he seems to be dismissing me, I got curious about what it looks like now. It has signicant differences.The new text is enclosed. Ill note key features for comment with ***#***.-- HARRISAbstract. Algebraic method for determining distribution of fac-tors within a polynomial factorization, which breaks through whatwas seen as a barrier from overinterpretations of Galois Theory.1. Advanced Polynomial Factorization Approached Determining the distribution of factors within irrational algebraic integers has long been considered impossible as it is not possible to dousing Galois Theory. However a simple technique through the intro-duction of more variables makes it possible. To highlight the standardbelief consider the algebraic integer roots of x^2 + x - 5.While you know that the algebraic integer roots are themselves fac-tors of 5, can either not have non unit factors of 5? How do you know?In looking to consider the distribution of algebraic integer factorswithin a factorization Ill be using what follow.The paper will show that given the factorization, in the ring of alge-braic integers,65x^3 - 12x + 1 = (a_1x + 1)(a_2x + 1)(a_3x + 1)one of the as is coprime to 5, using basic algebraic methods.First Ill need a simple lemma to generalize beyond factors of a poly-nomial that are themselves polynomials.***1***Lemma 1.1. Factorization Lemma:Given a factor g of a polynomial P(m), there exists r and c such thatg = r + cwhere r = 0, or varies as m varies, and c is a factor of the constantterm P(0) and is itself constant.Proof. Let m = 0, then g must be a factor of P(0), so at that pointc = g.(1) If when m does not equal 0, g = c, r = 0.(2) If when m does not equal 0, g <> c there must exist r whichvaries with m, i.e. r = g - c.As an example consider sqrt(m + 1) which is a non polynomial factorof m+1, and while there are an innity of irrational solutions considerthe rational solution at m=35.Then I have sqrt(35 + 1) = 6 = 5 + 1; therefore when m=35, g=6, r=5,and c=1.2. Primary ArgumentLetP(m) = f^2 ((m^3 f^4 - 3m^2 f^2 + 3m)x^3 - 3( - 1 + mf^2 )xu^2 + u^3 f)Here f is a non unit, non zero algebraic integer coprime to 3 and x,and u a non unit, non zero algebraic integer coprime to f. Note P(m)has a factor that is f^2.***2***That expression comes from expanding (v^3 +1)x^3 - 3vxy^2 +y^3 , usingthe substitutions v = - 1 + mf^2 , and y = uf.Now consider the factorizationP(m) = (a_1x + uf)(a_2x + uf)(a_3x + uf)where multiplying out shows thata_1 a_2 a_3 = m^3 f^6 - 3m^2 f^4 + 3mf^2 = f^2 (m^3 f^4 - 3m^2 f^2 + 3m)soa_1 a_2 a_3 = mf^2 (m^2 f^4 - 3mf^2 + 3)which shows that at least one of the as cannot be coprime to m, andat least one of the as must equal 0 when m = 0.Notice that the constant term P(0) isP(0) = f^2 (3xu^2 + u^3 f)and also that P(0)/f^2 = 3xu^2 + u^3 f, which is coprime to f.Then I have the factors of P(m), g_1, g_2, and g_3, where g_1 = a_1x+uf,g_2 = a_2x + uf, and g_3 = a_3x + uf.Since one of the as is not coprime to m, and indices are arbitrary, Ican choose that a1 is not coprime to m.g_1 = c_1 = ufwhich ts with f being a factor of the constant term. And in fact,exactly two of the as can equal 0, when m = 0, to get the factor f^2 inthe constant term P(0), as at m = 0g_1 g_2 g_3 = u^2 f^2 (3x + uf).Now as pointed out before P(m) has a factor that is f^2, and sepa-rating that factor off, gives a constant term coprime to f.Now before its separated off I have g_1 = a_1x + uf and after itsseparated off I know the constant term is coprime to f so it must betrue that g_1/f = a_1x/f + u, which at m = 0, gives g_1/f = u, asrequired, as it is coprime to f, and a factor of the contant term P(0).It may seem possible that some variable with a dependency on mdivides off, but that is refuted by considering that when f = 3 ALL***3***the as have a non-unit radical factor of 3 that is 3^{2/3} , without regard tom. And obviously if there were a variable dependency on m, it couldntswitch on or off dependent upon whether or not f is coprime to 3.Given that P(m) has a factor f^2 that separates off, two of the gsshould have a factor of f which would force two of the as to have afactor that is f.Therefore, that leaves one factor coprime to f.***4***Now letting m = 1, f = sqrt(5), where I can let u = 1, as its value isindependent of the as, I haveP(m) = (m^3 f^6 - 3m^2 f^4 +3m)x^3 - 3( - 1+mf^2 )xu^2 +u^3 = 65x^3 - 12x+1which may be more easily seen from using v = - 1 + mf^2 = 4, y=1with(v^3 + 1)x^3 - 3vxy^2 + y^3 = (a_1x + y)(a_2x + y)(a_3x + y).Therefore, with the factorization65x^3 - 12x + 1 = (a_1x + 1)(a_2x + 1)(a_3x + 1)one of the as is coprime to 5.***5***- ***1***: Note that the lemma has been changed to have 0 content. It is now simply a construction. It would help if a ring was specied so we could know which values of c count as factors.***2***: Here it is stated that f and u are non units, f coprime to 3 and x and u.***3***: Why are you talking about what happens when f=3? You said f is coprime to three so this situation cannot happen.***4***: Here you assign u=1. u is a non-unit, which excludes this assignment.***5***: Your conclusion contradicts the explicit factorization that was provided by W. Dale Hall. Since the numbers disagree, one of the above is likely to be a critical problem.Other problems include: what is x? What are v, y, m? If m is a variable (as it appears), then how can the as be coprime to it?If x is a variable, how can f be coprime to it?Considering a blunder I made earlier today, I may be missing the obvious, but these appear to be concrete problems that must be dealt with.-- Will Twentyman =a/b + b/c + c/a = n , including mention that (for n<200),> Except for n=142 and n=177, we have explicit solutions.I am pleased from a couple of elliptic-curve fans.another curve (from a third family besides the two discussed in thisthread) is a surjection on rational points. This allows another levelof descent, giving a more approachable search space. When translatedback (through two 3-isogenies) this gives us the lovely solution,apparently minimal, which happens to use positive integers only: a = 23381593234936372123623332652594607232679953206675800829 14801368818500874833339590707685028055162560266769298444575042 876 b = 6472270006492812673719886357710878031344481152384871782008 59980415730790019463760618736457807240738779916233265738309991 525 c = 3273877691398828904605661792745624282538951164849869225778 17123390124690992174499099412180521982402624674563255418744783 750I then appealed to Tom Womack for help with the last case; he has codedsome techniques for 4-descent, which again reduce the sizes of thenumbers being searched. (I dont claim to know how exactly he did this.)But in very little time his methods found a solution, again likelyto be minimal, and again using only positive values for a,b,c asrequired in the original problem: a = 44518868186006132682112995066685960920569055034715052011119463 0556966 436821631047458921363439516505891378648322116770968163708164 401573025159148773984893799821901869771134413368989401077170 b = 28864809268260953751234360351334673538914533564408357403583880 69273 117406748241229198230595283470644303272529734480463790129297 393707889262867159886114570461314560454492187347197854308700 c = 12678140381638229473517516094725992502350028810389611853152494 3372 845798336255598830049585497128339056967473477664040582804720 897007784402845395115328240333301067358938774092489803085043I nd these solutions to be really remarkable; even accounting forthe size-reducing tricks involved with descent, these are hard examplestofind without a very rened search! Hats off to these two gentlemen.Having now freed up some machines at work I have decided to extend thesearch space further to see what new challenges we can overcome. For the801 values of n from 200 to 1000, it is (conditionally, as usual) truethat the ranks are 0 (342 values of n), 1 (366), 2 (92), or 3 (only once,for n=484). We can unconditionally prove that these are at least lowerbounds for the ranks byfinding the 553 (independent) points. I havefound about a quarter of them, but am not optimistic that I have the mathematical tools and the spare CPU cycles tofind them all.Since the solutions found above use only positive integers, we now knowthat there are precisely 57 values of n under 200 for which the titleequation can be solved in positive integers. I dont know of any way tond out how many there are up to n=1000 withoutfinding a basis foreach elliptic curve. On the other hand, the list of 570 values of nup to 1000 for which the equation can be solved in integers is nowknown (subject to the usual deep conjectures of course). Ive attachedit below (hoping as usual I have made no typing errrrors).I have updated a few of the les related to this problem at http://www.math.niu.edu/~rusin/research-math/abcn/ dave3,5,6,9,10,13,14,15,16,17,18,19,20,21,26,29,30,31,35,36,38 , 40,41,44,47,51,53,54,57,62,63,64,66,67,69,70,71,72,73,74,76,77 ,83,84,86,87,92,94,96, 98, 99,101,102,103,105,106,107,108,109,110,112,113,116,117,119,120 , 122,123,124,126,127,128,129,130,132,133,136,142,143,145,147,14 8,149,151,154,155,156,158,159,160,161,162,164,166,167,172,174, 175,177,178,181,185,186,187,189,190,191,192,195,196,197,201,20 2,203,204,206,207,208,209,210,212,214,215,217,218,219,220,224, 229,230,233,235,236,237,242,243,244,245,246,248,250,253,255,25 6,261,262,263,267,268,269,270,272,273,274,275,278,280,281,288, 289,290,291,293,294,295,298,299,300,302,303,304,305,306,310,31 1,312,313,314,316,317,318,320,321,323,325,326,329,332,333,334, 335,336,339,340,344,346,349,350,351,354,357,358,360,364,366,36 8,369,370,372,373,375,376,379,380,381,383,384,386,388,391,394, 395,397,405,406,407,408,409,410,411,412,413,414,417,418,420,42 1,422,425,427,429,430,431,434,435,436,437,439,442,443,444,446, 450,451,452,454,455,456,458,467,468,469,470,471,476,478,483,48 4,489,493,495,497,501,504,505,506,507,510,511,512,515,517,518, 521,524,526,527,529,532,534,535,536,539,542,545,546,547,550,55 1,553,554,556,558,559,561,563,564,567,569,570,571,574,576,578, 579,580,581,584,585,587,588,589,590,591,593,594,595,596,597,59 8,599,600,602,606,607,609,611,612,613,614,616,617,618,619,621, 622,628,629,630,631,632,633,634,635,636,637,638,641,643,645,64 7,648,649,650,653,654,655,656,657,659,661,663,664,666,668,671, 672,675,676,677,680,681,682,683,685,686,690,692,693,694,695,69 6,697,699,700,701,703,707,708,710,711,713,714,715,717,718,719, 720,721,723,724,725,726,728,730,732,733,734,735,741,745,746,74 7,748,751,754,755,757,759,761,762,765,766,769,773,777,778,779, 781,782,783,784,787,789,790,796,797,798,799,800,802,803,806,80 9,810,811,813,814,816,818,819,820,821,824,825,827,831,832,833, 836,839,841,843,844,845,846,847,849,850,851,853,854,857,858,85 9,866,867,868,870,873,874,875,876,877,880,883,885,886,888,889, 891,892,893,895,897,899,900,901,902,903,905,907,908,909,910,91 1,914,917,918,919,920,922,923,924,925,927,929,932,933,934,935, 938,940,941,944,946,949,951,953,955,956,957,958,959,961,965,96 6,968,969,970,971,972,974,975,978,979,981,982,983,985,986,987, 989,990,991,992,994,995,997,998 = > > I was hoping some of you will be able to shed light on the process of > research. I am new to it and have been working at a problem for the > past 6 months or so without success. I am in the process of getting > an advanced degree in mathematics. > > How long do you spend on problems before say, you publish them ? > > Is it better to have several problems in the play at the same time ? > What do you do ? How many problems do you tackle ? Is it better to keep butting ones head against the wall or move onto > some other problem ? > > Any interesting experiences, ideas and suggestions would be welcome > especially from active mathematicians. > > > ArvindWell here is one advice on starting research in an area: Read researchpapers from those active in the area, but dont read monographs ordenitions and theorems.Reading multiple research papers is more work; elegant concepts andbest denitions often come later, after the original research hasbeen done. The rst proof of a theorem is often not the moststraightforward one. But approaching an area this way forces you tocreate your own understanding of concepts and their relationships. Andthe original papers often contain ideas that are later discarded in theinterest of consistency and economy of presentation, but may still bevery useful.Nemo =b_is below.> Then P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 -> 3(-1+mf^2 )x u^2 + u^3 f.>Now using b_1, b_2, b_3, w_1, w_2, and w_3, I have the factorization P(m)/f^2 = (b_1 x + u w_1)(b_2 x + u w_2)(b_3 x + u w_3)where w_1 w_2 w_3 = f, and> b_1 b_2 b_3 = (m^3 f^4 - 3m^2 f^2 + 3m),>> Only if your b_i are polynomial factors.>> Consider f(x)=x^2 + 2x + 1:>> If I factor f(x) = (x+1)(x+1) then b_1=b_2=x and b_1*b_2 = x^2.> Correction: b_1=b_2=1, 1*1 = 1> On the other hand, I can also factor it as f(x) = [ sqrt(x) + 1 ] [ (x >> sqrt(x) - x + 3 sqrt(x) - 4 + 4/(sqrt(x)+1)) + 1 ]>> Now b_1 = sqrt(x), b_2 = x sqrt(x) - x + 3 sqrt(x) - 4 + 4/(sqrt(x)+1)>> and b_1*b_2 = x^2 - x sqrt(x) + 3x - 3 sqrt(x) + (4 sqrt(x))/(x+sqrt(x))>> which is NOT x^2.> Correction: b_1= 1/sqrt(x), b_2=sqrt(x) - 1 + 3/sqrt(x) - 4/x + > 4/(xsqrt(x)+x)b_1*b_2=1 - 1/sqrt(x) + 3/x - 3/(x sqrt(x)) + 4/(x^2 sqrt(x)+x^2)> so b_1*b_2 <> 1> -- Will Twentyman =Let {a(k)} be any sequence of integers where the sum below converges.Let {b(j)} be a sequence such that:sum{j=0 to oo} b(j) x^j /j! =exp(A_n(x)),where A_n(x) = sum{j>=2, GCD(j-1,n)=1} a(j) x^j /j!,where this sum is over all integers j, j >= 2, wheren is coprime to (j-1), n = any xed positive integer.Then:b(j) is an integer sequence, and (the main result...):n divides b(n+1).Leroy Quet =I am going to repost this with all rs increased by 1, as to improvethe appearance of these equations.Let B(k) be a Bernoulli number, wheresum{k=0 to oo} B(k) x^k /k! = x/(e^x -1).Let H(k) be a harmonic number, H(k) = sum{j=1 to k} 1/j.Let, for r = any xed nonnegative integer, and for every positiveinteger m,c(m) = (H(2m+r-1) - H(r)) (2m+r-1)!/((2m-1)!(2m+2r)!),andd(m) = (H(2m+r) - H(r)) (2m+r)!/((2m)!(2m+2r+1)!)A little better-looking in ascii-art mode:c(m) =(H(2m+r-1) - H(r)) (2m+r-1)!- (2m-1)!(2m+2r)!d(m) = (H(2m+r) - H(r)) (2m+r)!- (2m)!(2m+2r+1)!Then:c(m) = 2 sum{k=1 to m} d(k) B(2m-2k) /(2m-2k)!andd(m) = 2 sum{k=1 to m} c(k) B(2m-2k+2) (4^(m+1-k) -1)/(2m-2k+2)!Ascii-art:c(m) = m d(k) B(2m-2k)2 / -- (2m-2k)! k=1and:d(m) = m c(k) B(2m-2k+2) m+1-k2 / -- (4 -1) (2m-2k+2)! k=1I am sorry if I posted these before. (The last one, anyway, does notlook familiar. And if I did post these before to sci.math, they haveprobably been long-forgotten anyhow...)LeroyQuet =Let H(0,m) = 1/m, for m = all positive integers;and let, for all n = positive integers,H(n,m) = sum{k=1 to m} H(n-1,k).(So, H(1,m) = H(m) = the m_th harmonic-number.)(Also, H(n,m) = binomial(m+n-1,n-1) (H(m+n-1) -H(n-1)).)(I have posted on this sum-of-sum-of... many times before.)Now, letG(0,n,m) = H(n,m) *m;and let, for all positive integers r,G(r,n,m) = sum{k=1 to m} G(r-1,n,k).(G(r,n,m) = binomial(r+n+m-1,r+n) (n*H(r+n+m-1) -n*H(r+n) +1), I think.)And, nally, let,for n+1 >= m,F(0,n,m) = H(n+1-m,m) *m *(-1)^m;and let, for all positive integers q,F(q,n,m) = sum{k=1 to m} F(q-1,n,k).(I have not come up with a closed-form for F as of currently.)Then:(m-1)! G(r,n,m) is always congruent to(m-1)! F(q,n,m) (-1)^m (mod{m+q+r})Leroy Quet => ...start with a dodecagon (12-gon) [...] make a path that moves> from vertex to vertex [...] visiting each vertex exactly one time. > And the path returns to the starting-point. But in this puzzle, > consecutive vertexes MAY be connected by a segment.So, I give a list of nonnegative integers below. As the path is drawn> (as opposed to after the path is completed), the n_th segment crosses> a(n) previously drawn segments, where a(n) is the n_th term of the> integer-list.The path starts at 12. And the rst segment goes from 12 to 8.> The list {a(n)}: 0, 0, 1, 0, 2, 2, 0, 3, 2, 2, 5, 2 > ...> I tried (but not hard) tofind an alternative path using this list.> But to do so seemed somewhat difcult. So this might be an> interesting puzzle.> 17 of the 10! paths that start off {12, 8} match that crossing-> count sequence, so solutions seem fairly rare - about 1 per 213000> paths - but on the other hand, there are perhaps O((n-3)!) possible > crossing-count sequences, so 17 could instead be an unusually high > prevalence, and this case no more rare than thousands of others. > I plan to look at this more andfind out, next weekend.> -jiwHmm...I wonder which sequences (if any), for the 12-gon, produce onesolution, but are interesting.If you, or anyone, happens tofind such a sequence, then it would beinteresting to post it to sci.math and rec.puzzles as a challenge forus all.:)Leroy Quet => Perfectly Innocent:> >Im wondering if its mathematically permissible, if space is> >homogeneous and isotropic, for a moving rod to experience a uniformexpansion or contraction during the time its not in its stationary> >frame of reference. No. The length of the rod is an ivariant. Applying 3 dimensional> analysis to a 4 dimensional length is inappropriate.The concept that space is homogeneous and isotropicis driven by mans survival instinct.As man is hardwired to try to be conservedhe has constructed a world, in whichconserved objects (Like he wants to be),vary in a medium that does not affect the conservationof the object (One that is homogenous and isotropic).Mans languages are based on objects (Nouns) varying in some way (Verbs),his religions propose the ultimate conserved object (The soul)varying in the ultimate, conserving medium (Heaven),and of course, mans ultimate rationalization (Physics)constructs a world where conserved objectsvary in homogeneous and isotropic media.As a Greek philosopher once said,You cannot step into the same river changes you.Regarding the posters questionif its mathematically permissible,an innite number of things are mathematically permissiblein maths, but these mathematically permissible operations do notnecessarily correlate very well with reality.--Tom Potter http://tompotter.us =Perfectly Innocent:> >Im wondering if its mathematically permissible, if space is> >homogeneous and isotropic, for a moving rod to experience a uniformexpansion or contraction during the time its not in its stationary> >frame of reference. No. The length of the rod is an ivariant. Applying 3 dimensional> analysis to a 4 dimensional length is inappropriate.The concept that space is homogeneous and isotropic> is driven by mans survival spherical homogenous cow is a cornerstone of physics nals. Physics nals are antithetical to survival. -- Uncle Alhttp://www.mazepath.com/uncleal/ (Toxic URL! Unsafe for children and most mammals)Quis custodiet ipsos custodes? The Net! =Im wondering if its mathematically permissible, if space is> homogeneous and isotropic, ...Can you prove that homogeneity and isotropy alone disallows this> possibility?> General Relativity is a projective geometry into four dimensions,> three physical and one time. You are whining about non-existent problems caused by your ignorance> of very basic phsyics - something that could be remedied by cracking> one elementary textbook. Show some personal initiative.Im not inquiring about General Relativity. Im aware that nosemi-Riemannian space has the geometry Ive described. But there areother possibilities such as Finsler geometry. I only requirehomogeneity and isotropy.My perspective is that of a mathematician, not a physicist. Myinterest is only in mathematical possibilities, not the way theuniverse really is.It is often said that the Lorentz transformation can be derived fromthe homogeneity and isotropy of space alone. Im looking for aconcrete counterexample to this claim.Obviously, if the claim is true, then Im wasting my time. But I amnot. I have already proven that the claim is false.See http://www.everythingimportant.org/relativity/ generalized.htmThere are more things in heaven and earth than are dreamt of in yourphilosophy.Eugene Shuberthttp://www.everythingimportant.org/relativity/http:// www.everythingimportant.org/relativity/simultaneity.htm =>> [...]>>Thus there are no burning issues today that force mathematicians to worry about >>these issues. There really are no difcult paradoxes in mathematics that are >>today left outstanding. You could try to dig deeper, and ask questions as to >>why truth tables always give the same tautologies as those given by some basic >>axioms along with modus ponens, but the search seems pointless and fruitless.> Not to focus on irrelevant details, but it doesnt involve much > digging to determine why truth tables give the same tautologies> as suitable sets of axioms and inference rules - this is a basic > result in mathematical logic that you can easilyfind proofs of> in books (one half is the (or a) Soundness Theorem, and one> half is the Completeness Theorem. The Completeness Theorem> is the non-trivial half - if I recall correctly it was rst proved > for predicate logic by Godel.)Or maybe its not irrelevant - if one were unaware of the Completeness> Theorem that might lead to some nervousness about what might> be missing from our deductive systems...> I had seen a proof of this statement, but the proof I saw used set theory and logic, so it was kind of pulling itself up by its shoelaces.-- Stephen Montgomery-Smithstephen@math.missouri.eduhttp:// www.math.missouri.edu/~stephen for a problem referenced by Martin Gardner, Chapter 11 in his> collection> entitled The 2nd Scientic American Book of Mathematical Puzzles &> Diversions,> Simon and Schuster, 1961.Gardner references Lewis Carroll telling that Carroll was fond of inventing> quaint and> enormously complicated problems of this sort. Eight are to be found in the> appendix of his Symbolic Logic.Gardner continues speaking of a large problem about judges not smoking> tobacco which was solved in 60s by Kemeny.Can you give me some help on where tond the text of the problem?> Symbolic Logic volume 1, Appendix Addressed to Teachers. => Im going by the one on the net at> http://home.ddc.net/ygg/etext/godel/godel3.htm> if that helps. It seems more or less legit to my eyes.Thats the one, but it doesnt render well outside of the Microsoftuniverse.>> Then there are primitive-recursive functions (just recursive in>> the paper). These are number-theoretic functions, taking numbers as>> arguments and returning numbers. These functions are not objects of the>> formal system P, so Godel does not assign Godel numbers to functions.>> There are no FUNCTIONS, but there are formulas (lower case) that represent>> functions in the formal system in a precise sense explained in the paper.>OK this is good because I was never sure exactly how to approach> these. Sometimes they seemed to be like a computer function as you> described, taking numbers as> arguments and returning numbers but other times it seemed less> clear. Does this include all those n Gl x type bits numbered 1 - 45> that Godel terms functions(relations).Yes, I think so. Those are mathematical objects.> If that is so when we see function 1: x/y as part of a formula or> another function do we replace it with a 1 or 0 to indicate the truth> or falsity of the statement.No.> I seem to recall this being done> elsewhere in the paper. Or do we replace it with (Ez)(z <= x & x => y.z) and then work out the Godel number from these symbols.> That seems> different from the number in, number out concept but would that be> one of the formulas (lower case) that represent> functions in the formal system in a precise sense explained in the> paper that you mentioned.Yes, exactly. The formula z <= x & x = y*z REPRESENTS the relationz= x/y in system P. Actually, the <= symbol is probably not in Psalphabet but is a shorthand a dened relation, so you would have toreplace it by its denition, for example ((Ew)(w+z = x)) & x = y*z.Its pretty easy tofind a representing formula for a relation, becausesystem P follows Principia Mathematica and it can talk about sets,ordered pairs, relations, functions, nite sequences etc. All you haveto do is transcribe your mathematical denition in the notation of P.Thus every mathematical object in the chain (1--45) has a representingformula.> Jesus this is getting complicated.No . Its not over, either. To proceed further you have to provethat your formula formally represents the original relation. And *that*means that, whenever you substitute the numerals of numbers that obey therelation, system P has a proof of the substituted formula; but if yousubstitute the numerals of number that dont obey the relation, sytemP has a proof of the negation of the substituted formula. For example,3= 6/2 and 4 != 6/2; sure enough, P proves ((Ew)(w+sss0 = ssssss0)) & sssss0 = ss0 * sss0and P also proves ~(((Ew)(w+ssss0 = ssssss0)) & ssssss0 = ss0 * ssss0)You have to convince yourself that this works in the general case.Note that this notion of representability does not assume that P isconsistent. If P is inconsistent, any formula represents everyrelation. Exercise.This notion of representability is crucial in the proof that the number17 Gen r is an UNDECIDABLE number. Neither a THEOREM nor the NEGATIONof a THEOREM.> Here is one of my main problems. Godel uses formulas such as> Q(x,y): ~{B[Sb(y 19|z(y))]}> Godel later talks about a relation sign q for this formula (by which I> presume he means what has become known as a Godel number)I think Q() is a mathematical relation, not a formula. The lower-case q,the relation sign, is either the formula, or the Godel number thereof.If it is a Godel number, Godel will probably call it a RELATION SIGN,not a relation sign.> meaning that> he has worked out the Godel number for Q. But in doing so, when we get> to the part of Q(x,y) that is Sb(y 19|z(y)) and we try to follow the> above steps wefind that y is a free variable and hence not a FORMULA,Thats ok, y was already a metamathematical variable in Q(x,y). BothQ() and Sb() are mathematical objects. What youre looking at is adenition of Q() in terms of Sb() and z() and the x PROVES yrelation (was that Bw() ? should be number 44).> and therefore Sb(y 19|z(y)) is undened as you point out and this> makes it impossible tofind the Godel number q for Q(x,y) which is> used later in the proof.No, its dened, in mathematics. Sb(y, 19|z(y)) is a number-theoreticfunction of one variable.The function Sb(), or the relation t= Sb(x,y,z), has a representingformula with 4 free variables. The function z(), or if you prefer therelation t= z(y), also has a representing formula, this one with twofree variables.By composing the two formulas appropriately, you get a representingformula for the function Sb(y, 19|z(y)). You then need to combine thatwith the representing formula of x PROVES y to obtain a formula q withtwo variables that represents the mathematical relation Q(x,y).> Would this be where we use the formula> representation of the fuction to represent Sb(y 19|z(y)) in the formal> version of Q(x,y). I could see how this might get around the problem> if it is possible but it seems that Sb() is sometimes a FORMULA and> other times not quite the same and there is no explicit instruction as> to which to use, unless Ive missed saomething.Again, read the paper slowly. I think Sb() is always anumber-theoretic function, but you have to make sure.> [ ... ] That is pretty much how I see the process going also if Sb() is> treated as a number in number out formula.Function, not formula. This nitpicking really matters.> [ how to code P(y) Ax ~(x PROVES Sb(y 19|z(y)) ) ]> OK so we can write out Exists t = Sb(y 19|z(y)) to get around the> problem? And presumably we expand the recursive formula deniton of> Sb(y 19|z(y)) in terms of the other functions it is dened as and we> get a loooooong string of formal symbols (Big Mess) that we can> Godelize into a number.which is p.> and then when we substitute in the numeral of> p to this formula we are replacing the occurences of y in the big> mess.Exactly. The resulting formula has Godel number r.> [ ... ] Which would mean that it is> not like a computer function where you can make recursive calls to the> functionYoure correct, this is not like recursive calls. The whole point is to*not* have any recursion. Godel constructs a sentence that says somethingof its own Godel number. If you perform the following computation,blah blah, the result has property blah. So you perform the indicatedcomputation and, sure enough, the result is the Godel number of If youperform the following computation, blah, blah, the result has propertyblah. A self-referential sentence. Indirectly self-referential,without any innite regress, but the representability theorem isstrong enough to make that stick.> a lot of things in this paper a lot clearer though it seems im still> a little behind.But thats normal. =This is what I know:LOG(A/B) = LOG(A) - LOG(B)You would then take the inverse of LOG(A) - LOG(B) to get the answer for A/B LN(B), it is approximately equal to (A/B) - 1.In nance, I have even see people use the value LN(A) - LN(B) as the periodreturn instead of (A/B) - 1.Note: LN = Natural LogQ. Is LN(A) - LN(B) a valid approximation for (A/B) - 1. Why do people useit to calculate period return?Jay => This is what I know:> LOG(A/B) = LOG(A) - LOG(B)> You would then take the inverse of LOG(A) - LOG(B) to get the answer for A/B> - LN(B), it is approximately equal to (A/B) - 1.Only if A and B are approximately equal, and therefore A/B is close to 1.Thats because ln(1+x) is approximately x when x is near 0. The line y=xis tangent to the curve y=ln(1+x) at the point (0,0).> In nance, I have even see people use the value LN(A) - LN(B) as the period> return instead of (A/B) - 1.> Note: LN = Natural Log> Q. Is LN(A) - LN(B) a valid approximation for (A/B) - 1. Why do people use> it to calculate period return?Maybe its a holdover from the days when people used log tables toperform such calculations.-- Dave SeamanJudge Yohns mistakes revealed in Mumia Abu-Jamal ruling. => Q. Is LN(A) - LN(B) a valid approximation for (A/B) - 1. Why do people use> it to calculate period return?I think you have it the other way around. LN(A) - LN(B) is likely the exact value you seek, and (A/B) - 1 is an approximation for it. Obviously (A/B) - 1 is easier to think about.The approximation works well only if (A - B)/B is small. This is essentially the tangent line approximation to ln(1+x) near 0: ln(1+x) = x + r(x), where the error term r(x) is much smaller than x if x is itself small. Putting the above together: ln(A) - ln(B) = ln(A/B) = ln (1 + (A-B)/B), and the latter is approximately (A-B)/B = (A/B) - 1, provided (A-B)/B is small. => This is what I know:> LOG(A/B) = LOG(A) - LOG(B)You would then take the inverse of LOG(A) - LOG(B) to get the answer for A/B - LN(B), it is approximately equal to (A/B) - 1.> In nance, I have even see people use the value LN(A) - LN(B) as the period> return instead of (A/B) - 1.Note: LN = Natural Log> Q. Is LN(A) - LN(B) a valid approximation for (A/B) - 1. Why do people use> it to calculate period return?> JayLet A = market value of a portfolio at time 0 (the beginning of theperiod)Let B = market value of a portfolio at time 1 (end of the period)NOTE: These are switched from your example.Assume that no cashows occur within the period. What is theperiodic return on your investment?For example, A = 100, B = 110. Return = (B/A)-1 = 0.10You earned 10% return in the period, because you gained 10 from astarting value of 100, correct?Not necessarily. By most conventions, this would be the holding period return, so the10% return is correct. However, in some investments -- usually xedincome assets -- the perspective is slightly different and therate/return/yield is calculated under the assumption of continuouscompounding. In these situations, the LN(B) - LN(A) result is thestated one. Here the return is 9.53%. The question asked here is nolonger How did my investment do in the period? Rather, it becomesAssuming that my money is continuously working, what rate gets meto the ending value from the starting value? The distinction issubtle.Note that you have the same underlying data (beginning mv, ending mv,no cashows), but the interpretation of the investment return resultsin different answers.(Sorry, the details of which nancial instruments use this escape meright now. I have drastically simplied the primary reason for theLN(x) convention, but the gist is there.)I dont see why anybody would calculate using the logarithm method,unless similar reasoning to the above is required. It is far easierto calculate the EMV/BMV - 1 return, which is what most peopleintuitively grasp. The continuous return assumption has its place,but not in many situations, I believe.Bye,Jay =i am ne thank you and you? huk? =I understand that a homeomorsme is a continuous bi-jective mapping betweensets.I learn that a homomorsme is some sort of mapping between groups. Thereare no continuous groups, though one author claims that to be so.What exactly is a homomorsme, and are there continuous groups?Peter => I understand that a homeomorsme is a continuous bi-jective mapping between> sets.Actually, a homeomorphism is a *bicontinuous* bijection. Its possibleto have a continuous bijection whose inverse is not continuous, but ahomeomorphism is required to be continuous in both directions.> I learn that a homomorsme is some sort of mapping between groups. There> are no continuous groups, though one author claims that to be so.> What exactly is a homomorsme, and are there continuous groups?> PeterIn general, a homomorphism is a mapping that preserves algebraicstructure of some kind. For example, there are group-homomorphisms,ring-homomorphisms, and eld-homomorphisms.And yes, there are such things as topological groups, and in such casesit makes sense to ask whether a homomorphism is continuous.-- Dave SeamanJudge Yohns mistakes revealed in Mumia Abu-Jamal ruling. => And yes, there are such things as topological groups, and in such cases> it makes sense to ask whether a homomorphism is continuous.And it makes sense to ask if it is both a homomorphism and a homeomorphism. => I understand that a homeomorsme is a continuous bi-jective mapping> between sets.homeomorphismand between topological spaces> I learn that a homomorsme is some sort of mapping between groups.homomorphism> There> are no continuous groups, though one author claims that to be so.> What exactly is a homomorsme, and are there continuous groups?There are topological groups (eg SO_3) which are groupswhich are also topological spaces and the group operations are continuous.A homomorphism from a group G to a group H is a map f:G -> Hsuch that f(ab) = f(a)f(b) for all a, b in G.-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html The League of Gentlemen =Peter> I understand that a homeomorsme is a continuous bi-jective mappingbetween> sets.> I learn that a homomorsme is some sort of mapping between groups. There> are no continuous groups, though one author claims that to be so.> What exactly is a homomorsme, and are there continuous groups?> PeterX-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Punge: Micro$oft = at 02:51 PM, Bouwman P. said:>I understand that a homeomorsme is a continuous bi-jective mapping>between sets.A homeomorphism is a continuous open bijection between topologicalspaces, that is, both the map and its inverse are continuous. Itmakes no sense to speak of a homeomorphism between sets with notopology specied.>I learn that a homomorsme is some sort of mapping between groups. A homomorphism in Group Theory is a function between groups thatpreserves the group operation, that is, H: {F, O_F, I_F} -> {G, O_G,I_G} maps the identity I_F into the identity I_G and, for x,y in F,H(O_F(x,y))=O_G(H(x),H(y)). >There are no continuous groups,Im not sure what you mean by that. There is something called atopological group, and there are continuous homomorphisms among them.-- Shmuel (Seymour => I remember a post where someone was wondering how I could keep> claiming that Im right with all these people posting disagreement,> and it seems to me that maybe explaining how might help.You see, my short proof of Fermats Last Theorem is awless with all> the disputes from mathematicians attacking the following statement:Given a factor g of a polynomial P(x), g=r+c, where c is a factor of> the constant term P(0) of the polynomial, given by the value of g at> x=0, and r=g-c.Thats it.Not true.Lots of people pointed that your denition of objects is curcular,> which obviously makes the whole proof invalid.You never addressed those objections. Will you address them now?> I bet you wont.> Because when you see objects mentioned in a reply, you go suddenly blind.In my original post you can see the linchpin of my proof of FermatsLast Theorem. Theres no excuse that Im ignored because Im sure bynow many of you have noticed that not only am I not ignored, but thereare people who at times try to make objections that at least soundmathematical. Here you see mention of objects which are numbers ina higher ring than algebraic integers, where the object ring is alsocomplete, whereas the ring of algebraic integers is awed.I didnt realize early on that mathematicians had such a aw as theproblem I found with algebraic integers, but when it was clear thatthe ring was incomplete, I went ahead and found the complete ring,which I call the object ring, as is my right as discoverer.Some of you may not understand or still mayfind it hard to believethat the ring is incomplete, so Im including the short argument,which proves that it is. Ill likely be including this argument inseveral posts, as there really is no more room for discussion, so Imready to put a sock in it.It is mathematics, after all.Ill use that result that I talked about before where here Ill have afactor g of the polynomial P(m) where P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f).Its in a special form so that I can factor it as P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)where the question is, how does f divide out?Clearly in the rst expression P(m) is a multiple of f^2.Well, a_1 a_2 a_3 = f^2(m^3 f^4 - 3m^2 f^2 + 3m), so at m=0, a_1 a_2 a_3 = 0which means that at least one of the as must equal 0.So picking a_1 as one that goes to 0, I pick for my g, g = a_1 x + uf, so at m=0, g = uf, so c=uf, and r=g-c, so in general r = a_1 x.You may wonder why mathematicians would argue against that g=r+c. Well, heres why.Going back to P(m), I notice that P(0)/f^2 = 3x u^2 + u^3 f = u^2(3x + uf)so if 3 and x are coprime to f, I have that P(0)/f^2 is coprime to f.More simply, you can look at it and see there are no more free fs todivide off, as the only remaining f you can see is blocked by 3x.So as g is a factor of P(m), dividing off f^2 divides off somethingfrom g, so lets call it w. Then I have g/w = r/w + c/w = a_1 x/w + uf/wand I know that g/w is a factor of P(m)/f^2, and I can just let m=0,again, and I get g/w = f/w, and as the constant term P(0)/f^2 = u^2(3x + uf)is coprime to f, I know that w = f.I know some want that w to be a factor of m, but consider g/w(m) = a_1 x/w(m) + uf/w(m) = a_1 x/w(m) + uf/w(0)where from before I know that w(0) = f, so I have a_1 x/w(m) + uf/w(m) = a_1 x/w(m) + uand subtracting a_1 x/w(m) from both sides gives uf/w(m) = u, so f = w(m), proving that w(m) equals f for all m.Its that simple. Now you can also see why mathematicians would argueagainst g=r+c if theyre ghting an FLT proof, as if they admit itstrue, its all over for them, as theres no more room to argue.So whats the big deal? The big deal is that with g=r+c, c is*constant* and because its a constant, you have that g/f = a_1 x/f + uis the new factor for all m, and not just m=0.So you see, if mathematicians admit that a factor g of a polynomialP(m) is g=r+c, splitting the factor up into a constant and potentiallyvarying portion, then they are admitting that I am correct.The battle is being fought out partly here on Usenet, on newsgroupswhere presumably people pride themselves on their rationality, theirknowledge, and their ability to determine the truth.How you all behave says a lot about what really goes on inside ofpeoples heads, all over the world. You could consider it a hugereality show, and a test unlike any other in human history that wasimpossible to give before the arrival of the Internet and Usenet.Worldwide networks made this test possible.I bid you all what may be your rst real welcome to the brave, newworld.James Harris =Your denition of object ring is still circular, thus your whole FLT proof is invalid.Once again you choose not to address this issue.Simple math question - does your object ring contain sqrt(2)?I bet you dont know. =You see, my short proof of Fermats Last Theorem is awless [...]>Great! If so, then why the hell dont you try to get it published insome journal?F. => Your denition of object ring is still circular, thus your whole FLT proof is invalid.> Once again you choose not to address this issue.Simple math question - does your object ring contain sqrt(2)?> I bet you dont know.Tut, tut. Dont you JSH does not answer questions?Gib =This could be the beginning of a beautiful relationship.Gib =I agree, and I also salute James even if I have read his postswheres hes quite explicit the immortal fame, money and interviewwith Opera are what hes after..You can review my writings www.adamskingdom.com anytime Ann.Sequel arriving soon,HercThis could be the beginning of a beautiful relationship.Gib> =......Simplicio: 2+2=5Salviati: 2+2=4Simplicio: 2+2=5Salviati: 2+2=4Simplicio: 2+2=5Salviati: 2+2=4Simplicio: 2+2=5Salviati: 2+2=4Simplicio: 2+2=5Salviati: 2+2=4Simplicio: 2+2=5Salviati: 2+2=4Simplicio: 2+2=5Salviati: 2+2=4Simplicio: 2+2=5Salviati: 2+2=4.........Salviati: 2+2=4Simplicio: 2+2=5Salviati: 2+2=4Simplicio: 2+2=5Salviati: 2+2=4Simplicio: 2+2=5Salviati: 2+2=4Simplicio: 2+2=5Salviati: 2+2=4Simplicio: 2+2=5Salviati: 2+2=4Simplicio: 2+2=5Salviati: 2+2=4Simplicio: 2+2=5Salviati: 2+2=4Simplicio: 2+2=5...... => Do you know about the noble character (irrational temper) of> Newton? Indeed, at the end he managed to impose his truths on us for> some centuries ahead (in Leibniz and other cases - by pulling the> strings of the Royal Society of which he was a president), but how did> that make him happy (a word that may very well be different from> fame:)? Nervous breakdowns were following one after the other. In> the meanwhile, he learned the bible by heart and parrot it by rote,> alas, couldnt decipher the coded Number. Something like that follows> inexorably when we lose touch with the virtuous mean (balance) between> No-ambition and Over-ambition. Even Aristotle had to learn it the hard> way, although it was him who formulated this rule of middle ground> happiness.is one of my favorite things to write about how Newton spent most of histime guring out numerology of the bible, you have any sources for moreon this?HercI like your style of self condence and tolerance, it is the opposite> of dogmatic, it gives freedom to yourself (and the people). Honestly, I> see no reason why should you be disappointed. Imagine for a moment that> your proof of Fermats theorem is awless from every possible point> of view (really:), isnt that enough to make you happy? If I were you> Id be the happiest person in the world (and on the Moon:). Or you> crave for some kind of social recognition, loads of money, or immortal> fame? Or maybe you desire the sweet intellectual property rights?> Luckily, your letters tell me otherwise.Should it make you anxious if not everyone accepts your truth (words)?> Do we need to impose our dream on everyone? What for? How does it> matter whether the nal proof will be ve or thousand pages long?> (asuming you are not an environmentalist preserving the trees:) You> probably know that what kind of proofs are deemed acceptable> depends on the point in space-time, it is a convention which changes> and grows in complexity to keep pace with the expansion of the universe> (of words in our mind:). I think the point of your exercise was to gain> condence in your abilities and to know thyself. And you did quite> well here :)One might say :) that I am speaking from a similar experience here. In> the course of the discussions in another newsgroup> (alt.dreams.castaneda, my home base:) I also had the chance to develop> a theory of my own. A strange one I admit but it made me happy. And I> did it spontaneously, in the last six months I took part in many> discussions (some of them were real battles; Seraph: You do not truly> of pages until I found the key, my key to the way out of the matrix> (labyrinth of reason).That is, I became an expert in word-crushing - that mysterious effect> of a word becoming less real in our mind and consciousness. Again the> universal laws of attraction and repulsion (from any Word) hold true:> adopting sequentially those two different points of view (good and> bad) on the word make it invariably shake in our mind until we reach> a fancy state of sober detachment and an unexpected feeling of freedom.Anotherfinding was that spontaneous improvisation is one certain way> to beat reason, some of the treatises (i.e. posts ranging from one to> annoying typos I can still read them with joy, if we are honest (with> oursouls) there is nothing to be ashamed of later.But do you think I will seek recognition or claim property rights? Or> money, success and personal prot? I couldnt care less about those> reasonable distractions (brakes), I did what I did as part of my> quest for the point of life (other abstract goal may also do the job)> and those writings I do not perceive as mine but as a product of> discussions, a joint work.That general theory of words (which I sub-consciously advertise> here:) helped me (the Sturdy Beggar) cleanse my sub-consciousness and> to free mysoul from the real chains of some unreal polluters.Words, basically. The most curious part is that after I found the point> of reason I followed he advice of Einstein and started asking> questions. And without noticing I crushed all the major theories in> my mind, i.e. they became considerably less real. You are probably 100%> and more convinced that the genes, viruses, our gradual evolution> from gases (to name just a few) are absolutely real and observable> facts, but I have found it was not so. The one certain thing I have> found them to be was words (or theories, i.e. galaxies of words).Ill spare you the details of that disappointment, and only tell you> one of my greatest realizations: modern doctors are no healers but> long-Latin-words memorisers and masters of words. Dark words usually,> they rst make us hear the sound of inevitability, the rest is done> by our true belief. Perhaps, you know about the incredibly high> positive correlation between poor health (some call it chronic> disease:) and the visits to the doctor? Are you sure which way the> direction of causality goes?More visits (doctors, medical books and TV information) lead to poorer> health or vice versa? Or is it an equivalence? Visits >Diseases or> Illnesses >Visits? Or is it more like:Visits==>Disease==>Visit==>Chronic Disease==>Visit==>... words as splinters in our mind, can it be? Take it as a joke, a> serious one :) What matters to me is that the novel approach and other>findings of the theory of words helped me develop what I may call> ability for independent thinking. It seems a trivial quality (or an> inherited character trait:) but I have discovered a systematic way to> develop it. I found my key (which is by no means universal) and for> that I needed only choice, imagination, sobriety and laughter, loads of> it.Basically, I believe that if Aristotle had known the word laughter or> just nished his alleged book on comedy (the funny lm the name of> the rose) he would have found the answer to the question and> probably revoked some of his scholastic writings. Just as I did, I> found the splinter in my mind that was driving me mad. And I didnt> stop there :) On my way to Everest I formulated (discovered or merely> reinvented the wheel:) some jolly rules of the Word and laws of> thermodynamic human nature. Like Know thyself rst (and not start> with an escape in Nature outside:). The funniest of them was that: by> stalking others we eventually stalk ourselves. Another one was the> axiom of choice and its most general version: our personal preferences> for the words.For example, nervous doctors describe some psychic phenomena as> brain states and neural networks, while we can, with the same success,> call them states of Buddha consciousness or Castanedian positions> (shifts) of the assemblage point. These are freely interchangeable> concepts aimed at explaining everything, even the magic perception of> reality (trance, dreams and many other real oddities). If we distance> ourselves a bit we may realize instead that they are just words that> are not so easy to prove rigorously. Unless we cut a living brain and> stick the white disk electrodes and would it be enough? Even then we> may not see the networks, though we may try to imagine them, that> usually works.It is sometimes called a theoretical proof of existence, like the> invisible rest of the super-stringed 10-dimensional universe in the> physical theory of everything, that teeny-weeny point to which the> initially six-dimensional sub-universe has supposedly shrunk. So is> this innitesimal dot real or not? Does it exist, is it there or is it> not?Both, say the masters of words, but it is a complex mystery, it is> there but you cannotfind it, its absolutely invisible, by assumption> and by denition (in other words like a god). So catch it if you can.> The theory is truly untestable, you see, but nevertheless true :) Why?> Because we know it, it has to be so. Moreover :), the formulas t> beautifully, therefore it cant be otherwise, the theoretical proof has> to be believed (when delivered, of course, for now they seem to be> waiting for something, like Newton, they are still in search of the> mathematics powerful enough to solve the equations).Even more, that they cannot make a ea is immaterial to the beautiful> argument of the origin of the universe and species. The hard evidence> might be missing but we have to believe, because there isnt any better> alternative (explanation) around. I didntfind it convincing and found> an alternative explanation that works for me, probably you have found> another one and I am sure there exist many private answers (separate> Creation realities). It now seems plain to me that that theory ought to be vacated in> Mark Twain - The Lowest Animal>As you might have guessed even in this group the ght is one of words> and for words; which camp will impose its words on the rest of the> gullible hearts and minds. This is a strange consequence of the theory> of evolution: when we believe it we perceive (see) the world through> the eyes of the predator, everything is a threat: other lower and> less favoured races and species, invisible viruses, rival thought,> you name it. It is part of the culture of fear. And quite> naturally scientists feel obliged to ght for the selection and> survival of their ttest dogma.Evolution is the law of policies: Darwin said it, Socrates endorsed> it, Cuvier proved it and established it for all time in his paper on> The Survival of the Fittest. These are illustrious names, this is a> mighty doctrine: nothing can ever remove it from its rm base, nothing> dissolve it, but evolution. Mark TwainEvolution, Morpheus, evolution. Like the dinosaur. A sad law of> policies for I think everyone has the right to defend the matrix of> words in his or her mind. We protect the words we like, a basic human> right of freedom of speech, thought and religion (belief). When I> noticed that spiritual and religious WORDS were constantly under attack> I decided it was a good idea to give spirituality a hand.In a way it is a choice for balance, I can hardly share the passionate> fanaticism with which some quasi-scientists seek to eradicate> spiritualism altogether: from schools, textbooks, press, news and> everyday life. I compare such a strategy to the Inquisition, it is like> a mirror image of it although the methods of the imposition of the> dogma have become much more rened and humane (e.g. compulsory> secular education, college admission tests, worshipping the> technological progress and machines in the media and ...). Whereas> the truth is that spiritualism and any other non-scientic thought has> the right to exist, it is as simple as that. It is called tolerance.I describe the tyrannical nature as anyone who tries to impose on> others his matrix of words (beliefs, thoughts, knowledge) by force.> I wouldnt like to be part of it, would you. On the other hand, I see> nothing wrong if you try to publish your proof somewhere, but do you> really need it? Youve done it already in this group. That you are> faced with criticism is to be expected (we all have our aws:),> especially in a science.skeptics group. Or is it> skeptical.scientists, I am not sure :)Id like to share a revelation that Ive head during my time here:> nobody cares about my theory too, but how do I care. Just like you I> learned a lot about human nature (mine:), and most of all about human> ego. Of course, we can choose to react differently (unhealthy) to the> criticism and become something like an angry scientist.Do you know about the noble character (irrational temper) of> Newton? Indeed, at the end he managed to impose his truths on us for> some centuries ahead (in Leibniz and other cases - by pulling the> strings of the Royal Society of which he was a president), but how did> that make him happy (a word that may very well be different from> fame:)? Nervous breakdowns were following one after the other. In> the meanwhile, he learned the bible by heart and parrot it by rote,> alas, couldnt decipher the coded Number. Something like that follows> inexorably when we lose touch with the virtuous mean (balance) between> No-ambition and Over-ambition. Even Aristotle had to learn it the hard> way, although it was him who formulated this rule of middle ground> happiness.There are many truths (even in the certain universe of mathematics)> and we are free to believe according to our choice, the story ends, we> wake up in our bed and believe whatever we want to believe. We may> even choose to believe that the Word is no God, i.e. not Good. And> not bad either :), but something and somewhat in between. What and> where?Best,> Ann>I remember a post where someone was wondering how I could keep> claiming that Im right with all these people posting disagreement,> and it seems to me that maybe explaining how might help.You see, my short proof of Fermats Last Theorem is awless with all> the disputes from mathematicians attacking the following statement:Given a factor g of a polynomial P(x), g=r+c, where c is a factor of> the constant term P(0) of the polynomial, given by the value of g at> x=0, and r=g-c.Thats it.Now if people were arguing with you trying to attack that when you> know its truth proves your case that you have a short proof of> Fermats Last Theorem, would you believe them?With polynomial factors I can show the statement easily enough as, for> instance, g = x+2 is a factor of x^2 + 3x + 2, and here r=x, and c=2,> and of course, at x=0, g=2. With non-polynomial factors, it gets a> little harder, but not impossible, to show. Still, why should I have> to give examples, when its so easy to prove?In the face of my short proof of Fermats Last Theorem, posters have> not disagreed with the truth of the statement with polynomial factors,> but have tried to cast doubt, disagree, or just ignore it for factors> in general.Remember a polynomial factor is something like x+2, where you have x> with a positive *integer* degree. Another example of a polynomial> factor is sqrt(2)x^2 + 3x + 7^{1/3}. And again, its a polynomial> because x has a positive *integer* degree. I generalized to a factor> of a polynomial.All I did was make a generalized statement about *any* factor of a> polynomial, not just polynomial factors, and proved it, as its easy> enough to do.So yes, I have absolute certainty that my proof of FLT is correct> because the linchpin is that statement above.When the mathematical proof of the statement is accepted, there is no> room for disagreement with my work.Now for those of you who are logical and rational people, such a> statement should seem easily checkable.Those of you who understand political realities might also understand> why mathematicians might try their best to ght a short proof of> Fermats Last Theorem found by an outsider like myself, even when they> can be shown to be arguing against math and logic itself, with such a> simple statement.Its about power. Its politics. Mathematicians have a society,> which they control. If they acknowledge the truth, it shakes up their> society, so mostly they ignore me, while a few make posts attacking.For me its just a weird experience to add to the rest, but it also> tells me a lot about how the real world works, and how easily people> can be convinced to disbelieve in things that should be obvious.My guess is that many of you believe theres some higher math or> some special rule, or incredibly hard to understand math thing that> makes it not true that given a factor g of a polynomial P(x), g=r+c,> where c is a factor of the constant term, given by the value of g at> P(0), and r=g-c.You may simply dismiss me without even paying attention to the> statement, or you may look at it and simply decide that you dont know> enough, that surely mathematicians couldnt ght such a thing, and if> they did, how could they get away with it?So I can watch you. And I can see what you believe in, really. I can> see how you think, and how its possible to control even large,> supposedly skeptical populations.Yup, you can learn a lot about people with a little math.>James Harris =>I remember a post where someone was wondering how I could keep>claiming that Im right with all these people posting disagreement,>and it seems to me that maybe explaining how might help.You see, my short proof of Fermats Last Theorem is awless with all>the disputes from mathematicians attacking the following statement:Given a factor g of a polynomial P(x), g=r+c, where c is a factor of>the constant term P(0) of the polynomial, given by the value of g at>x=0, and r=g-c.Thats it.Wrong on two counts. First, this is far from the only objectionpeople have. Second, nobody has objected to the statementabove - the objections have been to other versions of thatstatement, which were either false or so vaguely worded asto have an indeterminate truth value.I kinda suspect you left something out, because thestatement Given a factor g of a polynomial P(x), g=r+c, where c is a factor ofthe constant term P(0) of the polynomial, given by the value of g atx=0, and r=g-c.is utterly obvious (Pf: Let c = 1 and r = g - 1. QED.)You meant to assert a little more about c and r, right?>Now if people were arguing with you trying to attack that when you>know its truth proves your case that you have a short proof of>Fermats Last Theorem, would you believe them?With polynomial factors I can show the statement easily enough as, for>instance, g = x+2 is a factor of x^2 + 3x + 2, and here r=x, and c=2,>and of course, at x=0, g=2. With non-polynomial factors, it gets a>little harder, but not impossible, to show. Still, why should I have>to give examples, when its so easy to prove?In the face of my short proof of Fermats Last Theorem, posters have>not disagreed with the truth of the statement with polynomial factors,>but have tried to cast doubt, disagree, or just ignore it for factors>in general.Remember a polynomial factor is something like x+2, where you have x>with a positive *integer* degree. Another example of a polynomial>factor is sqrt(2)x^2 + 3x + 7^{1/3}. And again, its a polynomial>because x has a positive *integer* degree. I generalized to a factor>of a polynomial.All I did was make a generalized statement about *any* factor of a>polynomial, not just polynomial factors, and proved it, as its easy>enough to do.So yes, I have absolute certainty that my proof of FLT is correct>because the linchpin is that statement above.When the mathematical proof of the statement is accepted, there is no>room for disagreement with my work.Now for those of you who are logical and rational people, such a>statement should seem easily checkable.Those of you who understand political realities might also understand>why mathematicians might try their best to ght a short proof of>Fermats Last Theorem found by an outsider like myself, even when they>can be shown to be arguing against math and logic itself, with such a>simple statement.Its about power. Its politics. Mathematicians have a society,>which they control. If they acknowledge the truth, it shakes up their>society, so mostly they ignore me, while a few make posts attacking.For me its just a weird experience to add to the rest, but it also>tells me a lot about how the real world works, and how easily people>can be convinced to disbelieve in things that should be obvious.My guess is that many of you believe theres some higher math or>some special rule, or incredibly hard to understand math thing that>makes it not true that given a factor g of a polynomial P(x), g=r+c,>where c is a factor of the constant term, given by the value of g at>P(0), and r=g-c.You may simply dismiss me without even paying attention to the>statement, or you may look at it and simply decide that you dont know>enough, that surely mathematicians couldnt ght such a thing, and if>they did, how could they get away with it?So I can watch you. And I can see what you believe in, really. I can>see how you think, and how its possible to control even large,>supposedly skeptical populations.Yup, you can learn a lot about people with a little math.>James Harris************************David C. Ullrich =James Harris spewed forth in messageSo I can watch you. And I can see what you believe in, really. I can> see how you think, and how its possible to control even large,> supposedly skeptical populations.Yup, you can learn a lot about people with a little math.>James HarrisFor a person who SO loathes the mathematical community, you sure spend a lotof time and energy trying to get in!! Youre such a loser!Heres a hint of your pathetic nature: if youre listed in crank.com, yourea loser!!!!~Bhuvan => I remember a post where someone was wondering how I could keep> claiming that Im right with all these people posting disagreement,> and it seems to me that maybe explaining how might help.You see, my short proof of Fermats Last Theorem is awless with all> the disputes from mathematicians attacking the following statement:Given a factor g of a polynomial P(x), g=r+c, where c is a factor of> the constant term P(0) of the polynomial, given by the value of g at> x=0, and r=g-c.> No one is attacking this statement. Just what comes *after* it.> So yes, I have absolute certainty that my proof of FLT is correct> because the linchpin is that statement above.> When no one argues with your lynchpin, thats probably not where the perceived problem is. Go back and look at the counter-arguments.-- Will Twentyman =I remember a post where someone was wondering how I could keep> claiming that Im right with all these people posting disagreement,> and it seems to me that maybe explaining how might help.You see, my short proof of Fermats Last Theorem is awless with all> the disputes from mathematicians attacking the following statement:Given a factor g of a polynomial P(x), g=r+c, where c is a factor of> the constant term P(0) of the polynomial, given by the value of g at> x=0, and r=g-c.Thats it.Cant your statement be generalized as follows: Given a factor or non-factor g of a polynomial P(x), g=r+c,> where c is a factor of the constant term P(0) of the polynomial,> given by the value of g at x=0, and r=g-c.Nope. Its important that g is a factor of the polynomial. Itactually is an interesting result, though apparently subtle that givena factor g of a polynomial P(x), g=r+c, where c=g at x=0, and r=g-c,so r varies if g varies.I say its subtle as a lot of mathematicians have questioned me inthat area alone, including Barry Mazur of Harvard, interestinglyenough.The idea of separating out a factor of a polynomial based on anon-varying and varying element seems to test even top mathematiciansfor some peculiar reason. > That is, Given g and a polynomial P(x), then g=r+c,> where c is a factor of the constant term P(0) of the polynomial,> given by the value of g at x=0, and r=g-c.-- Bill HaleNope. Its important that g is a factor of the polynomial. Heresthe argument that connects everything together.Ill use the statement above except here Ill have a factor g of thepolynomial P(m) where P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f).The polynomial is in a special form so that I can factor it as P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)where the question is, how does f divide out?Clearly in the rst expression P(m) is a multiple of f^2.Well, a_1 a_2 a_3 = f^2(m^3 f^4 - 3m^2 f^2 + 3m), so at m=0, a_1 a_2 a_3 = 0which means that at least one of the as must equal 0.So picking a_1 as one that goes to 0, I pick for my g, g = a_1 x + uf, so at m=0, g = uf, so c=uf, and r=g-c, so in general r = a_1 x.You may wonder why mathematicians would argue against that g=r+c. Well, heres why.Going back to P(m), I notice that P(0)/f^2 = 3x u^2 + u^3 f = u^2(3x + uf)so if 3 and x are coprime to f, I have that P(0)/f^2 is coprime to f.More simply, you can look at it and see there are no more free fs todivide off, as the only remaining f you can see is blocked by 3x.So as g is a factor of P(m), dividing off f^2 divides off somethingfrom g, so lets call it w. Then I have g/w = r/w + c/w = a_1 x/w + uf/wand I know that g/w is a factor of P(m)/f^2, and I can just let m=0,again, and I get g/w = f/w, and as the constant term P(0)/f^2 = u^2(3x + uf)is coprime to f, I know that w = f.I know some want that w to be a factor of m, so Ill let the focusrest there for the moment.Basically Im facing people who spend a lot of effort to distract, andadd in side issues to try and cast doubt on the simple argument I gaveabove.Ultimately though, their position has to rest on the assumption of adependency on m for w. If I can get them to admit that then maybeprogress can be made.Its that simple. Now you can also see why mathematicians would argueagainst g=r+c if theyre ghting an FLT proof, as if they admit itstrue, its all over for them, as then they have to argue that aconstant factor of a polynomial actually has an m dependency.So whats the big deal? The big deal is that with g=r+c, c is*constant* and because its a constant, you have that g/f = a_1 x/f + uis the new factor for all m, and not just m=0.So you see, if mathematicians admit that a factor g of a polynomialP(m) is g=r+c, splitting the factor up into a constant and potentiallyvarying portion, then they are admitting that I am correct.James Harris =it is like a riddle for me. First I will assume that your witty remarkwas ironical, i.e. a genuine request for more information. Hence,welcome to the real facts (but in fact again words:).> Do you know about the noble character (irrational temper) of> Newton? Indeed, at the end he managed to impose his truths on us for> some centuries ahead (in Leibniz and other cases - by pulling the> strings of the Royal Society of which he was a president), but how did> that make him happy (a word that may very well be different from> fame:)? Nervous breakdowns were following one after the other. In> the meanwhile, he learned the bible by heart and parrot it by rote,> alas, couldnt decipher the coded Number. Something like that follows> inexorably when we lose touch with the virtuous mean (balance) between> No-ambition and Over-ambition. Even Aristotle had to learn it the hard> way, although it was him who formulated this rule of middle ground> happiness.is one of my favorite things to write about how Newton spent most of his> time guring out numerology of the bible, you have any sources for more> on this?HercWell, perhaps you would like to test me whether I could type Newtonbiography or Newton bible in a search engine. There we are, I willprove to you that I can, and shortly you will read the documentedevidence. Isnt it surprising that my intuition (jolly smile here:) wasright; yet does it qualify for knowledge?First Ill spend a couple of words on his happy life and toleranttemper, for I believe that the many things we think we are in life and theroles we play shouldnt be taken in isolation. Some positive psychologistsmay try to convince us to divide oursouls into different lives(professional, private, religious), supposedly that should make our lifeeasier. An illusion, these are intimately related and soon Ill deliverthe proof :)Then Ill get to your point and address the (unpublished) project of hislifetime - the study of the ancient texts. Note that by some randomcoincidence he was uent in Hebrew, Greek and Latin, a rare occurrencein the universe of mathematics and physics :) TRINITY: Dodge this!Moreover, Newtons writings on theological and biblical subjects aloneamount to about 1.3 million words, the equivalent of 20 of todaysstandard length books. My comments will be either normal text(unindented) or inside curly braces (or both). Life & Character Isaac Newton was born prematurely on Christmas day 1642 (4 January 1643, New Style) in Woolsthorpe... Much has been made of Newtons posthumous birth, his prolonged:) separation from his mother, and his unrivaled hatred of his stepfather. Until Hanna returned to Woolsthorpe in 1653 after the death of her second husband, Newton was denied his mothers attention, a possible clue to his complex character. Newtons childhood was anything but happy, and throughout his life he:) verged on emotional collapse, occasionally falling into violent > and vindictive attacks against friend and foe alike.{What do we learn from here: hatred, complex character, unhappychildhood, life verging on emotional collapse, violent,vindictive.Later we will encounter the words pleasure; money; emotional breakdown;furious; consumptive hatred; severe nervous disorder; deranged letters;discomposure in head, or mind, or both; enjoyed power and worldly success;he played it to his personal advantage; tyrannical and autocratic; hiscontrol over the lives and careers of younger disciples was all butabsolute; marshaled all the forces at his command; secretly; dominate;without rival; threatening to burn his mother and father; fame andrecognition; fear of criticism; his aim to humiliate Hooke in public wasabnormal; depression; mental illness; rage; irrational temper; the mostfearful, cautious and suspicious temper that his assistant has ever knew} But the turning point in Newtons life came in June 1661 when he left Woolsthorpe for Cambridge University. Here Newton entered a new world, one he could eventually call his own. {the dream ofanother mad philosopher, an extrovert escape from himself in Natureoutside. Regretfully we cannot hide from oursoul anywhere in cosmos,Aristotle tricked us here with the philosophy of science assumption ondependent and objective reality outside. While we dont know, the otherway round may also be worth exploring, i.e. starting instead from knowthyself and the dependent, subjective reality within. This was the nobleidea of Platos philosophic nature, rst we cure ourselves and then theindependent universe.}http://web.clas.u.edu/users/rhatch/pages/01- Courses/current-courses/08sr-newton.htm How did Newton view his accomplishments in light of his belief in appear to the world, but to myself I seem to have been only like a boy, playing on the seashore and diverting myself in now and then nding a smoother pebble or prettier seashell than ordinary, while the great ocean of truth lay all undiscovered before me. http://www.biblecodedigest.com/page.php/74A joking remark: { ...one of the strongest motives that lead men to art and science is escape from everyday life with its painful crudity and hopeless dreariness, from the fetters of ones own ever-shifting desires. A nely tempered nature longs to escape from the personal life into the world of objective perception and thought. Albert Einstein} In 1678, Newton suffered a serious emotional breakdown, and in the following year his mother died. Newtons response was to cut off contact with others and engross himself in alchemical research. These studies, once an embarrassment to Newton scholars, were not misguided musings but rigorous investigations into the hidden forces of nature. Newtons later insights in celestial mechanics can be traced in part to his alchemical interests {nearly three decades of alchemical research}. By combining action-at-a-distance and mathematics, Newton transformed the mechanical philosophy by:) adding a mysterious but no less measurable quantity, gravitational force.{Surprise, surprise but later another master of words showed that in factthere was no force, it was not the spoon that bent but only ourself,this mysterious but no less measurable quantity turned out an illusionof our imperfect eyesight, a byproduct of the bending space-time.} ... Newton was so furious with Hooke that he threatened to suppress Book III of the Principia altogether, nally denouncing science as an impertinently litigious lady. Newton calmed down and nally consented to publication. But instead of acknowledging:) Hookes contribution Newton systematically deleted every possible > mention of Hookes name. Newtons hatred for Hooke was consumptive. > In 1693, however, Newton suffered a severe nervous disorder {madness-t}, not unlike his breakdown of 1677-1678. The cause is open to interpretation... We only know Locke and Samuel Pepys received strange and seemingly deranged letters {Id love to read them, wouldnt you} that prompted concern for Newtons discomposure in head, or mind, or both. Whatever the cause, shortly after his recovery Newton sought a new position in London. During his London years Newton enjoyed power and worldly success...Newton was elected president of the Royal Society and was annually reelected until his death... He was knighted in 1705. {Dodge this:} Although his creative years had passed, Newton continued to exercise a profound inuence on the development of science. In:) > effect, the Royal Society was Newtons instrument, and he played it to his personal advantage. autocratic, and his control over the lives and careers of younger > disciples was all but absolute. Newton could not abide contradiction or controversy - his quarrels with Hooke provide singular examples. But in later disputes, as > president of the Royal Society, Newton marshaled all the forces at:) his command. For example, he published Flamsteeds astronomical observations - the labor of a lifetime - without the authors permission; and in his priority dispute with Leibniz concerning!!! the calculus, Newton enlisted younger men to ght his war of > words,{Let me repeat the last statement: TO FIGHT HIS WAR OF WORDS. Is it soimpossible that all those scientic and religious battles between mastersof words (including recent wars) have this single purpose: control overour credulous hearts and minds, ght for the gullible public opinion?Media, TV and propaganda wars that pre-determine the nal (real:)outcome. Notice, not necessarily a ght for the divine, independent,objective truth but for the private truth (knowledge) of say, Newton -tyrant absolute.} Newton enlisted younger men to ght his war of words, while behind the lines he secretly directed charge and countercharge. In the end, the actions of the Society were little more than:) extensions of Newtons will, and until his death he dominated the landscape of science without rival.http://web.clas.u.edu/users/rhatch/pages/01-Courses/ current-courses/08sr-newton.htm When examining his sins at age nineteen, Isaac listed:- > Threatening my father and mother Smith to burn them and the house over them.{A noble (Nobel) character and if he were to threaten me that way I willalso concede. To avoid his sound of inevitability I may accept to seeanything he tells me, the force of gravity, the uxions, theinnitesimal points, anything the dictator likes. Another piece of evidence comes from Isaacs list of sins referred to above. He lists one of his sins as:- > ... setting my heart on money, learning, and pleasure more than Thee ... which tells us that Isaac must have had a passion for learning. {not about himself, though, but reality outside, a prudent and safe strategy that eradicates nasty risk. Why vivisect ourselves when there are so many independent animals outside - test objects for the progress of mankind and divine research} He headed the text with a Latin statement meaning Plato is my friend, Aristotle is my friend, but my best friend is truth showing himself a free thinker from an early stage.{so far so good, the question is the attitude, what we do with thisindependent and free thinking. Do we use it for money, pleasure,success, fame and personal prot, do we try to impose it by force oneverybody else? Maybe, if we are very ambitious and nervous.} He was always pulled in two directions, there was something in his nature which wanted fame and recognition yet another side of him:) feared criticism and the easiest way to avoid being criticised was to publish nothing. Certainly one could say that his reaction to criticism was irrational, and certainly his aim to humiliate Hooke in public because of his opinions was abnormal. However, perhaps because of Newtons already high reputation, his corpuscular theory reigned until the wave theory was revived in the 19th century. Newton was at the height of his standing - seen as a leader of the university and one of the most eminent mathematicians in the world. After suffering a second nervous breakdown in 1693, Newton retired from research. The reasons for this breakdown have been discussed by his biographers and many theories have been proposed: chemical poisoning as a result of his alchemy experiments; frustration with his researches; the ending of a personal friendship with Fatio de Duillier, a Swiss-born mathematician resident in London; and:)==> problems resulting from his religious beliefs. Newton himself blamed lack of sleep but this was almost certainly:) :) a symptom of the illness rather than the cause of it. There seems little reason to suppose that the illness was anything other than > depression, a mental illness he must have suffered from throughout most of his life, perhaps made worse by some of the events we have just listed. {A depressed but very rich man, his youth goal of setting his heart on money was achieved with great success} Newton decided to leave Cambridge to take up a government position in London becoming Warden of the Royal Mint in 1696 and Master in 1699. However, he did not resign his positions at Cambridge until 1701. As Master of the Mint, adding the income from his estates, > we see that Newton became a very rich man. Given the rage that Newton had shown throughout his life when criticised, it is not surprising that he ew into an irrational > temper directed against Leibniz. ... We have given details of this controversy in Leibnizs biography is worth relating here is how Newton used his position as President of the Royal Society.:) In this capacity he appointed an impartial committee to decide the ofcial report of the committee (although of course it did:) :) not appear under his name) which was published by the Royal appeared in the Philosophical Transactions of the Royal Society. {Cunning, wasnt he.} Newtons assistant Whiston had seen his rage at rst hand. He Newton was of the most fearful, cautious and suspicious temper that I ever knew. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/ Newton.html Newton a theologist? Sir Isaac Newton and the Bible. By Professor Arthur B. Anderson. prophecies and predictions, subjects which had always been of interest to him. In addition to his scientic work (Newton would have said as a part of his scientic work), he devoted a substantial portion of his enormous energy to the study of the Bible and Biblical texts and history. He read the Bible daily throughout his life and Isaac Newton believed that the Bible is literally true in every respect. Throughout his life, he continually tested Biblical truth against the physical truths of experimental and theoretical science. He never observed a contradiction. In fact, he viewed his own scientic work as a method by which to reinforce belief in Biblical truth. He was a formidable Biblical scholar, was uent in the ancient languages, and had extensive knowledge of ancient history. He believed that each person should read the Bible, and through that reading, establish for himself an understanding of the universal truth it contains. Only one book of Newtons about the Bible was ever published. In 1733, six years after his death, J. Darby and T. Browne, published Observations Upon the Prophecies of Daniel and the Apocalypse of St. John. With his prodigious knowledge of ancient history and languages and his unequaled mental powers, Isaac Newton is the best qualied individual in this millennium to have written about the:) 12 and continued to be a special interest throughout his indicates that he, himself, is in awe of the words he has been given an opportunity to read. Isaac Newton concluded that it is intended that Revelation will be understood by very few until near the end of history, the time of judgment, and the beginning of the everlasting kingdom of the{Note that I am merely quoting Newtons wisdom, it doesnt mean I like anykind of dark and apocalyptic prophesies, which by the way, scientistsmake all the time: the inevitable death one day of the Earth, the Sun, thestars and a bunch of other hypotheses. This is part of the culture offear, to threaten our soul with a ludicrous thought scenario and preventus from thinking, i.e. accept all the other words that follow. Back toNewton:} This prophecy is called the Revelation, with respect to the Scripture of Truth, which Daniel was commanded to shut up and seal, till the time of the end. Daniel sealed it until the time of the end, and until that time comes, the Lamb is opening the seals: and afterward the two Witnesses prophesy out of it a long time in sackcloth, before they ascend up to heaven in a cloud. All of which is as much as to say, that the prophecies of Daniel and John should not be understood till the time of the end: but that some should prophesy out of it in an a'cted and mournful state for a long time, and that but darkly, so as to convert but few. But in the very end, the Prophecy should be so far interpreted so as to convince many. Then saith Daniel, many shall run to and fro, and knowledge shall be increased. For the Gospel must rst be preached in all nations before the great tribulation, and end of the world. The palm- bearing multitude, which came out of this great tribulation, cannot be innumerable out of all nations unless they be made so by the preaching of the Gospel before it comes. There must be a stone cut of the mountain without hands, before it can fall on the toes of the Image, and become a great mountain and ll the earth. An Angel must y through the midst of heaven with the everlasting Gospel to preach to all nations, before Babylon falls, and the Son of man reaps his harvest. The two prophets must ascend up to heaven in a cloud, before the kingdoms of this world become the kingdoms of Christ. Tis therefore a part of this Prophecy, that it should not be understood before the last age of the world; and therefore it makes for the credit of the Prophecy, that it is not yet understood. But if the last age, the age of opening these things, be now approaching, as by the great success of late Interpreters it seems to be, we have more encouragement that ever to look into these things. If the general preaching of the Gospel be approaching, it is for us and our posterity that these words mainly belong: In the time of the end the wise shall understand, but none of the wicked shall understand. Blessed is he that readeth, and they that hear the words of this Prophecy, and keep those things that are written therein (Daniel XII 4,10, Apoc. i 3). {And note the necessary pathos at the end, scientic books have similar inspiring conclusions that have greatest effect on a childs soul.} In conclusion: Sir Isaac Newton was totally correct in his Observations. If the greatest scientist who ever lived had no > problem believing the Bible, what excuse will evolutionists, atheists, agnostics, or other so called men of science have on Judgment Day!! http://www.reformation.org/newton.html Isaac Newton was a Christian who studied the Bible daily and believed that God created everything, including the Bible. He believed that the Bible was true in every respect. Throughout his life he continually tested biblical truth against the physical truths of experimental and theoretical science and never observed a contradiction, according to his many biographers. Newtons writings reected his belief that his scientic work was a method by which to reinforce belief in biblical truth. After he completed his monumental Philosophiae Naturalis Principia Mathematica, he began to devote more and more of his time to researching the Bible, eventually writing a book he believed unlocked the prophecies contained in Daniel and Revelation, two Bible books he viewed as intertwined. http://www.biblecodedigest.com/page.php/74like what we are doing now. Actually, I have yet to gure out what isthis urge to publish. Is it for the sweet intellectual property rights? Isit real? Look at me for example, a letter consisting mostly of othersquotes, what a shame :) Is there such a thing as intellectualcreativity? The secret to creativity is knowing how to hide your sources. Albert EinsteinBy the way, do you know his sources?Note that the one religious book that was published appeared six yearsafter his death, I guess he didnt want to damage his scienticreputation. The next Islamic quote shows that he was deep into the detailsof the subject matter :) SIR ISAAC NEWTON ON THE BIBLE corruption of the text of the New Testament concerning I John 5:7 Notable Corruptions of Scripture. Due to the prevailing environment against criticism, he felt it unwise to profess his beliefs openly and felt that printing it in England would be too dangerous. Newton sent a copy of this manuscript to John Locke requesting him to have it translated into French for publication in France. Two years later, Newton was informed of an attempt to publish a Latin translation of it anonymously. However, Newton did not approve of its availability in Latin and persuaded Locke to take steps to prevent this publication. http://cyberistan.org/islamic/newton1.html Newtons writings on theological andbiblical subjects alone amount to about 1.3 million words, the equivalentof 20 of todays standard length books. Other Researches. Throughout his career Newton conducted research in theology and history with the same passion that he pursued alchemy and science. Although some historians have neglected Newtons nonscientic writings, there is little doubt of his devotion to these subjects, as his manuscripts amply attest. Newtons writings > on theological and biblical subjects alone amount to about 1.3 million words, the equivalent of 20 of todays standard length books. Although these writings say little about Newtonian science,:) :) they tell us a good deal about Isaac Newton.did it, actually, many of his beliefs were our knowledge for a coupleof gullible centuries after his death.} Newtons research outside of science--in theology, prophecy, and > unite knowledge and belief, to reconcile the Book of Nature with the Book of Scripture. But for all the elegance of his thought and the boldness of his quest, the riddle of Isaac Newton remained. In the end, Newton is as much an enigma to us as he was, no doubt, to himself.http://web.clas.u.edu/users/rhatch/pages/01-Courses/ current-courses/08sr-newton.htmAnd here is another great man economist John Maynard Keynes (pronouncedCanes) who seems to have read those million words of his geniuspredecessor. According to him Newton regarded the entire universe(including the Bible) as a cryptogram. Two hundred years later, when Keynes became provost of Cambridge, he discovered the box and spent years going over the papers, which he estimated contained more than a million words about Newtons Bible research... Here is what Keynes had to say about Newton ... ... (Newton) looked on the whole universe and all that is in it as a riddle, as a secret which could be read by applying pure thought to certain evidence, certain mystic clues which God had laid about the world to allow a sort of philosophers treasure hunt to the esoteric brotherhood. He believed that these clues were to be found partly in the evidence of the heavens and in the constitution of elements (and that is what gives the false suggestion of his being an experimental natural philosopher), but also partly in certain papers and traditions handed down by the brethren in an unbroken chain back to the original cryptic revelation in Babylonia. He regarded the universe as a cryptogram set by the Almighty--just as he himself wrapt the discovery of the calculus in a cryptogram when he communicated with Leibnitz. By pure thought, by concentration of mind, the riddle, he believed, would be revealed to the initiate. He did read the riddle of the heavens. And he believed that by the same powers of his introspective imagination he would read the riddle of the Godhead, the riddle of past and future events divinely fore-ordained, the riddle of the elements and their constitution from an original undifferentiated rst matter, the riddle of health and of immortality. All would be revealed to him if only he could persevere to the end, uninterrupted, by himself, no one coming into the room, reading, copying, testing - all by himself, no interruption for Gods sake, no disclosure, no discordant breakings in or criticism, with fear and shrinking as he assailed these half-ordained, half-forbidden things, creeping back into the bosom of the Godhead as into his mothers womb. Voyaging through strange seas of thought alone,:) not as Charles Lamb, a fellow who believed nothing unless it:) :) was as clear as the three sides of a triangle. http://www.biblecodedigest.com/page.php/74And what did Keynes himself believe in. Well, he was a practical man whowanted us all dead in the long run. It is the highest good that allour greedy economies grow indenitely (and simultaneously) until the endof innity (or the burst of the bubble:). Another funny man:http://www.u-turn.net/2-4/keynes.htmlAnyway, from the above paragraphs it seems Newton had an interesting dream,he regarded the universe as a cryptogram (a matrix, maybe:). Let uscompare it with Einsteins universe, his introvert task was to freehimself from this prison, this optical delusion of our consciousness: A human being is a part of a whole, called by us _universe_, a part limited in time and space. He experiences himself, his thoughts and feelings as something separated from the rest... a kind of optical delusion of his consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Our task must be to free ourselves from this prison by widening our circle of compassion to embrace all living creatures and the whole of nature in its beauty. Albert EinsteinBest,Ann => I remember a post where someone was wondering how I could keep> claiming that Im right with all these people posting disagreement,> and it seems to me that maybe explaining how might help.You see, my short proof of Fermats Last Theorem is awless with all> the disputes from mathematicians attacking the following statement:Given a factor g of a polynomial P(x), g=r+c, where c is a factor of> the constant term P(0) of the polynomial, given by the value of g at> x=0, and r=g-c.> No one is attacking this statement. Just what comes *after* it.Given acceptance of the intriguing little result that given a factor gof a polynomial P(x), g=r+c, where c is a factor of the constant termP(0), given by c=g at x=0, and r=g-c, the proof follows easily enough. > So yes, I have absolute certainty that my proof of FLT is correct> because the linchpin is that statement above.> When no one argues with your lynchpin, thats probably not where the > perceived problem is. Go back and look at the counter-arguments.What Ive seen are repeated attempts at distractions.Ive given the math argument. If its wrong, people canfind an errorin it.Heres yet another look with a slight variation on the argument, withan emphasis on that portion where posters try to assert a false mdependency on key constants.Consider P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f.Now using b_1, b_2, b_3, w_1, w_2, and w_3, I have the factorization P(m)/f^2 = (b_1 x + u w_1)(b_2 x + u w_2)(b_3 x + u w_3)where w_1 w_2 w_3 = f, and b_1 b_2 b_3 = (m^3 f^4 - 3m^2 f^2 + 3m),and at m=0 P(0)/f^2 = 3xu^2 + u^3 f = u^2(3x + uf), so two of the bs must equal 0, which means P(0)/f^2 = w_1 w_2 u^2 (b_3 x + u w_3)which is P(0)/f^2 = u^2 (b_3 w_1 w_2 x + u f) = u^2(3x + uf)proving that w_1 w_2 must equal 1, as f is coprime to 3 from before,which leaves b_3 = 3.Essentially objections now come down to claiming that the ws aredependent on m, but consider that w_1 w_2 = 1, when m=0, here where s coprime to 3.But that was an arbitrary choice *I* made, so let f=3.Now w_1 w_2 = 3^{2/3} as long as m is coprime to 3, WITHOUT REGARD TOm.So those posters who try to convince you that the ws are actuallydependent on m, like being functions of m, must now also convince youthat the ws make a decision, rst looking to see if f=3 or have somenon-unit factor in common with 3, and THEN they decide if theyredependent on m.People can wafe trying to gure out who they are, but mathematicsis logical, which is why Ive emphasized that posters are acting on*social* not mathematical reasons.If they follow mathematical logic, then they face a social result thatso far has been more important to them in worrying about than thewonder of having a short proof of Fermats Last Theorem nallyavailable.So for social reasons they attack algebra itself, and many of you havebeen convinced. Which says a lot about you, and how you think.James Harris =I like your style of self condence and tolerance, it is the opposite> of dogmatic, it gives freedom to yourself (and the people). Honestly, I> see no reason why should you be disappointed. Imagine for a moment that> your proof of Fermats theorem is awless from every possible point> of view (really:), isnt that enough to make you happy? If I were you> Id be the happiest person in the world (and on the Moon:). Or you> crave for some kind of social recognition, loads of money, or immortal> fame? Or maybe you desire the sweet intellectual property rights?> Luckily, your letters tell me otherwise.No need to imagine it. It is we who are awed.You are right not to impugn his motives, however.> Should it make you anxious if not everyone accepts your truth (words)?> Do we need to impose our dream on everyone? What for? How does it> matter whether the nal proof will be ve or thousand pages long?> (asuming you are not an environmentalist preserving the trees:) You> probably know that what kind of proofs are deemed acceptable> depends on the point in space-time, it is a convention which changes> and grows in complexity to keep pace with the expansion of the universe> (of words in our mind:). I think the point of your exercise was to gain> condence in your abilities and to know thyself. And you did quite> well here :)When you say accepts your truth, you imply a subjective truth. Althoughsome truths may be subjective, this one is not: Jamess proof, indeed allof his thoughts that Ive ever encountered, are sublime, profound, objectiveTruths.And they are beautiful. This is more subjective, perhaps, but in Jamesswritings we see what are, according to St. Thomas Aquinas, the threeessential characteristics of beauty: integritas, consonantia, et claritas(wholeness, harmony, and radiance). But, to allude to another saint, weonly see them as through a glass darkly.> One might say :) that I am speaking from a similar experience here. In> the course of the discussions in another newsgroup> (alt.dreams.castaneda, my home base:) I also had the chance to develop> a theory of my own. A strange one I admit but it made me happy. And I> did it spontaneously, in the last six months I took part in many> discussions (some of them were real battles; Seraph: You do not truly> of pages until I found the key, my key to the way out of the matrix> (labyrinth of reason).James is misunderstood here because he is so out of place, so far abovethis rabble of mathematicians -- like a Nagual in New Jersey, despisedand feared but in tune with powers beyond suburban comprehension.> That is, I became an expert in word-crushing - that mysterious effect> of a word becoming less real in our mind and consciousness. Again the> universal laws of attraction and repulsion (from any Word) hold true:> adopting sequentially those two different points of view (good and> bad) on the word make it invariably shake in our mind until we reach> a fancy state of sober detachment and an unexpected feeling of freedom.Yes. The words are the problem. Jamess thoughts have the precision ofa laser, but to communicate with the masses he has to translate them intoour illogical, nebulous language. It must pain him like the sin of theworld pains God.> Anotherfinding was that spontaneous improvisation is one certain way> to beat reason, some of the treatises (i.e. posts ranging from one to> annoying typos I can still read them with joy, if we are honest (with> oursouls) there is nothing to be ashamed of later.James, too, is prolic in trying to pin down the ineffable with the bluntbeanbags of words.Im not trying to illuminate anything so complex, so Ill state it simply:God is here on sci.math. And his name is James Harris.-- | Jim Ferry | Center for Simulation |++ of Advanced Rockets || http://www.uiuc.edu/ph/www/jferry/ ++| jferry@[delete_this]uiuc.edu | University of Illinois | => I remember a post where someone was wondering how I could keep> claiming that Im right with all these people posting disagreement,> and it seems to me that maybe explaining how might help.You see, my short proof of Fermats Last Theorem is awless with all> the disputes from mathematicians attacking the following statement:Given a factor g of a polynomial P(x), g=r+c, where c is a factor of> the constant term P(0) of the polynomial, given by the value of g at> x=0, and r=g-c.Thats it.Not true.Lots of people pointed that your denition of objects is curcular,> which obviously makes the whole proof invalid.You never addressed those objections. Will you address them now?> I bet you wont.> Because when you see objects mentioned in a reply, you go suddenly blind.> In my original post you can see the linchpin of my proof of Fermats> Last Theorem. Theres no excuse that Im ignored because Im sure by> now many of you have noticed that not only am I not ignored, but there> are people who at times try to make objections that at least sound> mathematical. Here you see mention of objects which are numbers in> a higher ring than algebraic integers, where the object ring is also> complete, whereas the ring of algebraic integers is awed.I didnt realize early on that mathematicians had such a aw as the> problem I found with algebraic integers, but when it was clear that> the ring was incomplete, I went ahead and found the complete ring,> which I call the object ring, as is my right as discoverer.Some of you may not understand or still mayfind it hard to believe> that the ring is incomplete, so Im including the short argument,> which proves that it is. Ill likely be including this argument in> several posts, as there really is no more room for discussion, so Im> ready to put a sock in it.It is mathematics, after all.Ill use that result that I talked about before where here Ill have a> factor g of the polynomial P(m) where P(m) = f^2((m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f).Its in a special form so that I can factor it as P(m) = (a_1 x + uf)(a_2 x + uf)(a_3 x + uf)where the question is, how does f divide out?Clearly in the rst expression P(m) is a multiple of f^2.Well, a_1 a_2 a_3 = f^2(m^3 f^4 - 3m^2 f^2 + 3m), so at m=0, a_1 a_2 a_3 = 0which means that at least one of the as must equal 0.> Yes, this is OK when m = 0.> So picking a_1 as one that goes to 0, I pick for my g, g = a_1 x + uf, so at m=0, g = uf, so c=uf, and r=g-c, so in general r = a_1 x.> Oddly enough I do not disagree with this either.> You may wonder why mathematicians would argue against that g=r+c. Well, heres why.Going back to P(m), I notice that P(0)/f^2 = 3x u^2 + u^3 f = u^2(3x + uf)so if 3 and x are coprime to f, I have that P(0)/f^2 is coprime to f.More simply, you can look at it and see there are no more free fs to> divide off, as the only remaining f you can see is blocked by 3x.So as g is a factor of P(m), dividing off f^2 divides off something> from g, so lets call it w. Then I have g/w = r/w + c/w = a_1 x/w + uf/wand I know that g/w is a factor of P(m)/f^2, and I can just let m=0,> again, and I get g/w = f/w, and as the constant term P(0)/f^2 = u^2(3x + uf)is coprime to f, I know that w = f.I know some want that w to be a factor of m, but consider g/w(m) = a_1 x/w(m) + uf/w(m) = a_1 x/w(m) + uf/w(0)> Oh jesus christ. I see what youre thinking. You are thinking that uf/w(0) = c/w(0) is the constant term.Oh, man! You have gotten your variables confused. The constantc is the constant term *with respect to the variable x*.Thinking of g as a function of x, c = g(0). But w is not a function of x. It is a function of m.Thus w(0) means w(m = 0), not w(x = 0). There is noreason whatsoever for that term uf/w(0)in your expression above, when you are referring to m <> 0. In fact g is actually a function of two variables, mand x. Or you could think of g as a family of rst-degreepolynomials in the variable x, indexed by the variable m: g(x, m) = a1(m)* + u*f.The constant term c = u*f for all values of x and m. g(x, m)/w(m) = (a1(m)/w(m)) * x + u*f/w(m),and g(0, m) = u*f/w(m).This last equation is the important one. For m <> 0, you CANNOT say it is g(0, m)/w(m) = u*f/w(0)as you did above. The thing on the right is u*f/w(m). Am I getting through to you on this? g/w(m) = a_1 x/w(m) + uf/w(m) = a_1 x/w(m) + uf/w(0) You did *not* substitute 0 in for m everyplace, only inthat last term. Nor should you have done so. You wereright to leave it as w(m) in the term a_1 x/w(m). Youmistake was in putting in w(0) into that last term. Again:w = w(m) is a *function of m*, NOT a *function of x*. Bluntly put, you are making an understandable but pretty dumb mistake.> where from before I know that w(0) = f, True, but when m <> 0, there is no reasonto write u*f/w(0). The constant term is g(x = 0),not g(m = 0).> so I have a_1 x/w(m) + uf/w(m) = a_1 x/w(m) + u> No - it just doesnt work that way. See above.> and subtracting a_1 x/w(m) from both sides gives uf/w(m) = u, so f = w(m), proving that w(m) equals f for all m.> No - you have confused the two variables, x and m, andbefuddled yourself regarding the constant term - in generalyou have no reason to replace uf/w(m) by uf/w(0).> Its that simple. Now you can also see why mathematicians would argue> against g=r+c if theyre ghting an FLT proof, as if they admit its> true, its all over for them, as theres no more room to argue.So whats the big deal? The big deal is that with g=r+c, c is> *constant* and because its a constant, you have that> The part about g = r + c, your lemma, is not the issue atall. That is a triviality. You have misapplied it here byconfusing the variable m with the polynomial variable x, andgetting mixed up regarding the constant term.> g/f = a_1 x/f + uis the new factor for all m, and not just m=0.> No - it doesnt follow, as noted above.> So you see, if mathematicians admit that a factor g of a polynomial> P(m) is g=r+c, splitting the factor up into a constant and potentially> varying portion, then they are admitting that I am correct.> You are right about g = r + c, but that part is trivialanyway. The rest of what you have is incorrect as explainedabove.> The battle is being fought out partly here on Usenet, on newsgroups> where presumably people pride themselves on their rationality, their> knowledge, and their ability to determine the truth.> Actually I am encouraged that you have at least nally recognized that this part of your argument is a problem. Beforeyou were just waving your hands and saying that since thefactorization was of a certain form when m = 0, it must be of that same form when m <> 0. Your present argument is anattempt to bridge the gap. Obviously it fails. It is academic anyway. There are proofs that your mainconclusion is wrong. These are clear, rigorous proofs andyou have not refuted any of them at all. What this means is thatnot only does your central argument have an error, as described above, but also that IT CANNOT BE FIXED. Here as a reminder is one of several proofs: polynomial of the form P(x) = (v^3 + 1)*x^3 - 3*v*x*(u*f)^2 + (u*f)^3,where v = -1 + m*f^2, and m, u, and f are integers,with f prime and m coprime to f, then P(x)/f^2 can be factored in the form P(x)/f^2 = (b1*x + u)*(b2*x + u)*(b3*x + u*f) [1]where b1, b2, and b3 are algebraic integers.I say not. Let m = 1, f = 5, and u = 1. Thenv = 24, and v^3 + 1 = 13825 = 25*553. It is easily veried that P(x)/f^2 = 553*x^3 - 72*x + 5.If this is factored in the form [1] as claimed byHarris, then -u/b1 = -1/b1 is a root of Q(x) = P(x)/f^2. That is, Q(-1/b1) = 553*(-1/b1)^3 - 72*(-1/b1) + 5 = 0.Multiply through by b1^3: 5*b1^3 + 72*b1^2 - 553 = 0.The expression on the left is a *non-monic*polynomial in b1 with integer coefcients,and it is *irreducible* over the rationals.Therefore b1 cannot be an algebraic integer.Therefore your claim is false. about what really goes on inside of> peoples heads, all over the world. You could consider it a huge> reality show, and a test unlike any other in human history that was> impossible to give before the arrival of the Internet and Usenet.Worldwide networks made this test possible.I bid you all what may be your rst real welcome to the brave, new> world.> I could do without the gratuitous ponticating, especially in view of the rather dumb mistake you have made in your post. Nora B.James Harris =rock and hollered:I like your style of self condence and tolerance, it is the opposite>of dogmatic, it gives freedom to yourself (and the people). Honestly, I>see no reason why should you be disappointed. Imagine for a moment that>your proof of Fermats theorem is awless from every possible point>of view (really:), isnt that enough to make you happy? If I were you>Id be the happiest person in the world (and on the Moon:). Or you>crave for some kind of social recognition, loads of money, or immortal>fame? Or maybe you desire the sweet intellectual property rights?>Luckily, your letters tell me otherwise.1. Messages on Usenet are called *posts*, not letters.2. Three-hundred-line posts are seldom, if ever, read completely.3. Its polite to indicate who said what in a much more complete fashion 4. James Harris is, was, and probly always will be a fruitcake. HTH.(followups set: sci.skeptic)-- No collection of individuals is less vindictive than an audience at amateur theatricals. - P. G. Wodehouse, _The Intrusion of Jimmy_ =Words are like leaves, and where they most abound Much fruit of sense beneath is rarely found.Alexander Pope =And the wise men dont know how it fee-ee-ee-ee-ee-ee-eels To be thick As a brickIan Anderson (Jethro Tull) =the numerology is evident under ordinary analysis.This is done (one way) by analysing notable people in our society ratherthan characters of the bible though.Like what coincidence is it that Ronald Raegun introduced thestar wars program, a man called Ray Gun?And why is Hawking, a king!! our smartest? What coincidence is it?Why is Tiger Woods the best golfer?What did the famous person named Di do?The most famous billionaires name is Bill?I have a statistical test that does a bottom gure for the accumulationof coincidences at www.adamskingdom.comI may redo the test so it shows more prominent examples over a widertime frame..... anyway... and when my proof nally is veried, they wont just be words anymore.Hercit is like a riddle for me. First I will assume that your witty remark> was ironical, i.e. a genuine request for more information. Hence,> welcome to the real facts (but in fact again words:).>Do you know about the noble character (irrational temper) of> Newton? Indeed, at the end he managed to impose his truths on us for> some centuries ahead (in Leibniz and other cases - by pulling the> strings of the Royal Society of which he was a president), but how did> that make him happy (a word that may very well be different from> fame:)? Nervous breakdowns were following one after the other. In> the meanwhile, he learned the bible by heart and parrot it by rote,> alas, couldnt decipher the coded Number. Something like that follows> inexorably when we lose touch with the virtuous mean (balance) between> No-ambition and Over-ambition. Even Aristotle had to learn it the hard> way, although it was him who formulated this rule of middle ground> happiness.is one of my favorite things to write about how Newton spent most of his> time guring out numerology of the bible, you have any sources for more> on this?HercWell, perhaps you would like to test me whether I could type Newton> biography or Newton bible in a search engine. There we are, I will> prove to you that I can, and shortly you will read the documented> evidence. Isnt it surprising that my intuition (jolly smile here:) was> right; yet does it qualify for knowledge?First Ill spend a couple of words on his happy life and tolerant> temper, for I believe that the many things we think we are in life and the> roles we play shouldnt be taken in isolation. Some positive psychologists> may try to convince us to divide oursouls into different lives> (professional, private, religious), supposedly that should make our life> easier. An illusion, these are intimately related and soon Ill deliver> the proof :)Then Ill get to your point and address the (unpublished) project of his> lifetime - the study of the ancient texts. Note that by some random> coincidence he was uent in Hebrew, Greek and Latin, a rare occurrence> in the universe of mathematics and physics :) TRINITY: Dodge this!Moreover, Newtons writings on theological and biblical subjects alone> amount to about 1.3 million words, the equivalent of 20 of todays> standard length books. My comments will be either normal text> (unindented) or inside curly braces (or both). Life & Character Isaac Newton was born prematurely on Christmas> day 1642 (4 January 1643, New Style) in Woolsthorpe... Much has been made of Newtons posthumous birth, his prolonged> :) separation from his mother, and his unrivaled hatred of his> stepfather. Until Hanna returned to Woolsthorpe in 1653 after the> death of her second husband, Newton was denied his mothers> attention, a possible clue to his complex character. Newtons> childhood was anything but happy, and throughout his life he> :) verged on emotional collapse, occasionally falling into violent and vindictive attacks against friend and foe alike.{What do we learn from here: hatred, complex character, unhappy> childhood, life verging on emotional collapse, violent,> vindictive.Later we will encounter the words pleasure; money; emotional breakdown;> furious; consumptive hatred; severe nervous disorder; deranged letters;> discomposure in head, or mind, or both; enjoyed power and worldly success;> he played it to his personal advantage; tyrannical and autocratic; his> control over the lives and careers of younger disciples was all but> absolute; marshaled all the forces at his command; secretly; dominate;> without rival; threatening to burn his mother and father; fame and> recognition; fear of criticism; his aim to humiliate Hooke in public was> abnormal; depression; mental illness; rage; irrational temper; the most> fearful, cautious and suspicious temper that his assistant has ever knew} But the turning point in Newtons life came in June 1661 when he> left Woolsthorpe for Cambridge University. Here Newton entered a> new world, one he could eventually call his own. {the dream of> another mad philosopher, an extrovert escape from himself in Nature> outside. Regretfully we cannot hide from oursoul anywhere in cosmos,> Aristotle tricked us here with the philosophy of science assumption of> independent and objective reality outside. While we dont know, the other> way round may also be worth exploring, i.e. starting instead from know> thyself and the dependent, subjective reality within. This was the noble> idea of Platos philosophic nature, rst we cure ourselves and then the> independent universe.}http://web.clas.u.edu/users/rhatch/pages/01- Courses/current-courses/08sr-newton.htmHow did Newton view his accomplishments in light of his belief in> appear to the world, but to myself I seem to have been only like a> boy, playing on the seashore and diverting myself in now and then> nding a smoother pebble or prettier seashell than ordinary,> while the great ocean of truth lay all undiscovered before me.http://www.biblecodedigest.com/page.php/74A joking remark:> { ...one of the strongest motives that lead men to art and> science is escape from everyday life with its painful crudity and> hopeless dreariness, from the fetters of ones own ever-shifting> desires. A nely tempered nature longs to escape from the> personal life into the world of objective perception and thought.> Albert Einstein}> In 1678, Newton suffered a serious emotional breakdown, and in the> following year his mother died. Newtons response was to cut off> contact with others and engross himself in alchemical> research. These studies, once an embarrassment to Newton scholars,> were not misguided musings but rigorous investigations into the> hidden forces of nature.Newtons later insights in celestial mechanics can be traced in> part to his alchemical interests {nearly three decades of> alchemical research}. By combining action-at-a-distance and> mathematics, Newton transformed the mechanical philosophy by> :) adding a mysterious but no less measurable quantity, gravitational> force.{Surprise, surprise but later another master of words showed that in fact> there was no force, it was not the spoon that bent but only ourself,> this mysterious but no less measurable quantity turned out an illusion> of our imperfect eyesight, a byproduct of the bending space-time.}...Newton was so furious with Hooke that he threatened to suppress> Book III of the Principia altogether, nally denouncing science> as an impertinently litigious lady. Newton calmed down and> nally consented to publication. But instead of acknowledging> :) Hookes contribution Newton systematically deleted every possible mention of Hookes name. Newtons hatred for Hooke was> consumptive. > In 1693, however, Newton suffered a severe nervous disorder> {madness-t}, not unlike his breakdown of 1677-1678. The cause is> open to interpretation... We only know Locke and Samuel Pepys> received strange and seemingly deranged letters {Id love to read> them, wouldnt you} that prompted concern for Newtons> discomposure in head, or mind, or both.Whatever the cause, shortly after his recovery Newton sought a new> position in London.During his London years Newton enjoyed power and worldly> success...Newton was elected president of the Royal Society and> was annually reelected until his death... He was knighted in 1705.{Dodge this:}Although his creative years had passed, Newton continued to> exercise a profound inuence on the development of science. In> :) > effect, the Royal Society was Newtons instrument, and he played> it to his personal advantage.autocratic, and his control over the lives and careers of younger disciples was all but absolute.Newton could not abide contradiction or controversy - his quarrels> with Hooke provide singular examples. But in later disputes, as president of the Royal Society, Newton marshaled all the forces at> :) his command. For example, he published Flamsteeds astronomical> observations - the labor of a lifetime - without the authors> permission; and in his priority dispute with Leibniz concerning> !!! the calculus, Newton enlisted younger men to ght his war of words,{Let me repeat the last statement: TO FIGHT HIS WAR OF WORDS. Is it so> impossible that all those scientic and religious battles between masters> of words (including recent wars) have this single purpose: control over> our credulous hearts and minds, ght for the gullible public opinion?> Media, TV and propaganda wars that pre-determine the nal (real:)> outcome. Notice, not necessarily a ght for the divine, independent,> objective truth but for the private truth (knowledge) of say, Newton -> tyrant absolute.}Newton enlisted younger men to ght his war of words, while> behind the lines he secretly directed charge and countercharge. In> the end, the actions of the Society were little more than> :) extensions of Newtons will, and until his death he dominated the> landscape of science without rival.http://web.clas.u.edu/users/rhatch/pages/01-Courses/ current-courses/08sr-newton.htm> When examining his sins at age nineteen, Isaac listed:- > Threatening my father and mother Smith to burn them and the> house over them.{A noble (Nobel) character and if he were to threaten me that way I will> also concede. To avoid his sound of inevitability I may accept to see> anything he tells me, the force of gravity, the uxions, the> innitesimal points, anything the dictator likes. Another piece of evidence comes from Isaacs list of sins referred> to above. He lists one of his sins as:- > ... setting my heart on money, learning, and pleasure more than Thee ...which tells us that Isaac must have had a passion for> learning. {not about himself, though, but reality outside, a> prudent and safe strategy that eradicates nasty risk. Why vivisect> ourselves when there are so many independent animals outside -> test objects for the progress of mankind and divine research}He headed the text with a Latin statement meaning Plato is my> friend, Aristotle is my friend, but my best friend is truth> showing himself a free thinker from an early stage.{so far so good, the question is the attitude, what we do with this> independent and free thinking. Do we use it for money, pleasure,> success, fame and personal prot, do we try to impose it by force on> everybody else? Maybe, if we are very ambitious and nervous.}> He was always pulled in two directions, there was something in his> nature which wanted fame and recognition yet another side of him> :) feared criticism and the easiest way to avoid being criticised was> to publish nothing.Certainly one could say that his reaction to criticism was> irrational, and certainly his aim to humiliate Hooke in public> because of his opinions was abnormal. However, perhaps because of> Newtons already high reputation, his corpuscular theory reigned> until the wave theory was revived in the 19th century.Newton was at the height of his standing - seen as a leader of the> university and one of the most eminent mathematicians in the> world.After suffering a second nervous breakdown in 1693, Newton retired> from research. The reasons for this breakdown have been discussed> by his biographers and many theories have been proposed: chemical> poisoning as a result of his alchemy experiments; frustration with> his researches; the ending of a personal friendship with Fatio de> Duillier, a Swiss-born mathematician resident in London; and> :)==> problems resulting from his religious beliefs.Newton himself blamed lack of sleep but this was almost certainly> :) :) a symptom of the illness rather than the cause of it. There seems> little reason to suppose that the illness was anything other than depression, a mental illness he must have suffered from throughout> most of his life, perhaps made worse by some of the events we have> just listed. {A depressed but very rich man, his youth goal of setting his> heart on money was achieved with great success} Newton decided to leave Cambridge to take up a government position> in London becoming Warden of the Royal Mint in 1696 and Master in> 1699. However, he did not resign his positions at Cambridge until> 1701. As Master of the Mint, adding the income from his estates, we see that Newton became a very rich man. Given the rage that Newton had shown throughout his life when> criticised, it is not surprising that he ew into an irrational temper directed against Leibniz. ...We have given details of this controversy in Leibnizs biography> is worth relating here is how Newton used his position as> President of the Royal Society.:) In this capacity he appointed an impartial committee to decide> the ofcial report of the committee (although of course it did> :) :) not appear under his name) which was published by the Royal> appeared in the Philosophical Transactions of the Royal> Society. {Cunning, wasnt he.}Newtons assistant Whiston had seen his rage at rst hand. He Newton was of the most fearful, cautious and suspicious temper> that I ever knew.http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/ Newton.html> Newton a theologist? Sir Isaac Newton and the Bible.> By Professor Arthur B. Anderson. prophecies and predictions, subjects which had always been of> interest to him.In addition to his scientic work (Newton would have said as a> part of his scientic work), he devoted a substantial portion of> his enormous energy to the study of the Bible and Biblical texts> and history. He read the Bible daily throughout his life andIsaac Newton believed that the Bible is literally true in every> respect. Throughout his life, he continually tested Biblical> truth against the physical truths of experimental and theoretical> science. He never observed a contradiction. In fact, he viewed> his own scientic work as a method by which to reinforce belief> in Biblical truth.He was a formidable Biblical scholar, was uent in the ancient> languages, and had extensive knowledge of ancient history. He> believed that each person should read the Bible, and through that> reading, establish for himself an understanding of the universal> truth it contains. Only one book of Newtons about the Bible was ever published. In> 1733, six years after his death, J. Darby and T. Browne,> published Observations Upon the Prophecies of Daniel and the> Apocalypse of St. John.With his prodigious knowledge of ancient history and languages> and his unequaled mental powers, Isaac Newton is the best> qualied individual in this millennium to have written about the> :) 12 and continued to be a special interest throughout his> indicates that he, himself, is in awe of the words he has been> given an opportunity to read.Isaac Newton concluded that it is intended that Revelation will> be understood by very few until near the end of history, the time> of judgment, and the beginning of the everlasting kingdom of the{Note that I am merely quoting Newtons wisdom, it doesnt mean I like any> kind of dark and apocalyptic prophesies, which by the way, scientists> make all the time: the inevitable death one day hypotheses. This is part of the culture of> fear, to threaten our soul with a ludicrous thought scenario and prevent> us from thinking, i.e. accept all the other words that follow. Back to> Newton:} This prophecy is called the Revelation, with respect to the> Scripture of Truth, which Daniel was commanded to shut up and> seal, till the time of the end. Daniel sealed it until the> time of the end, and until that time comes, the Lamb is> opening the seals: and afterward the two Witnesses prophesy> out of it a long time in sackcloth, before they ascend up to> heaven in a cloud. All of which is as much as to say, that> the prophecies of Daniel and John should not be understood> till the time of the end: but that some should prophesy out> of it in an a'cted and mournful state for a long time, and> that but darkly, so as to convert but few. But in the very> end, the Prophecy should be so far interpreted so as to> convince many. Then saith Daniel, many shall run to and fro,> and knowledge shall be increased. For the Gospel must rst be preached in all nations before> the great tribulation, and end of the world. The palm-> bearing multitude, which came out of this great tribulation,> cannot be innumerable out of all nations unless they be made> so by the preaching of the Gospel before it comes. There must> be a stone cut of the mountain without hands, before it can> fall on the toes of the Image, and become a great mountain> and ll the earth. An Angel must y through the midst of> heaven with the everlasting Gospel to preach to all nations,> before Babylon falls, and the Son of man reaps his> harvest. The two prophets must ascend up to heaven in a> cloud, before the kingdoms of this world become the kingdoms> of Christ. Tis therefore a part of this Prophecy, that it should not be> understood before the last age of the world; and therefore it> makes for the credit of the Prophecy, that it is not yet> understood. But if the last age, the age of opening these> things, be now approaching, as by the great success of late> Interpreters it seems to be, we have more encouragement that> ever to look into these things. If the general preaching of the Gospel be approaching, it is> for us and our posterity that these words mainly belong: In> the time of the end the wise shall understand, but none of> the wicked shall understand. Blessed is he that readeth, and> they that hear the words of this Prophecy, and keep those> things that are written therein (Daniel XII 4,10, Apoc. i> 3). {And note the necessary pathos at the end, scientic books have> similar inspiring conclusions that have greatest effect on a> childs soul.} In conclusion: Sir Isaac Newton was totally correct in his> Observations. If the greatest scientist who ever lived had no problem believing the Bible, what excuse will evolutionists,> atheists, agnostics, or other so called men of science have on> Judgment Day!!> http://www.reformation.org/newton.html Isaac Newton was a Christian who studied the Bible daily and> believed that God created everything, including the Bible. He> believed that the Bible was true in every respect. Throughout his> life he continually tested biblical truth against the physical> truths of experimental and theoretical science and never observed> a contradiction, according to his many biographers. Newtons> writings reected his belief that his scientic work was a> method by which to reinforce belief in biblical truth. After he> completed his monumental Philosophiae Naturalis Principia> Mathematica, he began to devote more and more of his time to> researching the Bible, eventually writing a book he believed> unlocked the prophecies contained in Daniel and Revelation, two> Bible books he viewed as intertwined. http://www.biblecodedigest.com/page.php/74>like what we are doing now. Actually, I have yet to gure out what is> this urge to publish. Is it for the sweet intellectual property rights? Is> it real? Look at me for example, a letter consisting mostly of others> quotes, what a shame :) Is there such a thing as intellectual> creativity? The secret to creativity is knowing how to hide your sources.> Albert EinsteinBy the way, do you know his sources?Note that the one religious book that was published appeared six years> after his death, I guess he didnt want to damage his scientic> reputation. The next Islamic quote shows that he was deep into the details> of the subject matter :) SIR ISAAC NEWTON ON THE BIBLE corruption of the text of the New Testament concerning I John 5:7> Notable Corruptions of Scripture. Due to the prevailing> environment against criticism, he felt it unwise to profess his> beliefs openly and felt that printing it in England would be too> dangerous. Newton sent a copy of this manuscript to John Locke> requesting him to have it translated into French for publication> in France. Two years later, Newton was informed of an attempt to> publish a Latin translation of it anonymously. However, Newton did> not approve of its availability in Latin and persuaded Locke to> take steps to prevent this publication.http://cyberistan.org/islamic/newton1.html Newtons writings on theological and> biblical subjects alone amount to about 1.3 million words, the equivalent> of 20 of todays standard length books. Other Researches. Throughout his career Newton conducted research in theology and> history with the same passion that he pursued alchemy and> science. Although some historians have neglected Newtons> nonscientic writings, there is little doubt of his devotion to> these subjects, as his manuscripts amply attest. Newtons writings on theological and biblical subjects alone amount to about 1.3> million words, the equivalent of 20 of todays standard length> books. Although these writings say little about Newtonian science,> :) :) they tell us a good deal about Isaac Newton.did it, actually, many of his beliefs were our knowledge for a couple> of gullible centuries after his death.} Newtons research outside of science--in theology, prophecy, and unite knowledge and belief, to reconcile the Book of Nature with> the Book of Scripture. But for all the elegance of his thought and> the boldness of his quest, the riddle of Isaac Newton remained. In> the end, Newton is as much an enigma to us as he was, no doubt, to> himself.http://web.clas.u.edu/users/rhatch/pages/01-Courses /current-courses/08sr-newton.htmAnd here is another great man economist John Maynard Keynes (pronounced> Canes) who seems to have read those million words of his genius> predecessor. According to him Newton regarded the entire universe> (including the Bible) as a cryptogram. Two hundred years later, when Keynes became provost of Cambridge,> he discovered the box and spent years going over the papers, which> he estimated contained more than a million words about Newtons> Bible research... Here is what Keynes had to say about Newton ... ... (Newton) looked on the whole universe and all that is in it> as a riddle, as a secret which could be read by applying pure> thought to certain evidence, certain mystic clues which God had> laid about the world to allow a sort of philosophers treasure> hunt to the esoteric brotherhood. He believed that these clues> were to be found partly in the evidence of the heavens and in> the constitution of elements (and that is what gives the false> suggestion of his being an experimental natural philosopher),> but also partly in certain papers and traditions handed down by> the brethren in an unbroken chain back to the original cryptic> revelation in Babylonia. He regarded the universe as a> cryptogram set by the Almighty--just as he himself wrapt the> discovery of the calculus in a cryptogram when he communicated> with Leibnitz. By pure thought, by concentration of mind, the> riddle, he believed, would be revealed to the initiate. He did read the riddle of the heavens. And he believed that by> the same powers of his introspective imagination he would read> the riddle of the Godhead, the riddle of past and future events> divinely fore-ordained, the riddle of the elements and their> constitution from an original undifferentiated rst matter, the> riddle of health and of immortality. All would be revealed to> him if only he could persevere to the end, uninterrupted, by> himself, no one coming into the room, reading, copying,> testing - all by himself, no interruption for Gods sake, no> disclosure, no discordant breakings in or criticism, with fear> and shrinking as he assailed these half-ordained, half-forbidden> things, creeping back into the bosom of the Godhead as into his> mothers womb. Voyaging through strange seas of thought alone,> :) not as Charles Lamb, a fellow who believed nothing unless it> :) :) was as clear as the three sides of a triangle. http://www.biblecodedigest.com/page.php/74And what did Keynes himself believe in. Well, he was a practical man who> wanted us all dead in the long run. It is the highest good that all> our greedy economies grow indenitely (and simultaneously) until the end> of innity (or the burst of the bubble:). Another funny man:> http://www.u-turn.net/2-4/keynes.htmlAnyway, from the above paragraphs it seems Newton had an interesting dream,> he regarded the universe as a cryptogram (a matrix, maybe:). Let us> compare it with Einsteins universe, his introvert task was to free> himself from this prison, this optical delusion of our consciousness: A human being is a part of a whole, called by us _universe_, a part> limited in time and space. He experiences himself, his thoughts and> feelings as something separated from the rest... a kind of optical> delusion of his consciousness. This delusion is a kind of prison for> us, restricting us to our personal desires and to affection for a few> persons nearest to us. Our task must be to free ourselves from this> prison by widening our circle of compassion to embrace all living> creatures and the whole of nature in its beauty.> Albert EinsteinBest,> Ann> =some of my writing....************************************** INTRODUCTIONJennifers mating call was in her ofce, she blurtsout Im 21, Japanese phoneme for me. In a teaching tuteMichelle was bragging about always nishing work early andchatting and says I can do 2 things at once. A weeklater I gure out I can have any woman I choose and INext day at uni, shes with the in crowd, Im sure everyoneknows, but she sits by herself behind me very still. Sheknows I dont know if she knows.I notice how much shorter she is, but she has a betterV taper than me. Shes into phrenology and evolution,chemistry major, noticed her iq when she describes herher and get a date, at uni we notice each other in theJapanese for yes. In a micro teaching session werole play molecules and we end up holding a piece ofstring bond between our ngers, she notices my indexnger has the only trimmed ngernail and smiles, myrst reassurance of destiny. Weeks of calls and nodate, she has a 6 year boyfriend (shes only 22), lucky guyshes so sultry. Invites me to the nal day drinks, chatwith a few under my belt but she leaves, never saw heragain, she wore a ower in her hair, thought it was for me.Last month I moaned out aloud couple times at Jennifer justas thunder roars, hardly think of Michelle, yesterday afterhalf hour silence I whisper Michelle and a gust of wind shakesthe windows.INTRODUCTIONthe news media doesnt determine if you are a prodogy, it is apart of nature, you have to understand somewhat how fate works.If its my fate to marry someone, then you can deduce that nomatter what, in the mean time I cannot die. Does this make meimmortal, if I jump in front of a train will I survive? I wont but itdoes mean that it is impossible for me to make the decision tojump, for that is the event that would breach my fate. I cant tellyou for certain if you are a prodogy, but if it is your fate to seeme again then denying this will only further confound you, fatecannot be escaped. Hating me for a time was just part of yourfate. The news media reects whatever is in mymind or my recent events, my life runs a parallel to the eventsof the world, not just media, weather, people, machines, everything.Can you interpret media and the environment? This may be only myskill, but if I am right you share a focus of the worlds events and therewill be some train of stimulus effecting you to come to me.We are mortal like everyone we know, but now we cannot die, andwe the closest thing to being gods.Below is my introduction to a book, Im planning to compile my writingstogether, and ask my father to do a biography, not for a few years, Iwant my destiny to be realized rst. We are similar, both broad -great compliment, we are the two most intelligent of our species,can you deny our possibility, the future of our race? I want to love you.Most of all I want to hold you.This is my true story, a story about the reality of prophecy, the truthof fate. My fate or destiny, merely months away now, is a woman.These writings are more about the underpinnings of the universe, to makecredible a conventionally unscientic belief. As an introduction to themysterious workings of purpose in a random universe, I will attempta story, a story of how this woman and I met.First to introduce myself, a computer programmer and recreationalmathematician. This book is unlike any other, rst I am not a writerin the conventional sense, I ll notebook upon notebook with hazydiagrams and brief descriptions I later cannot decipher myself, second,as a mathematician all my writing serves a single purpose, to prove.Take my words with an open mind and your understanding of theworld will change, for I will show you that numerology is a science.Jennifer, my destiny, caught my attention with her name over twoyears ago when I met her in a realty shop, and read her businesscard, the name rhyming with a movie character I adored Jessica Six,the movie - Loguns Run. What has a movie to do with prophecy,indeed the prophecy of some couple? We are no ordinary couple butI will leave this detail for the body of this book. Every couplethat comes together does so as each of their fate, but the drivingforce bringing Jennifer and I together is of a new nature. You seeJennifer is very beautiful, I can barely type to write - she hasbright green eyes, and I am very intelligent.Only a few who have been close to me have insight to my intelligence,I shouldnt boast, there are other people with much superior mentalskills to mine in areas, detail of the nature of my mind will againbe detailed in course. The movie Loguns Run, a fantastic movie,refreshed my drive for monogamy, true love. There are many inuencesacting upon us each day that can actually affect the type of decisionswe make, like me deciding to visit the realty store again. Movies,songs, advertising, friends and strangers, everything in the worldcan play inuences, and anything in the world can play a part inprophecy. The movie Loguns Run, the characters and actors names,looks, birthdates, script, all run to a celestial clockwork, itwas my destiny to see the movie shortly before meeting Jennifer.It is difcult to explain several things together at once, Im notjust intelligent in that I get everything right, and Jennifer is not justbeautiful in that she is awless, like the shark that hasnt changed inmillions of years, we are the rst perfect people.It is difcult to understand fate, like the entire world is a directedplay, events come together with twists as if they were written beforehand. At certain times many things t into place, and you mustwonder how the contributors made these things t without some higherpurpose controlling their actions. I mentioned a movie charactersname that rhymed with Jennifers, there is an actress I saw yearsbefore meeting Jennifer who looks like her, the rst time I saw theactress on television I said Im going to marry her.Many people can accept fate as part of our world, but dont grasp thecomplexity involved. My life has had hundreds of occurrences whichall had to happen for my prophecy to be meaningful, seeing certainmovies, postcards, books, hearing certain songs, thinking certainthoughts at the right time, doing particular things, all complex and alllaid out for me before I was born. The end result of one act of destinyrelies on the myriad of happenings throughout ones life, and all theevents that one interacts with. This imposes a rigid view of the world,where every action is predetermined, perhaps there is some marginof freedom in our actions that dont affect our destinies.I have already lost most scientically minded readers with my ideasleading to divine purpose of furthering of our species, the acceptedview of random chemical, biological and social interaction responsiblefor life is complete and sound. I have some thoughts on the overallmethodology of science to address the need for a more open viewto the nature of the universe. There are many unresolved issues incurrent science, say you imagine a line from near your person andextend it out towards the sky into space. Where does the line endup? It goes past many stars and galaxies, then supposedly intoemptiness. Does the universe nish at a certain region, enclosedwithin nothing, or extend innitely? Most people imagine the bigbang as an event that manufactured matter into an empty universe.Thirteen billion years ago there was nothing, twelve billion years agoeverything was created. This isnt true, there was no thirteen billionyears ago, time itself, together with what we know as the universestarted with the big bang. We dont readily comprehend that beforethe big bang is meaningless because we live about a third of the wayinto the universes life, time may continue forever but supposedlyeverything will just be a random soup, life wont exist anyway. Theway everything in the universe behaves is different both at differenttimes and at different scales, say for a star compared to an atom.We dont know if the laws of physics will hold throughout time, theymay certainly be extended in future.The progress of science is analogous to deciphering instructions, thisimplies to me a period of construction, and even one day when we turnit on. For instance, extra sensory perception has never been proven,yet a moment may pass in the future that not only alters current scienticviews but alters our nature of existence, making esp conceivable. Therecould be growth points as part of the universes life affecting the natureof how things work.Most people believe in some non scientic view, a standardreligion or their own beliefs, yet any examination or test of these views,or indeed their audience by a skeptic refutes them. This makes meact of observation interrupts the experiment. Could not this andother laws of physics that work at micro scales manifest themselves intoour reality? The models of matter at small scales do not resembleour tangible world, bizarre in that they accurately predict whathappens, but described as not what is really there. Could our visibleworld also work in ways we can see but not comprehend?occur together at the same time, something we are unaccustomedto in our observable world. Perhaps if these occurrences manifestedat observable scale, one of my parallel thoughts could be the locationof a person, if it were our destiny to meet then this thought wouldbecome real. The bubble of non observation around an event couldexplode to include human comprehension, it may be in our lifetimesthat testing of things like espfinds it valid. The universe may bewinding up to one set of laws, and after a pivotal moment, unwindto another. I am not saying these things are true, merely that stringentviewpoints of science are not always correct, and certainly not complete.Scientists make principles from observations of their environment, it isnot mathematically sound for them to discount theories just because theydont t current theory. The line between fantasy and reality is blurred.Reaching a complete understanding of the universe involves theoriesof information, not matter. Our knowledge of events, plus facts thattranscend time, like a triangle always has three sides, the currentlyunexplainable fact of our sentience and feeling are all part of theuniverse. The fact that information now travels around our planet atthe speed of light is a part of the universe. Information takes manyforms, from a simple message in speech, to every activated neuronon our retina, to a symbolic natural formation like a river runninginto a sea, to the representation of a physical object.The foundation of information is the mere number, it ishard to dene information but it works as a kind of parallel.A complete theory of the universe would have to involve the parallelsor duality that occur in many ways: the non natural way small matterbehaves manifesting into reality, the environments solitary existenceinteracting with its manifestation in our minds, the parallels ofbroadcast ction leading reality and describing history, the creationof life from two lives coming together. The universe is inhabited withsocial creatures, does not thi 7ï WEcf !.854|èÿÿ car drives by of remarkable metallicgreen colour. There is a synchronized duality. If I ponder my gift androll two dice a seven appears, but I cant repeatedly roll sevens in frontof an audience, its not my fate to prove my sevenness. The astonishingukes of my life only occur for purpose, I can ick a piece of paperinto place with precision, but this is a coordination outside of my body.Ongoing examples dont prove anything, I will mention one last coincidenceas part of this introduction, it occurred on the seventh of the seventh,two thousand. I found an old CD that night, Reactivate 10 which mademe think of turning on my pursuit for Jennifer again, I should mentionJennifers number is 10. I repeatedly played a song, some people believein the 5th dimension, others think the sixth will show the way to theseventh... ecstasy, then I stopped when I realized it nished on theinitial of my surname - X T C. I noticed the time, three thirty, beinghalf of seven, then I slept.Sound plays an important role because prophecy is more powerful thanwe can imagine, its not just some predicted story, it borders on the beliefthat all of history is predetermined, and unraveling the past and future relieson interpretation of our language, for in a sense language is not made by man, justdiscovered. Having deciphered and written many computer programs, I amaccustomed to making up functional words, this has given me the abilityto decipher a fraction of spoken languages as being purposeful to thenature of the universe. I will not take the storm out of the body of my booknow, it will better explain the keys and secrets that await us.of meaning. In a purposeful universe words, names and symbolsare all linked in a complex network. Pythagarus didnt invent thecircle, most people can comprehend that it transcends time, butthe culmination of an entire language, the associated derivationsfrom people, places, animals and biology, chemistry, all jargonand all symbolic objects, is difcult to believe that it is more thana historically developed facet. Movie makers carefully make upnames to add a dimension to each character, the script is calculated,very carefully at certain points to ensure ow in each scene andeverything ties together at the nish. Could not the universe itselfplay out life, the story of the most advanced man, each sceneculminating into events to lead him to the most advanced woman?The climax would be the progress of life, something we all know toexist but not accommodated in any laws of physics.doesnt count for the existence of a concept, a desire.it is intricate, a structure of meaning and purpose thatdrives our physical world.Such a pity, the wooden axle never able to bear its required load, thetribe returned to the old technology, the proven rolling logs method,the spark of ingenuity destined to remain one mans dream. We livein a strange universe, we think of moving ourselves and other objectsand shortly after things happen, its a comfortable understanding, buthow does it work? Things didnt always happen, and things dont existwhen they are made. If the tribe failed in its invention, another tribe wouldtake its place, in a sense the wheel existed before it was made, andwill exist after they are all destroyed. We have one understanding ofourselves with our subcomposition, but what wefind arent things thatmove, physics describes matter like a magic show, vanishing, blurring,jumping, doubling. Time is the mystery, on one hand a universe witha beginning and ending of time, just a still object we ow through, andanother still world that we cant even touch, void of all events yetvisible, described as the platonic world, knowledge.But things do have a temporal existence, our reality the physical world,and there is only one universe, how it all blends together, this is my story. =Your denition of object ring is still circular, thus your whole FLT proof is invalid.> Once again you choose not to address this issue.Simple math question - does your object ring contain sqrt(2)?> I bet you dont know.Tut, tut. Dont you JSH does not answer questions?GibThe answer is that sqrt(2) is an object.What Ive seen are repeated attempts at distractions.Ive highlighted key methods with a math argument. If its wrongfindan error in it.I emphasize that the argument is *short* but the implications are hugefor mathematicians.Heres yet another look with a slight variation and an emphasis onthat portion where posters try to assert a false m dependency on keyconstants.Consider P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f.Now using b_1, b_2, b_3, w_1, w_2, and w_3, I have the factorization P(m)/f^2 = (b_1 x + u w_1)(b_2 x + u w_2)(b_3 x + u w_3)where w_1 w_2 w_3 = f, and b_1 b_2 b_3 = (m^3 f^4 - 3m^2 f^2 + 3m),and at m=0 P(0)/f^2 = 3xu^2 + u^3 f = u^2(3x + uf), so two of the bs must equal 0, which means P(0)/f^2 = w_1 w_2 u^2 (b_3 x + u w_3)which is P(0)/f^2 = u^2 (b_3 w_1 w_2 x + u f) = u^2(3x + uf)proving that w_1 w_2 must equal 1, as f is coprime to 3 from before,which leaves b_3 = 3.Essentially objections now come down to claiming that the ws aredependent on m, but consider that w_1 w_2 = 1, when m=0, here where s coprime to 3.But that was an arbitrary choice *I* made, so let f=3.Now w_1 w_2 = 3^{2/3} as long as m is coprime to 3, WITHOUT REGARD TOm.My guess is that certain mathematicians relied on the possibility thatpeople would read into my use of m=0 believing its a special case,but its unlikely *they* thought it was. They are mathematiciansafter all. What better people to fool others about mathematics, evenphysicists, than experts in mathematics, which is what mathematiciansare?So those posters who try to convince you that the ws are actuallydependent on m, like being functions of m, must now also convince youthat the ws make a decision, rst looking to see if f=3 or have somenon-unit factor in common with 3, and THEN they decide if theyredependent on m.People can wafe trying to gure out who they are, but mathematicsis logical, which is why Ive emphasized that posters are acting on*social* not mathematical reasons.James Harris => >I remember a post where someone was wondering how I could keep>claiming that Im right with all these people posting disagreement,>and it seems to me that maybe explaining how might help.You see, my short proof of Fermats Last Theorem is awless with all>the disputes from mathematicians attacking the following statement:Given a factor g of a polynomial P(x), g=r+c, where c is a factor of>the constant term P(0) of the polynomial, given by the value of g at>x=0, and r=g-c.>No one is attacking this statement. Just what comes *after* it.> Given acceptance of the intriguing little result that given a factor g> of a polynomial P(x), g=r+c, where c is a factor of the constant term> P(0), given by c=g at x=0, and r=g-c, the proof follows easily enough.And yet I keep failing to follow it, and when I explained why you keep failing to explain it, even though Ive repeatedly pointed out the post where I stated where I dont follow the proof.-- Will Twentyman => the numerology is evident under ordinary analysis.This is done (one way) by analysing notable people in our society rather> than characters of the bible though.Like what coincidence is it that Ronald Raegun introduced the> star wars program, a man called Ray Gun?And why is Hawking, a king!!Hes not a king.> our smartest? What coincidence is it?Zero. Why only apply the word king to the most well-knownperson in one particular subeld of physics? What aboutthe rest of the sciences and other elds of human endeavor?For instance...Why is Tiger Woods the best golfer?> You mean, why is Tiger Woods the king of golf? I dontknow, why? And how about before Tiger? And after? WasJack Nicklaus a failure because of the absence of Woodin his name? Wouldnt Wood also suggest he should bea carpenter or a woodcarver? What about other peoplenamed Woods? Are they all good golfers?> What did the famous person named Di do?So being called Di implies that youll die, and notbeing called Di implies that you wont? Cool! Imgolden!> The most famous billionaires name is Bill?Therefore you expect Bill Clinton and actor BillPullman to become billionaires? What do you make ofAdnan Khoshogi?Your numerology does serve a useful purpose in illustratingin a very clear way the fundamental logical fallacy innumerology (as well as most psychic phenomena, such asJohn Edwards talking to the dead): In a large sample pool,you canfind a lot of random hits. If you ignore the missesand only focus on the hits, your results show an amazingnumber of successes.Psychics such as Edwards succeed because there is a verystrong psychological effect of doing precisely that: rememberingthe hits and forgetting the misses. Ive read a number ofaccounts from people actually at a John Edwards show asopposed to watching the edited version on TV. If youobjectively study the unedited tapes or keep scorewhen youre sitting in the audience, youll noticethings like Is there signicance with the letterJ? R? N? S? T? V? W? B? K?. In one amusing case he actuallywent through 20 LETTERS before getting the hit, yet theaudience still was amazed that he got the letter EXACTLYRIGHT! The subjective and objective results are completelyat odds.Im not going to tell you not to mess with numerology if itprovides you amusement. I will tell you that you reallyshould talk with a psychiatrist. They really can helpstop the torture you keep talking about. Wouldnt you liketo stop hearing those things or to turn the volume down?Please consider getting help. None of us on the internet,hundreds or thousands of miles away, can help you. Butthere are people in your town who can. - Randy =Your denition of object ring is still circular, thus your whole FLT proof is invalid.> Once again you choose not to address this issue.Simple math question - does your object ring contain sqrt(2)?> I bet you dont know.Tut, tut. Dont you JSH does not answer questions?GibThe answer is that sqrt(2) is an object.Very good.> What Ive seen are repeated attempts at distractions.Thats what you think you see.But in reality your denition of object ring is still circular, thus your whole FLT proof isstill invalid.So you cannot say nobody found an error in my FLT proof. =>> >> Your denition of object ring is still circular, thus your whole FLT proof is invalid.>> Once again you choose not to address this issue.>> >> Simple math question - does your object ring contain sqrt(2)?>> I bet you dont know.>> >> Tut, tut. Dont you JSH does not answer questions?>> >> GibThe answer is that sqrt(2) is an object.Prove it. Use your denition and PROVE IT.You have not provided a coherent denition of object. Many times Ihave pointed out the problems with yoru denition, and you have NEVERreplied. The denition that appears in your website is circular: Objects are members of commutative rings where any unit and its multiplicative inverse are units in all possible commutative rings in which either and all integers are members, where no member is a factor of an object for which it is not a factor in all possible commutative rings that include all integers in which it and that object are members. If we drop the circularity by dening object ring as: An object ring is a commtuative ring R such that: (i) if u is a unit in R, and v is its multiplicative inverse, then any commutative ring containing the integers and either u or v must satisfy that u (if u is there) is a unit; or v (if v is there) is a unit. (ii) If a and b are elements of R such that a is a factor of b in R, then in any commutative ring which contains the integers, a, and b, also satises that a is a factor of b. An object is an element of an object ring.This denition is not circular, but under this denition NOTHING isan object and nothing is an object ring, because you have placed norestrictions on the ring structures you are considering. A morereasonable interpretation of what you mean, which I have offered manytimes (and you have never replied) is: An object ring is a subring R of the complex numbers which satises the following two properties: (i) If u in R is a unit in R, then u is a unit in any subring of C containing the integers and u; equivalently, u is a unit in Z[u]. (ii) If a and b are in R, and a is a factor of b in R, then a is a factor of b in any subring of C which contains the integers, a, and b; equivalently, a is a factor of b in Z[a,b]. An object is a complex number which is an element of an object ring.Under this denition, the ONLY objects are the integers.To prove that sqrt(2) is NOT an object under this proposed (and neverchallenged) denition, simply note that in any ring containingsqrt(2) and 2, 2 divides 2*sqrt(2); however, 2 does not divide2*sqrt(2) in Z[2,2*sqrt(2)] = Z[2*sqrt(2)]. Therefore, sqrt(2) cannotbe an element of any object ring, and so sqrt(2) is NOT an object. [.snip.] such a reasoner as Mr. Smith? I answer as a deceased friend of mine used to answer on like occasions - A mans capacity is no measure of his power to do mischief. Mr. Smith has untiring energy, which does something; self-evident honesty of conviction, which does more; and a long purse, which does most of all. He has made at least ten publications, full of gures few readers can critize. A great many people are staggered to this extend, that they imagine there must be the indenite something in the mysterious all this. They are brought to the point of suspicion that the mathematicians ought not to treat all this with such undisguised contempt, at least. -- A Budget of Paradoxes, Vol. 2 p. 129 by Augustus de Morgan sci.skeptic, posted by Gib leaves, and where they most abound> Much fruit of sense beneath is rarely found.Alexander PopeWell, that pretty much nishes the Britannica, not tomention the OED. Marvel Comics, anyone?-- Bob C.(without the spaces, of course) The most exciting phrase to hear in science, the one that heralds new discoveries, is not Eureka! but Thats funny... - Isaac Asimov => the numerology is evident under ordinary analysis.This is done (one way) by analysing notable people in our society rather> than characters of the bible though.Like what coincidence is it that Ronald Raegun introduced the> star wars program, a man called Ray Gun?And why is Hawking, a king!!Hes not a king.our smartest? What coincidence is it?Zero. Why only apply the word king to the most well-known> person in one particular subeld of physics? What about> the rest of the sciences and other elds of human endeavor?> For instance...>Why is Tiger Woods the best golfer?>You mean, why is Tiger Woods the king of golf? I dont> know, why? And how about before Tiger? And after? Was> Jack Nicklaus a failure because of the absence of Wood> in his name? Wouldnt Wood also suggest he should be> a carpenter or a woodcarver? What about other people> named Woods? Are they all good golfers?What did the famous person named Di do?So being called Di implies that youll die, and not> being called Di implies that you wont? Cool! Im> golmm-124 Any idea what is the analytical solution forF(x) - 0.8F(0.2+1.1x) = 1/(x+3.0)F(innity)=0I know,F(x)=Sum[(0.8^i)/(x_i+3.0)], x_(i+1)=0.2+1.1x_i, i=0..innitybut, does it look familiar for anybody?More general, what would be a good reference for discrete equations likeF(x) - H(x)F(a+bx) = K(x) Any idea what is the analytical solution for>F(x) - 0.8F(0.2+1.1x) = 1/(x+3.0)>F(innity)=0The xed point of the mapping x -> 1/5 + 11/10 x being -2, letsrst do a change of variables y = x + 2, G(y) = F(x):G(y) - 4/5 G(11/10 y) = 1/(y+1)The general solution for y > 0 will allow G to be arbitrary on, say, the interval [1, 11/10); given G(y_0) = g_0 with y_0 in this interval,and y_n = (11/10)^n y_0, G(y_n) = g_n where g_{n+1} = 5/4 (g_n - 1/(y_n+1))and thus g_n = (5/4)^n g_0 - sum_{j=0}^{n-1} (5/4)^(n-j)/(y_j+1)But arranging for g_n -> 0 will be tricky. Its convenient to write z_n = 1/(y_n+1), so z_n -> 0 as n -> innity.Even though we have an explicit formula for z_n, well write it ascoming from a recursion. So we have the discrete dynamical systemg_{n+1} = 5/4 (g_n - z_n)z_{n+1} = 10/11 z_n/(1 - z_n)which has a hyperbolic xed point at the origin. What we want is the stable manifold, a curve through (0,0) with theproperty that if (g_0, z_0) is on this curve, (g_n, z_n) -> (0,0) as n -> innity.Suppose [ and I think there are theorems to back this up ]the stable manifold is given by a function g = h(z) that is analyticat z=0, so we have a convergent Taylor series g = sum_{j=1}^innity h_j z^jNow we want h to satisfy the functional equation h(10/11 z/(1-z)) = 5/4 (h(z) - z) Expanding everything in powers of z, from the coefcient of each power z^n we will get an equation to determine h_n in terms of the previous ones: according to Maple, 10 /10 100 2 -- h[1] z + |-- h[1] + h[2]| z + 11 11 121 / /10 200 1000 3 |-- h[1] + h[2] + - h[3]| z + 11 121 1331 / /10 10000 300 3000 4 5 |-- h[1] + -- h[4] + h[2] + - h[3]| z + O(z ) = 11 14641 121 1331 / 2 3 4 (5/4 h[1] - 5/4) z + 5/4 h[2] z + 5/4 h[3] z + 5/4 h[4] z 5 + O(z )and thus 968 713416 2739975304 h[1] = 11/3, h[2] = , h[3] = , h[4] = - 123 21771 16064579I believe (from looking at it numerically with 19 terms instead of 20)that this series has radius of convergence approximately 1/10. So thesecoefcients wouldnt be of much use near y = 1 (or z = 1/2), but theywould provide a good numerical approximation near, say, y = 100 (or z = 1/101). I doubt that the function h(z) can be written in closed form.>I know,>F(x)=Sum[(0.8^i)/(x_i+3.0)], x_(i+1)=0.2+1.1x_i, i=0..innityThats incorrect. > > Any idea what is the analytical solution for> > F(x) - 0.8F(0.2+1.1x) = 1/(x+3.0)> F(innity)=0> > I know,> > F(x)=Sum[(0.8^i)/(x_i+3.0)], x_(i+1)=0.2+1.1x_i, i=0..innity> > but, does it look familiar for anybody?> > More general, what would be a good reference for discrete equations like> > F(x) - H(x)F(a+bx) = K(x)> Author: Goldberg, Samuel.Title: Introduction to difference equations, withillustrative examples from economics, psychology, and sociology.Publication info: New York, Science Editions, 1961 [c1958]-- Julian V. ^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ Science knows only one commandment: contribute to science. -- Bertolt Brecht, Galileo. >Any idea what is the analytical solution for>>F(x) - 0.8F(0.2+1.1x) = 1/(x+3.0)>F(innity)=0>> The xed point of the mapping x -> 1/5 + 11/10 x being -2, lets> rst do a change of variables y = x + 2, G(y) = F(x):>> G(y) - 4/5 G(11/10 y) = 1/(y+1)>> The general solution for y > 0 will allow G to be arbitrary on, say, the> interval [1, 11/10); given G(y_0) = g_0 with y_0 in this interval,> and y_n = (11/10)^n y_0, G(y_n) = g_n where> g_{n+1} = 5/4 (g_n - 1/(y_n+1))> and thus> g_n = (5/4)^n g_0 - sum_{j=0}^{n-1} (5/4)^(n-j)/(y_j+1)>> But arranging for g_n -> 0 will be tricky.>> Its convenient to write z_n = 1/(y_n+1), so z_n -> 0 as n -> innity.> Even though we have an explicit formula for z_n, well write it as> coming from a recursion. So we have the discrete dynamical system>> g_{n+1} = 5/4 (g_n - z_n)> z_{n+1} = 10/11 z_n/(1 - z_n)>> which has a hyperbolic xed point at the origin.> What we want is the stable manifold, a curve through (0,0) with the> property that if (g_0, z_0) is on this curve, (g_n, z_n) -> (0,0)> as n -> innity.>> Suppose [ and I think there are theorems to back this up ]> the stable manifold is given by a function g = h(z) that is analytic> at z=0, so we have a convergent Taylor series>> g = sum_{j=1}^innity h_j z^j>> Now we want h to satisfy the functional equation> h(10/11 z/(1-z)) = 5/4 (h(z) - z)h(10z/(11-z)) = 5/4 (h(z) - z)( after z = 1/(y+1), h(z) = G(y) )Boundary condition now is h(0)=0, so you proposed Taylor approximation nearzero:> Expanding everything in powers of z, from the coefcient of each power> z^n we will get an equation to determine h_n in terms of the previous> ones: according to Maple,> 10 /10 100 2> -- h[1] z + |-- h[1] + h[2]| z +> 11 11 121 />> /10 200 1000 3> |-- h[1] + h[2] + - h[3]| z +> 11 121 1331 />> /10 10000 300 3000 4 5> |-- h[1] + -- h[4] + h[2] + - h[3]| z + O(z ) 11 14641 121 1331 />> 2 3 4> (5/4 h[1] - 5/4) z + 5/4 h[2] z + 5/4 h[3] z + 5/4 h[4] z>> 5> + O(z )>> and thus> 968 713416 2739975304> h[1] = 11/3, h[2] = , h[3] = , h[4] = -> 123 21771 16064579>> I believe (from looking at it numerically with 19 terms instead of 20)> that this series has radius of convergence approximately 1/10. So these> coefcients wouldnt be of much use near y = 1 (or z = 1/2), but they> would provide a good numerical approximation near, say, y = 100> (or z = 1/101). I doubt that the function h(z) can be written in> closed form.>>I know,>>F(x)=Sum[(0.8^i)/(x_i+3.0)], x_(i+1)=0.2+1.1x_i, i=0..innity>> Thats incorrect.Other than a typo (x_0 instead of x) I thinkF(x_0)=Sum[(0.8^i)/(x_i+3.0)], x_(i+1)=0.2+1.1x_i, i=0..innityis a correct solution. It can be directly checked by plug-in into theinitial FE.In Mathematica I calculate it ask[x_] = 1/(x + 3.0);g[x_] = 0.2 + 1.1x;f[x_] := NSum[(0.8^i) k[Nest[g, x, i]], {i, 0, [Innity]}];The only problem that it takes signicant amount of time to calculate F(x)by that recursion. I was looking for a closed form solution (if such exist)to increase the computation speed.Vadym Any idea what is the analytical solution for >F(x) - 0.8F(0.2+1.1x) = 1/(x+3.0)>F(innity)=0> The xed point of the mapping x -> 1/5 + 11/10 x being -2, lets> rst do a change of variables y = x + 2, G(y) = F(x):> G(y) - 4/5 G(11/10 y) = 1/(y+1)> The general solution for y > 0 will allow G to be arbitrary on, say, the> interval [1, 11/10); given G(y_0) = g_0 with y_0 in this interval,> and y_n = (11/10)^n y_0, G(y_n) = g_n where> g_{n+1} = 5/4 (g_n - 1/(y_n+1))> and thus> g_n = (5/4)^n g_0 - sum_{j=0}^{n-1} (5/4)^(n-j)/(y_j+1)>> But arranging for g_n -> 0 will be tricky.>> Its convenient to write z_n = 1/(y_n+1), so z_n -> 0 as n -> innity.> Even though we have an explicit formula for z_n, well write it as> coming from a recursion. So we have the discrete dynamical system>> g_{n+1} = 5/4 (g_n - z_n)> z_{n+1} = 10/11 z_n/(1 - z_n)>> which has a hyperbolic xed point at the origin.> What we want is the stable manifold, a curve through (0,0) with the> property that if (g_0, z_0) is on this curve, (g_n, z_n) -> (0,0)> as n -> innity.>> Suppose [ and I think there are theorems to back this up ]> the stable manifold is given by a function g = h(z) that is analytic> at z=0, so we have a convergent Taylor series>> g = sum_{j=1}^innity h_j z^j>> Now we want h to satisfy the functional equation> h(10/11 z/(1-z)) = 5/4 (h(z) - z)> h(10z/(11-z)) = 5/4 (h(z) - z)> ( after z = 1/(y+1), h(z) = G(y) )Oops, yes youre right: it should be z_n = 10 z/(11 - z). Sorry about that slip-up. So now the Taylor series coefcients start 88 5896 2058584 h[1] = 11/3, h[2] = , h[3] = --, h[4] = --, 123 21771 16064579 90906635864 h[5] = - 1302050192529and moreover it looks like the radius of convergence is now close to 1.For example, to calculate F(0) = G(2) = h(1/3), the power series (for 19 terms) gives 1.3136893661154333 approximately. Since this is only approximate and the recursion is unstable, we would expect that starting with this value the terms will eventually diverge, but (using 17 digit arithmetic) I found the g_n decreasing in absolute value until n = 91.So this is quite a good approximation.>I know,> >F(x)=Sum[(0.8^i)/(x_i+3.0)], x_(i+1)=0.2+1.1x_i, i=0..innity>> Thats incorrect.> > Other than a typo (x_0 instead of x) I think> > F(x_0)=Sum[(0.8^i)/(x_i+3.0)], x_(i+1)=0.2+1.1x_i, i=0..innity> > is a correct solution. It can be directly checked by plug-in into the> initial FE.Oops again. I take that back. Yes, youre right.I still dont think theres a closed-form solution, but there is a transformation of the series that may be useful. If a = .8 and b = 1.1, we can write your series as G(y_0) = sum_{n=0}^innity a^n/(b^n y_0 + 1)Expand the summand in a series in powers of 1/(b^n y_0), and you getG(y_0) = sum_{n=0}^innity sum_{m=1}^innity a^n (-1)^(m+1)/(b^n y_0)^m = sum_{m=1}^innity sum_{n=0}^innity a^n (-1)^(m+1)/(b^(mn) y_0^m) = sum_{m=1}^innity (-1)^(m+1)/(1-a/b^m) y_0^m) which will converge more rapidly than the original series if y_0 is large.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 =I would like to know whether Maple V (or later) includes proceduresthat support Boolean function denition (in terms of and, or, and notoperators)and Boolean function manipulation (eg. expansion and evaluation ofBooleanexpressions). What version of Maple do I need and what syntax do I usetocarry out such algebraic manipulations? Is such functionality alsoavailablein other packages such as Matlab or Mathematica?Neil =Maple used to have a symbolic logic package, but some time ago this was removed; it doesnt appear to have been reinstated into Maple 9. A logic package was available through the Maple Applications Center (http://www.mapleapps.com), but I just checked the site to discover that it is no longer available. I dont know why Maple doesnt support symbolic logic - Ifind it very annoying. MuPAD does however. I dont know about Mathematica.You can do some Boolean work in Maple with evalb.- Alasdair> > I would like to know whether Maple V (or later) includes procedures> that support Boolean function denition (in terms of and, or, and not> operators)> and Boolean function manipulation (eg. expansion and evaluation of> Boolean> expressions). What version of Maple do I need and what syntax do I use> to> carry out such algebraic manipulations? Is such functionality also> available> in other packages such as Matlab or Mathematica?> > > Neil-- =computer. I looked for both ./congure or Makele in both linuxdirectory and shared directory which I have got after untar the lesthat I downloaded from mupad website. However, I could notfind anyles like that. Could you tell me how I can install this software inmy computer? Could you tell me how I can install this software in> my computer? Just follow the README: Either (preferred form) use an rpm or untarboth share_252.tgz and bin_linux_252.tgz in a directory supposed tohold the installation. Then, add the share/bin directory to your$PATH.-- +--+ +--+| |+-|+ Christopher Creutzig (ccr@mupad.de) =spherical coordinates or cylindrical coordinates. I tried tofind thefunction using: info(),but could notfind it. Could you help me? Hey all,I am having some problems importing the procedures on one module to be usedunderstanding, I should be using with(module), but Im not sure where itshould be located in the module. Should I write a proc to be executed uponloading that calls with(module)? is there a better way? if anyone needs anymore details about this problem, just let me know. - Chris =|>I am having some problems importing the procedures on one module to be used|>in another, and I was hoping someone out there could give me a hand. > moduleB:= module() export pB; pB:= proc(x) moduleA:-pA(x) + 1 end proc end module:or > moduleC:= module() export pC; pC:= proc(x) use moduleB in pB(x)+1 end use end proc end module:You could also say > moduleD:= module() export pD; use moduleC in pD:= proc(x) pC(x)+1 end proc end use end module:but that will only work if moduleC is dened before moduleD is.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 =Im not a mathematician, hopefully you can help me with my question:I was looking at some equations in a text book that were written insummation notation and felt that they would be clearer if rewritten asmatrix operations. In the course of rewriting, it struck me that thereare actually many families of summation notation equations that can bereasonably translated into matrix/vector operations. The reverse is alsotrue; many basic matrix operations can be easily translated into summationstyle symbology. This especially comes to my mind in the case of tryingto write computer procedures for linear algebra; you could (although itsnot always the most efcient way) code many matrix operations as a seriesof control loops iterating over matrix subscripts, which seems to meclosely related to standard summation notation.A simple example of what Im getting at is that sum(i=1)(i=n)(Ai*Bi) canbe easily rewritten as A dot B provided you make the intuitivesubstitution that vector A is the n-dimensional vector [A1 A2 ... An] andB likewise. This becomes matrix operation if you bother to keep track ofrows versus columns.Conversely, I could convert a simple dot product to the more cumbersomesum notation, something which it seems I am implicitly doing if I try tocode a dot product as a for(i=0, i Im not a mathematician, hopefully you can help me with my question: > > I was looking at some equations in a text book that were written in > summation notation and felt that they would be clearer if rewritten > as matrix operations. In the course of rewriting, it struck me that > there are actually many families of summation notation equations that > can be reasonably translated into matrix/vector operations. The > reverse is also true; many basic matrix operations can be easily > translated into summation style symbology. This especially comes to > my mind in the case of trying to write computer procedures for linear > algebra; you could (although its not always the most efcient way) > code many matrix operations as a series of control loops iterating > over matrix subscripts, which seems to me closely related to standard > summation notation. > > A simple example of what Im getting at is that sum(i=1)(i=n)(Ai*Bi) > can be easily rewritten as A dot B provided you make the intuitive > substitution that vector A is the n-dimensional vector [A1 A2 ... > An] and B likewise. This becomes matrix operation if you bother to > keep track of rows versus columns. > > Conversely, I could convert a simple dot product to the more > cumbersome sum notation, something which it seems I am implicitly > doing if I try to code a dot product as a for(i=0, i control loop. > > What I want to know is: has this relationship and the interconversion > of these families of equations been formalized in any signicant > way? Is there any symbolic formalism for taking an equation in > summation notation and rewriting it as standard matrix/vector > operations (or vice versa, although this way is easier to gure out > intuitively at least for me)? > > Like I said, Im not a mathematician, so if this is a nonsensical > question and you can explain specically why it is nonsensical that > would also interest me. >There are programming languages in which the n-dimensional vector is abasic data type, with all the right operations on it.APL was the rst such language. Its modern version is J:http://www.jsoftware.com/J is a highly developed well thought out formalism for what you areasking for, and much more.You can download a J interpreter from the J Software web site, and alsopapers and books showing applications of J in various areas of mathematics.Nemo > What I want to know is: has this relationship and the interconversion> > of these families of equations been formalized in any signicant> > way?> There are programming languages in which the n-dimensional vector is a> basic data type, with all the right operations on it.>> APL was the rst such language. Its modern version is J:> http://www.jsoftware.com/>> J is a highly developed well thought out formalism for what you are> asking for, and much more.>> You can download a J interpreter from the J Software web site, and also> papers and books showing applications of J in various areas ofmathematics.For purely mathematical purposes, most of the elementary notions of linear(or multilinear) algebra are denable without using bases and matrices atall; ee.g. linear mappings, kernels, traces, determinants, and transposes.But for the real world, I cant improve on Nemos answer.(I was pretty amazed by APL when I played with it at the University ofToronto in the early 70s. Source code for an APL interpreter was in one ofthe popular programming magazines 2 or 3 years ago, but I dont recall thedetails.)Larry = Does anybody know whether these software packages are still availablefor purchase in forms that will operate under Windows XP or other Windowsoperating system? J. Ogilvie Does anybody know whether these software packages are still available>for purchase in forms that will operate under Windows XP or other Windows>operating system?> J. Ogilvie>John -Yes, you can still purchase it, although I am unsure of the address to contact. If you post to the maxima news group, someone can point you to the source from which to buy commercial macsyma and pdease. I have no problem running it under xp. If you have trouble getting an answer, please contact me. I am sure I can track down the necessary address.Dick Fell =John -Dick> Does anybody know whether these software packages are still available>for purchase in forms that will operate under Windows XP or other Windows>operating system?> J. Ogilvie> =Im using Maple v. 7, and am stumped trying to gure out how to use subs (orsomething equivalent) with matrices.Here is an example of the problem Im having:> with(linalg):> mat:=array([[a,b],[c,d]]); [a b] mat := [ ] [c d]> subs(a=0.05,mat); matIn other words, substituting a=0.05 into the relevant element(s) of the matrixmat isnt working.How can I do this? Seems like it should be straightforward, but Ill be dangedif I can gure it out. Im using Maple v. 7, and am stumped trying to gure out how to use subs(or> something equivalent) with matrices.>> Here is an example of the problem Im having:>> with(linalg):> mat:=array([[a,b],[c,d]]);>> [a b]> mat := [ ]> [c d]>> subs(a=0.05,mat);>> mat> In other words, substituting a=0.05 into the relevant element(s) of thematrix> mat isnt working.>Use: subs(a=0.05,evalm(mat)); > In other words, substituting a=0.05 into the relevant element(s) of the>matrix>> mat isnt working.>Use: subs(a=0.05,evalm(mat));>>Makes sense, now that you mention it. Use: subs(a=0.05,evalm(mat)); By the way: What is the point in having evalm in the rst place?In each and every example I have seen so far, evalm was justnecessary to make Maple do anything at all with an expression whereit was completely obvious (and easy to deduce for a computer) thatthe computation should be done on matrices.-- +--+ +--+| |+-|+ Christopher Creutzig (ccr@mupad.de) =does anybody know if there is an interface from SciLab to MuPAD? iwant to adress MuPAD from SciLab. (exchange parameter and have MuPADcalculate thing for SciLab).can i somehow adress MuPAD from outside ?i do know of the MuPAD - SciLab link but i want to include MuPAD in SciLab and not vice versa.thanx bent system,> which runs in any computer with Java. You can play it online.>> www.SymbMath.com > >Having said all that to show that I need to prove metric invariance ->my real problem is that I have a stereographic projection and I want to>be sure that when I say points A and B are closer than A and C in the>complex plane that this information is preserved through the>stereographic projection.> > Its not true. If it were, the stereographic projection f (from the> sphere to the complex plane) would map a circle centred at A to a> circle centred at f(A). The stereographic projection maps circles to> circles (and straight lines in the case of circles through the north> pole), but does not preserve the centres.> Intuitively though I think the relative positions of the centres arepreserved. So if A is closer to the origin than B in the complex plane,this is still true if you consider the distance around the sphere fromthe south pole (f(0)) to the projected points f(A) and f(B). Perhaps my relating this concept to the metric was incorrect?I think what I was thinking goes something like this:1. The metric measures the shortest distance between two points in aspace [this assumption may or may not be correct; appreciate help hereif Im wrong]. As a result using the metric you are able to say thedistance between A and B is shorter than distance between A and C.2. If I have a mapping function that maps between two spaces with twodifferent metrics (which is denitely the case in the stereographicprojection) it might be possible that as a result of the mapping thestatement the distance between A and B is shorter than distance betweenA and C is no longer true for distances f(A)f(B) and f(A)f(C).I think my initial example showed that statement (2) is a valid concern,although in the example the spaces had the same metric.Does this help explain my train of thought?Russell. =is it possible to work with two-sided polynomials in MuPAD? By two-sided polynomials I mean something like:>> a := 3*z + 2 + 4*z^-1 + 2*z^-2; Namely, I would like to get coefcients of >> a*x; 2 3 / x1 x2 x3 x4 x5 x6 (a0 + z a1 + z a2 + z a3) | x0 + -- + -- + -- + -- + -- + -- | | z 2 3 4 5 6 | z z z z z /that are associated with nonpositive powers of z. The COEFF command does not work, is there any alternative way?Best ragards,Zdenek =I took me some time (I am a MuPAD newbie) but nally I succeeded. Here is my solution: >> a := _plus(a.i*z^i $ i=0..3); 2 3 a0 + z a1 + z a2 + z a3>> x := _plus(x.i*z^-i $ i=0..10); x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x0 + -- + -- + -- + -- + -- + -- + -- + -- + -- + z 2 3 4 5 6 7 8 9 10 z z z z z z z z z>> y := revert([coeff(poly(numer(a*x),[z]),i) $ i = 2..10]);[a0 x0 + a1 x1 + a2 x2 + a3 x3, a0 x1 + a1 x2 + a2 x3 + a3 x4, a0 x2 + a1 x3 + a2 x4 + a3 x5, a0 x3 + a1 x4 + a2 x5 + a3 x6, a0 x4 + a1 x5 + a2 x6 + a3 x7, a0 x5 + a1 x6 + a2 x7 + a3 x8, a0 x6 + a1 x7 + a2 x8 + a3 x9, a0 x7 + a1 x8 + a2 x9 + a3 x10, a0 x8 + a1 x9 + a2 x10](well, I omitted the last two terms)Zdenek Hurak> > is it possible to work with two-sided polynomials in MuPAD? By two-sided> polynomials I mean something like: a := 3*z + 2 + 4*z^-1 + 2*z^-2;> > Namely, I would like to get coefcients of> > a*x;> > 2 3 / x1 x2 x3 x4 x5 x6 > (a0 + z a1 + z a2 + z a3) | x0 + -- + -- + -- + -- + -- + -- |> | z 2 3 4 5 6 |> z z z z z /> > that are associated with nonpositive powers of z. The COEFF command does> not work, is there any alternative way?> > Best ragards,> Zdenek glasses.--Edward CaruthersMercury481Ralph HertleFirst Name --You may know a lot about the cosmos but you cant spell to save your soul.The word is prodigy not prodogy.--NormDePloomTheKidJohn LFirst Name --Basically...Your Off Topic posts are; unwarranted, annoyining and idiotic.utilities.If you cant focus on that topic...GET LOST ! Your posts are more akin to to help you with yourproblem.Are you taking your Prozac ? You should stay on the regiment...Itll help youwith your problem.--Hold, Ill think of it in just a nanosecond or twoMercury481David H. LipmanGreg Evans-- >Ah yes, the things we learn when we least expect it. Feels good to take> control, doesnt it? *back rub hugs*back rub hugs, my favourite> do 10 years of weight training and having strong arms means> YOU do the massage!!HercHaa! Have been in the Air Force for 12 years, Have Rank, Guess who gives theorders.--Apostatepatty-anne-leaSee You In Hell My Friend. --This is a good point. We may not be further evolving, but that doesntmeanselection isnt going on.In the last forty years, the welfare state has had some very powerfulpositive side effects. Women have on the whole been able to shag whothey like, without need for a mans money. (The latest no faultdivorce laws allow women mens money basically regardless.)I believe the meteoric rise in height is partly due to womens freedomto shag about. Also the majority of people were poor in the 19thcentury, and sleeping with rich men failed to work, as they could justdeny parentage.One other thing you never hear these days is Older men are moreattractiveObviously they never were, but with money coming from the state, womenno longer have to butter up the wealthy.--ChrisLee S. BillingsIansertec-- That would be nice if I knew where the top and bottom were..or even wherethe left to right borders were..then I could do that. Not sure if theresan easy way tofind that out though with a triangle.--scribe2bMatt Giwercliff86Chris-- How about if we just called it the empty set?--Rich ShewmakerThe PervertTerry Wildergecko-- >> I merely asked how, IYO, mathematics relates to belief>> or nonbelief in supernatual beings.because in mathematics you only believe what you KNOW is true. And so it goes, possibly as long as the cosmos itself. OK, Mr. Only Agnostics Know Which Side Is Up, what belief is it that you are trying to inject into every atheists veins, so you can claim that we/they all believe things we dont know? Are you prepared to believe that I know that I dont believe in any gawds? Are you able to get a faint glimpse of the possibility that I could hold no belief in any gawds, while not making any rash claims I cant demonstrate the truth of (much less ask anyone else to believe as much or as little as I do)? Is my not being so smug as to troll atheist groups chiding the posters for their na.95ve faith in a strawman religious stance I posit for them, enough grounds for you to count me among the question-begging atheist fundamentalists? Or have you even heard of agnostic atheists? Does it make you feel all warm and secure, just before you fall off to sleep, to know for sure that you havent angered any gawds by scofng at them? Just in case, you know, you die before you wake?--David H. LipmanApostatemalcolm burtonChris-- Answers Below.1 Wally Anglesea see angel2 Edward Caruthers car3 Xcott Craver no cot4 Mitch Dickson rst cause5 Lawrence & Bobbie fence bird6 J.y.n.x tempt, act, silence!7 Hold, Ill think of it in just a nanosecond or two open mind8 Mercury481 boiling atmosphere9 sertec sir technical10 Scribe2b scribes11 Tim Kozusko mount koziosko12 CNote see note13 Greg Neill nil14 Shanx show then space16 Matt Giwer give away17 Someone one bar18 John L loo19 Rich Shewmaker rich showmaker20 See You In Hell My Friend. depends21 Rust attacks metal22 Chas chase23 Saad Malik well said24 Greg Evans even25 Ben Sauvin save27 beavith animal28 Odysseus odysey29 malcolm burton button30 G=EMC^2 Glazier glacier31 Wanda wand dissapears32 Lee S. Billings leave33 Impmon impossible for man34 PlanetaryMatrix space dimension35 Kurst cursed36 gecko primitive37 Roundtable round38 cliff86 jump off a cliff39 The Pervert inadequate40 raven1 raving41 Gary Rockley rock42 NormDePloom normal circumstances43 First Name rst person perspective44 TheKid child45 David H. Lipman tell off46 patty-anne-lea pat47 Ian I Agree48 Chris nice49 Terry Wilder empty will do50 Apostate agnostic atheist =Your denition of object ring is still circular, thus your whole FLT proof is invalid.> Once again you choose not to address this issue.Simple math question - does your object ring contain sqrt(2)?> I bet you dont know.Tut, tut. Dont you JSH does not answer questions?GibThe answer is that sqrt(2) is an object.Very good.What Ive seen are repeated attempts at distractions.Thats what you think you see.But in reality your denition of object ring is still circular, thus your whole FLT proof is> still invalid.> So you cannot say nobody found an error in my FLT proof.Well I didfind a problem with the denition of the object ring thatId given, and Ive updated it.However, you still seem to not understand what a mathematical proos.It is a perfect argument that begins with a truth and proceeds bylogical steps to a conclusion which then must be true.So its impossible tofind an error in a proof.However, a would-be discoverer *can* make errors in describing aproof, or think they see a proof where none exists, and potentiallythat can be found out by starting at the beginning of the proof, andproceeding through it checking each step to make certain that it is alogical one.James Harris => >I remember a post where someone was wondering how I could keep>claiming that Im right with all these people posting disagreement,>and it seems to me that maybe explaining how might help.You see, my short proof of Fermats Last Theorem is awless with all>the disputes from mathematicians attacking the following statement:Given a factor g of a polynomial P(x), g=r+c, where c is a factor of>the constant term P(0) of the polynomial, given by the value of g at>x=0, and r=g-c.>No one is attacking this statement. Just what comes *after* it.> Given acceptance of the intriguing little result that given a factor g> of a polynomial P(x), g=r+c, where c is a factor of the constant term> P(0), given by c=g at x=0, and r=g-c, the proof follows easily enough.And yet I keep failing to follow it, and when I explained why you keep > failing to explain it, even though Ive repeatedly pointed out the post > where I stated where I dont follow the proof.Yet, in your previous reply you said:No one is attacking this statement. Just what comes *after* it.And readers can see your statement above including what preceded itfor context.For those readers whofind it bizarre that a poster would say onething in one post, make a reply contradicting themselves in anotherpost where their previous posts were included, I have to say, welcometo Usenet.Im putting back the short math argument which the poster did deleteout.Is he a mathematician really? It is Usenet, so who knows. But if heis then his behavior is that much more fascinating.Consider P(m)/f^2 = (m^3 f^4 - 3m^2 f^2 + 3m) x^3 - 3(-1+mf^2 )x u^2 + u^3 f.Now using b_1, b_2, b_3, w_1, w_2, and w_3, I have the factorization P(m)/f^2 = (b_1 x + u w_1)(b_2 x + u w_2)(b_3 x + u w_3)where w_1 w_2 w_3 = f, and b_1 b_2 b_3 = (m^3 f^4 - 3m^2 f^2 + 3m),and at m=0 P(0)/f^2 = 3xu^2 + u^3 f = u^2(3x + uf), so two of the bs must equal 0, which means P(0)/f^2 = w_1 w_2 u^2 (b_3 x + u w_3)which is P(0)/f^2 = u^2 (b_3 w_1 w_2 x + u f) = u^2(3x + uf)proving that w_1 w_2 must equal 1, as f is coprime to 3 from before,which leaves b_3 = 3.Essentially objections now come down to claiming that the ws aredependent on m, but consider that w_1 w_2 = 1, when m=0, here where s coprime to 3.But that was an arbitrary choice *I* made, so let f=3.Now w_1 w_2 = 3^{2/3} as long as m is coprime to 3, WITHOUT REGARD TOm.So those posters who try to convince you that the ws are actuallydependent on m, like being functions of m, must now also convince youthat the ws make a decision, rst looking to see if f=3 or have somenon-unit factor in common with 3, and THEN they decide if theyredependent on m.People can wafe trying to gure out who they are, but mathematicsis logical, which is why Ive emphasized that posters are acting on*social* not mathematical reasons.James Harris =>jstevh opined:>So its impossible tofind an error in a proof.Of the many things you repeat ad nauseum, this is one of the stupidest. => However, 7ï WEcf !.856Àèÿÿ seems easier to reword prove if mn not = nm then Mcap N is not {e}, but neither wording any luck :(B4E => Seeks hints for:If M and N normal subgroups of G but M cap N is {e}, prove mn=nm all m> in M, n in N> Show that mnm^{-1}n^{-1} is in M and in N.-- Stephen Montgomery-Smithstephen@math.missouri.eduhttp:// www.math.missouri.edu/~stephen => Seeks hints for:If M and N normal subgroups of G but M cap N is {e}, prove mn=nm all m> in M, n in N> Show that mnm^{-1}n^{-1} is in M and in N.Ahh! Now I see. Very interesting indeed :-). Easy to show, usingdenition of normal subgroup... thus mnmn is in M cap N but M cap Nis {e} means mnmn is e means mn is (mn) but (mn) is nm so nmis mn... very nice, many thank yous! =Thats my problem: how tofind this M in a systemetic/non-ad-hoc way?Strangs test book does not show how to do this... can you help me? Atleast tell me the key word of this method that I can search on?-lalalaJulien similar in a general/systematic way?They cant be similar, they dont have the same trace.> How tofind the transform matrix M such that M^(-1)*A*M=B?> In a systematic way you can which can transform [4 1;> 0 4]to [4 0;> 0 4]?Why?No. First matrix has a minimal polynomial = (X-4)^2 and hence cannot be> diagonalized. =>Thats my problem: how tofind this M in a systemetic/non-ad-hoc way?>Strangs test book does not show how to do this... can you help me? At>least tell me the key word of this method that I can search on?The key phrase is Jordan Form, which youllfind in Strang. Also calledJordan Canonical Form. But this is not numerically stable. The othermethod that was suggested, to solve the system A M - M B = 0 for the entries of M, is not a bad idea I think. It doesnt even requirefinding any eigenvalues. If A and B are n x n, you have a system ofn^2 equations in n^2 unknowns (but sparse, since each equation has at most 2n-1 nonzero entries). Expressed in a fancier way, the coefcient matrix is A otimes I - I otimes B^T where otimes is the Kronecker productand ^T means transpose. The matrices A and B are similar iff there is a nonsingular matrix in the kernel of that n^2 x n^2 matrix.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 =I am reading Conway & SLOANEs Sphere Packings, Lattices and Groups> in this book, he expresses a 24 dimensional vector in MOG format> Where can ifind some information about reading this format?> In the book they express the vector likeu = |-|> |-2 2 | 4 0 | 0 0 |> | 2 2 | 0 0 | 0 0 |> | 2 2 | 0 0 | 0 0 |> | 2 2 | 0 0 | 0 0 |> |-|Whats u?Its a vector!This is just an alternative way of writing vectors in R^24.They could have written this as (-2, 2, 2, ...., 0, 0, 0).But writing vectors in the above format makes it easy to checkif theyre in the Leech lattice (as long as youve mastered-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html The League of Gentlemen =I read that chapter, but maybe I did not understand good enough.For example if I have a Golay code 1100....I write them in the MOG format,and I decide 0,1,omg,bar(omg) and try to see whether it is Hexacodeso that I can tell whether it is Golay Code,but how to apply to the leech lattice?> I am reading Conway & SLOANEs Sphere Packings, Lattices and Groups> in this book, he expresses a 24 dimensional vector in MOG format> Where can ifind some information about reading this format?> In the book they express the vector likeu = |-|> |-2 2 | 4 0 | 0 0 |> | 2 2 | 0 0 | 0 0 |> | 2 2 | 0 0 | 0 0 |> | 2 2 | 0 0 | 0 0 |> |-|Whats u?Its a vector!This is just an alternative way of writing vectors in R^24.> They could have written this as (-2, 2, 2, ...., 0, 0, 0).> But writing vectors in the above format makes it easy to check> if theyre in the Leech lattice (as long as youve mastered => I read that chapter, but maybe I did not understand good enough.> For example if I have a Golay code 1100....> I write them in the MOG format,> and I decide 0,1,omg,bar(omg)> and try to see whether it is Hexacode> so that I can tell whether it is Golay Code,> but how to apply to the leech lattice?Please dont top-post.To see if a vector with even coordinate in in Leechcheck that the sum of the coordinates is a multiple of 8;then check whether the coordinate positions of coordinatesnot divisible by 4 form a word in the Golay code.There is a similar test for vectors with odd coordinates.-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html The League of Gentlemen =how can I show that the polynomial>T^8 - 40 T^6 + 352 T^4 - 960 T^2 + 576 in Q[T]>is irreducible over Q ist?>(According to MuPAD (an algebra system) that is the case, but I need a >calculation (that is a proof.)).>PS: I got the polynomial by the following MuPAD-calculation>(In case it helps...):fac := [ T+a*sqrt(2)+b*sqrt(3)+c*sqrt(5)> $ a in [-1,1] $ b in [-1,1] $ c in [-1,1] ];>p := _mult ( i $ i in fac );>f:= expand (p);>g := poly (f, [T], Dom::Rational);> A much different approach denotes the eight real roots by r1, r2, ..., r8. All of these are bounded in absolute valueby sqrt(2) + sqrt(3) + sqrt(5) < 1.5 + 1.8 + 2.3 < 6.The numerical value of the polynomial is prime when T = 11.If the polynomial f(T) has a factorization f1(T) * f2(T)where f1, f2 are monic of degrees d1 and d2 and integer coefcients, then |f1(11)| >= (11 - 6)^d1 |f2(11)| >= (11 - 6)^d2 .Unless d1 = 0 or d2 = 0, both |f1(11)| and |f2(11)| exceed 1.But these are integers whose product is a prime. (f(5) = -37799 is the negative of a prime -- you may be able to use T = 5 rather than T = 11 in a variation of this proof.)-- Spammers: I dont want a small digital camera to post photos of a large, lowweight, penis on a re-nanced Nigerian domain site. Peter-Lawrence.Montgomery@cwi.nl Home: San Rafael, California Microsoft Research and CWI =I want to solve a classical overdetermined system of equations, min_{x}||Ax-b||_2^2 but I have some missing values (without structure) in A. Howcan I solve it?E.g. in matlab notation[2 3; 4 ?? ; 4 5][ x1 x2]=[2 ;1 ;3] => [2 3; 4 ?? ; 4 5][ x1 x2]=[2 ;1 ;3]Am I missing something? The top and bottom rows give you a standardlinear equation, so solve it (x1 = 1/2, x2 = 1). Plug them in,multiply, notice that 1 = ?? - 2, so ?? = 3. QEI.-- [mdw] =I want to solve a classical overdetermined system of equations, min_{x}> ||Ax-b||_2^2 but I have some missing values (without structure) in A. How> can I solve it?> E.g. in matlab notation[2 3; 4 ?? ; 4 5][ x1 x2]=[2 ;1 ;3]Same way, in principle. Maple could probably solve it,using for instance a for the unknown parameter, but theresult is no doubt some huge ugly expression in a.If A = [2 3; 4 a; 4 5] and b = [2; 1; 3] thenA*A = [36, 26 + 4a; 26 + 4a, 34 + a^2]andA*b = [20; 21+a]The least-squares solution is the solution to thelinear system (A*A)x = (A*b)x, which you could solvesymbolically in terms of a. Its going to be a fair amountof messy algebra.Actually its not so complex as all that since thissystem is 2x2. Heres what Matlab says is the solution:ans =[ 1/2*(67-55*a+8*a^2)/(137+5*a^2-52*a)][ -(-59+11*a)/(137+5*a^2-52*a)] - Randy => Im stuck on the following problem (from the book Function Theory of> One Complex Variable by Krantz):> > True or false: Let f be holomorphic on the unit disk. Let f^2 be a> holomorphic polynomial on the unit disk. Then f is also a holomorphic> polynomial on the unit disk.Why should that be true? What about sqrt(z) or something? Well, that doesnt quite work ... hmmm ... beats me ... hmmm ... =Im stuck on the following problem (from the book Function Theory of> One Complex Variable by Krantz):True or false: Let f be holomorphic on the unit disk. Let f^2 be a> holomorphic polynomial on the unit disk. Then f is also a holomorphic> polynomial on the unit disk.Why should that be true? What about sqrt(z) or something? Well, that> doesnt quite work ... hmmm ... beats me ... hmmm ...I wonder if holomorphic polynomial means the same as polynomial.In which case suppose f^2 = x + 2, to get rid of nasty singularities in theunit disk when we pick a branch for f = sqrt(x + 2).-- chip =>Im stuck on the following problem (from the book Function Theory of>One Complex Variable by Krantz):> >True or false: Let f be holomorphic on the unit disk. Let f^2 be a>holomorphic polynomial on the unit disk. Then f is also a holomorphic>polynomial on the unit disk.False - the square root of any polynomial with no zero in the disk isa counterexample, as has been pointed out.Its interesting that its true with the plane in place of the disk: If f is an entire function and f^2 is a polynomial then f is apolynomial.>Mike************************David C. Ullrich => Im stuck on the following problem (from the book Function Theory of>> One Complex Variable by Krantz):>> True or false: Let f be holomorphic on the unit disk. Let f^2 be a>> holomorphic polynomial on the unit disk. Then f is also a holomorphic>> polynomial on the unit disk.>> Why should that be true? What about sqrt(z) or something? Well, that>> doesnt quite work ... hmmm ... beats me ... hmmm ...I wonder if holomorphic polynomial means the same as polynomial.It means what you meant by polynomial when you asked. Lets take z = x + iy. The reason for the terminology is to distinguish between polynomials in z and polynomials in x and y;if I say P(z) = x that would count as a polynomial for somepurposes, but not here. >In which case suppose f^2 = x + 2, to get rid of nasty singularities in the>unit disk when we pick a branch for f = sqrt(x + 2).-- chip>************************David C. Ullrich <3f2d3fbb$20$fuzhry+tra$mr2ice@news.patriot.net> =>Let me tell you why Im asking about this. In college, my rst love>was philosophy. The thing about philosophy is that, as a subject, it>leads to just about every other subject. I got turned on to math>because of just such a journey. But where philosophy classes were>always concerned with logic and proof (I had, to put it mildly, an>excellent philsophy teacher) my math classes were all about taking the>course just so you can graduate. I hope to almightly Odin I never hear>the phrase will this be on the test? ever again for the rest of my>life. The text books I encountered were in the same vein. Then I>discovered the greats: Euclid, Archimedes, Descartes. I didnt>understand most of what I read, but what I did get was that these>books, starting with a few axioms (the fewer the better), proved every>mathematical claim in the rest of the book. The modern approach of,>Heres the formula you need for the test, couldnt compare. I only>got as far as the beginnings of calculus after that. I would have>pursued more math by reading the classics but then I discovered>programming and.... Thats really another story. Now I want to try again. To be clear, when I say Im looking for a>modern book or set of books, its because I want something that is as>rigorus as Euclid tried to be (The Elements were THE standard textbook>until the last century if I remember correctly) but in line with>todays mathematics. As I understand it, Euclid did make unstated>assumptions and outright errors, and quite frankly Id rather study>geometry post Descartes. Using coordinates seems much easier than>using a compass and a straight edge. Knowing that irrational numbers>were a huge sticking point with mathematitions for thousands of years>and being passing familiar with the number line--Im sure if I dont>say passing familiar somebody here will say, You have to know vector>tensor shmelaculus in 15 triad synergies to really understand the>number line. Its not even called that, its called the real>torticular space. or something to that effect--the synthetic approach>help everybody.>Im a bit late to this thread (and I missed the earlier ones, apologies ifIm repeating); however, I would recommend Michael Spivaks _Calculus_(not _Calculus on Manifolds_) as a nice starting point for going fromrst axioms to calculus and some real analysis. I always found himwonderfully readable.-Davis =Euclids Elements are great. They start from a small number of axiomsand then prove everything from then on with those few axioms. Buttheyre out of date, we now know Euclid made errors and ommisions.Plus Id like to use graphs, coordinates, algebra, etc. There areplenty of new texts with all of that. What I cant seem tofind is atext that starts from a few axioms and then proves everything fromthose few axioms.Thats all I want. A book or set of books that start from the mathbasics, algebra, geometry, and trig, and then to calculus onward. Butproving everything along the way. There have got to be modern bookslike that right? => Euclids Elements are great. They start from a small number of axioms> and then prove everything from then on with those few axioms. But> theyre out of date, we now know Euclid made errors and ommisions.> Plus Id like to use graphs, coordinates, algebra, etc. There are> plenty of new texts with all of that. What I cant seem tofind is a> text that starts from a few axioms and then proves everything from> those few axioms.Thats all I want. A book or set of books that start from the math> basics, algebra, geometry, and trig, and then to calculus onward. But> proving everything along the way. There have got to be modern books> like that right?Pretty much every mathematics book states denitions (axioms) and provesconsequences thereof.However, not all denitions are as rich (have as many consequences) as thedenition of euclidean geometry.-mb =Euclids Elements are great. They start from a small number of axioms> and then prove everything from then on with those few axioms. But> theyre out of date, we now know Euclid made errors and ommisions.> Plus Id like to use graphs, coordinates, algebra, etc. There are> plenty of new texts with all of that. What I cant seem tofind is a> text that starts from a few axioms and then proves everything from> those few axioms.Thats all I want. A book or set of books that start from the math> basics, algebra, geometry, and trig, and then to calculus onward. But> proving everything along the way. There have got to be modern books> like that right?Yes, right, but in many and varied books.For geometry see Part III of Coxeters Introduction to Geometry.For trig (yuk!) see books on real analysis for the denitions of thecircular functions that begin with axioms that dene the reals, say AFirst Course in Mathematical Analysis by J C Burkill.For algebra, say A Survey of Modern Algebra by Birkhoff and Mac Lane.For axiomatics generally see Introduction to Logic and to theMethodology of Deductive Sciences by Tarski.Three cheers for Euclid anyway, eh?GC-- =Let me tell you why Im asking about this. In college, my rst lovewas philosophy. The thing about philosophy is that, as a subject, itleads to just about every other subject. I got turned on to mathbecause of just such a journey. But where philosophy classes werealways concerned with logic and proof (I had, to put it mildly, anexcellent philsophy teacher) my math classes were all about taking thecourse just so you can mm-125 When I did A level maths at school, about 30+ years ago, in myanalytical geometry course I was taught a subject called ïinversionÍ.As I recall, one defined a point O (centre of inversion?) and usedthis to transform a curve. Any point P on the curve was transformedto PÍ, such that O, P, & PÍ were on a straight line and OP*PPÍ = k^2(k = radius of inversion?).The one result I can remember is that a circle can be transformed intoa straight line, and vice versa.I have never come across tar as the beginnings of calculus after that. I would havepursued more math by reading the classics but then I discoveredprogramming and.... Thats really another story. Now I want to try again. To be clear, when I say Im looking for amodern book or set of books, its because I want something that is asrigorus as Euclid tried to be (The Elements were THE standard textbookuntil the last century if I remember correctly) but in line withtodays mathematics. As I understand it, Euclid did make unstatedassumptions and outright errors, and quite frankly Id rather studygeometry post Descartes. Using coordinates seems much easier thanusing a compass and a straight edge. Knowing that irrational numberswere a huge sticking point with mathematitions for thousands of yearsand being passing familiar with the number line--Im sure if I dontsay passing familiar somebody here will say, You have to know vectortensor shmelaculus in 15 triad synergies to really understand thenumber line. Its not even called that, its called the realtorticular space. or something to that effect--the synthetic approachhelp everybody. => Let me tell you why Im asking about this. In college, my rst love> was philosophy. The thing about philosophy is that, as a subject, it> leads to just about every other subject. I got turned on to math> because of just such a journey. But where philosophy classes were> always concerned with logic and proof (I had, to put it mildly, an> excellent philsophy teacher) my math classes were all about taking the> course just so you can graduate. I hope to almightly Odin I never hear> the phrase will this be on the test? ever again for the rest of my> life. The text books I encountered were in the same vein. Then I> discovered the greats: Euclid, Archimedes, Descartes. I didnt> understand most of what I read, but what I did get was that these> books, starting with a few axioms (the fewer the better), proved every> mathematical claim in the rest of the book. The modern approach of,> Heres the formula you need for the test, couldnt compare. I only> got as far as the beginnings of calculus after that. I would have> pursued more math by reading the classics but then I discovered> programming and.... Thats really another story.> How depressing. I hope you are reassured that your course was badlytaught, not that modern mathematics is not as rigorous as Euclid.> Now I want to try again. To be clear, when I say Im looking for a> modern book or set of books, its because I want something that is as> rigorus as Euclid tried to be (The Elements were THE standard textbook> until the last century if I remember correctly) but in line with> todays mathematics. As I understand it, Euclid did make unstated> assumptions and outright errors, and quite frankly Id rather study> geometry post Descartes. Using coordinates seems much easier than> using a compass and a straight edge. Knowing that irrational numbers> were a huge sticking point with mathematitions for thousands of years> and being passing familiar with the number line--Im sure if I dont> say passing familiar somebody here will say, You have to know vector> tensor shmelaculus in 15 triad synergies to really understand the> number line. Its not even called that, its called the real> torticular space. or something to that effect--the synthetic approach> help everybody. => Euclids Elements are great. They start from a small number of axioms> and then prove everything from then on with those few axioms. But> theyre out of date, we now know Euclid made errors and ommisions.> Plus Id like to use graphs, coordinates, algebra, etc. There are> plenty of new texts with all of that. What I cant seem tofind is a> text that starts from a few axioms and then proves everything from> those few axioms.Thats all I want. A book or set of books that start from the math> basics, algebra, geometry, and trig, and then to calculus onward. But> proving everything along the way. There have got to be modern books> like that right?Maybe you should consider category theory.It is quite abstract, and you need to know a great deal of mathematics toreally appreciate it - but still it is a nice exercise to prove a lot ofstuff from four very basic axioms.The rst two chapters of McLarty - Elementary Categories, Elementary Toposesare quite readable.Martin-- Tout ce quil y a de b.8eb.90te. -- Grothendieck =I have the following three parametric equations:y_{0}=b_{0}*u+c_{0}*v+d_{0}*u**2+f_{0}*v**2+e_{0}*u *v;y_{1}=b_{1}*u+c_{1}*v+d_{1}*u**2+f_{1}*v**2+e_{1}*u*v;y_{2 }=b_{2}*u+c_{2}*v+d_{2}*u**2+f_{2}*v**2+e_{2}*u*v;Is there an implicit equation that, when the above equations are substitutedinto, equal zero? =>I have the following three parametric equations:>y_{0}=b_{0}*u+c_{0}*v+d_{0}*u**2+f_{0}*v**2+e_{0}* u*v;>y_{1}=b_{1}*u+c_{1}*v+d_{1}*u**2+f_{1}*v**2+e_{1}*u*v;>y _{2}=b_{2}*u+c_{2}*v+d_{2}*u**2+f_{2}*v**2+e_{2}*u*v;Is there an implicit equation that, when the above equations are substituted>into, equal zero?Sure. You want to eliminate two variables from three equations.I chose some random values for the bs etc., and eliminated u and vusing Maple. This gives a polynomial with total degree 4 in thethree ys. I suppose that might be the general pattern, in whichcase there are (I think) 35 coefcients to work out, each of whichis some polynomial in the bs etc. I suspect its rather a ghastlything even to display, much less compute, but it can be done.You should note, though, that despite your subject line, a genericsuch arrangement describes a _surface_, not a curve.dave =Could someone explain this to me...1) if aCould someone explain this to me...1) if a2) if a<=b then aor something might be nice...The problem here is with the equals:The case where a=b, in rule 1 is no problem (a isnt less than b)but in case 2, if a=b, then the true statement, a<=b (less than ORequal to) would imply a Could someone explain this to me...1) if a 2) if a<=b then a or something might be nice...JoshThe problem you are having is understanding the word OR Here is an example: I will go to the movies tonight OR I will study math.This statement is true if ONE OF THE TWO or Both are true.Now in a <= b means either a Integrate { sin^2(x) + cos^2(x)*ln(cos(x)) } / { 0.25*sin^2(2x) }> indenitely wrt x.(where sin^2(x) refers to sin(x) squared, etc.)It would be of immense help if anyone could integrate the above> function. The reason is because I am currently working on some> integration techniques, and I have tried them on many integrals and> they all seem to work. However, every integral which I have tried my> own developed techniques on could be solved by some other technique> anyway (ie substitution or by parts or some other standard method).Therefore, I tried to think of an integral which could not be> calculated anayltically using any technique I know of, and came up> with the above nasty looking integral. My own developed techniques can> solve it, but no other technique I am aware of can solve it, unless I> am missing something obvious.If someone can solve this integral(giving their method if possible)> then that would greatly help my current research. I am aware that> people frequently post integral requests onto this group so I> apologise in advance for any annoyance and thank anyone who tries to> help me.This looks quite straightforward.int [{sin^2(x) + cos^2(x)*ln(cos(x)) } / { 0.25*sin^2(2x)} ] dxstep 1. Use the double angle formula sin(2x) = 2 sin(x) cos(x)= int [{sin^2(x) + cos^2(x)*ln(cos(x)) } / { sin^2(x)cos^2(x) } ] dxstep 2. Break the fraction and simplify= int [ 1 / cos^2(x) + { ln(cos(x)) } / sin^2(x) ] dxstep 3. The rst part is easy; integrate the second by parts:= int [1 / cos^2(x)] - ln(cos(x))*cot(x) - int [ tan(x)*cot(x) ] dxAre you sure this is not a homework problem? => I want to ask if the Fourier Series Expansion is unique to each > function?>> >> For continuous bounded functions f on [-Pi, Pi), yes.>> >> Even for L^1 functions if one identies>> function equal Lebesgue almost everywhere.measures, for which you can assert actual uniqueness, and beyond that are >distributions, for which the same can be said. Hell, why stop there?I was getting tired.> Let P be the space of trigonometric polynomials (just a vector space,> no topology). Let P be the space of all linear functionals on P. Then> every element of P has a unique Fourier series expansion... (and now> _every_ trig series has become a Fourier series, so this is as far> as we can go. Im gonna be famous for this, the worlds most general> notion of Fourier series (on the circle).)Fine, but what bearing does this have on the uniqueness claims being made? => I want to ask if the Fourier Series Expansion is unique to each >> function?> > For continuous bounded functions f on [-Pi, Pi), yes.> > Even for L^1 functions if one identies> function equal Lebesgue almost everywhere.>>measures, for which you can assert actual uniqueness, and beyond that are >>distributions, for which the same can be said. >> >> Hell, why stop there?I was getting tired.Thats not good enough, sorry.>> Let P be the space of trigonometric polynomials (just a vector space,>> no topology). Let P be the space of all linear functionals on P. Then>> every element of P has a unique Fourier series expansion... (and now>> _every_ trig series has become a Fourier series, so this is as far>> as we can go. Im gonna be famous for this, the worlds most general>> notion of Fourier series (on the circle).)Fine, but what bearing does this have on the uniqueness claims being made???? Of course I was being a little silly, but I dont follow thequestion - the relevance is the same as for L^1 functions,measures and distributions: An element of P is determinedby its Fourier series.************************David C. Ullrich =>>measures, for which you can assert actual uniqueness, and beyond that are >>distributions, for which the same can be said. >> >> Hell, why stop there?I was getting tired.Thats not good enough, sorry.> >> Let P be the space of trigonometric polynomials (just a vector space,>> no topology). Let P be the space of all linear functionals on P. Then>> every element of P has a unique Fourier series expansion... (and now>> _every_ trig series has become a Fourier series, so this is as far>> as we can go. Im gonna be famous for this, the worlds most general>> notion of Fourier series (on the circle).)Fine, but what bearing does this have on the uniqueness claims being made???? Of course I was being a little silly, but I dont follow the> question - the relevance is the same as for L^1 functions,> measures and distributions: An element of P is determined> by its Fourier series.My question arose because the P, P business is a triviality (linear functionals are determined by their values on a basis) that in no way implies the previous uniqueness results (and doesnt help the OP at all), whereas with continuous functions, L^1 functions, measures, and distributions, the uniqueness claims are not trivial, and as we progress down the line we obtain results that imply the preceding ones. That was how I viewed the discussion (although I may have been unclear), so when you didnt stop there I thought you had dived off a cliff. Hence my quesion. =>My question arose because the P, P business is a triviality (linear >functionals are determined by their values on a basis) that in no way >implies the previous uniqueness results (and doesnt help the OP at all), >whereas with continuous functions, L^1 functions, measures, and >distributions, the uniqueness claims are not trivial, and as we progress >down the line we obtain results that imply the preceding ones. That was how >I viewed the discussion (although I may have been unclear), so when you >didnt stop there I thought you had dived off a cliff. Hence my quesion.Ok, that makes sense. And yes, of course when I started this my> intention was to dive off a slightly silly cliff.I am glad that you appear to be in good health after the encounter with the cliff.> But come to think of> it my amazing result about P does imply the previous results,> because continuous functions, ... distributions on T are all> elements of P! (A distribution is an element of P in the> same sense as, say, an L^1 function is a distribution...)One needs to know that, say, the natural map from> D (distributions) into P is 1-1, which follows from the> fact that trigonometric polynomials are dense in the> test functionsYes, it is true that if we rst prove the result we are interested in, then recast it in in P,P garb, and then rederive it, we still have a proof. > Yup, the more I think about it the more clear it is that Im> gonna be famous for this.Heres what I know about that: People have become famous for far less. So to you I bid godspeed and send you off with the hope that fame and fortune will not have the deleterious effects we see so often in celebrities. =Is information theory considered as main stream math ?> Or, at least, main stream applied math ?The mathematical aspects of information theoryare considered as mathematics.-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html The League of Gentlemen =Is information theory considered as main stream math ?> Or, at least, main stream applied math ?Or, is information theory considered as a branch of statistics or> electrical engineering ?> in the MSC... 62B1068P30, 68Q3094A17 =So, are there any good math in information theory ?or did information theory contribute some fundamental new results to generalmath, or is it only consided as application eld of math ?> Is information theory considered as main stream math ?> Or, at least, main stream applied math ?The mathematical aspects of information theory> are considered as mathematics. => So, are there any good math in information theory ?> or did information theory contribute some fundamental new results to> general math, or is it only consided as application eld of math ?Please dont top-post.I am particularly interested in error-correcting codes, whichare certainly part of information theory and are related tostructures such as lattices (e.g. the Leech lattice) whichare related to sporodic groups, algebraic numbe theory, modular formsetc.-- Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html The League of Gentlemen => So, are there any good math in information theory ?> or did information theory contribute some fundamental new results to> general math, or is it only consided as application eld of math ?There are different branches of information theory,eg classical Shannon theory, as opposed toChaitin/Kolmogorov algorithmic information theory.Both involve good math, in my opinion.For algorithmic information theory,look at any of Chaitins books.(One used to be on-line.)Its an interesting, and probably important, theory.For example, Chaitin applies his information theoryto get a new take on Godels work.-- Timothy Murphy tel: +353-86-233 6090 =I cant seem to construct one. Is there such a mapping? (R is the realnumbers) => I cant seem to construct one. Is there such a mapping? (R is the real> numbers)The Cantor-Bernstein theorem is your friend. All you need is aninjection f: R^n -> R and another going the opposite direction, but thelatter is trivial. To construct the injection f, try aninterleave-the-digits trick. You can choose your decimal representationsto avoid any that terminate in all 9s.-- Dave SeamanJudge Yohns mistakes revealed in Mumia Abu-Jamal ruling. =But this doesnt seem to be possible, R^n has dimension n and R hasdimension 1.You cant create an injection here because of different dimensions, howeversurjection is easy.Am I missing something?> I cant seem to construct one. Is there such a mapping? (R is the real> numbers)The Cantor-Bernstein theorem is your friend. All you need is an> injection f: R^n -> R and another going the opposite direction, but the> latter is trivial. To construct the injection f, try an> interleave-the-digits trick. You can choose your decimal representations> to avoid any that terminate in all 9s.>-- > Dave Seaman> Judge Yohns mistakes revealed in Mumia Abu-Jamal ruling.> =But this doesnt seem to be possible, R^n has dimension n and R has> dimension 1.It doesnt *seem* possible, but naive intuition is not agood guide when it comes to innities.> You cant create an injection here because of different dimensions, however> surjection is easy.Because of different dimensions is an appeal to intuition,not a proof. Indeed, for a somewhat related question, you caneasily show that N^n and Q^n are both countable, hence injectonto N despite the difference in dimension. Dimension turnsout not to be very important here.Am I missing something?Youre probably thinking too much inside the box. AFAIKthere are no *continuous* injections from R^n -> R, but wedont require one here. Think a bit about Daves hint onterleaving digits. => But this doesnt seem to be possible, R^n has dimension n and R has> dimension 1.> You cant create an injection here because of different dimensions, however> surjection is easy.Am I missing something?Discontinuous maps need not preserve dimension. Of course there is noCONTINUOUS bijection R^n onto R. => AFAIK> there are no *continuous* injections from R^n -> R.Why is this?-- Stephen Montgomery-Smithstephen@math.missouri.eduhttp:// www.math.missouri.edu/~stephen => But this doesnt seem to be possible, R^n has dimension n and R has> dimension 1.> You cant create an injection here because of different dimensions,however> surjection is easy.Am I missing something?>Apparently you are. Dave has correctly pointed out that a bijection doesexist.Raising the dimensions issue is a red herring. Dimensions only matter inthe context of linear algebra.If you meant to ask if there is a linear isomorphism between R^n and R (asreal vector spaces), then the answer to that question is no.Curiously enough, if R^n and R (for nite dimension n) were considered asvector spaces over Q (the rationals), then they would be linearlyisomorphic. To show this, it sufces to place a basis for R^n in 1-to-1correspondance with a basis for R (over Q). all,I am facing with the following crazy integral... please help me judge:a) Does it exist/converge at all?b) Is it possible to write the explicit form out? c) If it is integrable then how to evaluate it?The integral is:Integrate[exp(-0.25*w^2)*exp( _i_ *w*t)/(-w^2+ _i_ *w+1), w from -infto inf, t is parameter]My bag of tricks such as residue theory, change variables, etc. failedfor this integral... please help me out of the swamp!-Losmnd all,I am facing with the following crazy integral... please help me judge:a) Does it exist/converge at all?> b) Is it possible to write the explicit form out? > c) If it is integrable then how to evaluate it?The integral is:Integrate[exp(-0.25*w^2)*exp( _i_ *w*t)/(-w^2+ _i_ *w+1), w from -inf> to inf, t is parameter]Yes, it converges, at least when t is real. Indeed, for each real w and t,we have|exp(-0.25*w^2)*exp( _i_ *w*t)/(-w^2 + _i_ *w + 1)| = = exp(-0.25*w^2)/sqrt(w^4-w^2+1) <= 1/sqrt(w^4 - w^2 + 1),and the integral of 1/sqrt(w^4 - w^2 + 1) when w goes from -innityto +innity obviously converges.Jose Carlos Santos integral... please help me judge:>a) Does it exist/converge at all?>b) Is it possible to write the explicit form out? >c) If it is integrable then how to evaluate it?>The integral is:>Integrate[exp(-0.25*w^2)*exp( _i_ *w*t)/(-w^2+ _i_ *w+1), w from -inf>to inf, t is parameter]>My bag of tricks such as residue theory, change variables, etc. failed>for this integral... please help me out of the swamp!For (a), the answer is trivially yes, assuming that _i_is the imaginary unit. The terms multiplying exp(-0.25*w^2)are uniformly bounded.As for evaluating it, one can write w = 2z + 2_i_t, obtaining 2*exp(-t^2)*int exp(-z^2)/Q(z) dz,where Q is a quadratic. The limits are rather odd, but the integral is unchanged if the limits are moved to the real line. Write Q in terms of partial fractions,and use that the integral of the exp(-z^2)/(z+c) is thecomplex error function if c is not real. If c is purely imaginary, it is the proportional to the ratioof the tail error function to the density.-- This address is for information only. I do not claim that these viewsare those of the Statistics Department or of Purdue University.Herman Rubin, Department of Statistics, Purdue University => Integrate[exp(-0.25*w^2)*exp( _i_ *w*t)/(-w^2+ _i_ *w+1), w from -inf> to inf, t is parameter] Do you mean i*(w+1) or i*w+1? (not that it makes that much of a difference) > My bag of tricks such as residue theory, change variables, etc. failed> for this integral... please help me out of the swamp! Small bag of tricks try Fourier analysis. Also Abs[Integral[F(x)dx]]<=Integral[Abs[F(x)]dx] so your integral exists =|I found that all metal means can also have a proportional reduction of|its rectangles like Phis golden rectangle.|Then with the input of another poster (John Burglund) who found that x|can be any value greater than zero that is plugged into this equation|will do the same thing and have proportional reducing rectangles as|well!This is a (known) general property of rectangles where the ratio betweenthe lengths of the sides is given by a continued fraction.|Where x can = transcendental or irrational or rational or|integer values.I found the following difcult to read:|Another fact also is if a rational x of n decimal length like as an|example x = .141. Then its decimal length (n) = 3 is plugged into the|equation the outcome will be m = an irrationalOk so far.|and 1/m = ms decimal|expansion after (n) decimal length in m and 1/m.Apparently what you mean here is that the decimal expansions of m andof 1/m are the same after the rst n decimal places. The way you haveit written made it seem like you were saying either that 1/m = m, whichisnt true, or that 1/m was the portion of the decimal expansion of m.|This holds true for Phi and the metal means where the match of m = 1/m|decimal expansion is from the rst decimal place because x = an|integer with zero decimal places.This is true for a very simple reason; m = x + 1/m, so if x has a decimalexpansion which stops at a given point, the decimal expansions of m and1/m will agree beyond that point.Keith Ramsay =>I have a system of PDEs: > >f_x = g_t>f_t = (c^2)g_x> >f= f(x,t)>g= g(x,t)>c=c(t)> > I dont have any particular c(t) in mind; just so long as its > NOT constant. > How can Ifind general solutions to the PDEs for f, g and c?If you dont really care what c(t) is, you can replace the second equation > by (f_t/g_x)_x = 0, i.e. f_{xt} g_x = f_t g_{xx}, and dene > c(t) = sqrt(f_t/g_x) (of course youll want f_t/g_x > 0, but hopefully> that will be true at least in some region).Maple 9 thenfinds three families of solutions. Two are not of interest > because f_t or g_x is 0, but the third may be of some interest although> its not very general:f(x,t) = c1 x + c3 x^2/2 + F1(t), g(x,t) = c2 + c1 t + (c3 t + c4) x> where c^2(t) = F1(t)/(c3 t + c4). Here c1, c2, c3, c4 are arbitrary> constants and F1 is an arbitrary function; of course we want > F1(t)/(c3 t + c4) > 0, which would cause a singularity at t=-c4/c3 if > c3 <> 0 unless F1(-c4/c3) = 0. Thus with c3 = 1, c4 = 0, > F1(t) = t^2/2 + t^4/2 + t^6/6, we get solutionsf(x,t) = c1 x + x^2/2 + t^2/2 + t^4/2 + t^6/6> g(x,t) = c2 + c1 t + x t> c(t) = 1+t^2I also found another interesting family of polynomial solutions:> c(t) = c[0] + c[1] t> 2 3 2 4> f(x, t) = 2/3 b[2] c[1] x t + 1/2 a[2] c[1] t 2> + 2 c[0] c[1] b[2] x t 2 3 2> + (4/3 c[0] c[1] a[2] + 1/3 b[1] c[1] ) t + a[2] x 2 2 2> + 2 c[0] b[2] x t + (c[0] a[2] + c[0] c[1] b[1]) t 2> + a[1] x + c[0] b[1] t + a[0]> 2 4 3 2> g(x, t) = 1/6 b[2] c[1] t + 2/3 c[0] c[1] b[2] t + b[2] x 2 2> + 2 a[2] x t + c[0] b[2] t + b[1] x + a[1] t + b[0]> Robert Israel israel@math.ubc.ca> Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia > Vancouver, BC, Canada V6T 1Z2Wow, Robert! Thats amazing. I was able to guess f(x,t) = c1 x + c3 x^2/2 + F1(t) g(x,t) = c2 + c1 t + (c3 t + c4) xbut your 4th order polynomial solution is truly awesome.I have never used anything like Maple or similar programs but Imguessing that you assumed a general 4th order polynomial solution forf and g and then allowed Maple to solve for (f_t/g_x)_x = 0. Is thatright?Do you have any insights about other solutions to these equations?Eugene Shuberthttp://www.everythingimportant.org =>Wow, Robert! Thats amazing. I was able to guess >f(x,t) = c1 x + c3 x^2/2 + F1(t) >g(x,t) = c2 + c1 t + (c3 t + c4) x>but your 4th order polynomial solution is truly awesome.>I have never used anything like Maple or similar programs but Im>guessing that you assumed a general 4th order polynomial solution for>f and g and then allowed Maple to solve for (f_t/g_x)_x = 0. Is that>right?IIRC I assumed polynomials fourth order in x and in t for f and g and afourth order polynomial in t for c(t), substituted in to the equations andsolved the mess of equations you get for the coefcient of each x^i t^j.Actually, I could have done better:assume f(x,t) = f0(x) + x f1(t) + f2(t) [where we can take f0(0)=0] g(x,t) = g0(x) + x g1(t) + g2(t) [ where say g0(0) = 0]The rst equation says f0(x) + f1(t) = x g1(t) + g2(t)Then we must have f0(x) = a1 x + a0, g1(t) = a1, a0 + f1(t) = g2(t).The second equation says x f1(t) + f2(t) = c(t)^2 (g0(x) + g1(t))so g0(x) = b1 x + b0, f1(t) = c(t)^2 b1, f2(t) = c(t)^2 (b0 + g1(t)).Now g0(x) = b1 x^2/2 + b0 x f0(x) = a1 x^2/2 + a0 x g1(t) = a1 t + a3 f2(t) = int c(t)^2 (b0 + a1 t + a3) dt + a4 f1(t) = b1 int c(t)^2 dt + a5 g2(t) = a0 t + int f1(t) dt + a6This solution is actually a superposition of the following:(corresponding to a0) f(x,t) = x, g(x,t) = t(corresponding to a1) f(x,t) = x^2/2 + int c(t)^2 t dt, g(x,t) = x t(corresponding to a3 and b0) [ seems I didnt have an a2 ] f(x,t) = int c(t)^2 dt, g(x,t) = x(corresponding to a4) f(x,t) = 1, g(x,t) = 0(corresponding to a5) f(x,t) = x, g(x,t) = t(corresponding to a6) f(x,t) = 0, g(x,t) = 1(corresponding to b1) f(x,t) = x int c(t)^2 dt, g(x,t) = x^2/2 + int(int c(t)^2 dt) dtRobert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israelUniversity of British ColumbiaVancouver, BC, Canada V6T 1Z2 =Robert,I cant tell you how delighted I am with your answer.Im guessing that you tried a solution of the form:f(x,t) = f0(x) + x f1(t) + f2(t)g(x,t) = g0(x) + x g1(t) + g2(t)and then realized you had uncovered a superposition of severalsolutions because of the linearity off_x = g_t f_t = (c^2)g_x If any other ideas occur to you related to this problem (anything atall), or if you think of more solutions that may be added to thegeneral form, please let me know.Eugene Shuberthttp://www.everythingimportant.org =>I have a system of PDEs: >f_x = g_t>f_t = (c^2)g_x>f= f(x,t)>g= g(x,t)>c=c(t)Another useful family of solutions isf(x,t) = - i/k exp(i k x) g1(t) g(x,t) = exp(i k x) g1(t)where g1(t) + k c(t)^2 g1(t) = 0Of course you can take real and imaginary parts to get real solutionsinvolving sin(kx) and cos(kx).It may be hard to solve the DE for g1 explicitly if c(t) is given, butif you dont really care what c(t) is you can choose the g1 rstand then get c(t); of course you want g1(t)/g1(t) < 0 so c willbe real. One nice example is c(t) = exp(rt), g1(t) = a1 J_0(exp(rt)/r) + a2 Y_0(exp(rt)/r)where J_0 and Y_0 are Bessel functions of the rst and second kind. Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 => (corresponding to a0)> f(x,t) = x, g(x,t) = t(corresponding to a5)> f(x,t) = x, g(x,t) = tIn the joy of reading your post carefully (making sure every line was as delightful as it appeared) I didnt notice this duplication. I realize its not a problem. So a5=0.Eugene Shuberthttp://www.everythingimportant.org =Faggot ass trolls ght it out, get ur pay per view herewww.sting.com =collier.com says...Well, Jesus should have received gods Y-chromosome if Yahweh is a male > :). So if we ever locate an artifact with Jebus blood on it then we will > have part of sky-daddys genetic code too, presuming of course that the > new testament isnt a bunch of bull.How long after death can DNA be detected?Actually, it starts breaking down hours after death unless processes act to preserve it. However, there are a lot of natural processes which can slow this. > A lock> of hair from Jefferson was used to determine that he was probably > related to Sally Hemmings heirs.There may well have been something done to preserve the hair. But then again, while DNA starts to break down immediately, there might have been long enough fragments to get a pretty certain result in the case of jefferson. I ask because I wondered if DNA testing was ever attempted on the> samples of the Turin ShroudI know that theyve scraped off part of it that were supposed to be blood and found that in fact they are paint pigments. That doesnt seem to phase the true believers. > that were examined by carbon-14 dating> in 1988. At the time, it was determined that the shroud was> probably manufactured in the 13th century, contemporaneous with> Dante Alighieri.Right. I think the real challenge would befinding a bona de relic. If Jesus was real then why is it that we cantfind any real, reliable artifacts that we know belonged to him?-- _____________________________________________________ Quibbler (quibbler247atyahoo.com)It is fashionable to wax apocalyptic about the threat to humanity posed by the AIDS virus, mad cowdisease, and many others, but I think a case can be made that faith is one of the worlds great evils, comparable to the smallpox virus but harder to eradicate. -- Richard Dawkins => bobcrowley@optusnet.com.au says...can switch off now. Occasionally they are accurate to the point whereI can relate to the person involved what they were actually saying.At other times its been embarassingly wrong, although close to thetruth.Now yesterday I was lying on the couch when I seemed to receive animage of a young bloke, slim but t looking, grinning at he looked atthe computer screen. He had a large cigar in his hand (left I think)and was wearing a singlet or t-shirt and something like army trousers. He had fairly short hair, and it seemed to be fair. He was obviouslythinking about what to write in reply, I assume, to one of my items. And that is about all the detail I was given.Does this t you or any other readers of this stream?Just curious.Bob Crowley. => mike420@ziplip.com says...> Can we do a paternety test, preferably on Jerry Springer, so > we canfind out if God was really Jesus Christs father,> as Jesus claims? Well, Jesus should have received gods Y-chromosome if Yahweh is a male > :).Not necessarily. Assuming for the moment that such a person as Jesusexisted and that he was the Son of God (whatever that is), he couldhave a perfectly ordinary Y-chromosome for the simple fact that thebible also makes the claim that God created Adam without the use of ahuman father. Now, supposing that both Adam and Jesus existed and thatboth were created directly by God as the bible claims, there is noreason why Jesus Y-chromosome should be any more special than that ofAdam. ;) =diarmidlogan@yahoo.com says...> mike420@ziplip.com says...> Can we do a paternety test, preferably on Jerry Springer, so > we canfind out if God was really Jesus Christs father,> as Jesus claims? Well, Jesus should have received gods Y-chromosome if Yahweh is a male > :).Not necessarily. Assuming for the moment that such a person as Jesus> existed and that he was the Son of God (whatever that is), he could> have a perfectly ordinary Y-chromosome for the simple fact that the> bible also makes the claim that God created Adam without the use of a> human father.Yeah, but did Genesis claim that Adam was the son of god or only begotten son of god, etc?> Now, supposing that both Adam and Jesus existed and that> both were created directly by God as the bible claims, there is no> reason why Jesus Y-chromosome should be any more special than that of> Adam. ;)Yeah, mebbe :) Im just pointing out what the modern biological view of being a son implies. -- _____________________________________________________ Quibbler (quibbler247atyahoo.com)It is fashionable to wax apocalyptic about the threat to humanity posed by the AIDS virus, mad cowdisease, and many others, but I think a case can be made that faith is one of the worlds great evils, comparable to the smallpox virus but harder to eradicate. -- Richard Dawkins => mike420@ziplip.com says...> Can we do a paternety test, preferably on Jerry Springer, so > we canfind out if God was really Jesus Christs father,> as Jesus claims? Well, Jesus should have received gods Y-chromosome if Yahweh is a male > :).Not necessarily. Assuming for the moment that such a person as Jesus> existed and that he was the Son of God (whatever that is), he could> have a perfectly ordinary Y-chromosome for the simple fact that the> bible also makes the claim that God created Adam without the use of a> human father. Now, supposing that both Adam and Jesus existed and that> both were created directly by God as the bible claims, there is no> reason why Jesus Y-chromosome should be any more special than that of> Adam. ;)Except that the babble never claimed God fathered Adam. It doesclaim he fathered Jesus.jwk =bobcrowley@optusnet.com.au says...> bobcrowley@optusnet.com.au says...can switch off now. Occasionally they are accurate to the point where> I can relate to the person involved what they were actually saying.At other times its been embarassingly wrong, although close to the> truth.Now yesterday I was lying on the couch when I seemed to receive an> image of a young bloke, slim but t looking, grinning at he looked at> the computer screen. He had a large cigar in his hand (left I think)> and was wearing a singlet or t-shirt and something like army trousers.> He had fairly short hair, and it seemed to be fair. He was obviously> thinking about what to write in reply, I assume, to one of my items. > And that is about all the detail I was given.Does this t you or any other readers of this stream?Just curious.Ummm. Bits and pieces perhaps. I dont smoke, am not left handed, am pretty slim, have short hair, slightly tan but not too dark complexion, dont wear army trousers, do wear t-shirts a lot, still fairly young, but not 18. So Id say that youre about 50-50 on right and wrong. But probably many of those things are fairly typical of large groups of people. There are no details that seemed uncanny like sunburned insteps and calluses from wearing sandals too much this summer, black calculator watch on left wrist, collection of soda cans from several day, gun-metal gray oval glasses, just barely 6 feet tall (maybe a quarter inch shy). -- _____________________________________________________ Quibbler (quibbler247atyahoo.com)It is fashionable to wax apocalyptic about the threat to humanity posed by the AIDS virus, mad cowdisease, and many others, but I think a case can be made that faith is one of the worlds great evils, comparable to the smallpox virus but harder to eradicate. -- Richard Dawkins = > Not necessarily. Assuming for the moment that such a person as Jesus> existed and that he was the Son of God (whatever that is), he could> have a perfectly ordinary Y-chromosome for the simple fact that the> bible also makes the claim that God created Adam without the use of a> human father. Now, supposing that both Adam and Jesus existed and that> both were created directly by God as the bible claims, there is no> reason why Jesus Y-chromosome should be any more special than that of> Adam. ;)Well of course we male chauvinists have turned everything upside down.First God created Eve, than he cloned her, cut of a piece from one ofher X-chomosomes (resulting in a Y), and thus created adam.Jesus chromosomes are therefor equal to Marys, as God did the sametrick again. Of course this explains why (according to Johns Gospel)Jesus was such a Fag. He was a woman born in a mans body.Think for yourselfAtheist#1107AmstelveenThe Netherlands (Aug 5, 1950)P.S. Je zus is Dutch for your sister => bobcrowley@optusnet.com.au says...> bobcrowley@optusnet.com.au says...the computer screen. He had a large cigar in his hand (left I think)> and was wearing a singlet or t-shirt and something like army trousers.> He had fairly short hair, and it seemed to be fair. He was obviously> thinking about what to write in reply, I assume, to one of my items. > And that is about all the detail I was given.Does this t you or any other readers of this stream?Just curious.Ummm. Bits and pieces perhaps. I dont smoke, am not left handed, am > pretty slim, have short hair, slightly tan but not too dark complexion, > dont wear army trousers, do wear t-shirts a lot, still fairly young, but > not 18. So Id say that youre about 50-50 on right and wrong. But > probably many of those things are fairly typical of large groups of > people. There are no details that seemed uncanny like sunburned insteps > and calluses from wearing sandals too much this summer, black calculator > watch on left wrist, collection of soda cans from several day, gun-metal > gray oval glasses, just barely 6 feet tall (maybe a quarter inch shy). > No ... the thing that seemed most most incongruous was the cigar,unless it was something that looked like a cigar, but wasnt. If youdont smoke it wasnt you.Probably best to forget it. Ill put it down to a distraction, dreamor whatever. Just that it seemed a bit uncanny since Ive only beendebating these issues for a month.And in any case I did get onto this group to ask about Chaos Maths. It was just that I was diverted by the title, and couldnt resisthaving a go.Bob Crowley. => mike420@ziplip.com says...> Can we do a paternety test, preferably on Jerry Springer, so > we canfind out if God was really Jesus Christs father,> as Jesus claims? Well, Jesus should have received gods Y-chromosome if Yahweh is a male > :).Not necessarily. Assuming for the moment that such a person as Jesus> existed and that he was the Son of God (whatever that is), he could> have a perfectly ordinary Y-chromosome for the simple fact that the> bible also makes the claim that God created Adam without the use of a> human father. Now, supposing that both Adam and Jesus existed and that> both were created directly by God as the bible claims, there is no> reason why Jesus Y-chromosome should be any more special than that of> Adam. ;)Except that the babble never claimed God fathered Adam. It does> claim he fathered Jesus.Actually the bible says that God created both Adam and Jesus so hewould be the father of both. I fail to see what distinction you aremaking between the two unless you are now saying that God had sex withMary. => As for the mention of Jews, that, too, is easily explained.> rears its pin-shaped head, Jews top the list as an obvious target> group. As noted jackass Fred Reed admits, his random clicking on> websites turned up no hard evidence that Jews had taken over> American research activities, but thats because Jews typically> disguise themselves using false names to better inltrate and> destroy American society. Heck, points out Fred, even a guy named> Miller could be a secret Jew! Fred probably had a few dozen> paragraphs on the Protocols of the Elders of Zion written up for the> whenever you burn a ag, a kitten dies public service> advertisements.It sure sounded like he was complimenting Jews and Asians.It sure sounded like he was complimenting Jews and Asians.I believe I said the racism was thinly disguised not so blatantlyobvious even Joseph Hertzlinger can spot it.Of course he was complimenting Asians. As any good racist knows,Asians are generally smarter and more industrious than Caucasians.compliments Jews, unless Sally Chen is supposed to be Jewish. IfFred was complimenting the Jewish people, it was only to point outtheir cleverness in disguising their last names from real Americansand in hiding behind the liberal taboo against forcing students tostate their religion to get into Harvard.If you dont spot anything off-key about phrases like read like a NewDelhi phone book, colonies of Indians, and Jews are doing lotsmore than their share of research or the endless cited lists of nameschosen for their foreign sound, you have my congratulations. Yourcandy-coated dream world must be a beautiful place to live and work!Hell, dont you see anything odd about the fact that all of Fredsevidence involves lists of mostly Indian and Asian-sounding names(with a few Arabic and Greek ones tossed in), yet for some strangereason Jews make a unwarranted and jarring appearance in paragraphthree?If I was babbling on about poverty in India and started interjectingthings like not only is India highly populated, but there sure are alot of Jews around these days and speaking of crowded trains, haveyou noticed how many people in the entertainment industry seem to beJews, would it zip right over your head?Anyway, I call Ban on Politics, at least for me. Im done with thisthread.-- Kevin =>> It sure sounded like he was complimenting Jews and Asians....>compliments Jews, unless Sally Chen is supposed to be Jewish. If>Fred was complimenting the Jewish people, it was only to point out>their cleverness in disguising their last names from real Americans>and in hiding behind the liberal taboo against forcing students to>state their religion to get into Harvard.The fact that they got into Harvard without any special assistance isa compliment.>If you dont spot anything off-key about phrases like read like a New>Delhi phone book, colonies of Indians, and Jews are doing lots ^^^^^^^^^^^^^^^^^^^>more than their share of research or the endless cited lists of names ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^Sounds like a compliment.lojbab-- lojbab lojbab@lojban.orgBob LeChevalier, Founder, The Logical Language Group(Opinions are my own; I do not speak for the organization.)Articial language Loglan/Lojban: http://www.lojban.org =Youre being remarkably dense Nora Baron.Think about it.Math is not a popularity contest. Its not a fashion show. The truthmatters.I want readers to imagine what its like having real mathematiciansshadowing you replying to your posts trying to confuse people.Youre trying to confuse people, Nora, and thats very wrong.Its math after all. It should be simple enough. When youre wrong,you accept it.After all, you dont get to have a math argument thats wrong todaythat becomes right tomorrow.Math doesnt work that way.Remember, the proof wraps up *innite* complexity, and Ive justgiven you a small taste of the with the earlier manipulations.Most of you dont have a snowballs chance in hell of understanding itwith mathematicians working to confuse you.Think about it. If it gives you a headache, well, if it does, it justdoes.The proof can be found by going to the linkhttp://groups.msn.com/AmateurMath and keep in mind that althoughmathematicians may lie, Proofs do not.And I note that Nora Baron may not care about the truth, but aboutsocial issues. That is, her brain may be quite dead set against thetruth, as itd upset the status quo.However I hope that some of you prize that quality of mathematicswhere truth is independent of social issues. If a proof being truemeant that all of humanity would die, youd just have fun with thetime you have left because youd realize that no matter what all ofhumanity thought, no matter what they did, they couldnt change thetruth.Proofs dont give a damn about society, social issues, or whether ornot someone can handle the truth.Proofs are just true.And for some of you, I thought that was a large part of why you likedmathematics.James Harris =Youre being remarkably dense Nora Baron.Think about it.Math is not a popularity contest. Its not a fashion show. The truth> matters.I want readers to imagine what its like having real mathematicians> shadowing you replying to your posts trying to confuse people.Youre trying to confuse people, Nora, and thats very wrong.Its math after all. It should be simple enough. When youre wrong,> you accept it.After all, you dont get to have a math argument thats wrong today> that becomes right tomorrow.Math doesnt work that way.Remember, the proof wraps up *innite* complexity, and Ive just> given you a small taste of the with the earlier manipulations.Most of you dont have a snowballs chance in hell of understanding it> with mathematicians working to confuse you.Think about it. If it gives you a headache, well, if it does, it just> does.The proof can be found by going to the link> http://groups.msn.com/AmateurMath and keep in mind that although> mathematicians may lie, Proofs do not.And I note that Nora Baron may not care about the truth, but about> social issues. That is, her brain may be quite dead set against the> truth, as itd upset the status quo.However I hope that some of you prize that quality of mathematics> where truth is independent of social issues. If a proof being true> meant that all of humanity would die, youd just have fun with the> time you have left because youd realize that no matter what all of> humanity thought, no matter what they did, they couldnt change the> truth.Proofs dont give a damn about society, social issues, or whether or> not someone can handle the truth.Proofs are just true.And for some of you, I thought that was a large part of why you liked> mathematics.James HarrisIm sorry but I scanned the above twice for any refutation of Norasproof that b1 is not an algebraic integer. Could you please address thisissue. Nothing you say above has any relevance to either your proof orNoras. After all, this is mathematics as you say yourself. Would it notbe better to keep the discussion focused on mathemetical points? Is b1an algebraic integer? If so why? Where did Nora go wrong in the unlikelyevent she did?Chuck-- ... The times have been, That, when the brains were out, the man would die. ... Macbeth Chuck Simmons chrlsim@earthlink.net => Youre being remarkably dense Nora Baron.Did you ever notice that Nora Baron spelled backwards is . . . =>Youre being remarkably dense Nora Baron.[etc, etc...]posting time; language seems subtly different. -- Thomas Wasell | All intelligent species own cats. wasell@bahnhof.se | =>>Youre being remarkably dense Nora Baron.> Did you ever notice that Nora Baron spelled backwards is . . .Also, skizziks spelt backwards is skizziks.Gib =>Youre being remarkably dense Nora Baron.>Did you ever notice that Nora Baron spelled backwards is . . .Also, skizziks spelt backwards is skizziks.GibPlease could anyone else a-waiting to reply do us the courtesy of trimmingAlt.ction.original out of any further replies? =Please remove AFO from your crosspost, newbie wit. --Robert>Youre being remarkably dense Nora Baron. *****Amateurs, dilettantes, hacks, cowboys, clonesThe streets groan with little Caesars,Napoleons and cuntsWith their building blocks and their tinyplastic phonesCounting on their ngers, with crumbsdown their fronts -Nick Cave => Youre being remarkably dense Nora Baron.> Uh-oh. Looks like the Challenge is not going to be answered. > Think about it.Math is not a popularity contest. Its not a fashion show. The truth> matters.> Couldnt agree more. Thats why I posted.> I want readers to imagine what its like having real mathematicians> shadowing you replying to your posts trying to confuse people.Youre trying to confuse people, Nora, and thats very wrong.> People, if *I* have confused you, I apologize.> Its math after all. It should be simple enough. When youre wrong,> you accept it.> Good principle. You should give it a whirl.> After all, you dont get to have a math argument thats wrong today> that becomes right tomorrow.Math doesnt work that way.Remember, the proof wraps up *innite* complexity, and Ive just> given you a small taste of the with the earlier manipulations.Most of you dont have a snowballs chance in hell of understanding it> with mathematicians working to confuse you.Think about it. If it gives you a headache, well, if it does, it just> does.The proof can be found by going to the link> http://groups.msn.com/AmateurMath and keep in mind that although> mathematicians may lie, Proofs do not.And I note that Nora Baron may not care about the truth, Nora Baron cares absolutely and deeply about the truth.That was my reason for posting: to get to the bottom ofthings. You have refused to move in that direction. > but about> social issues. That is, her brain may be quite dead set against the> truth, as itd upset the status quo.However I hope that some of you prize that quality of mathematics> where truth is independent of social issues. If a proof being true> meant that all of humanity would die, youd just have fun with the> time you have left because youd realize that no matter what all of> humanity thought, no matter what they did, they couldnt change the> truth.Proofs dont give a damn about society, social issues, or whether or> not someone can handle the truth.Proofs are just true.> By now you should be able to recognize a tautology when you see one, or, in this case, write one. The problem is, what you think is a proof is not. It has an error.It has been pointed out. It is NOT about your trivial lemmaregarding g, c, and r. It is the bit about the form of the factorization when m = 0 generalizing in the only wayyou can imagine to factorizations when m <> 0. You havetried several ways to circumvent this in the last few days.All have failed, and your are back to your original bogusargument. Where your argument breaks down is actually not veryimportant. Several people have now posted rigorous,complete proofs that your main conclusion is wrong. Itdoesnt matter really how you derived it. As long asyou keep claiming the same thing, you have an error.That was the point of my post. As long as you keep failing tofind an error in the several disproofs of your claims, you cannot keep saying you have a proof and that no one has found an error. Mathematicians read sci.math. Perhaps the editor towhom you most recently sent your Advanced PolynomialFactorization paper reads sci.math, or perhaps hisreviewers or the grad student he assigns to refereeyour paper. Perhaps the reviewer will read your postand see that you have not made any effort to refute myargument. On the other hand I think they mayfind what I have given pretty convincing. If so they will send the paper back to you soon with a short note, perhaps saying politely ...not suitable for our journal or some such. Is that what you want? Continuing failure in your questfor recognition? If so, you are following the right strategy. Finally, it is obvious that if you had found an errorin my post, you would have pounced on it. There is everyreason for you to do so. It would make no sense at allfor you to withhold it. Certainly you have noticed errorsin statements by other posters in the past and you did not hesitateto announce them. I must therefore conclude that you havenot found an error in what I have posted. You are concedingthat there is an unanswered counterargument. To continueto maintain that you have a proof, in the face of that, isdishonest and unethical. Bottom line: if you had an actual refutation of my argumentand those of others, you would give it. Obviously you dont. Nora B.> And for some of you, I thought that was a large part of why you liked> mathematics.> James Harris => Youre being remarkably dense Nora Baron.Think about it.Math is not a popularity contest. Its not a fashion show. The truth> matters.Right. So lets stick to the math. Heres Noras:>> I say not. Let m = 1, f = 5, and u = 1. Then>> v = 24, and v^3 + 1 = 13825 = 25*553. It is >> easily veried that >> P(x)/f^2 = 553*x^3 - 72*x + 5.>>If this is factored in the form [1] as claimed by>>Harris, then -u/b1 = -1/b1 is a root of Q(x) = P(x)/f^2. >>That is,>> Q(-1/b1) = 553*(-1/b1)^3 - 72*(-1/b1) + 5 = 0.>>Multiply through by b1^3:>> 5*b1^3 + 72*b1^2 - 553 = 0.>>The expression on the left is a *non-monic*>>polynomial in b1 with integer coefcients,>>and it is *irreducible* over the rationals.>>Therefore b1 cannot be an algebraic integer.So, just sticking to the math and not worrying about popularitycontests, which mathematical statement here is wrong?The truth matters. Is something here untrue? Which statement? - Randy => Youre being remarkably dense Nora Baron.Comma after dense. Im also not sure about the narrative voice in this piece. It seems to be a fusion of rst and second person. Is the protagonist relating the tale, or is the reader supposed to identify with Nora Baron?> Think about it.Okay, Im getting a sense of second-person storytelling here, and Ill come right out and say its extremely difcult to pull this off. Well see how well you manage it.> Math is not a popularity contest. Its not a fashion show. The truth> matters.A bit surreal. Is this an extract from a longer piece?> I want readers to imagine what its like having real mathematicians> shadowing you replying to your posts trying to confuse people.Aha! Somebody wants something. Always a good ctional starting point. Perhaps this is where your story engine is kicking in. Id think about making this the opening paragraph; nothing much seems to have happened up to now.> Youre trying to confuse people, Nora, and thats very wrong.Repetition of confuse people, but its good to see you developing some conict here.> Its math after all. It should be simple enough. When youre wrong,> you accept it.After all, you dont get to have a math argument thats wrong today> that becomes right tomorrow.Math doesnt work that way.I think you could cut these three paras without losing anything.> Remember, the proof wraps up *innite* complexity, and Ive just> given you a small taste of the with the earlier manipulations.Im not sure what youre trying to achieve with this ashback. Perhaps it needs to be eshed out more. As it is, Im losing the thread of your tale.> Most of you dont have a snowballs chance in hell of understanding it> with mathematicians working to confuse you.Think about it. If it gives you a headache, well, if it does, it just> does.Were seeing the problems with second-person narration again. Id advise you to redraft this in rst person, or third person limited.> The proof can be found by going to the link> http://groups.msn.com/AmateurMath and keep in mind that although> mathematicians may lie, Proofs do not.Intriguing idea, but in the event you everfind a paper publisher for this, that hyperlinks going to be a problem. My feeling is that its better for a story to be completely self-contained.> And I note that Nora Baron may not care about the truth, but about> social issues. That is, her brain may be quite dead set against the> truth, as itd upset the status quo.Sorry, this just doesnt follow on from the story up to now. Perhaps some early foreshadowing would help.> However I hope that some of you prize that quality of mathematics> where truth is independent of social issues. If a proof being true> meant that all of humanity would die, youd just have fun with the> time you have left because youd realize that no matter what all of> humanity thought, no matter what they did, they couldnt change the> truth.Inconsistent voice: you seem to have gone from second person singular mode of address to second person plural.> Proofs dont give a damn about society, social issues, or whether or> not someone can handle the truth.Proofs are just true.And for some of you, I thought that was a large part of why you liked> mathematics.Overall, I felt you should have shown more and told less. Cut back on the infodump and try to work some more drama in. I also didnt get a feel for your protagonist -- you need to engage us with your characters if youre to sustain our interest in your ction.-- Huwhttp://huw.hexlibris.com => [snipped]Oops, my apoligies to the residents of the cross-posted groups for inicting this ction critique, in which Im sure they have no interest, on them.Please set follow-ups appropriately.-- Huwhttp://huw.hexlibris.com => Thomas Wasell had these two cents to give:unusual> posting time; language seems subtly different.Its a little suspect, but I wont lose any sleep over it. This post> ogs the same website JSH has previously advertised, and uses the same> *symbols* for *emphasis* as his other *posts*. If it is a forgery, Mr.> Harris can complain to the appropriate parties and have the offending> so into the> killle s/he goes.>Where he will join original Mr. Harris ?Goran => Youre being remarkably dense Nora Baron.Comma after dense. [...]> Youre trying to confuse people, Nora, and thats very wrong.Repetition of confuse people, Yeah, but at least he got the commas right. We should givecredit where credit is due: Some people in other groups tendto say sometimes hes never got _anything_ right; theyregonna have to change their tune.>but its good to see you developing some >conict here.Hes been developing conicts for years. (Oh, thats notwhat you meant. Ill get the hang of this...)>> Its math after all. It should be simple enough. When youre wrong,>> you accept it.>> >> [...]Overall, I felt you should have shown more and told less. Cut back on >the infodump and try to work some more drama in. I also didnt get a >feel for your protagonist -- you need to engage us with your characters >if youre to sustain our interest in your ction.Sorry again. Its just too hard to resist replying to a reply to himthat contains the word ction.************************David C. Ullrich =(follow-ups reset)door:^ ^ > Youre being remarkably dense Nora Baron.^ ^ Did you ever notice that Nora Baron spelled backwards is . . .You bas***d, I didnt notice that. I suffer from aibohphobia.Andy--No, you claim the magpie is to blame for all the worlds ills, based on your ignorance of magpies. (4a7391c12e538ef306d33d71c9482221@TeraNews) Malcolm@malcsplace.com =Wickedly good. ::snipping unrelated groups::> [snipped]Oops, my apoligies to the residents of the cross-posted groups for > inicting this ction critique, in which Im sure they have no > interest, on them.Please set follow-ups appropriately. =>Youre being remarkably dense Nora Baron.> [etc, etc...]posting time; language seems subtly stuff, youll see its justa mix-n-match job.address (as you noted), went through a different news host and posted at anon-JSH time (as you also noted).Id say this is almost certainly not JSH posting (though is his words).--Michael BrownAdd michael@ to emboss.co.nz - My inbox is always open =>Youre being remarkably dense Nora Baron.> [etc, etc...]posting time; language seems subtly different. JSH has posted in alt.ction.original that it isnt = >Why is there something called Abstract Algebra when algebra already >is abstract without calling some particular kind of algebra, >Abstract?Abstract or modern algebra is called abstract or modern because its moreabstract and modern than algebra. >Are any of those other algebras in reach for someone who currently >maintains some Intermediate Algebra knowledge, and who has only faint >memories of Calculus (three semesters) with which he struggled many >many years ago?Here have a go at it. Set theory will be of greater comfort thancalculus. 3.1.3. Denition. A group (G,*) is a nonempty set G together with a binary operation * on G such that the following conditions hold: (i) Closure: For all a,b in G, a*b, written ab, is in G. (ii) Associativity: For all a,b,c in G, a(bc) = (ab)c. (iii) Identity: Theres an identity element e in G such that ea = a = ae, for all a in G. (iv) Inverses: For each a in G theres inverse element a^-1 in G with aa^-1 = e = a^-1 a. a(b + c) = ab + ac. (v) Additive identity: R contains an additive identity 0, such that for all a in R, a + 0 = a and 0 + a = a. (vi) Additive inverses: For each a in R, the equation a + x = 0 has a solution x in R, the additive inverse of a, denoted by -a. The commutative ring R is called a commutative ring with identity when R contains an identity element 1 /= 0 with for all a in R, 1a = a.Also, from your knowledge of algebra give some examples of a ring.Acknowledgment: www.math.niu.edu/~beachy/abstract_algebraAbstract Algebra, 2nd Edition 1996, by John A. Beachy- =In sci.math, Steven:> You have n towels of different sizes placed randomly into a single pile(> after doing your laundry). The one move you are allowed to make is to take a> stack of towel off the top of the pile and place this stack onto the bottom> of the pile. The goal is to get the towels into a pile from the largest> sized towel to the smallest sized towel. What is the least number of moves> required?> This is an odd way of doing laundry; are you sure youre notmisstating the problem? For example, one could hypothesizetwo piles A and B, with an initial state of all towels in arandom order on A, and operations allowing movement ofsome towels from the top of A to the bottom of A or B, and fromthe top of B to the bottom of A or B. One can therefore solvethe problem without much difculty by simply doing a variantof a pick sort; rotate the towel in A just over the smallesttowel into As bottom then pick up the single smallest towel andput it on B; then rotate the towel over the next-smallesttowel in A to As bottom and pick that towel and shove it underthe one towel in B, and so on.A variant of the above would rotate towels from the bottomto the top or simply pulling out towels from the middleof A, which is probably the simplest solution anyway(although the least interesting).Still another variant would pull chunks out of the middle,using a single pile. This is indeed a pick sort and issolvable in at most N moves, and can take at least Nmoves if the pile starts up upside down. (Of coursein that case one might simply ip the entire pile instead.)Other users have commented at some length as to what onedoes with the pile after removal (if not ipped, theuser is simply rotating the accumulator pile; if ipped,the problem is solvable in at most N moves); this is probablywhat you had in mind.Still another slightly more interesting variant wouldallow a user to move single towels into 3 piles, withthe requirement that no towel ever be placed over anothersmaller towel (except during the initial removal from thedryer, perhaps). This is in fact a variant of the classicHanoi Towers puzzle and may require at most 2^N-1 moves.A variant of that would allow pickup of an ordered pilefrom the original stack, as opposed to single towels.Of course it would be a very inconvenient method ofdoing laundry... :-)Just remember the quarters for the washing machine.... :-)-- #191, ewill3@earthlink.netIts still legal to go .sigless. =You got a point there. I for my part hate the reverse numbering of thesigngroups (J. P. Olivier uses it).On the other hand, sometimes you got to think new.I suggest my alternativ notation by two good reasons:1) By the same reason that letters are used in algebra.2) I introduce the idea of counting the functions of the stems,referring to letters are thereby far more conveinent.For instance in Evans numbering the frequency of sign 2 is 19. In thealphabetic notation you got A frequency 19.Moreover my kind of notation forms minatures (thumbnails) of thesigngroups. Evans: A01) 18 1 13 12 2. Hagen: A01) ZUTBA(g).Hagen>> SNIP>> So if any other editor can offer me to get my discovery launched in a>> periodical, cosidering it must be a quick and honest co-operation, he>> shall be very much welcome to contact me about a publication of my>> alphabetic notation of the signs.May I give you an advice, Ole ?.. >If you are re-publishing somewhere the content of your book, could you>use the NUMBERING of the SIGNS adopted by almost every specialist>since Evans proposed it ?..>Your own alphabetic notation of the signs is a real problem for>those who want to understand your work !.. (I am talking by http://www.gvdnet.dk/~hagen/faistos.htm => There are many math-puzzles and problems (covering many math topics),> printed in math journals and on-line, which are written for solvers at> a wide-variety of levels.Like the American Mathematical Monthly for example, its nearlyguaranteed to be subscribed to at any local university and its reallywell done.For logical and mathematical trickery try Raymond M Smullyans books.> (A little off-topic: What percentage of professional mathematicians,> in practice, actually do MORE mathematics on vacation than when> ofcially working?)I quote the above line because its so funny. Im still only astudent, but i revel in winter break and summer break when i cannally get lots of work in!!This poster is very correct though, I was probably too preachy. Readfun math books, histories of math, puzzles or just try guring stuffout. A good book to inspire the physics/science in you is calledConsider a Spherical Cow: A Course in Environmental Problem Solvingby John Harte which plays on estimating and back of the envelopecalculations which can be fun.Good luck!Kevin => First of all, thank you all very much for the responses, very> sounds like maybe I should start with Set Theory and Logic rst in> order to move onto Modern Algebra, Calculus,etc. Are there any> recommended texts that are best for Set Theory and Logic that I could> learn by myself? And also, how good are the Schaums Outlines in> general? I am kind of on a limited budget here, all my money goes into> math and computer books, so the Schaums Outlines seem like a good> deal, but Im not sure whether or not their quality is great though.The best way to get around the limited budget is the library - preferablya university or college library. Even if you are not allowed to check out books at such a library, you can always go inside,find a book, sitat a desk and read. As far as Calculus goes, you really do not need set theory and logic.Most Calculus courses are not proof-oriented nowadays. Nonetheless,eventually you do hit the line between courses that are taught withoutproof and those that are nothing but proof. And for that you mightwant books like Devlins _Sets, Functions, and Logic_ or _How to Readand Do Proofs_ by Solow. I also think there is something to gain fromEuclids Elements in terms of gaining skill with understanding proofs. Its also pleasant to read, in my opinion.I once owned the previously mentioned book on knots and surfacesand I think it is very accessible to those without a lot of math background. Good luck,Hugh =If you read and understood most of the concepts and exercises in L. Gersteinbook, then you should immediately jump into Calculus by Spivak and Abstractalgebra by, maybe, J. Gallian. I would worry about set theory and logiclater. Most undergraduates never take any logic or set theory, save thelittle they pick up here and there in there courses. If you feel relativelycomfortable with writing and reading proofs, then get crackin at realmath. As for Shaums outlines, yes I have found them to be, in general,very good and helpful.You will use proof by contradiction quite extensively in Real analysis.Pick up the Calculus and A.A. books and study, study, study! KeepGersteins book close by so you can refer to it when needed. Be relentlessand patient when things get confusing. You will learn it.LurchFirst of all, thank you all very much for the responses, very>sounds like maybe I should start with Set Theory and Logic rst in>order to move onto Modern Algebra, Calculus,etc. Are there any>recommended texts that are best for Set Theory and Logic that I could>learn by myself? And also, how good are the Schaums Outlines in>general? I am kind of on a limited budget here, all my money goes into>math and computer books, so the Schaums Outlines seem like a good>deal, but Im not sure whether or not their quality is great though. A textbook that I liked a lot was:Introduction to Mathematical Structures and Proof> by Larry Gerstein I found it very easy to read on my own. It starts out> with the usual stuff: set theory, logic, proof by> induction, proof by contradiction and it is all> explained very well.adam> => First of all, thank you all very much for the responses, very> sounds like maybe I should start with Set Theory and Logic rst in> order to move onto Modern Algebra, Calculus,etc. Thats a safe way to go, because 1/ you get a broader perspective of whatyoure learning and 2/ youll need it at a point anyway. A lot of problemsdealing with multiple-variable functions are solved with basic topology, forinstance.One thing you should try to learn as soon as possible (and its possible rightnow) is to conduct proofs. First, its much more interesting than justanswering questions and doing calculations, second, youll need it at a pointtoo, third, it allows you to understand what youre studying better, and isvery useful when doing exercises (youll often end up using a techniqueyouve seen in a proof of a theorem or property).> And also, how good are the Schaums Outlines in> general? I am kind of on a limited budget here, all my money goes into> math and computer books, so the Schaums Outlines seem like a good> deal, but Im not sure whether or not their quality is great though.>Theyre very good in general. As I already said, the main feature missing iscomplete proofs to all theorems (or at least those accessible to anyundergrad). But as Hugh stated, your closest college library can be veryhelpful. Itll probably have books with complete proofs. Sam-- People sometimes ask me if it is a sin in the Church of Emacs to use vi. Using a free version of vi is not a sin; its a penance. - Richard Stallman => First of all, thank you all very much for the responses, very> sounds like maybe I should start with Set Theory and Logic rst in> order to move onto Modern Algebra, Calculus,etc. Are there any> recommended texts that are best for Set Theory and Logic that I could> learn by myself? And also, how good are the Schaums Outlines in> general? I am kind of on a limited budget here, all my money goes into> math and computer books, so the Schaums Outlines seem like a good> deal, but Im not sure whether or not their quality is great though.> John K.You willfind many good study sources on the Internet, free.Start athttp://www.ams.org/mathweb/mi-books.htmland play with Google for an hour; you willfind more than you can possibly read.Some online sources are pretty rough, often unedited notes, but if you spend a bit of time searching, you willfind materials just as good as the expensive books from the bookstore.In fact, it would be great if somebody took the time to compile a list of recommended introductory online sources for various areas of math.NX-Cise: tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Punge: Micro$oft =>Now, with my background, I am>sort of limited as to what I can study (understand). But, I would>like to learn more math that doesnt require as many prereqs, are>there any suggestions as to another branch of math I can teach>myself?Youll need some Set Theory, but you may be able to pick it up on they. In no particular order:Geometry. I know that youve had a course by that name, but itsprobably strong on calculations and weak on theory, with lots ofapparent proofs that arent valid. Look for a good book on SyntheticGeometry. If you canfind a copy and it isnt too difcult, tryavailable as a Dover reprint.Number Theory. While advanced topics require tools you dont have, youshould be able to handle introductory texts.Real Analysis, not including Measure Theory. If you canfind a copy,Apostols book is excellent. Some people contend, with good reasons,that this should be taught prior to Calculus.Once you have some Real Analysis under your belt, Vector Spaces. Ifound Halmoss Finite Dimensional Vector Spaces to be easy. As thetitle suggests, Halmos restricts himself to an easy but stillinteresting subset of the eld. If you are frustrated by the narrowscope and can handle something more advanced, look for books onAlgebra. There are survey texts that cover groups, rings and elds.Topology. I like Kelleys General Topology; some mayfind it dated.>Is it too hard to teach yourself Calculus?That depends on both your background and on how your mind works. Somepeople do well only in a classroom setting, some do well only on theirown, and some ourish in either. Note: once you get up to speed, check the local universities. Someschools will allow outside students to audit classes, participate incolloquia, etc. Even if you learn well on your own, contact with othermathematically minded people can be helpful, both for advice ondifcult topics and for advice on other areas you might ndinteresting.-- Shmuel (Seymour J.) Metz, SysProg and tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Punge: Micro$oft = at 03:23 PM, john.mitchell@autodesk.com (John Mitchell) said:>For Calculus, I started on my own with Calculus for the Common Man.I tried it when I was young and found it impossible to understand;copy of Thomas, which used a rigorous Epsilon-Delta approach, and Irealized that they other seemed wrong because it was wrong.>Theres a wonderful series by Newman (?) called The World of>Mathematics containing writings of many famous mathematicians and>physicists. I think youdfind it fascinating.I spent many enjoyable hours with that book. At the time it was infour volumes in a slip cover; I dont know whether its been reprintedas on big book.-- Shmuel (Seymour J.) Metz, SysProg and JOATto tanbanso@iinet.net.auX-CompuServe-Customer: YesX-Coriate: admin@interspeed.co.nzX-Ecrate: tanandtanlawyers.comX-Punge: Micro$oft =>Are there any>recommended texts that are best for Set Theory and Logic that I could>learn by myself? You might want to start with Halmoss Naive Set Theory, then tacklesomething more advanced.-- Shmuel (Seymour J.) Metz, => at 03:23 PM, john.mitchell@autodesk.com (John Mitchell) said:>For Calculus, I started on my own with Calculus for the Common Man.I tried it when I was young and found it impossible to understand;>copy of Thomas, which used a rigorous Epsilon-Delta approach, and I>realized that they other seemed wrong because it was wrong.another book. The one I was familiar with many years ago was used incalculus sequences for engineers, and managed to give its rigorousdenition of a derivative before its rigorous denition of alimit.>Theres a wonderful series by Newman (?) called The World of>>Mathematics containing writings of many famous mathematicians and>>physicists. I think youdfind it fascinating.I spent many enjoyable hours with that book. At the time it was in>four volumes in a slip cover; I dont know whether its been reprinted>as on big book.Larry(this space unintentially left blank ..... =>another book. The one I was familiar with many years ago was used in>calculus sequences for engineers, and managed to give its rigorous>denition of a derivative before its rigorous denition of a>limit.>Huh? I recall Thomas having an entire chapter on limits before discussing derviatives. And when he did fudge something (e.g., the denition of the Riemann integral), he noted that a rigorous denintion requires more details.-- Stephen J. Herschkorn herschko@rutcor.rutgers.edu => at 03:23 PM, john.mitchell@autodesk.com (John Mitchell) said:> >For Calculus, I started on my own with Calculus for the Common Man.I tried it when I was young and found it impossible to understand;> copy of Thomas, which used a rigorous Epsilon-Delta approach, and I> realized that they other seemed wrong because it was wrong.Could be, I have no recollection of the material, but I do recalllearning the basic concepts of derivative and integral from thebook. I later took a Calculus course in high school. We used Thomas,and I enjoyed it. Another one I like is Courant and JohnsIntroduction to Calculus and Analysis. Its alive with connectionsto physics and geometry, unlike the bone-dry texts that were used whenI was teaching.Again, I think the O.P. should peruse several different books andeither read several or choose the one that he likes best. Later on, hecan read others.John Mitchell =>another book. The one I was familiar with many years ago was used in>calculus sequences for engineers, and managed to give its rigorous>denition of a derivative before its rigorous denition of a>limit.> Huh? I recall Thomas having an entire chapter on limits before > discussing derviatives. And when he did fudge something (e.g., the > denition of the Riemann integral), he noted that a rigorous > denintion requires more details.If you mean Calculus and Analytic Geometry by George B. Thomas,Addison-Wesley, he does dene the derivative of a function beforeformally introducing the notion of a limit (at least in my 3rdedition). In fact, he rst denes the slope of (the graph of) offunction somewhat informally, then uses that to motivate thedenition of derivative, and nally uses the denition ofderivative to motivate a more rigorous study of limits. The wholediscussion is preceded by a discussion of velocities and ratesintended to motivate the denitions that follow. I, for one, thinkthis is pedagogically more effective than starting with a rigorousdenition of limit and moving on from there. The modern textbooksthat were used when I was teaching dispensed with motivationaltogether.John Mitchell =>>another book. The one I was familiar with many years ago was used in>>calculus sequences for engineers, and managed to give its rigorous>>denition of a derivative before its rigorous denition of a>>limit.>>Huh? I recall Thomas having an entire chapter on limits before >discussing derviatives. And when he did fudge something (e.g., the >denition of the Riemann integral), he noted that a rigorous >denintion requires more details.As I said, it was many years ago (upwards of 30), and there may bemultiple books or multitple Thomases, but the book I recall gave theusual limit denition of derivative BEFORE the denition of a limit.Larry(this space unintentially left blank ..... www.geocities.com/erniespage... =I am having trouble understanding this problem. How can a sequence ontegers such as {1,2,3,1,2,3,.......} have 1, 2, and 3 as limits? Ithought that the limit of a sequence was unique. How does | x_n - L | < e,for n > N, work if the sequence just keeps cycling? Shouldnt the distancebe ever decreasing as n --> oo ? If say 3 is the limit, then| x_n - 3 | < e will hold only for 3, not 2 or 1.Maybe this is a bad example because it is monotonically increasing, but myconfusion is in any sequence which has repeating elements.TIALurch =I am having trouble understanding this problem. How can a sequence of>integers such as {1,2,3,1,2,3,.......} have 1, 2, and 3 as limits? It doesnt. It has them as limit points, which is something different.> I>thought that the limit of a sequence was unique. When it exists, a limit exists. However, some people say that x_0 is alimit point of the sequence if there is a SUBsequence that convergesto x_0. So maybe thats what is confusing you? what I accept as reality. Calvin (Calvin and Hobbes) I am having trouble understanding this problem. How can a sequence of> integers such as {1,2,3,1,2,3,.......} have 1, 2, and 3 as limits?It doesnt. It has 1, 2, and 3 as subsequential limits. =I am having trouble understanding this problem. How can a sequence of>integers such as {1,2,3,1,2,3,.......} have 1, 2, and 3 as limits?It doesnt. It has them as limit points, which is something different.>I know them as cluster points, the difference between a limit, a cluster pointis the sequence is respectively eventually, frequentlyarbitrarly close to the limit, cluster point> I>thought that the limit of a sequence was unique.When it exists, a limit exists. However, some people say that x_0 is a> limit point of the sequence if there is a SUBsequence that converges> to x_0. So maybe thats what is confusing you?>Yes, each cluster point is a limit of some subsequence and visa versa.Also technically limit points are cluster points. =I am working out of Rudins Principles of mathematical analysis. Therst two exercises from Chp. 2 are:1) Construct a bounded set of real numbers with exactly 3 limit points.2) Construct a compact set of real numbers whose limit points form acountable set.Does my sequence t the bill?x_n = {1,2,3,1,2,3,......}It is bounded since d(p,q) < 3, using the standard metric, for all p in Xand some q in X . It is closed since it contains all of its limit points.If it is closed and bounded, then it is compact. The set {1,2,3} is clearlycountable. Therefore, the sequence should t the two aforementionedrequirements. Right?Is this correct?TIALurchI am having trouble understanding this problem. How can a sequence of> integers such as {1,2,3,1,2,3,.......} have 1, 2, and 3 as limits? I> thought that the limit of a sequence was unique. How does | x_n - L | for n > N, work if the sequence just keeps cycling? Shouldnt thedistance> be ever decreasing as n --> oo ? If say 3 is the limit, then> | x_n - 3 | < e will hold only for 3, not 2 or 1.Maybe this is a bad example because it is monotonically increasing, but my> confusion is in any sequence which has repeating elements.TIALurch => I am working out of Rudins Principles of mathematical analysis. The> rst two exercises from Chp. 2 are:1) Construct a bounded set of real numbers with exactly 3 limit points.>{ 1/n, 1 + 1/n, 2 + 1/n | n in N }. BTW, its not a compact set.> 2) Construct a compact set of real numbers whose limit points form a> countable set.>Whew, Rudin wants me to think non-trivial. Does this work? { 1/m + 1/mn | n,m in N }No, its not compact.Exercise: make it compact by adding a some points.> Does my sequence t the bill?> x_n = {1,2,3,1,2,3,......}>No, its not a set, its a sequence. A limit point or accumulation pointof a set is different than a cluster point of a sequence which alas is alsoknow as _a_ limit point of the sequence or the limit of a subsequence.Yes, 3 is the limit of the subsequence 3,3,3,3... of 1,2,3,1,2,3,... butits not a limit point of { 1,2,3 }. That set has no limit points. Why?Find and read Rudins denition of a limit point of a _set_.> It is bounded since d(p,q) < 3, using the standard metric, for all p in X> and some q in X . It is closed since it contains all of its limit points.> If it is closed and bounded, then it is compact. The set {1,2,3} is clearly> countable. Therefore, the sequence should t the two aforementioned> requirements. Right?>No, sequences arent compact, nor closed; those are property of sets.How does Rudin use countable? As nite or equinumerous to N, or justequinumerous to N. Did Rudin in 2) mean innitely countable? =>I am working out of Rudins Principles of mathematical analysis. The>rst two exercises from Chp. 2 are:1) Construct a bounded set of real numbers with exactly 3 limit points.2) Construct a compact set of real numbers whose limit points form a>countable set.Does my sequence t the bill?>x_n = {1,2,3,1,2,3,......}It is bounded since d(p,q) < 3, using the standard metric, for all p in X>and some q in X . It is closed since it contains all of its limit points.>If it is closed and bounded, then it is compact. The set {1,2,3} is clearly>countable. Therefore, the sequence should t the two aforementioned>requirements. Right?Is this correct?He probably meant to say countably innite (or the book he is usingdenes countable to be nite).In that case, consider the closure (in RxR, of course) of{(1/n,1/m)|m,n natural numbers}. The cluster points (I dont likelimit points, it causes all the confusion we have seen in this thread)are (0,0), {(0,1/n)}, and {(1/n,0)}. TIA>Lurch> I am having trouble understanding this problem. How can a sequence of>> integers such as {1,2,3,1,2,3,.......} have 1, 2, and 3 as limits? I>> thought that the limit of a sequence was unique. How does | x_n - L | <>e,>> for n > N, work if the sequence just keeps cycling? Shouldnt the>distance>> be ever decreasing as n --> oo ? If say 3 is the limit, then>> | x_n - 3 | < e will hold only for 3, not 2 or 1.>> Maybe this is a bad example because it is monotonically increasing, but my>> confusion is in any sequence which has repeating elements.>> TIA>> Lurch>Larry(this space unintentially left blank ..... => I am working out of Rudins Principles of mathematical analysis. The> rst two exercises from Chp. 2 are:1) Construct a bounded set of real numbers with exactly 3 limit points.2) Construct a compact set of real numbers whose limit points form a> countable set.Does my sequence t the bill?> x_n = {1,2,3,1,2,3,......}It is bounded since d(p,q) < 3, using the standard metric, for all p in X> and some q in X . It is closed since it contains all of its limit points.> If it is closed and bounded, then it is compact. The set {1,2,3} is clearly> countable. Therefore, the sequence should t the two aforementioned> requirements. Right?Is this correct?Answering 1) and 2) is not as simple as 1,2,3. First, any set with a limit point must be innite; this is right on the same page in Rudin where limit point is dened. Second, note that Rudin denes a set to be countable iff it is in 1-1 correspondence with the positive integers. So nite sets are not countable with this denition. You need to read these denitions a lot more carefully. =Why isnt a sequence a set, albeit an innite set?> Well a set and a sequence are two different things.A set is (informally speaking) a collection of objects. So {1,2,3} is a set. Each element is counted only once, in other words your set {1,2,3,1,2,3,1,2,3 . . .} is really the set {1,2,3}. It has exactly three elements.A sequence is a function from the natural numbers N to a set S. Instead of writing the sequence as f(1), f(2), f(3), etc., we write s_1, s_2, s_3, etc.For example you have a sequence 1, 2, 3, 1, 2, 3, 1, 2, 3 . . . Thats a function N->{1,2,3}.But the set {1,2,3} only has three elements. > Also, I was reading Knopps book Innite series and in that book he> states that any sequence such as {0,1,0,1,0,....} has ipso facto 0 and 1> as limit points. Maybe, I am just confusing cluster (accumulation) point> with limit point, I dont know.Same thing, different names. It would be helpful if you would tell us what Knopps exact denitions are for these terms so we can see if he is making some subtle distinction in his denitions. A lot of topology books use a lot of these denitions in different ways, for example some books use countable to mean countably innite while others use countable to mean nite or countably innite. Its important to make sure you understand what the author means. =Why isnt a sequence a set, albeit an innite set?If you know and understand the denition of a sequence, youll have your answer.> Also, I was reading Knopps book Innite series and in that book he> states that any sequence such as {0,1,0,1,0,....} has ipso facto 0 and 1> as limit points. Maybe, I am just confusing cluster (accumulation) point> with limit point, I dont know.If youre doing exercises in Rudin, doesnt it make sense to use the denitions in Rudin? Rudins denition of limit point is quite standard; you wont see the word sequence in that denition. =Im not a mathematician, hopefully you can help me with my question:I was looking at some equations in a text book that were written insummation notation and felt that they would be clearer if rewritten asmatrix operations. In the course of rewriting, it struck me that thereare actually many families of summation notation equations that can bereasonably translated into matrix/vector operations. The reverse is alsotrue; many basic matrix operations can be easily translated into summationstyle symbology. This especially comes to my mind in the case of tryingto write computer procedures for linear algebra; you could (although itsnot always the most efcient way) code many matrix operations as a seriesof control loops iterating over matrix subscripts, which seems to meclosely related to standard summation notation.A simple example of what Im getting at is that sum(i=1)(i=n)(Ai*Bi) canbe easily rewritten as A dot B provided you make the intuitivesubstitution that vector A is the n-dimensional vector [A1 A2 ... An] andB likewise. This becomes matrix operation if you bother to keep track ofrows versus columns.Conversely, I could convert a simple dot product to the more cumbersomesum notation, something which it seems I am implicitly doing if I try tocode a dot product as a for(i=0, i Im not a mathematician, hopefully you can help me with my question: > > I was looking at some equations in a text book that were written in > summation notation and felt that they would be clearer if rewritten > as matrix operations. In the course of rewriting, it struck me that > there are actually many families of summation notation equations that > can be reasonably translated into matrix/vector operations. The > reverse is also true; many basic matrix operations can be easily > translated into summation style symbology. This especially comes to > my mind in the case of trying to write computer procedures for linear > algebra; you could (although its not always the most efcient way) > code many matrix operations as a series of control loops iterating > over matrix subscripts, which seems to me closely related to standard > summation notation. > > A simple example of what Im getting at is that sum(i=1)(i=n)(Ai*Bi) > can be easily rewritten as A dot B provided you make the intuitive > substitution that vector A is the n-dimensional vector [A1 A2 ... > An] and B likewise. This becomes matrix operation if you bother to > keep track of rows versus columns. > > Conversely, I could convert a simple dot product to the more > cumbersome sum notation, something which it seems I am implicitly > doing if I try to code a dot product as a for(i=0, i control loop. > > What I want to know is: has this relationship and the interconversion > of these families of equations been formalized in any signicant > way? Is there any symbolic formalism for taking an equation in > summation notation and rewriting it as standard matrix/vector > operations (or vice versa, although this way is easier to gure out > intuitively at least for me)? > > Like I said, Im not a mathematician, so if this is a nonsensical > question and you can explain specically why it is nonsensical that > would also interest me. >There are programming languages in which the n-dimensional vector is abasic data type, with all the right operations on it.APL was the rst such language. Its modern version is J:http://www.jsoftware.com/J is a highly developed well thought out formalism for what you areasking for, and much more.You can download a J interpreter from the J Software web site, and alsopapers and books showing applications of J in various areas of mathematics.Nemo =>Draw a path consisting of connected straight>>line-segments within the nonagon, so that each segment starts/ends on>>the vertexes of the 9-gon, and where each vertex is visited exactly>>once, and where the 1st and last segment are connected.> >>Now the path must be such that there are a total of exactly 13 crossings >>inside the nonagon.> >> This rule generates the right sequence:>> Starting at an arbitrary vertex, visit alternately the second and third vertex>> from the last, when youfind that you would be revisiting a vertex, skip to the>> next empty one.--les ducs dEnron!>http://members.tripod.com/~american_almanacas there are only three (well-known) star possibilites,>other than the simple enneagon:>skip every-other vertex (nine total crossings);>skip over two (three trigona, each line crosses four others);>skip over three (each line crosses six others, but>combinatorics isnt my specialty; I just drew them .-)And which of these gives exactly 13 crossings?-- Patrick Hamlyn posting from Perth, Western AustraliaWindsurng capital of the Southern HemisphereModerator: polyforms group (polyforms-subscribe@egroups.com) => I dont grok the difculty,> as there are only three (well-known) star possibilites,> other than the simple enneagon:> skip every-other vertex (nine total crossings);> skip over two (three trigona, each line crosses four others);> skip over three (each line crosses six others, but> combinatorics isnt my specialty; I just drew them .-)>Draw a path consisting of connected straight>line-segments within the nonagon, so that each segment starts/ends on>the vertexes of the 9-gon, and where each vertex is visited exactly>once, and where the 1st and last segment are connected.> >[...] such that there are a total of exactly 13 crossings>inside the nonagon.This rule generates the right sequence:> Starting at an arbitrary vertex, visit alternately the second and > third vertex from the last, when youfind that you would be > revisiting a vertex, skip to the next empty one.its correct, there are 467 other permutations (with 1 xed inplace) that generate 13 crossings, eg 135268497, 139752846, etc.most of which probably ignore your rule. Here are the numbersof permutations that give various crossing counts, per program: 9. 2 10. 54 11. 108 12. 252 13. 468 14. 450 15. 198 16. 576 17. 396 18. 168 19. 144 20. 162 21. 90 23. 18 24. 18 27. 2The program considers the 3106 permutations that start with1 and have no adjacencies (like 1 next to 9 or 2) but it doesnot exclude paths followed in reverse, so for example it distinguishes 2 9-crossing and 2 27-crossing paths even though they look the same when drawn.1. Is there an easy way (aside from enumeration) to showthat no 22- and 25-crossing paths exist?2. In a 23-crossing path I drew, I dont see offhand how to relate the permutation (148372596) to line intersections since some steps that skip 3 nodes cross 5 lines, and others cross 6.3. There is a straightforward way to tell from a permutation whether the path has an axis of symmetry (for example, 136824795 does, 135268497 doesnt).Is there an easy way to tell if all paths of a givencrossing count are symmetri 7ï WEcf !.85ÃLèÿÿ no adjacencies (like 1 next to 9 or 2) but it does> not exclude paths followed in reverse, so for example it > distinguishes 2 9-crossing and 2 27-crossing paths even though > they look the same when drawn.> 1. Is there an easy way (aside from enumeration) to show> that no 22- and 25-crossing paths exist?2. In a 23-crossing path I drew, I dont see offhand how to > relate the permutation (148372596) to line intersections > since some steps that skip 3 nodes cross 5 lines, and others > cross 6.3. There is a straightforward way to tell from a > permutation whether the path has an axis of symmetry > (for example, 136824795 does, 135268497 doesnt).> Is there an easy way to tell if all paths of a given> crossing count are symmetric?> -jiwThe path-counting is very interesting.And so are your questions.I wonder about the cases for other m-gons.Which number of crossings, for an m-gon, has the most solutions?What is the most possible number of crossings for an m-gon?And, inspired by James Waldbys question 1, which numbers (< maxnumber of crossings) of crossings has no solutions for an m-gon?LeroyQuet =...> of permutations that give various crossing counts, per program:> 9. 2> 10. 54[etc.]> 21. 90> 23. 18> 24. 18> 27. 2...> 1. Is there an easy way (aside from enumeration) to show> that no 22- and 25-crossing paths exist?... [for closed paths with no adjacent-vertex arcs]> I wonder about the cases for other m-gons.Which number of crossings, for an m-gon, has the most solutions?What is the most possible number of crossings for an m-gon?And, inspired by James Waldbys question 1, which numbers (< max> number of crossings) of crossings has no solutions for an m-gon?I have run my program for a few other m, with results shown belowfor min and max numbers of crossings, and non-existent crossing-counts,with some obvious patterns developing! m min max max missing-crossing-counts 5 5 5 m - 6 7 7 m+1 - 7 7 14 2m 9 12 13 8 9 17 2m+1 - 9 9 27 3m 22 25 2610 10 31 3m+1 -11 11 44 4m 39 42 4312 12 49 4m+1 --jiw =...> I have run my program for a few other m, with results shown below> for min and max numbers of crossings, and non-existent crossing-counts,> with some obvious patterns developing! m min max max missing-crossing-counts> 5 5 5 m -> 6 7 7 m+1 -[etc]> 12 12 49 4m+1 -Following is corrected and slightly expanded table. n min max max missing 5 5 5 n - 6 7 7 n+1 - 7 7 14 2n 9 12 13 8 9 17 2n+1 - 9 9 27 3n 22 25 2610 10 31 3n+1 -11 11 44 4n 39 42 4312 11 49 4n+1 -13 12 65 5n 60 63 64A graph of crossing-counts frequency for m=7 to 13 appears athttp://pat7.com/jp/counts.plot.jpg while the data and setup for the plot are in http://pat7.com/jp/counts.dat andhttp://pat7.com/jp/counts.gnuplot-jiw => why x top-posting?Because it should be possible to read any message, inisolation, from top to bottom. Top-posting breaks that.> often, it [non-top-posting] just means that one has> to use the scroll-down, instead of immediately seeing> what the reply is, thus engaging the concept> of context (or memory) in the OP and his reply.Youre supposed to snip the parts of the quote that youre notreplying to. That negates this argument. =Why does the following implication never holds true?P is trueQ is falseP => Q is falseWhen I think about it, it doesnt really feel right. Why cant I provethat =