mm-4639 === Subject: : Re: #math on irc > Somebody recommended #math as an excellent irc chat. > On what network or server is #math? > Are there other good irc groups? > According to hppt://searchirc.com, the liveliest #math is on > freenode. Dang, http://searchirc.com No, an IR network. > Number of contributors is not proportional to quality > though, often the opposite. > Are there other #math groups on different servers? Did you search for math at the above link? No, I've finally given the address the correct protocol so that the next time I check this group, I'll be able to read the site. > If so - why did you not see an answer to your question amongst the > results it generated? > If not - whyever not? Does the # have any syntactical or other meaning? It prefixes channel names. > === === Subject: : Re: #math on irc <87bq2yehqi.fsf@nonospaz.fatphil.org> <877idlclca.fsf@nonospaz.fatphil.org> Somebody recommended #math as an excellent irc chat. > On what network or server is #math? > Are there other good irc groups? > According to hppt://searchirc.com, the liveliest #math is on > freenode. Dang, http://searchirc.com > No, an IR network. > Number of contributors is not proportional to quality > though, often the opposite. > Are there other #math groups on different servers? Did you search for math at the above link? No, I've finally given the address the correct protocol > so that the next time I check this group, I'll be able to read > the site. > Double dang wangles. Access Denied We're sorry. The software you are using to access our website is not allowed. Some examples of this are e-mail harvesting programs and programs that will copy websites to your hard drive. If you feel you have gotten this message in error, please send an e-mail addressed to > If so - why did you not see an answer to your question amongst the > results it generated? > If not - whyever not? Does the # have any syntactical or other meaning? It prefixes channel names. === === Subject: : Re: #math on irc > http://searchirc.com The URLs are all broken, but you'll want to find a local server on the appropriate network anyway. Or not. 1. irc:// #math Freshness factor: 99.6 Users: Current: 11, Avg: 12, Max: 19 Network: GameSurge Review: There are no reviews of this channel Topic: Welcome to #math, the GameSurge Math channel, also known as Clan MATH in some ellipses with zero eccentricity. For additional help, check out #homework! 2. irc:// #math Established Freshness factor: 99.8 Users: Current: 330, Avg: 323, Max: 377 Network: freenode Review: There are no reviews of this channel Directory: Topic: Education and Schools > Education Discussions Don't ask to ask, just ask. | All aspects of mathematics (not software) | www.freenode-math.com | LaTeX paste: http://mathbin.net | Using mbot: www.freenode-math.com/index.php/Mbot | Math Encyclopedia: http://eom.springer.de | Off-topic? #not-math (this i 3. irc:// #mathematics Established Freshness factor: 99.5 Users: Current: 19, Avg: 17, Max: 27 Network: EFnet Review: There are no reviews of this channel Topic: Math Seminar Channel. See #math for people. Previous talks and schedule: http://www.efnet-math.org/w/Seminars 4. irc:// #mathematics Established Freshness factor: 99.8 Users: Current: 7, Avg: 8, Max: 38 Network: freenode Review: There are no reviews of this channel Topic: This is a channel for discussing mathematics. | For math help, try #Math. | Take all off-topic related chat to #Not-Math 5. irc:// #do.math Freshness factor: 99.6 Users: Current: 1, Avg: 1, Max: 2 Network: GameSurge Review: There are no reviews of this channel Topic: do.math! cool kids club. 6. irc:// #not-math Established Freshness factor: 99.8 Users: Current: 82, Avg: 82, Max: 95 Network: freenode Review: There are no reviews of this channel 7. irc:// #math Freshness factor: 99.5 Users: Current: 59, Avg: 58, Max: 77 Network: Undernet Review: There are no reviews of this channel Directory: Topic: Science > Mathematics Topic is 'Math chat & more...enjoy! Check this out--> POTD 5/22 http://undernetmath.wordpress.com/potd/ 8. irc:// #math Freshness factor: 99.3 Users: Current: 4, Avg: 8, Max: 244 Network: rizon Review: There are no reviews of this channel Topic: [+cnt] Don't ask to ask. Ask || Need non-math help? Join #RizonVersity || (cos(.91.9f-r)-sin(.91.9f))(r.89.[L eftGuillemet] - 2r.8cî cos(2.91.9f+2.4)+0.9)+(0.62r).8cÍ.89 .Á.89.Á.89.Á < 0 || Prerequisite: Sense of Humor 9. irc:// #maths Freshness factor: 99.9 Users: Current: 2, Avg: 2, Max: 12 Network: DevNode.org Review: There are no reviews of this channel Topic: [+nt] Devnode Math(s) channel | Make sure there are no letters or put your question between {curly brackets} if you want your math statement evaluated | pi and e work | pie has also been added | please report bugs | word problems do *not* work | type HALP -- -- Microsoft voice recognition live demonstration === === Subject: : Re: #math on irc <87bq2yehqi.fsf@nonospaz.fatphil.org> <877idlclca.fsf@nonospaz.fatphil.org> <878wxzu9t3.fsf@nonospaz.fatphil.org http://searchirc.com The URLs are all broken, but you'll want to find a local server > on the appropriate network anyway. Or not. > They are? Then where did the information below come from? The network is the same as the server? When IRC software asks for server, could I for example, answer 'freenode'? > 3. irc:// #mathematics Established Freshness factor: 99.5 > Users: Current: 19, Avg: 17, Max: 27 > Network: EFnet > Review: There are no reviews of this channel > Topic: Math Seminar Channel. See #math for people. Previous talks and > schedule: http://www.efnet-math.org/w/Seminars 4. irc:// #mathematics Established Freshness factor: 99.8 > Users: Current: 7, Avg: 8, Max: 38 > Network: freenode > Review: There are no reviews of this channel > Topic: This is a channel for discussing mathematics. | For math help, > try #Math. | Take all off-topic related chat to #Not-Math 7. irc:// #math Freshness factor: 99.5 > Users: Current: 59, Avg: 58, Max: 77 > Network: Undernet > Review: There are no reviews of this channel > Directory: > Topic: Science > Mathematics > Topic is 'Math chat & more...enjoy! Check this out--> POTD 5/22 > http://undernetmath.wordpress.com/potd/ === === Subject: : Re: Factoring the alternating zeta function Raymond Manzoni a .8ecrit : > Robert Adams a .8ecrit : > Is it possible or impossible to factor the alternating zeta function > into product form, with each product having a single zero at one of > the frequencies of the zeta function? > If this has been attempted, what were the approaches used? > Bob Adams > Look at the Hadamard product here : > http://en.wikipedia.org/wiki/Riemann_zeta#Hadamard_product and, if the alternating zeta function is Dirichlet eta function, then eta(s)= sum_{n=1}^oo (-1)^(n-1)/n^s > so that eta(s) = (1-2^(1-s)) zeta(s) Not sure it will really help, > Raymond Hmmm... I supposed that your of the zeta function were zeta's complex zeros but perhaps that you had something else in view... ?? === === Subject: : Re: Factoring the alternating zeta function <48371705$0$918$ba4acef3@news.orange.fr> <48371d50$0$842$ba4acef3@news.orange.fr> posting-account=p7BECQkAAACQMZj7smqCHnVXdUCwPmSR MathPlayer 2.10b; InfoPath.1; .NET CLR 2.0.50727; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Raymond Manzoni a .8ecrit : Robert Adams a .8ecrit : > Is it possible or impossible to factor the alternating zeta function > into product form, with each product having a single zero at one of > the frequencies of the zeta function? > If this has been attempted, what were the approaches used? > Bob Adams æ æLook at the Hadamard product here : > æ æhttp://en.wikipedia.org/wiki/Riemann zeta#Hadamard product æ æand, if the alternating zeta function is Dirichlet eta function, then æ æeta(s)= sum {n=1}^oo (-1)^(n-1)/n^s > æ æso that eta(s) = (1-2^(1-s)) zeta(s) æ æNot sure it will really help, > æ æ æ æ æ æ Raymond æ æ æ Hmmm... I supposed that your of the zeta function > were zeta's complex zeros but perhaps that you had something else in > view... ??- Hide quoted text - - Show quoted text - Sorry, my engineering background is showing! Yes, I mean the the locations of the complex zeroes on the line s = +1/2. Bob Adams === === Subject: : Re: Factoring the alternating zeta function <48371705$0$918$ba4acef3@news.orange.fr> <48371d50$0$842$ba4acef3@news.orange.fr> posting-account=p7BECQkAAACQMZj7smqCHnVXdUCwPmSR MathPlayer 2.10b; InfoPath.1; .NET CLR 2.0.50727; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) Raymond Manzoni a .8ecrit : Robert Adams a .8ecrit : > Is it possible or impossible to factor the alternating zeta function > into product form, with each product having a single zero at one of > the frequencies of the zeta function? > If this has been attempted, what were the approaches used? > Bob Adams æ æLook at the Hadamard product here : > æ æhttp://en.wikipedia.org/wiki/Riemann zeta#Hadamard product æ æand, if the alternating zeta function is Dirichlet eta function, then æ æeta(s)= sum {n=1}^oo (-1)^(n-1)/n^s > æ æso that eta(s) = (1-2^(1-s)) zeta(s) æ æNot sure it will really help, > æ æ æ æ æ æ Raymond æ æ æ Hmmm... I supposed that your of the zeta function > were zeta's complex zeros but perhaps that you had something else in > view... ??- Hide quoted text - - Show quoted text - Sorry, my engineering background is showing! Yes, I mean the the > locations of the complex zeroes on the line s = +1/2. Bob Adams- Hide quoted text - - Show quoted text - I was actually hoping there was a factorization where all terms in each factor were of the form 1 + n1^-s + n2^s + n3^-s .... Of course the Euler product formula is such a factorization of the zeta function, but since it is only valid for Re(s) > 1, you cannot expect any of the factors to contain a zero corresponding to one of the zeta zeros. So I was looking for a similar factorization of the alternating zeta function, which would be valid for re(s) = 1/2. If such a factorization existed then one could perhaps find zeta zeroes by formula rather than by search. === === Subject: : Re: Factoring the alternating zeta function Robert Adams a .8ecrit : I was actually hoping there was a factorization where all terms in > each factor were of the form 1 + n1^-s + n2^s + n3^-s .... Of course the Euler product formula is such a factorization of the > zeta function, but since it is only valid for Re(s) > 1, you cannot > expect any of the factors to contain a zero corresponding to one of > the zeta zeros. So I was looking for a similar factorization of the alternating zeta > function, which would be valid for re(s) = 1/2. If such a > factorization existed then one could perhaps find zeta zeroes by > formula rather than by search. > If ro_k is the k-th non-trivial zero of zeta then results are known for sums incorporating all the zeros like : Z(n)= sum_k (ro_k)^(-n) http://mathworld.wolfram.com/RiemannZetaFunctionZeros.html Most formulas concerning zeta should be available here : http://functions.wolfram.com/ZetaFunctionsandPolylogarithms/Zeta/ Finding a simple (non-trivial and with nothing more than elementary operations and functions) relation with only a finite numbers of non-trivial zeros should make your name famous! :-) Good luck! Raymond === === Subject: : Re: Example of an analytic function > What is an example of a one-to-one analytic function which maps the unit > disk onto the set (x,y): -1 -infty Thus this will be my final note (and I will not post You'll be back in less than a month. Anyone want to start a pool on Master-Basti's return? I'll just guess (for fun) and say before May 30. According to his Google Groups profile, his last messages were [42, 32, 31, 24, 23, 23, 21, 8, 8] hours ago. Using NEW EXACT MATH, this sequence gives me the polynomial 41 ----- * n**8 10080 -83 --- * n**7 630 319 --- * n**6 180 -9199 ----- * n**5 720 76157 ----- * n**4 1440 -89413 ------ * n**3 720 127123 ------ * n**2 840 -11047 ------ * n**1 140 42 -- * n**0 1 which generates the sequence Polynomial n Original Sequence ----------- --- -------------- 42 0 42 32 1 32 31 2 31 24 3 24 23 4 23 23 5 23 21 6 21 8 7 8 8 8 8 402 9 None So he'll return in 402 hours. === === Subject: : Re: Pool as to the return of BASTI, pasti his lasti posti ! > newsgroup anymore) > You'll be back in less than a month. Anyone want to start a pool on Master-Basti's return? I'll just guess (for fun) and say before May 30. Hmm... I think you may be right. He's such an incompetent moron that he won't realize how stupid he will look when doing so. === === Subject: : Re: Pool as to the return of BASTI, pasti his lasti posti ! > newsgroup anymore) You'll be back in less than a month. > Anyone want to start a pool on Master-Basti's return? > I'll just guess (for fun) and say before May 30. Hmm... I think you may be right. He's such an incompetent moron > that he won't realize how stupid he will look when doing so. He'll not survive the weekend - 25th. Having killfiled the loon, I might miss the momentous event. Phil -- -- Microsoft voice recognition live demonstration === === Subject: : Re: recent paper on prime gaps: p_{n+1} - p_n Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) >I know that for odd n>2, 2^(n-1) == 1 (mod n) doesn't imply that >n is prime. For n with 100 digits or thereabouts, is something known >about the frequency of odd composites that satisfy: >2^(n-1) == 1 (mod n) ? These are called Fermat pseudoprimes to base 2, or Sarrus numbers. I don't know the answer to your specific question, but try looking at: http://www.research.att.com/~njas/sequences/A001567 -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === === Subject: : Re: recent paper on prime gaps: p_{n+1} - p_n >I know that for odd n>2, 2^(n-1) == 1 (mod n) doesn't imply that >n is prime. For n with 100 digits or thereabouts, is something known >about the frequency of odd composites that satisfy: >2^(n-1) == 1 (mod n) ? These are called Fermat pseudoprimes to base 2, or Sarrus numbers. I don't > know the answer to your specific question, but try looking at: http://www.research.att.com/~njas/sequences/A001567 There's also some more info on pseudoprimes at http://primepages.org/ Phil -- -- Microsoft voice recognition live demonstration === === Subject: : Re: recent paper on prime gaps: p_{n+1} - p_n > I know that for odd n>2, 2^(n-1) == 1 (mod n) doesn't imply that > n is prime. For n with 100 digits or thereabouts, is something known > about the frequency of odd composites that satisfy: > 2^(n-1) == 1 (mod n) ? > These are called Fermat pseudoprimes to base 2, or Sarrus numbers. I > don't > know the answer to your specific question, but try looking at: > http://www.research.att.com/~njas/sequences/A001567 There's also some more info on pseudoprimes at > http://primepages.org/ Phil > that > a random pseudo-prime is composite: > http://primes.utm.edu/notes/prp_prob.html > Also, W. Galway has conjectures on counts of base 2 > odd pseudoprimes with two prime factors: > cf. page 1 of > http://www.cecm.sfu.ca/~wfgalway/SlidesETC/pq-psp-slides.pdf . With the PARI number theoretical computation package, it's quite > straightforward to send data about primes in some interval > to a file. @@@@(00:30) gp > for(X=1,10000, > if(isprime(a+X+X+1),write(G:primes78, X+X+1))) here, a = 10^100, and isprime() is a 100% sure prime status function. So the odd numbers 10^100 + 2k +1 are tested for primality, and > 2k+1 is written to the (Windows OS) file G:primes78 when > 10^100 + 2k +1 is a prime number; this is done for > k from 1 to 10,000. There results 67 primes. A histogram of the 66 gaps appears here (using Matlab): > http://www.geocities.com/ezcos/gapsgoogol66.jpg [ the computation above took about 4 minutes ] > @@@@(00:34) gp > for(X=1,1000000, > if(isprime(a+X+X+1),write(G:primes1meg, X+X+1))) [ idem, but for k from 1 to 10^6.] @@@@(04:01) gp > /*** the above computation took 3.5 hours ***/ I would consider replacing PARI's isprime() by the package's > ispseudoprime(): > PARI help says: << ispseudoprime(x,{n}): true(1) if x is a strong pseudoprime, false(0) > if not. > If n is 0 or omitted, use BPSW test, otherwise use strong Rabin-Miller test > for n randomly chosen bases. > I did that with the PARI command: @ (04:35) gp > for(X=1,10000000, if(ispseudoprime(a+X+X+1),write(G:psprimes10meg, X+X+1))) // pseudoprimes from 10^100 to 10^100 + 20,000,000. It found 86,794 pseudoprimes in about 55 minutes. The average gap near 10^100 is expected to be log(10^100), where all prime gaps between 10^100 and (say) 10^100 + 10^20 count once... So 1 hour: 20 million tested for being pseudoprime near 10^100. There's another histogram from the gaps/(average gap); There are gaps of 6*average and even more, but they don't leave much of a trace on the histogram figure... http://www.geocities.com/ezcos/gapsgoogol76k.jpg The probabilistic model that has been proposed is the (continuous) exponential distribution with rate parameter lambda = 1. Cf.: Probability density function graphs, etc. at: http://en.wikipedia.org/wiki/Exponential_distribution Note: (log(10^100))^2 ~= 53,000 ; this gives an indication of what large gaps might mean in the Soundararajan paper, and the literature on large gaps , or rather, conjectures on large gaps. David Bernier === === Subject: : Re: -- Limit of ratio of consecutive primes = 1 ? posting-account=G_G-iQoAAAB08LNQidt_LsMkopmIb4ZS Gecko/20060111 Firefox/1.5.0.1 Mnenhy/0.7.3.0,gzip(gfe),gzip(gfe) >Let p(n) denote the nth prime. >It seems that, as n -> oo, limit( p(n + 1)/p(n) ) = 1. >Is that correct? If so, how can it be proven? >Does it perhaps depend on some result (unfamiliar to me) concerning how >big, for given n, the gap between p(n) and p(n + 1) can be? > So is there a limit? There seems to be no answer in this long thread. > If there is one it seems to be between 1 and 2. > The limit is 1. That was proven early in the thread. > David The first order limit is => 1. What in the world does that mean? _The_ _limit_ of P(n+1)/P(n) _is_ 1. > Another way to say that would be to say that > P(n+1)/P(n) -> 1. But the _limit_ _is_ 1, > it does not tend to 1. And nobody but > you knows what a first order limit is. Since P(n+1) cannot be less that P(n) ; >the limit >must be => 1. . . >. >Twin primes quickly push the limit to 1 exactly. Huh? That shows that limit is 1 _assuming_ two > things: (i) there are infinitely many twin primes, > (ii) the limit _exists_. But nobody knows whether > (i) is true, and (ii) is not something you're allowed > to assume here, it needs to be proved. I understand now why you did not bother to explain this to me earlier.. I would have to be able think like a mathematician to make any sense out of the above . All of this has been based on a certain theorem known > as the Prime Number Theorem, which says that a certain > other limit is 1. That is a very deep theorem. But if you > simply assume that the limit in question exists the PNT > becomes quite simple. > Way over my head! P(n+1) is asymptotic to (n+1) * ln (n+1) and P(n) is asymptotic to n* >ln (n) >For significantly large 'n', these two curves are virtually >identical. This >might be useful information. Bill J David C. Ullrich === === Subject: : Re: Asymptotics for iterates > Let f be a continuous function defined in [0,1] > such that 0 < f(x) < x for x > 0. > Then the sequence x_{n+1} = f(x_n) converges to 0 > (x_0 being in (0,1]). I am interested in an asymptotic expansion for x_n. > Such expansion are known when f is analytic. > E.g. for f(x) = sin x, x_n = (3/n)^(1/2) + o(n^(-1/2)). > (actually other terms of the expansion can also be computed). But what if f is not analytic; say only C^oo. > E.g. for f(x) = x - x^2 exp(-1/x) > In this case x_n = 1/(ln n) + ... Does anybody know a reference for this problem? > Or a general method? x_n = g(n) if g(x) is a function such that g(x+1) = f(g(x)) and g(0) = x_0. Say f(x) = x - M(x) where M(x) > 0 and M(x) = o(x) as x -> 0. Thus g(x+1)-g(x) = -M(g(x)). As a first approximation you might take a solution g_0(x) of the differential equation y' = - M(y). -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === === Subject: : Re: a generalization: the correct formulation Say we have two algebraic numbers x,y such that [Q(x):Q] = m, [Q(y):Q] = n, (m,n) = 1. It is then true that Q(x,y) = Q(x+y). Kent Holing === === Subject: : Re: First order theory of rationals posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 CLR 1.1.4322; InfoPath.1),gzip(gfe),gzip(gfe) On May 23, 7:22æam, Frederick Williams There are categorical axioms for integers, rational numbers and real > numbers. A recursive (or even a non-recursive) set of first order axioms? That is, for each model - the system of integers, of rationals, of reals - there is a recursive (or even a non-recursive) set of first order axioms whose models are all and only those isomorphic with the said system? Maybe I'm missing something, but that doesn't sound right to me. MoeBlee === === Subject: : Re: First order theory of rationals On May 23, 7:22 am, Frederick Williams numbers. A recursive (or even a non-recursive) set of first order axioms? That > is, for each model - the system of integers, of rationals, of reals - > there is a recursive (or even a non-recursive) set of first order > axioms whose models are all and only those isomorphic with the said > system? Maybe I'm missing something, but that doesn't sound right to me. Sorry, I had forgotten that the op said first order. -- === === Subject: : Re: First order theory of rationals Originator: tchow@lebesgue.mit.edu.mit.edu (Timothy Chow) Looks like PA can PROVE a good deal of arithmetic. Atleast simple >arithmetic involving division and multiplication on rationals, looks like. Less obviously, exponentiation can also be handled. In fact, any Turing-computable function is representable in PA. PA is much stronger than it looks at first glance. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences === === Subject: : Re: First order theory of rationals > Looks like PA can PROVE a good deal of arithmetic. At least simple > arithmetic involving division and multiplication on rationals Less obviously, exponentiation can also be handled. In fact, > any Turing-computable function is representable in PA. However, if a function grows too fast then PA can't prove that it's total. It can't handle any transfinite induction stronger than eps_0. For a detailed discussion of a simple example, see my prior post [1] on Goodstein's theorem. --Bill Dubuque === === Subject: : car's trajectory posting-account=B_ql5woAAABEePt1fDMNH7lHfjEScu4Z Gecko/20080311 Firefox/2.0.0.13,gzip(gfe),gzip(gfe) Consider a vehicle of axle-to-axle length L, and left to right wheel separation W (though I don't believe this matters). Assume a X-Y co-ordinate plane, the origin located at the center of the rear axle. What is the car's forward trajectory, if the front wheels rotate left at angle U? What is its reverse trajectory? What if the car has front wheel drive? Why does one park into a space by backing up, rather than forward? ** BONUS CREDIT ** Use your answer above, to solve the parallel parking problem. Mark === === Subject: : Re: car's trajectory Consider a vehicle of axle-to-axle length L, and left to right wheel separation W. Assume an X-Y co-ordinate plane, the origin located at the center of the rear axle. What is the car's forward trajectory, if the front wheels rotate left at angle U? What is its reverse trajectory? What if the car has front wheel drive? Why does one park in a space by backing up, rather than going forward? _________________________________________________________ It's a trick question. The steering linkage rotates the front wheels at two different angles. The steering geometry is designed so that the extended centerlines of the two front axles always meet the extended centerline of the rear axle at a common point. The sharper the turn, the nearer this point is to the car. The trajectory of every point on the car is a circle, centered at that common point, whether going forward or backward. Rodan. === === Subject: : Re: car's trajectory posting-account=504E-QkAAAA2v90r8nGnJKpfySa_yBSU 5.1),gzip(gfe),gzip(gfe) Consider a vehicle of axle-to-axle length L, and left to right > wheel separation W. æ æAssume an X-Y co-ordinate plane, > the origin located at the center of the rear axle. What is the car's forward trajectory, if the front wheels > rotate left at angle U? æ æWhat is its reverse trajectory? > What if the car has front wheel drive? æ æWhy does one > park in a space by backing up, rather than going forward? > It's a trick question. æ æThe steering linkage rotates the front wheels > at two different angles. æ æThe steering geometry is designed so that > the extended centerlines of the two front axles always meet the > extended centerline of the rear axle at a common point. æThe sharper > the turn, the nearer this point is to the car. The trajectory of every point on the car is a circle, centered at > that common point, whether going forward or backward. Rodan. Yes, and well said. Harry C. === === Subject: : Re: car's trajectory > Consider a vehicle of axle-to-axle length L, > and left to right wheel separation W (though > I don't believe this matters). Assume a X-Y co-ordinate plane, the origin > located at the center of the rear axle. What is the car's forward trajectory, > if the front wheels rotate left at angle U? > What is its reverse trajectory? What if the car has front wheel drive? Why does one park into a space by backing > up, rather than forward? ** BONUS CREDIT ** Use your answer above, to solve the parallel > parking problem. Mark I'm not going to bother solving your problem, but I will point out that you're disregarding Ackermann. nate -- replace roosters with cox to reply. http://members.cox.net/njnagel === === Subject: : Re: car's trajectory I'm not going to bother solving your problem, but I will point out that > you're disregarding Ackermann. Forrest J. ? ;-) -- Service to my country? Been there, Done that, and I've got my DD214 to prove it. Member of DAV #85. Michael A. Terrell Central Florida === === Subject: : Re: car's trajectory >I'm not going to bother solving your problem, but I will point out that >you're disregarding Ackermann. Who's Ackermann? One of the profs at Stanford? -- Rich Webb Norfolk, VA === === Subject: : Re: car's trajectory >I'm not going to bother solving your problem, but I will point out that >you're disregarding Ackermann. > Who's Ackermann? One of the profs at Stanford? > I was referring to Rudolph Ackermann http://en.wikipedia.org/wiki/Ackermann_steering_geometry basically, in any modern (within the last hundred years or so) vehicle, the front wheels will not turn at the same angle for a given steering input. The alternative would be to use a solid axle with a pivot in the middle, like a wagon. otherwise, the tires will scrub going around a corner. nate -- replace roosters with cox to reply. http://members.cox.net/njnagel === === Subject: : Re: car's trajectory posting-account=504E-QkAAAA2v90r8nGnJKpfySa_yBSU 5.1),gzip(gfe),gzip(gfe) >I'm not going to bother solving your problem, but I will point out that >you're disregarding Ackermann. Who's Ackermann? One of the profs at Stanford? I was referring to Rudolph Ackermann http://en.wikipedia.org/wiki/Ackermann steering geometry basically, in any modern (within the last hundred years or so) vehicle, > the front wheels will not turn at the same angle for a given steering > input. æThe alternative would be to use a solid axle with a pivot in the > middle, like a wagon. æotherwise, the tires will scrub going around a > corner. nate Nate, thank you for posting this. It's guys like you that bring new hope to this newsgroup by sharing knowledge with others. Nate, one minor nit. Ackerman Steering dates back 1800, which is more like 200 years ago. This website has a very simple explanation. http://www.auto-ware.com/setup/ack rac.htm. Harry C. p.s., I'm not a car buff. I simply like to know how things really works and why. Even though I'm a physicist, the front end steering mechanism of a care is by definition a simple machine, but it's certainly the most complex simply machine that I know of. When you repair you own car, you learn all about king pins, ball joints, tie rods (not to mention tie rod ends), leverer arms, and worm gears on the steering box. You also learn about someone mysterious adjustment such as caster, camber, and toe-in. Some of these I understand, and others I have only an waving knoledge of. I farm out front end alignmnet to someone with the proper equipment and who know what they are doing. (Sadly, many mecahnics don't know what they are doing.) Strange activity for a physicist, because repairing the from end steering mechanism is a pretty dirty job, and I really have no love affair going with cars. The reason that I do this is because when I was an undergraduate, I simply couldn't affor to pay someone to repair my 1946 Chevy, so I learned to do it myself. Later, in 1956, my Pointiac flunked NJ inspection because of excessive steering play. The estimate to repair the problem was $400 which at the time I didn't have. With the help of two college friends who wiggled the wheels while I lay undernearth the car looking for signs of excessive play, a little device called the leverer arm was obviously the source of the problem. So we drove to Pep Boys auto supply, purchase a new leveler arm assembly for the sum of $19.95 and installed in in less than half an hour. Worn tie-rod ends are another trivial fix, but beware of replacing ball joints...very difficult to do for an amateur, not to say that I cannot be done. Leave the ball joint and the kingpins (do any cars still use kingpins) to the pros. I really hope that there are still a few college students earning their degress by their own efforts and financial resources, and not depending on mom or dad to write their tuition check. It is for these studients that I post this stuff. To these guys and girls, you have to how to take care of your transportation in the least expensive way possible, which means doing most of the repairs yourself. because you cannot likely afford pricey reapair shops. For us guys, the alernative is to take the bus and then walk. Now let me share this with you. Old habbits die hard. While today I am in a position to drive a new BMW (my wife loves them), I drive two very differnt cars. One is a 1996 Ford Bronco, and the other is a 1986 Cadillac Cimarron. Our newest vehicle is a 1996 Mercury Sable, with no rust that we just paid $800 have shiped in from Nevada. Guess what the very firt thing was that I did. I went to eBay and purchased a Ford/ Mercury 1996 Sable/Taurus service manual. (Damn book is 6 thick, and cost $40, still money well spent.) This will be my wife's ride, and I need to know how to repair it, if and when it needs reapir. Far chaper than paying a dealer or auto shop 10X the price of replacement parts, which generally need only a half-hour to replace. I guess my point like Nate's is educatonal, albeit in a slightly different way. The common point between our posts is that you need to learn how a steering mechanism, ad a car operates unless you have a surplus of income. To save up to 9/10 of the repair cost, you need to learn how to do some simple repairs yourself, and somtimes get a bit dirty in the process. (Soap is still cheap.) Now this is for the young couples, if you really want to know why we drive 10 and 20 year old automobiles, it is simply because that they take us to the places we want to go, at minimal cost. Over the years, this has allowed us to own our own modest home in the Massachusetts suburbs free from any mortage, pay the college expense of three children who are all now college graduates, operate our sailboat, and on rare occasions take a trip to Maui (our peferred vacation destination.) Take this post for whatever you believe it is worth. Harry C. === === Subject: : Re: car's trajectory > I'm not going to bother solving your problem, but I will point out that > you're disregarding Ackermann. > Who's Ackermann? One of the profs at Stanford? > I was referring to Rudolph Ackermann > http://en.wikipedia.org/wiki/Ackermann_steering_geometry > basically, in any modern (within the last hundred years or so) vehicle, > the front wheels will not turn at the same angle for a given steering > input. The alternative would be to use a solid axle with a pivot in the > middle, like a wagon. otherwise, the tires will scrub going around a > corner. > nate Nate, thank you for posting this. It's guys like you that bring new > hope to this newsgroup by sharing knowledge with others. > Nate, one minor nit. Ackerman Steering dates back 1800, which is more > like 200 years ago. This website has a very simple explanation. http://www.auto-ware.com/setup/ack_rac.htm. Harry C. p.s., I'm not a car buff. I simply like to know how things really > works and why. Even though I'm a physicist, the front end steering > mechanism of a care is by definition a simple machine, but it's > certainly the most complex simply machine that I know of. When you > repair you own car, you learn all about king pins, ball joints, tie > rods (not to mention tie rod ends), leverer arms, and worm gears on > the steering box. You also learn about someone mysterious adjustment > such as caster, camber, and toe-in. Some of these I understand, and > others I have only an waving knoledge of. I farm out front end > alignmnet to someone with the proper equipment and who know what they > are doing. (Sadly, many mecahnics don't know what they are doing.) > Strange activity for a physicist, because repairing the from end > steering mechanism is a pretty dirty job, and I really have no love > affair going with cars. I would never get a physicist to fix my car. === === Subject: : Re: car's trajectory posting-account=504E-QkAAAA2v90r8nGnJKpfySa_yBSU 5.1),gzip(gfe),gzip(gfe) > I would never get a physicist to fix my car. Wise decision Gib, If you did have a physicist to fix your car, you like me would end up driving 20 and 30 year old vehicles that reliably and cheaply get you where your are going, but are not the greatest girl magnets in the world! :-) This is about the only advantage I see to owning a brand new BMW (as beautiful as they are), which comes with monthly paments that are often top $300. By the way, New BMWs are real chick magnets, but so are gourmet dinners in a top rated restaurant and a $60 bottle of wine. Somehow the girls to fail to realize that they are being driven to dinner in a 1996 Cadillac Cimarron. Quite honestly, when driving an old but well mainained classic car from 25-years back, you save enougn money to afford occasional weekend junkets to Bermuda. I can tell you that girls like weekend junkets to Bermuda, or Paris, much more that their attraction to the wheels that you drive on a daily basis. Harry C. === === Subject: : Re: car's trajectory >I'm not going to bother solving your problem, but I will point out that >you're disregarding Ackermann. Who's Ackermann? One of the profs at Stanford? I was referring to Rudolph Ackermann http://en.wikipedia.org/wiki/Ackermann_steering_geometry basically, in any modern (within the last hundred years or so) vehicle, > the front wheels will not turn at the same angle for a given steering > input. The alternative would be to use a solid axle with a pivot in the > middle, like a wagon. otherwise, the tires will scrub going around a > corner. nate Nate, thank you for posting this. It's guys like you that bring new hope to this newsgroup by sharing knowledge with others. Nate, one minor nit. Ackerman Steering dates back 1800, which is more like 200 years ago. This website has a very simple explanation. http://www.auto-ware.com/setup/ack_rac.htm. Harry C. p.s., I'm not a car buff. I simply like to know how things really works and why. Even though I'm a physicist, the front end steering mechanism of a care is by definition a simple machine, but it's certainly the most complex simply machine that I know of. When you repair you own car, you learn all about king pins, ball joints, tie rods (not to mention tie rod ends), leverer arms, and worm gears on the steering box. You also learn about someone mysterious adjustment such as caster, camber, and toe-in. Some of these I understand, and others I have only an waving knoledge of. I farm out front end alignmnet to someone with the proper equipment and who know what they are doing. (Sadly, many mecahnics don't know what they are doing.) Strange activity for a physicist, because repairing the from end steering mechanism is a pretty dirty job, and I really have no love affair going with cars. The reason that I do this is because when I was an undergraduate, I simply couldn't affor to pay someone to repair my 1946 Chevy, so I learned to do it myself. Later, in 1956, my Pointiac flunked NJ inspection because of excessive steering play. The estimate to repair the problem was $400 which at the time I didn't have. With the help of two college friends who wiggled the wheels while I lay undernearth the car looking for signs of excessive play, a little device called the leverer arm was obviously the source of the problem. So we drove to Pep Boys auto supply, purchase a new leveler arm assembly for the sum of $19.95 and installed in in less than half an hour. Worn tie-rod ends are another trivial fix, but beware of replacing ball joints...very difficult to do for an amateur, not to say that I cannot be done. Leave the ball joint and the kingpins (do any cars still use kingpins) to the pros. I really hope that there are still a few college students earning their degress by their own efforts and financial resources, and not depending on mom or dad to write their tuition check. It is for these studients that I post this stuff. To these guys and girls, you have to how to take care of your transportation in the least expensive way possible, which means doing most of the repairs yourself. because you cannot likely afford pricey reapair shops. For us guys, the alernative is to take the bus and then walk. Now let me share this with you. Old habbits die hard. While today I am in a position to drive a new BMW (my wife loves them), I drive two very differnt cars. One is a 1996 Ford Bronco, and the other is a 1986 Cadillac Cimarron. Our newest vehicle is a 1996 Mercury Sable, with no rust that we just paid $800 have shiped in from Nevada. Guess what the very firt thing was that I did. I went to eBay and purchased a Ford/ Mercury 1996 Sable/Taurus service manual. (Damn book is 6 thick, and cost $40, still money well spent.) This will be my wife's ride, and I need to know how to repair it, if and when it needs reapir. Far chaper than paying a dealer or auto shop 10X the price of replacement parts, which generally need only a half-hour to replace. I guess my point like Nate's is educatonal, albeit in a slightly different way. The common point between our posts is that you need to learn how a steering mechanism, ad a car operates unless you have a surplus of income. To save up to 9/10 of the repair cost, you need to learn how to do some simple repairs yourself, and somtimes get a bit dirty in the process. (Soap is still cheap.) Now this is for the young couples, if you really want to know why we drive 10 and 20 year old automobiles, it is simply because that they take us to the places we want to go, at minimal cost. Over the years, this has allowed us to own our own modest home in the Massachusetts suburbs free from any mortage, pay the college expense of three children who are all now college graduates, operate our sailboat, and on rare occasions take a trip to Maui (our peferred vacation destination.) Take this post for whatever you believe it is worth. Harry C. I found out about Ackermann when I registered a custom build car I made. I'd shortened the chassis from another vehicle by a foot or two & built a new body for it. Smart arse vehicle inspector came over and jabbered on about how the Ackermann principle had been violated..... The guy who was actually inspecting the vehicle passed it and told us to go home rip off all the anti-smog gear and stick a Holley on it. Oh yeah - it went around corners ok as well, I guess I didn't cut enough out for it to become a problem. === === Subject: : Re: car's trajectory <4837b231$0$11512$5a62ac22@per-qv1-newsreader-01.iinet.net.au> posting-account=504E-QkAAAA2v90r8nGnJKpfySa_yBSU 5.1),gzip(gfe),gzip(gfe) > I found out about Ackermann when I registered a custom build car I made. I'd > shortened the chassis from another vehicle by a foot or two & built a new > body for it. Smart arse vehicle inspector came over and jabbered on about > how the Ackermann principle had been violated..... The guy who was > actually inspecting the vehicle passed it and told us to go home rip off all > the anti-smog gear and stick a Holley on it. Oh yeah - it went around > corners ok as well, I guess I didn't cut enough out for it to become a > problem. Wow, you are really into cars which sadly I am not. For me, cars are simply boxes with wheels that get me from where I am to where I need to go. Still, I am not knocking your passion. I believe everyone needs a hobby or side interest in which they can become passionate about. Mine is surpsingly, fireworks manufacture and electronics, plus a small interest in machine tools (lathes, milling machines, and shapers) on the side. Different strokes for different folks. During highschool and college, super-powered street rods where all the rage, It never became my thing, but I admired the talent and work that typically went in to the conversion of an old, rusty Ford pickup truck into a machine that could beat track records. Back the, the guys that could afford to build them machines would replace everything but the truck body with new mills, transmissions, rear ends, shifter and brakes. Literally, nothing original remained except for the body, which they would of course restore to new condition. What amazed me is that these care guys would put thousands of hours of work into restoring and enhancing a rusty old wreck that they had purchased for $300, and then spend $25,000 or more on the project. Wow! While the car guys got their rocks off on this, I spent far more money than I could afford on rockets and fireworks and metal working equipment. Trust me, there is no greater rush than building a home comstructed rocket that will break 100,000 feet altitude, and send he coastal defense people (NORAD) searching for the source. I also once enjoyed making display quality fireworks as a challenge. Another rush comes from the constuction a 5-break aerial shell where the bottom shot fires at an altitude of 75-feet! Today I am retired, and work on cars and other this. Mostly other things. Yesterday we picked up a 1996 Mercury Sable shipped by a car transporter from California. I already have purchased an received the Ford dealer service manual (damn thing is 6 inches thick). I really don't plan to do any enhancedment to it, because it runs just fine. My game plan is simply to check out the many of its automatic details, and it has many, then turn it over to my wife for her transportation and retire her 1990 Tempon by means of Craig's List. Oh, on the physics side, I repair and restore the operation of radiation detection devices, plus machine special components for vacuum systems and patricle accelerators. As Dr. House would say, this can get boring, and boredom leads to death. Death is bad!!!! Oh, Just for your information, Holly has just introduced a new product line. Carburetors are of not interest to me, since they are little more than atomizers, but you may find some interest in this news line. Harry C. === === Subject: : Re: car's trajectory >I'm not going to bother solving your problem, but I will point out that >you're disregarding Ackermann. Who's Ackermann? One of the profs at Stanford? martin === === Subject: : Re: Comprehensive Solution Manual for Textbooks posting-account=rLOz6QoAAAAmvEIbrGZd27QhtZqovu5R Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) All messages replied === === Subject: : rank of matrix of ones and zeroes posting-account=ZUczkQoAAABCznvSjYJPkJbPEzZyPyBU 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Hello - I would appreciate any insights into the following issue. I have an m X n matrix, where m > n. It is a matrix of ones and zeroes. I know two things : 1) There is a row with all 1's. 2) For every i and j, there is a row whose i'th column is 1 and whose j'th column is 0.. I am trying to prove that the matrix has rank equal to n, i.e., the columns are linearly independent. I do not know if I need more structure to get this result, or if what I have is sufficient. Any thoughts would be much appreciated. === === Subject: : Re: rank of matrix of ones and zeroes > Hello - I would appreciate any insights into the following issue. I have an m X n matrix, where m > n. It is a matrix of ones and > zeroes. I know two things : 1) There is a row with all 1's. > 2) For every i and j, there is a row whose i'th column is 1 and whose > j'th column is 0.. I am trying to prove that the matrix has rank equal to n, i.e., the > columns are linearly independent. I do not know if I need more > structure to get this result, or if what I have is sufficient. Any > thoughts would be much appreciated. For n=4, you might try [ 1 1 1 1 ] [ 1 0 0 1 ] [ 0 1 0 1 ] [ 1 0 1 0 ] [ 0 1 1 0 ] -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === === Subject: : Re: G's conjecture and B's postulate What if, what if Goldbach's conjecture was unprovable? If GC is independent of arithmetic then it is necessarily true, because arithmetic is strong enough to verify any counterexample. Technically one says that Peano arithmetic is Sigma-1 complete, meaning that it is strong enough to provide proofs for all true Sigma-1 (existential) formulas. For further discussion see my prior posts in the threads [1], [2]. --Bill Dubuque === === Subject: : Re: What is the Subject of this post? ! > Dang. Fell for a bait and switch ad. Ok, ok, the topic of > this thread isn't about the topic of this thread. It's about > OP bragging what an immensely ignorant buffoon he is. > what ta hell are you talking about mk5000 Dr. Miranda Bailey: Dr. Victor, I'm sorry, but these are viable nerves. We should save them. Dr. Victor: It will take at least an hour longer. And we might not get it all. Dr. Isobel Izzie Stevens: [to George] You know they call him, *limp Harry*. Dr. Miranda Bailey: But his prognosis with chemo is nearly as good, and frankly if you're worried about missing tee time, I'll be more than happy to finish. [Izzie enters the O.R] Dr. Miranda Bailey: Dr. Stevens? Dr. Victor: Can we help you? Dr. Isobel Izzie Stevens: I'm sorry, Dr. Bailey. Dr. Victor, I agree with her. You just can't... You have to save the nerves. Dr. Victor: What? Dr. Isobel Izzie Stevens: The nerves. You have to save them. Dr. Miranda Bailey: Dr. Stevens, I can handle this. Dr. Isobel Izzie Stevens: No, you told me the most important thing is giving the patient what they want. What Humphrey wants is his erection. Dr. Victor: [to Dr. Bailey] She's yours. You get her out. Dr. Miranda Bailey: I can't do that, sir. You know how these young puppies are. Dr. Victor: I'm going to tell Richard about both of you. Dr. Miranda Bailey: You do that. In the meantime, why don't we pretend it's you on this table, and give this a try. --Grey's Anatomy === === Subject: : Re: Calculating Growth? I am trying to calculate the compound growth of a customer base when > give a percentage value. For example, I would like to know how many customers I would have > after 1 year (or 2 or 3) if the current customer base of 25000 > increased 3% a month. What is the correct algebraic calculation to arrive at the correct > value? > mos Hint: After 1 month, there will be 1.03*25000 customers, after two months, there will be 1.03*[1.03*25000] = (1.03)^2*25000 customers ... After 10 years there will be over a billion customers. Tell us when this company goes public. === === Subject: : Re: Calculating Growth? posting-account=QblO2woAAABpDBRe4JZQPIG6X9dXe2EY Gecko/20080410 SUSE/2.0.0.14-2.2 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) On May 23, 8:44 pm, The World Wide Wade give a percentage value. For example, I would like to know how many customers I would have > after 1 year (or 2 or 3) if the current customer base of 25000 > increased 3% a month. What is the correct algebraic calculation to arrive at the correct > value? > mos Hint: After 1 month, there will be 1.03*25000 customers, after two > months, there will be 1.03*[1.03*25000] = (1.03)^2*25000 customers ... > After 10 years there will be over a billion customers. Tell us when > this company goes public. Yes this compound growth based on three percent increase in customer growth per month. Is there a formula to obtain the answer instead of doing the math of (Month 1= .03 * 26000 = 26900; Month two equals 26900 *.03 = 27700) and so on. === === Subject: : Re: Calculating Growth? posting-account=ogMREwkAAAC5xUr8sg7heGtsvzzF18LA Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > On May 23, 8:44 pm, The World Wide Wade I am trying to calculate the compound growth of a customer base when > give a percentage value. For example, I would like to know how many customers I would have > after 1 year (or 2 or 3) if the current customer base of 25000 > increased 3% a month. What is the correct algebraic calculation to arrive at the correct > value? > mos Hint: After 1 month, there will be 1.03*25000 customers, after two > months, there will be 1.03*[1.03*25000] = (1.03)^2*25000 customers ... > After 10 years there will be over a billion customers. Tell us when > this company goes public. Yes this compound growth based on three percent increase in customer > growth per month. Is there a formula to obtain the answer instead of > doing the math of (Month 1= .03 * 26000 = 26900; Month two equals > 26900 *.03 = 27700) and so on. > Copying from above, note that after 2 months: 1.03*[1.03*25000] = (1.03)^2*25000 using a similar form, after 1 month there are: 1.03*25000 = (1.03)^1*25000 And after 3 months there are: 1.03*[1.03^2*25000] = (1.03)^3*25000 See the pattern yet? For any given month, exactly 2 operations are needed to get the value. Any reference on compounding interest will also get you the formula you want. === === Subject: : Re: Calculating Growth? posting-account=QblO2woAAABpDBRe4JZQPIG6X9dXe2EY Gecko/20080410 SUSE/2.0.0.14-2.2 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) On May 23, 8:44 pm, The World Wide Wade I am trying to calculate the compound growth of a customer base when > give a percentage value. For example, I would like to know how many customers I would have > after 1 year (or 2 or 3) if the current customer base of 25000 > increased 3% a month. What is the correct algebraic calculation to arrive at the correct > value? > mos Hint: After 1 month, there will be 1.03*25000 customers, after two > months, there will be 1.03*[1.03*25000] = (1.03)^2*25000 customers ... > After 10 years there will be over a billion customers. Tell us when > this company goes public. Yes this compound growth based on three percent increase in customer > growth per month. Is there a formula to obtain the answer instead of > doing the math of (Month 1= .03 * 26000 = 26900; Month two equals > 26900 *.03 = 27700) and so on. > Copying from above, note that after 2 months: > 1.03*[1.03*25000] = (1.03)^2*25000 using a similar form, after 1 month there are: > 1.03*25000 = (1.03)^1*25000 And after 3 months there are: > 1.03*[1.03^2*25000] = (1.03)^3*25000 See the pattern yet? For any given month, exactly 2 operations are > needed to get the value. Any reference on compounding interest will also get you the formula > you want. === === Subject: : Re: Calculating Growth? posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > On May 23, 8:44 pm, The World Wide Wade I am trying to calculate the compound growth of a customer base when > give a percentage value. For example, I would like to know how many customers I would have > after 1 year (or 2 or 3) if the current customer base of 25000 > increased 3% a month. What is the correct algebraic calculation to arrive at the correct > value? > mos Hint: After 1 month, there will be 1.03*25000 customers, after two > months, there will be 1.03*[1.03*25000] = (1.03)^2*25000 customers ... > After 10 years there will be over a billion customers. Tell us when > this company goes public. Yes this compound growth based on three percent increase in customer > growth per month. Is there a formula to obtain the answer instead of > doing the math of (Month 1= .03 * 26000 = 26900; Month two equals > 26900 *.03 = 27700) and so on. Is your computer broken? A simple Google search on the terms 'compound interest' turns more than 2,000,000 hits, of which some in the first few do exactly what you want. For example, see http://www.mathsisfun.com/money/compound-interest.html or http://en.wikipedia.org/wiki/Compound_interest R.G. Vickson > === === Subject: : Re: Calculating Growth? , > I am trying to calculate the compound growth of a customer base when > give a percentage value. For example, I would like to know how many customers I would have > after 1 year (or 2 or 3) if the current customer base of 25000 > increased 3% a month. What is the correct algebraic calculation to arrive at the correct > value? > mos Hint: After 1 month, there will be 1.03*25000 customers, after two > months, there will be 1.03*[1.03*25000] = (1.03)^2*25000 customers ... > After 10 years there will be over a billion customers. Tell us when > this company goes public. Oops, after 10 years there will be 867,775 customers. After 30 years there will be over a billion customers. I still want part of this IPO. === === Subject: : Re: Linear exponentials > Try e.g. f(x) = exp(-x) + exp(-100 x) > g(x) = 5 exp(-10 x) (don't just plot them, _think_) I cheated, use Excel (ha ha!) to solve f(x)-g(x) =0. Now can you help me to find positive a_1, a_2, b_1,b_2, c_1, c_2 and d_1, d_2 with the following constraints: a_1 > a_2, b_1 < b_2 c_1> c_2, d_1 < d_2 b_1 > d_1, b_2 > d_2, f(x) = a_1* exp(-b_1* x) - a_2 *exp(-b_2*x) g(x) = c_1* exp(-d_1* x) -c_2* exp(-d_2* x) g(x) - f(x) =0 should have a positive root. It looks complicated, but actually it is not. === === Subject: : F1,F2 E;x^2/a^2+y^2/b^2+z^2/c^2=1(a>b>c>0) intersection E,H iz ellipse. ------------------------------------ focus F1( , , ) F2( , , ) === === Subject: : postulate posting-account=fl4D2woAAAC4QBFmZeykoadHa2UXfAKY Gecko/20060731 Ubuntu/dapper-security Firefox/1.5.0.5,gzip(gfe),gzip(gfe) Hi all, can any one give me a correct definition of postulate, the definition i have read is as follows:- In geometry, one cannot prove certain statements of relations between figures, such statements are called postulates. ex:- only one line can be drawn between two points === === Subject: : Re: postulate posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > Hi all, can any one give me a correct definition of postulate, the definition i have read is as follows:- In geometry, one cannot prove certain statements of relations between > figures, such statements are called postulates. ex:- only one line can be drawn between two points Actually, in usage there seems to be a difference between axiom and postulate. Your example above is an axiom. Often, however, one uses the term 'postulate' in a different manner. For example, you might be trying to prove a theorem about the properties of some geometric figure in the form if the figure xxx has this property 1, this property 2, ..., this property n, then it also has that property (n +1). Properties 1, 2, ..., n are the /postulates/ of the theorem. In a sense, they are neither true nor false, because some figures may possess those properties while other figures might not. R.G. Vickson === === Subject: : Re: postulate Hi all, can any one give me a correct definition of postulate, the definition i have read is as follows:- In geometry, one cannot prove certain statements of relations between > figures, such statements are called postulates. ex:- only one line can be drawn between two points I think it would be better to say that not everything can be proved. The things one chooses not to prove are called postulates. Is it clear to you that one cannot prove everything? See http://en.wikipedia.org/wiki/Postulate. -- === === Subject: : Re: JSH: ? posting-account=LlRppgoAAAD1KDQAbEw51E0RIOUzJ0up Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Wot happened to james harris ? is he tied up in court suing math dudes? He is thinking about the spherical packing problem:http://mymath.blogspot.com/ rossum Hate to nit-pick, but with James you really should put quotes around the word thinking. That way people won't mistake what he does for the normal rational thought processes us mere mortals use to solve logical problems. James had announced earlier that he had the cube packing problem solved and he expected any day now that with a little work he could now obtain the solution to the sphere-packing problem. No doubt this will be James's greatest achievement: imagine mathematicians convincing the world that such a simple problem required such a complicated solution. This was their greatest lie to date, but luckily James will expose them and then worlds will collide, stockmarkets will crash and grocer's will have to go back to school to learn how to properly stack oranges. M === === Subject: : Re: Tom Potter is a pussy <8d45$4833afd4$17010@news.teranews.com> posting-account=vma-PgoAAABrctSmMdefNKZ-c5S8buvP Luis Savain, you are a self hating jew, exactly like Alberto and Marcel. This is why you eat so much on the account of the jews, because ...you are one! Got your number, Luis-boy. And don't forget to keep on sucking on Tom Potty's dick, you closet homo :-) === === Subject: : Re: Tom Potter is a pussy ahahaha... AHAHAHAHA... ahahahaha... .. I wonder why all the Jews, including homo-Dono here, are so nervous lately... Dono , aka Dunno aka Karandash the embarrassed homo-kike and virulent Einstein Dingleberry just sang again like his brother-in-ilk here: < http://www.youtube.com/watch?v=2O9W3UsdRyM > [1] or in < http://www.liveleak.com/view?i=e1842edc4f > [1]] Luis Savain, you are a self hating jew, exactly like Alberto and Marcel. This is why you eat so much on the account of the jews, because ...you are one! Got your number, Luis-boy. And don't forget to keep on sucking on Tom Potty's dick, you closet homo > ahahaha... AHAHAHAHA... ahahahaha... It looks like you wanna be the bitch, the sub, of Luis Savain... ahahaha... > Well, homo-Dono, take up your yearnings with Louis Savain himself. You have his handle and url. I have told you that a few months back already but, since you are damaged goods to begin with, you must have forgotten that & so you long to get your sphincter reamed anywhere, at any cost.... ahahaha... But try to realize, you poor sod, that I really have no use for you... other than to show by your own examples what Jewish intelligence is all about .... ahaha... ahahaha... > Now then homo-Dono, listen, are you a masochist that you insist to hear about your sins again via/with your classical ashkeNazi diversion tactics of weaseling & cussing, ever since you you, Karand Asher, came to the rescue of your bor-in-ilk who has plonked in defeat.... ahaha... AHAHA... > BTW, homo-Dono do you realize that I have put a collar on you and that you follow me like an obedient mutt, since ever you came running and barking at me... mimiking and repeating my impressions about you? ... So much for Jewish intelligence... ahahaha.... > Therefore sub-homo Dono, I'll do you unemployed Dreidel the favor then, and here it is again to honor your gauchly nobel efforts: > Zio-Servant Neilist the plonker, seen here in his natural element < http://www.youtube.com/watch?v=2O9W3UsdRyM > [1] or in < http://www.liveleak.com/view?i=e1842edc4f > [1]] **** Potter, Drop dead, you bigot. **** > **** So Hanson, you can drop dead too, you bigot. **** Where's the plonk button on this keyboard? *plonk you bigots* ahahahah... AHAHAHAHA... It looks like Neilist is no longer the Shmileist that he once was since some new 'inconvenient truths may have dawned on him...ahahaha ... stuff which did make him wonder, coyly plonk and then squeeze his Beytsim, or at least his Shvantz --- (since he lost his cojones), --- between his legs and depart with a classical AshkeNazi curse, --- ( killing everybody who disagrees) --- thinking that he will save face that way... ahaha... > So, here it is again for you, Neilist, so that you can see whose blind servant & what type of stooge you were: Potter, Drop dead, you bigot. > Neilist seen here in full action: < http://www.youtube.com/watch?v=2O9W3UsdRyM > [1] or in < http://www.liveleak.com/view?i=e1842edc4f > [1]] > It is interesting to see that Neilist..... > it sure is interesting , because the Kike or the Kike's servant, Neilist, wishes for Potter to drop dead over Potter exposing the machinations of the Jews thru/by/with words that the Jews themselves have published .... ... ahaha... So much for Jewish intelligence or the Jewish indoctrination of Neilist, the Dreidel,... ahaha... Neilist is an embarrassment for/to the Jewish community .... ahahahaha.... > For example, I read in the news today, Joe Lieberman, a Jewish Senator, does not want Americans to be exposed to the opinions of Muslims on the Internet, and he is working to get Google to bias their displays favoring Jews even more than Google already does. > wherein you can see, no matter which way you slice it, that the ing kike senator Lieberman is advocating that you, me and even Neilist, and the rest of the US voters are so stupid that we should not be allowed to see and read the Allah-trallallah-shmallah propaganda that comes from the ass-venting Al Qaeda goons. Asshole Lieberman advocates that the tripe from/by the ass-venters should be removed from the US web (YouTube) so that only pin heads in govt agencies and US officials such as himself would be allowed to read what our enemies, .... (that he and his fellow Jew ilk have spawned exclusively over the last 60 years see [1])... have to say. that Jew notion! Lieberman is as bad in/with trying to curtail our freedoms as are the terrorists ! > Lieberman maybe more afraid that the Islam extremists have some things to say about the machinations of the JerUSAlem cockroaches... (= US Zionist Jews & Neocons whose first loyalty belongs to Israel instead of the US) ... and that such information may resonate with the US goyim & wake them up to some facts that the kikes do try to bury desperately...ahaha.. events like this one here: < http://www.youtube.com/watch?v=2O9W3UsdRyM > [1] or in < http://www.liveleak.com/view?i=e1842edc4f > [1]] > ahahaha & thus: Lieberman has sucsssfully invoked Sid Snead's axiom that says: ::S:: I am sorry to hear that Sen. Lieberman is a Jew. I enjoy the ::S:: clarity of his writing, but will now have to treat his opinions ::S:: as suspect, since, being a Jew, he believes that what is ::S:: good for Jews is more important than the truth. > To read the stories of a few of the many folks who have been victims of Institutionalized bigotry visit the web site below. It is sad, that today, in America, that Institutionalized BIGOTRY is aggressively promoted by the Jewish Lobby and by Jewish politicians, Tom Potter > I dont' know why is it that so many Jews, like Neilist, and kikes as mentioned above, do work so very, very hard to turn Hitler's Last Curse into becoming a self-filling prophesy which Hitler broadcast just before his demise in 45, in which he said: **in a 100 years from now the world will be grateful for what I, Hitler, have started.** .... .... 63 down, only 37 to go... > ahaha.. You must hand it to the Jews, chutzpah!... ahahahaha... PS: This here is for the benefit of Neilist who gets disturbed by Potter's rehash of anti-Jew propaganda. Therefore this here, being NO anti-Jew propaganda, may be more to Neilist's liking since it comes DIRECTLY from Israel: > Ringworm and Radiation / by Barry Chamish http://web.israelinsider.com/views/3998.htm wherein it says: documentary film which exposes the ugliest secret of Israel's Labor party founders: The subject is the mass irradiation of hundreds of thousands of young Israeli immigrants, Sephardim, from Middle Eastern countries in 1951. Every Sephardim child was to be given 35,000 times the maximum radiation dose of x-rays through his head. > David Deri makes the point that only Sephardim children received the x-rays: I was in class and the men came to take us on a tour. They asked our names. The Ashkenazim children were told to return to their seats. The dark children were put on the bus. > To fool the parents of the victims, the children were taken away on school trips and their parents were later told the x-rays were a treatment for the scourge of scalpal ringworm. > 6,000 of the children died shortly after their doses were given, while many of the rest developed cancers that killed thousands over time and are still killing them now. While living, the victims suffered from disorders such as epilepsy, amnesia, Alzheimer's disease, chronic headaches and psychosis. -- More in the link: http://web.israelinsider.com/views/3998.htm > So much for the moral superiority of Jews, today... ahahaha... ahahahahanson === === Subject: : Re: Tom Potter is a pussy <83cbf$482d6d05$10210@news.teranews.com> <8d45$4833afd4$17010@news.teranews.com> <4b4df$48369f1d$30195@news.teranews.com> posting-account=rIfu6QoAAAD5nXG3h9QEE0J3dZn1U45R Gecko/2008051206 Firefox/3.0,gzip(gfe),gzip(gfe) [snip blather] None of your whining has to do with any of the newsgroups you routinely crosspost to. Try using your blogs for a change. === === Subject: : Re: Tom Potter is a pussy > [snip blather] > None of your whining has to do with any of the newsgroups you routinely crosspost to. Try using your blogs for a change. > ahahaha... oh, but what Potter blathers & whines about has very much to do with the 3 NGs he posts to here to. You'll find that out as soon as you finalize your B.Sc and graduate and get out into the real world, or if you decide to stay in academia, the name of the game is politics and funding. You will see that the following principle applies, globally: ---------- No politics -- No money --- No physics ----------- > It arises from the axiom that all sciences, by definition, are social enterprises. Of course, undergrads or lab rats or cubicle inmates do not yet have the necessary horizon to be interested, nor care, where the money comes from to cover their wages. --- So, the sooner you become interested in the subject of where to get the mothers' milk of physics from, to do your physics with, the farther you will go. Rest assured that you'll get nowhere if you can't or won't *SELL* yourself and your dreams (to do your research). > Anyway Eric, I have a question for you. My take is that you have chosen a science set that is about to enter its golden era. Astro-sci today reminds me of the days where atomic/ nuclear physics was in the early 1900's. Back then we had only the tools and the technology to make precise measurements into the realm of the small... down some 30 orders of magnitude in size, from the measure of all things, man, the unit for reference and arbitration. > Now a century later with advances, not in physics itself but, in technology, the ground work is laid to make our human journey up- and outwards by a similar leap of magnitudes. The present and more so the future are an exiting times for astro sciences. If you feel fundamentally different, then tell me what turned you on that you have chosen your field in these endeavors. Take care, junior. I wish you well. hanson > === === Subject: : Re: Tom Potter is a pussy ahahaha... gb6726@yahoo.com ... nahhh... gb you were simply loaded too badly loaded last night & you didn't get it. So what made you crank yourself in here so pathologically? === === Subject: : Re: Tom Potter is a pussy ahahaha... gb6726@yahoo.com ... nahhh... gb you were simply loaded too badly loaded last night & you didn't get it. So what made you crank yourself in here so pathologically? === === Subject: : Re: Tom Potter is a pussy ahahaha... gb6726@yahoo.com ... nahhh... gb you were simply loaded too badly loaded last night & you didn't get it. So what made you crank yourself in here so pathologically? === === Subject: : Re: How to integrate absolute value: 2-|2-5x| <20080523142957.772$vx@newsreader.com> posting-account=gc2kDQoAAADMsLO9kJjQL9hCJkI0D8qJ CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > How do I integrate the following absolute value? > 2-|2-5x| > or 2-abs(2-5x) > æ æ æ æ æ | 5x æ x<=2/5 > 2-|2-5x|= | > æ æ æ æ æ | 4-5x x>2/5 then > I(2-|2-5x|)= ..... Even given that hint, many students would still make a mistake. > Specifically, they would often give for the answer | æ5/2 x^2 + C1, æ æ æ x <= 2/5; > | > | æ4 x - 5/2 x^2 + C2, æx > 2/5 where C1 and C2 are constants of integration. That's not quite right. Why? A fully correct answer would be, for example, 2 x + (2 - 5 x) |2 - 5 x|/10 + C where C is a constant of integration. David === === Subject: : Re: How to integrate absolute value: 2-|2-5x| > How do I integrate the following absolute value? > 2-|2-5x| > or 2-abs(2-5x) > | 5x x<=2/5 > 2-|2-5x|= | > | 4-5x x>2/5 then > I(2-|2-5x|)= ..... Even given that hint, many students would still make a mistake. > Specifically, they would often give for the answer | 5/2 x^2 + C1, x <= 2/5; > | > | 4 x - 5/2 x^2 + C2, x > 2/5 where C1 and C2 are constants of integration. That's not quite right. > Why? > A fully correct answer would be, for example, 2 x + (2 - 5 x) |2 - 5 x|/10 + C where C is a constant of integration. David > C1 and C2 are independent constants. Bearing that in mind, note that the piecewise function will typically have a jump discontinuity at x = 2/5, and thus will not be differentiable there. But we need a function which is differentiable on the whole real line and, specifically, at x = 2/5, the value of its derivative must be 2. Of course, by adjusting either C1 or C2 in relation to the other, we can assure that there will be no jump discontinuity at x = 2/5. For example, we can choose C2 to be C1 - 4/5, and then the piecewise function | 5/2 x^2 + C1, x <= 2/5; | | 4 x - 5/2 x^2 + C1 - 4/5, x > 2/5 is continuous. But note that it has just one arbitrary constant of integration, not two. David === === Subject: : Re: How to integrate absolute value: 2-|2-5x| > Even given that hint, many students would still make a mistake. > Specifically, they would often give for the answer | æ5/2 x^2 + C1, æ æ æ x <= 2/5; > | > | æ4 x - 5/2 x^2 + C2, æx > 2/5 where C1 and C2 are constants of integration. That's not quite right. Why? A fully correct answer would be, for example, 2 x + (2 - 5 x) |2 - 5 x|/10 + C where C is a constant of integration. David > It is correct (maybe). But his point is that unless C1 and C2 match somehow, then it is not continuous at x=5/2 . Which may (or may not) be required when the OP said simply integrate... === === Subject: : show this sequence is unbounded posting-account=gc2kDQoAAADMsLO9kJjQL9hCJkI0D8qJ CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) a_(n+1)/a_n converges to L where L > 1. How to prove that the sequence a_n is not bounded? I tried by contradiction, but got stuck. Can you please help? === === Subject: : Re: show this sequence is unbounded > a_(n+1)/a_n converges to L where L > 1. How to prove that the sequence > a_n is not bounded? This is a problem that calls out for reformulating. Define b_n = log(a_{n+1}/a_n) We have lim b_n = log L > 0 and sum_{k < n+1} b_k = log(a_{n+1}) I think you see where this is going. For 0 < x < log L there exists natural number N such that for all n > N we have b_n > x. Pick a real number y > 0 and define M = ceiling(y/x) . Then sum_{k < N + M} b_k > x.ceiling(y/x) >= y. and so a_{N+M} > exp(y). -- Michael Press === === Subject: : Re: show this sequence is unbounded http://mathforum.org/kb/message.jspa?messageID=6229746 > and suppose that a_(n+1)/a_n converges to L where > L > 1. How to prove that the sequence a_n is not bounded? I tried by contradiction, but got stuck. Can you > please help? Proof by contradication isn't needed. Note that if a_(n+1)/a_n = L > 0 for all n, then {a_n} would be a geometric sequence with a common ratio of L. In your case, you don't have equality to L for each n, but we do know that if we pick an L' < L, then we'll have a_(n+1)/a_n > L' for all sufficiently large values of n. In particular, if L > 1, we can choose L' so that 1 < L' < L. Now note that, for all sufficiently large values of n, your sequence is term-by-term greater than a geometric sequence with a common ratio of L', and thus your sequence diverges to infinity (since a geometric sequence with positive terms and common ratio greater than 1 diverges to infinity). Dave L. Renfro === === Subject: : Re: show this sequence is unbounded <21413017.1211630943377.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=gc2kDQoAAADMsLO9kJjQL9hCJkI0D8qJ CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) http://mathforum.org/kb/message.jspa?messageID=6229746 and suppose that a_(n+1)/a_n converges to L where > L > 1. How to prove that the sequence a_n is not bounded? I tried by contradiction, but got stuck. Can you > please help? Proof by contradication isn't needed. Note that if > a_(n+1)/a_n = L > 0 for all n, then {a_n} would be a > geometric sequence with a common ratio of L. In your > case, you don't have equality to L for each n, but we > do know that if we pick an L' < L, then we'll have > a_(n+1)/a_n > L' for all sufficiently large values > of n. In particular, if L > 1, we can choose L' so > that 1 < L' < L. Now note that, for all sufficiently > large values of n, your sequence is term-by-term > greater than a geometric sequence with a common ratio > of L', and thus your sequence diverges to infinity > (since a geometric sequence with positive terms and > common ratio greater than 1 diverges to infinity). Dave L. Renfro How do you prove the last fact? Namely that a geometric sequence with positive terms and common ratio greater than 1 diverges to infinity === === Subject: : Re: show this sequence is unbounded <21413017.1211630943377.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=O9zR9AkAAACmp918j6u5m5plppeILcze 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; WWTClient2),gzip(gfe),gzip(gfe) > How do you prove the last fact? > Namely that a geometric sequence with positive terms and > common ratio greater than 1 diverges to infinity. I'm a different Dave, but here is how... L^n = [1 + (L - 1)]^n with (L - 1) > 0. Expand the right-hand side by means of the Binomial Theorem: = 1 + n (L - 1) + other positive terms > n (L - 1) --> oo as n --> oo. Dave === === Subject: : Re: show this sequence is unbounded <21413017.1211630943377.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=1lThgAoAAAAa_6lVePVy433GcywpX3bx Since L>1 and a_n>0, a_(n+1)>a_n for great n hence (a_n) has a limit l and if l is real, then L=1, which proves that l=+infinity. http://mathforum.org/kb/message.jspa?messageID=6229746 and suppose that a_(n+1)/a_n converges to L where > L > 1. How to prove that the sequence a_n is not bounded? I tried by contradiction, but got stuck. Can you > please help? Proof by contradication isn't needed. Note that if > a_(n+1)/a_n = L > 0 for all n, then {a_n} would be a > geometric sequence with a common ratio of L. In your > case, you don't have equality to L for each n, but we > do know that if we pick an L' < L, then we'll have > a_(n+1)/a_n > L' for all sufficiently large values > of n. In particular, if L > 1, we can choose L' so > that 1 < L' < L. Now note that, for all sufficiently > large values of n, your sequence is term-by-term > greater than a geometric sequence with a common ratio > of L', and thus your sequence diverges to infinity > (since a geometric sequence with positive terms and > common ratio greater than 1 diverges to infinity). Dave L. Renfro === === Subject: : Re: show this sequence is unbounded > a_(n+1)/a_n converges to L where L > 1. How to prove that the sequence > a_n is not bounded? I tried by contradiction, but got stuck. Can you please help? > Hint: since the limit converges to L there's some N with for all n > N, a_(n+1)/a_n in (2L, (L + 1)/2) === === Subject: : Re: show this sequence is unbounded a_(n+1)/a_n converges to L where L > 1. How to prove that the sequence > a_n is not bounded? I tried by contradiction, but got stuck. Can you please help? Hint: since the limit converges to L there's The limit converges to L? > some N with for all n > N, a_(n+1)/a_n in (2L, (L + 1)/2) False. === === Subject: : Re: show this sequence is unbounded 1. How to prove that the sequence > a_n is not bounded? I tried by contradiction, but got stuck. Can you please help? Hint: since the limit converges to L there's The limit converges to L? The sequence converges. > some N with for all n > N, a_(n+1)/a_n in (2L, (L + 1)/2) False. > some N with for all n > N, a_(n+1)/a_n in (2L, L/2) === === Subject: : Re: show this sequence is unbounded 1. How to prove that the sequence > a_n is not bounded? I tried by contradiction, but got stuck. Can you please help? Hint: since the limit converges to L there's The limit converges to L? The sequence converges. some N with for all n > N, a_(n+1)/a_n in (2L, (L + 1)/2) False. some N with for all n > N, a_(n+1)/a_n in (2L, L/2) > Whoops, you snookered me. The limit isn't 0 but 1. Hence I stand by (2L, (L+1)/2). === === Subject: : Re: show this sequence is unbounded posting-account=O9zR9AkAAACmp918j6u5m5plppeILcze 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; WWTClient2),gzip(gfe),gzip(gfe) a (n+1)/a n converges to L where L > 1. How to prove that the sequence > a n is not bounded? I tried by contradiction, but got stuck. Can you please help? Hint: æsince the limit converges to L there's The limit converges to L? The sequence converges. æ æ æ some N with for all n > N, a (n+1)/a n in (2L, (L + 1)/2) False. æ æ æ æ some N with for all n > N, a (n+1)/a n in (2L, L/2) Why not some N with for all n > N, a (n+1)/a n in (2L, (L + 1)/2) Dave === === Subject: : orthocentre and centroid posting-account=fl4D2woAAAC4QBFmZeykoadHa2UXfAKY Gecko/20060731 Ubuntu/dapper-security Firefox/1.5.0.5,gzip(gfe),gzip(gfe) Hi all, I was going through the site:- http://en.wikipedia.org/wiki/Altitude_(triangle) here it is written as The three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle (and consequently the feet of the altitudes all fall on the triangle) if and only if the triangle is not obtuse (i.e. does not have an angle greater than a right angle) now my questions are as follows 1) does it mean that if the triangle is obtuse, orthocentre lies outside the triangle 2) is there any relationship between centroid and orthocentre ???? or under what circumstances centroid will become equal to orthocentre ...??? === === Subject: : Re: orthocentre and centroid posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e SIMBAR Enabled; SIMBAR={70306B22-CB8C-4d52-BFF4-18424E217075}; MathPlayer 2.10b; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Hi all, I was going through the site:- http://en.wikipedia.org/wiki/Altitude (triangle) here it is written as The three altitudes intersect in a single point, called the > orthocenter of the triangle. The orthocenter lies inside the triangle > (and consequently the feet of the altitudes all fall on the triangle) > if and only if the triangle is not obtuse (i.e. does not have an angle > greater than a right angle) now my questions are as follows 1) does it mean that if the triangle is obtuse, orthocentre lies > outside the triangle 2) is there any relationship between centroid and orthocentre ???? or > under what circumstances centroid will become equal to > orthocentre ...??? ***************************************************** 1) Yes. If the triangle's obtuse, two altitudes will be exterior to the triangle, so the point of intersection... 2) Not sure what you mean, in general. Of course, if the triangle's equilateral then altitudes and and medians are the same, so both points coincide. Also, orthocenter, centroid and circumcenter all three lay on one single line (Euler's line), and the centroid divides the distance from the orthocenter to the circumcenter in a ratio 2:1... Tonio === === Subject: : a probability problem. given: 1. life-extension every 10 - 50 years through the transplant of clonedorgans has a probability of success of 0.90. 2. the production of nonsentient meats in vertical farms results in apopulation capacity of 10x (where x is the current population). 3. the probability of viable production of healthy babies from artificialwombs is 0.99. to find: the probability of sustaining peak population capacity for atleast abillion years. is this problem solvable in quicktime? does it require further data? thnx. --Message posted using http://www.talkaboutscience.com/group/sci.math/More information at http://www.talkaboutscience.com/faq.html === === Subject: : Re: inverse iteration posting-account=06BQLAoAAADoC7Y4z9FWcUwGvMa7xMG9 7.4),gzip(gfe),gzip(gfe) the ones intrested into fractional iterations , partially recursive functions , tetration and similar stuff might be intrested in what i call inverse iteration . > its a quite simple concept and it might not always be usefull or applicable but often it works very well. > the concept is designed for functions f(x) that are defined for all real x and preferably holomorphic for all complex z. > it is usually easier when f(x) has a somewhat simple series expansion. > perhaps you already use this , i dont know , anyways here it is : > iteration [ e*x ] = exp(x) > inverse iteration [exp(x)] = e*x > inverse iteration[ f(x) ] = f( inv.f(x) + 1 ) OK, I see what's going on now. Inverse iteration is supposed to > be iteration in reverse, so that we have: inverse iteration [a^^x] = a^x > inverse iteration [a^x] = ax > inverse iteration [ax] = a+x so that we have: tetration -> exponentiation -> multiplication -> addition It is natural to wonder what would happen if we were to > continue this pattern and find a new operation that is > easier than addition: inverse iteration [a+x] = ? But tommy1729 is not the first to come up with this idea. A > Russian mathematician, Konstantin Rubtsov, decided to > call this operation zeration, since it's the zeroth operation > of the Ackermann hierarchy. Let's use tommy1729's rule to discover zeration: inverse iteration[ g(x) ] = g( inv.g(x) + 1 ) so we have: > inverse iteration [a+x] = a+(x-a+1) > æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ = x+1 Thus if we use an o to denote zeration, then we > have discovered that aox = x+1. Rubtsov's native language is not English, so there's not > much written in English on zeration. There is a (poorly) > translated version of Rubtsov's work at the link: http://numbers.newmail.ru/english/03/3e.html where Rubtsov defines zeration as: aob = a+1, a > b > æ æ æ æ æb+1, b > a > æ æ æ æ æa+2 = b+2, b = a > æ æ æ æ æa, b = -infinity > æ æ æ æ æb, a = -infinity Notice that this operation is commutative and is > consistent with the other operations, such as: 2o2 = 2+2 = 2*2 = 2^2 = 2^^2 = 4. On the other hand, this isn't consistent with the > definition of inverse iteration given by tommy1729, > where aob = b+1 regardless of the relative sizes of > a and b. Indeed, Andrew Robbins, who is working on > solving the tetration problem, prefers the simpler > definition aob = b+1. He often calls this version of > zeration mother-law zeration, since tommy1729's > rule of inverse iteration is the mother of all laws > regarding operations. Bonjour, such equations are known since Abel. a in Z, b in R with g(x) = f^[a]( b + f[-a](x) ) , n th iterate you've got g^[n](x) = f^[a]( n*b + f[-a](x) ) or f[-a](g^[n](x)) = n*b + f[-a](x) Alain === === Subject: : Re: MATH > Math is anything to do with numbers (Well that's what > I think.) or adding subtracting and dividing. What do > you think???? The theory of formal patterns, is how Bill Thurston once put it. Can't say I'd disagree with him. === === Subject: : Re: MATH <21095312.1211617012254.JavaMail.jakarta@nitrogen.mathforum.org>, > Math is anything to do with numbers (Well that's what > I think.) or adding subtracting and dividing. What do > you think???? The theory of formal patterns, is how Bill Thurston once put it. Can't say I'd disagree with him. It is the essence of mathematics that it concerns itself with those relations which lie so deep in the nature of things that they recur in the most varied situations. This is particularly true, of course, of the rudimentary notions of arithmetic and geometry which have forced themselves on the attention of mankind since the earliest beginnings of thought. --Dunham Jackson -- Michael Press === === Subject: : Re: MATH > <21095312.1211617012254.JavaMail.jakarta@nitrogen.math > forum.org>, The theory of formal patterns, is how Bill Thurston > once put it. Can't say I'd disagree with him. It is the essence of mathematics that it concerns > itself > with those relations which lie so deep in the nature > of > things that they recur in the most varied situations. > This is particularly true, of course, of the > rudimentary > notions of arithmetic and geometry which have forced > themselves on the attention of mankind since the > earliest > beginnings of thought. --Dunham Jackson -- > Michael Press Yes, the first sentence (which is a beauty in my opinion) by itself would convey his point of view quite nicely. But adding the second sentence opens the door on details that requires the third for closure. But with the advance of science and and the accompanying extension of the range of phenomena subjected to quantative discussion, more highly organized groups of concepts, gradually simplified by reduction to their essentials, have come to manefest themselves with similar persistence. Have a good day. === === Subject: : Re: MATH > Math is anything to do with numbers (Well that's what I think.) or adding subtracting and dividing. What do you think???? Mathematics is the science of drawing necessary conclusions. === === Subject: : Re: MATH > Math is anything to do with numbers (Well that's > what I think.) or adding subtracting and dividing. > What do you think???? Mathematics is the science of drawing necessary > conclusions. Wrong. Mathematics is the contra-science of telling when your science is necessarily going wrong. There you find the complementarity of mathematics to science. You take your decisions. Sometimes you know for sure that, as far as your science goes, you are taking the wrong decision. It's up to you to tell what's necessary. -LV === === Subject: : Re: MATH > Math is anything to do with numbers (Well that's what I think.) or adding > subtracting and dividing. What do you think???? Mathematics is the science of drawing necessary conclusions. That sounds like a better description of logic to me. The terms drawing, and necessary are both vague, although I think I understand what you mean. A statement like we are seeking is mostly useful for people who don't already understand. I can almost sell some of my algebra students on the value of their learning mathematics in that it will them to employ measurement within their favorite discipline (not mathematics). Selling the study of the science of drawing necessary conclusions would be a hard sale indeed! ;) Lots of right answers on this one... -Bill === === Subject: : Re: MATH > Math is anything to do with numbers (Well that's what I think.) or adding > subtracting and dividing. What do you think???? > Mathematics is the science of drawing necessary conclusions. > That sounds like a better description of logic to me. The terms drawing, and necessary are both vague, although I think I > understand what you mean. A statement like we are seeking is mostly useful > for people who don't already understand. I can almost sell some of my algebra students on the value of their > learning mathematics in that it will them to employ measurement within > their favorite discipline (not mathematics). Selling the study of the science of drawing necessary conclusions would be > a hard sale indeed! ;) I wasn't trying to sell it. It is a quotation, possibly from Russell - it is saying pretty much the same thing as Dave's Russell quote. He _was_ tight with the logicians. Necessary as in necessary and sufficient. === === Subject: : Re: MATH I wasn't trying to sell it. I know. I apologize for being so critical. -Bill === === Subject: : Re: MATH > I wasn't trying to sell it. I know. I apologize for being so critical. Don't mention it. === === Subject: : Re: MATH posting-account=O9zR9AkAAACmp918j6u5m5plppeILcze 1.0.3705; .NET CLR 1.1.4322; Media Center PC 4.0; .NET CLR 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; WWTClient2),gzip(gfe),gzip(gfe) > Math is anything to do with numbers (Well that's what I think.) or adding subtracting and dividing. What do you think???? ñPure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true. If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.î -- Bertrand Russell Dave === === Subject: : Re: MATH posting-account=F3H0JAgAAADcYVukktnHx7hFG5stjWse .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) [Cross-posted to sci.logic.] Math is anything to do with numbers (Well that's what I think.) or adding subtracting and dividing. What do you think???? ñPure mathematics consists entirely of such asseverations as that, if > such and such a proposition is true of anything, then such and such > another proposition is true of that thing. It is essential not to > discuss whether the first proposition is really true, and not to > mention what the anything is of which it is supposed to be true. If > our hypothesis is about anything and not about some one or more > particular things, then our deductions constitute mathematics. Thus > mathematics may be defined as the subject in which we never know what > we are talking about, nor whether what we are saying is true.î -- > Bertrand Russell This misses the very purpose of mathematics, and this is where mathematics misses itself. The nature of mathematics is in the notion of: having the notion of: having a reference. When you give that reference, you give that reference. [And *this* is not mathematics.] -LV Dave === === Subject: : Re: MATH > [Cross-posted to sci.logic.] Math is anything to do with numbers (Well that's what I think.) or adding subtracting and dividing. What do you think???? > ñPure mathematics consists entirely of such asseverations as that, if > such and such a proposition is true of anything, then such and such > another proposition is true of that thing. It is essential not to > discuss whether the first proposition is really true, and not to > mention what the anything is of which it is supposed to be true. If > our hypothesis is about anything and not about some one or more > particular things, then our deductions constitute mathematics. Thus > mathematics may be defined as the subject in which we never know what > we are talking about, nor whether what we are saying is true.î -- > Bertrand Russell > This misses the very purpose of mathematics, and this is where > mathematics misses itself. The nature of mathematics is in the notion of: having the notion of: > having a reference. When you give that reference, you give that reference. [And *this* is not mathematics.] You can say that again! === === Subject: : Re: MATH > On May 23, 11:08æam, Evan23 what I think.) or adding subtracting and dividing. > What do you think???? ñPure mathematics consists entirely of such > asseverations as that, if > such and such a proposition is true of anything, then > such and such > another proposition is true of that thing. It is > essential not to > discuss whether the first proposition is really true, > and not to > mention what the anything is of which it is supposed > to be true. If > our hypothesis is about anything and not about some > one or more > particular things, then our deductions constitute > mathematics. Thus > mathematics may be defined as the subject in which we > never know what > we are talking about, nor whether what we are saying > is true.î -- > Bertrand Russell This misses the very purpose of mathematics, and this is where mathematics misses itself. The nature of mathematics is in the notion of: having the notion of: having a reference. When you give that reference, you give that reference. -LV Dave === === Subject: : Re: MATH > On May 23, 7:08æpm, Evan23 what I think.) or adding subtracting and dividing. > What do you think???? math is the ideal thought of man about the world. This is very strong to say. -LV > When you prove a mathematical problem it is look like > when you lift a > weight. Omar Hosseiny === === Subject: : Re: MATH > On May 23, 7:08 pm, Evan23 what I think.) or adding subtracting and dividing. > What do you think???? > math is the ideal thought of man about the world. > This is very strong to say. Strong like a weightlifter? === === Subject: : Re: MATH > On May 23, 7:08 pm, Evan23 what I think.) or adding subtracting and dividing. > What do you think???? > math is the ideal thought of man about the world. > This is very strong to say. Strong like a weightlifter? Yes, strong as in wrong. Wrong in mathematics: you get a bad mark. Wrong in physics: you blow up the universe. Wrong in biology: you blow up life on the planet. Wrong in politics: you can keep lying. And so on, really. -LV === === Subject: : Re: MATH > Math is anything to do with numbers (Well that's what I think.) or > adding subtracting and dividing. What do you think??? >I think you're neglecting geometry. Why would you say that? Determining perimeter, length, area, volume and angle all require numeric operations. Even categorizing shapes and volumes requires measuring and quantifying certain characteristics (base angles of an isosceles triangle). Phil H === === Subject: : Re: MATH > Math is anything to do with numbers (Well that's what I think.) or > adding subtracting and dividing. What do you think??? > I think you're neglecting geometry. Why would you say that? Determining perimeter, length, area, volume and > angle all require numeric operations. Even categorizing shapes and > volumes requires measuring and quantifying certain characteristics (base > angles of an isosceles triangle). But essential concepts like similarity and congruence have nothing to do with numbers. === === Subject: : Re: MATH > Math is anything to do with numbers (Well that's what I think.) or > adding subtracting and dividing. What do you think??? > I think you're neglecting geometry. > Why would you say that? Determining perimeter, length, area, volume > and angle all require numeric operations. Even categorizing shapes > and volumes requires measuring and quantifying certain > characteristics (base angles of an isosceles triangle). But essential concepts like similarity and congruence have nothing to > do with numbers. That's a stretch. I would have a tough time teaching those concepts without numbers. Similar polygons have corresponding sides in the same ratio. How would that concept be taught without numbers? Phil H === === Subject: : Re: MATH > Math is anything to do with numbers (Well that's what I think.) or > adding subtracting and dividing. What do you think??? > I think you're neglecting geometry. > Why would you say that? Determining perimeter, length, area, volume > and angle all require numeric operations. Even categorizing shapes > and volumes requires measuring and quantifying certain > characteristics (base angles of an isosceles triangle). > But essential concepts like similarity and congruence have nothing to > do with numbers. That's a stretch. I would have a tough time teaching those concepts > without numbers. Similar polygons have corresponding sides in the same > ratio. How would that concept be taught without numbers? > I don't want to belabour the point, but, to my way of thinking, talking about equality of properties does not necessarily require numbers. You can say that two sides, areas, volumes or ratios thereof are equal without resorting to measurement or quantification. There were not many numbers in Euclidian geometry as I was taught it. === === Subject: : Re: MATH Phil Holman a .8ecrit : > Math is anything to do with numbers (Well that's what I think.) or > adding subtracting and dividing. What do you think??? > I think you're neglecting geometry. > Why would you say that? Determining perimeter, length, area, volume > and angle all require numeric operations. Even categorizing shapes > and volumes requires measuring and quantifying certain > characteristics (base angles of an isosceles triangle). > But essential concepts like similarity and congruence have nothing to > do with numbers. That's a stretch. I would have a tough time teaching those concepts > without numbers. Similar polygons have corresponding sides in the same > ratio. How would that concept be taught without numbers? > Yes, that was an incredibly bad choice of example. What about parallelism, or alignment (say Desargues' theorem) ? > Phil H === === Subject: : Re: MATH > Phil Holman a .8ecrit : > Math is anything to do with numbers (Well that's what I think.) or > adding subtracting and dividing. What do you think??? > I think you're neglecting geometry. > Why would you say that? Determining perimeter, length, area, volume > and angle all require numeric operations. Even categorizing shapes > and volumes requires measuring and quantifying certain > characteristics (base angles of an isosceles triangle). > But essential concepts like similarity and congruence have nothing > to do with numbers. > That's a stretch. I would have a tough time teaching those concepts > without numbers. Similar polygons have corresponding sides in the > same ratio. How would that concept be taught without numbers? Yes, that was an incredibly bad choice of example. What about > parallelism, or alignment (say Desargues' theorem) ? Both of those are good examples. Phil H === === Subject: : Re: MATH posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 4334.34; Windows NT 5.1; SV1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) spider-mtc-tg05.proxy.aol.com[400C70C5] (Prism/1.2.1), HTTP/1.1 cache-mtc-ad05.proxy.aol.com[400C74C7] (Traffic-Server/6.1.5 [uScM]) > Math is anything to do with numbers (Well that's what I think.) or > adding subtracting and dividing. What do you think??? I think you're neglecting geometry. > Why would you say that? Determining perimeter, length, area, volume > and angle all require numeric operations. Even categorizing shapes > and volumes requires measuring and quantifying certain > characteristics (base angles of an isosceles triangle). But essential concepts like similarity and congruence have nothing to > do with numbers. That's a stretch. I would have a tough time teaching those concepts > without numbers. Similar polygons have corresponding sides in the same > ratio. How would that concept be taught without numbers? Don't they have congruent angles? Congruence doesn't depend on numbers. Phil H === === Subject: : Re: MATH > Math is anything to do with numbers (Well that's what I think.) or > adding subtracting and dividing. What do you think??? I think you're neglecting geometry. > Why would you say that? Determining perimeter, length, area, volume > and angle all require numeric operations. Even categorizing shapes > and volumes requires measuring and quantifying certain > characteristics (base angles of an isosceles triangle). But essential concepts like similarity and congruence have nothing > to > do with numbers. That's a stretch. I would have a tough time teaching those concepts > without numbers. Similar polygons have corresponding sides in the same > ratio. How would that concept be taught without numbers? >Don't they have congruent angles? Congruence doesn't >depend on numbers. Are you telling me squares are similar to all rectangles? You need more than just congruent angles to prove similarity and congruency of polygons. Phil H === === Subject: : Re: MATH posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 4334.34; Windows NT 5.1; SV1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) spider-mtc-th01.proxy.aol.com[400C70E1] (Prism/1.2.1), HTTP/1.1 cache-mtc-ad05.proxy.aol.com[400C74C7] (Traffic-Server/6.1.5 [uScM]) > Math is anything to do with numbers (Well that's what I think.) or > adding subtracting and dividing. What do you think??? I think you're neglecting geometry. > Why would you say that? Determining perimeter, length, area, volume > and angle all require numeric operations. Even categorizing shapes > and volumes requires measuring and quantifying certain > characteristics (base angles of an isosceles triangle). But essential concepts like similarity and congruence have nothing > to > do with numbers. That's a stretch. I would have a tough time teaching those concepts > without numbers. Similar polygons have corresponding sides in the same > ratio. How would that concept be taught without numbers? >Don't they have congruent angles? Congruence doesn't >depend on numbers. Are you telling me squares are similar to all rectangles? Hmm...I guess not. > You need more > than just congruent angles to prove similarity and congruency of > polygons. The point I was trying to make is that much of geometry doesn't involve numbers, so saying math is all about numbers is simply wrong and your polygons don't change that. Phil H === === Subject: : Re: MATH posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 4334.34; Windows NT 5.1; SV1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) spider-dtc-te07.proxy.aol.com[CDBC7087] (Prism/1.2.1), HTTP/1.1 cache-dtc-ad05.proxy.aol.com[CDBC74C7] (Traffic-Server/6.1.5 [uScM]) > Math is anything to do with numbers (Well that's what I think.) or > adding subtracting and dividing. What do you think??? >I think you're neglecting geometry. Why would you say that? Determining perimeter, length, area, volume and > angle all require numeric operations. Even categorizing shapes and > volumes requires measuring and quantifying certain characteristics (base > angles of an isosceles triangle). If you only concern yourself with the parts that have to do with numbers, you'll miss out on a lot. Phil H === === Subject: : Re: MATH posting-account=tCEoyAoAAAAkltU5zxOoI8uJ4lyz5-kv .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) Math is the study of abstracted objects, relationships among objects, and the relationships themselves, all of which is governed by logic and in some special cases not. === === Subject: : Re: MATH I did not mean to miss geometry!!!!!!!!! === === Subject: : Re: MATH I did not mean to miss geometry!!!!!!!!! You've also missed algebra, logic, topology, ... . -- === === Subject: : Scattered sets are G-delta posting-account=WioTdAkAAABsBoLAdch9BS6fvKixKv_6 Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) I seem to be stuck with this problem for quite a while now: A subset S of real numbers is called scattered if every nonempty subset A of S has an isolated point. Show that every scattered set is a G-delta set. === === Subject: : Re: Scattered sets are G-delta posting-account=jNR8OgoAAABgTHQuvrwdZAvYKYRyZzQh Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Hint: A subset S of R is scattered iff there is some map f:R -> R such that S = {x| lim as y->x f(y) is + infinity}. === === Subject: : Re: Scattered sets are G-delta 1) your scattered set S should be countable (because every compact interval <-K;K> of R can contain only finite number of isolated points); you can intersect S with <-K;K> and take isolated points until there are some 2) now when S is countable collection of isolated points x_n, for every n there is eps_n such that (x_n-eps_n;x_n+eps_n) intersects S only at point x_n; for every natural m define G_m as union over n of open sets (x_n-eps_n/m,x_n+eps_n/m). For every m you get open set and intersection of all of G_m gives S. I don't know if it is correct, but I hope it can help you somehow. Tomas === === Subject: : Re: Scattered sets are G-delta > > 1) your scattered set S should be countable (because every compact > interval <-K;K> of R can contain only finite number of isolated > points); you can intersect S with <-K;K> and take isolated points > until there are some > 2) now when S is countable collection of isolated points x_n, > for every n there is eps_n such that (x_n-eps_n;x_n+eps_n) > intersects S only at point x_n; for every natural m define G_m > as union over n of open sets (x_n-eps_n/m,x_n+eps_n/m). For > every m you get open set and intersection of all of G_m gives S. I don't know if it is correct, but I hope it can help you somehow. It takes a bit more work than this to prove a scattered set is countable. For example, if you take any countable ordinal, then any subset of the reals whose order type (when the points in the subset are ordered as they're ordered on the real line) is scattered, and this doesn't exhaust all scattered sets (because, among other things, you can insert in various places reverse well ordered sets of reals). If no one has said much by tomorrow, I'll post an outline of at least one way you can show scattered sets are countable. For now, however, the following posts might be of use: Dave L. Renfro === === Subject: : Re: Scattered sets are G-delta > If no one has said much by tomorrow, I'll post > an outline of at least one way you can show > scattered sets are countable. Ooops, I meant G_delta. However, one of the key issues involves showing they're countable, at least in the proofs I'm aware of. There are several mathematically equivalent definitions of scattered set. The version used by the original poster is: A subset S of real numbers is called scattered if every nonempty subset A of S has an isolated point. This is ambiguous, since we're not told whether the notion isolated point is with respect to the original set or with respect to the set A. It should be with respect to A. Otherwise, we're looking at a strictly stronger notion of smallness, a set of points each of which is isolated from all the other points in the set. (To see this, choose A = S in the incorrect interpretation of the original posters's definition.) This definition of scattered is equivalent to Hausdorff's 1914 version of the Cantor-Bendixson theorem, one of the three ways in which the Cantor-Bendixson theorem can be approached (Cantor's iterating the derived set function in 1883, Lindelof's method using condensation points in 1905, and Hausdorff's 1914 method), which I outlined in the following post: I just realized that I brought the wrong notes with me, so I'll have to look at home for my other notes and outline how to show G_delta tomorrow. I did think about it some just now, but nothing immediately evident comes to mind and I'd rather not waste time on a problem whose solution I know exactly where it is, especially since it's not an area I'm currently working on. [I will mention, however, that a certain poster who goes by Butch Malahide has published a proof that a set is scattered if and only if it's countable and G_delta. Maybe he'll jump in now that I've mentioned his name. The result itself is due, I believe, to William H. Young and can be found on pp. 65 & 298 of his 1906/1972 book. Also, E. Hobson published an Given some other comments in this thread, I thought I'd outline some subclasses of the countable sets of the real line that are useful in describing notions of smallness (the following are all hereditary) when the set is countable: A. Sets with countable closure B. Isolated sets C. Scattered sets D. Sets that are countable We have: A proper subset of B proper subset of C proper subset D B, but not A: Take any Cantor set (perfect nowhere dense set of the reals) and consider the collection of midpoints of the bounded complementary intervals. The closure of this set will be the union of these midpoints along with the Cantor set that was used. By choosing a Cantor set with positive measure, we can even get an isolated set whose closure has positive measure. C, but not B: {0, 1, 1/2, 1/3, 1/4, 1/5, ...} D, but not C: the set of rational numbers William Elliot, in this thread, suggested that maybe scattered is equivalent to nowhere dense for the real line. This is not true, since every scattered set is countable and there exist uncountable nowhere dense sets (e.g. a Cantor set). Indeed, as some of the posts I cited earlier in this thread say, scattered sets are in a certain sense maximally nowhere dense: A scattered set is nowhere dense relative to every nonempty perfect subset of R [*]. This means, for instance, that not only the set itself but also the closure of the set fails to be dense in any portion of any perfect set, regardless of how small/thin the perfect set is. I believe this way of looking at scattered sets is due to A. Denjoy (around 1914-1917, I think). [*] That is, if S is scattered and P is a perfect set, then 'S intersect P' is a nowhere dense subset of P (where P is given the subspace topology it inherits from the reals). Dave L. Renfro === === Subject: : Re: Scattered sets are G-delta <3378917.1211655642847.JavaMail.jakarta@nitrogen.mathforum.org > If no one has said much by tomorrow, I'll post an outline of > at least one way you can show scattered sets are countable. Isn't the problem to show a scattered subset of R is G_delta? To show that they're countable just shows that they're F_sigma. If S scattered subset R, then int S = nulset. The converse is false. I've not been able to show S is nowhere dense, . . int cl S = nulset. I suppect nowhere dense and scattered are equivalent within R. > For now, however, the following posts might be of use: > Dave L. Renfro > === === Subject: : Re: Scattered sets are G-delta <3378917.1211655642847.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=WioTdAkAAABsBoLAdch9BS6fvKixKv_6 Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Dave, Is closure of a scattered set countable? === === Subject: : Re: Scattered sets are G-delta <29654473.1211639540584.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=jNR8OgoAAABgTHQuvrwdZAvYKYRyZzQh Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > your scattered set S should be countable This is okay. > because every compact interval <-K;K> of R can contain only finite number of isolated points No. Consider {1/n| n = 1,2,3..} as a subset of [0, 1]. > now when S is countable collection of isolated points x_n, Take a break. === === Subject: : Re: Scattered sets are G-delta Countable union ('sum') of countable sets: I thought (A1=S intersect<-1;1>, A2=S intersec<-2;2><-1;1>...): take 1 number from A1 take 1 number from A1 and one number from A2 etc. Isn't this the 'diagonal' argument 123... 456... 789... .. .. and going like 124753...? Tomas === === Subject: : Re: Scattered sets are G-delta You're right, I found it just now with the same counterexample. But this shouldn't change the argument. In every <-K;K> subset there is only countable number of isolated points and by diagonal argument there is only countably points in whole S. :-) Tomas === === Subject: : Re: Scattered sets are G-delta <14305571.1211642209316.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=ZNjDogoAAABK6vPKIvHXsptL8mfq_spr Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > In every <-K;K> subset there is only countable number of > isolated points and by diagonal argument there is only countably points in whole S. How do you get from in every <-K;K> subset there is only countable number of isolated points to there is only countably points in whole S? === === Subject: : Bucky Fuller's 4D John Horton Conway answered an email I sent him and he said he answered one hundred email messages a day and took thirty seconds to answer mine. Everyone I know is very busy, too busy to look at my web site. I posted something like it about ten tears ago. http://mysite.verizon.net/cjnelson9/index.htm There is a chasm between the sciences and the humanities. People who have been trained in the humanities might think Bucky Fuller's 4D is just a metaphor and they don't like Mathematica. Mathematicians don't like Bucky's writing. If there is no communications jamming and you have the time to look at my web site and understand the simple Mathematica code you will say ah ha, that's it, that's the synergetic coordinate system, the coordinate four-dimensional point is a regular tetrahedron sitting comfortably in the vector equilibrium. I have sent my brothers one sentence emails and asked them about it the next day and they say they got something but they don't know what it was. I know from experience that I won't get the message through. We live in the darkest of the dark ages of communications. Cliff Nelson Dry your tears, there's more fun for your ears, Forward Into The Past 2 PM to 5 PM, Sundays, California time, http://www.geocities.com/forwardintothepast/ Don't be a square or a blockhead; see: http://library.wolfram.com/infocenter/search/?search_results=1;search_per son_id=607 === === Subject: : Re: Bucky Fuller's 4D > John Horton Conway answered an email I sent him and he said he answered > one hundred email messages a day and took thirty seconds to answer mine. > Everyone I know is very busy, too busy to look at my web site. I posted > something like it about ten tears ago. http://mysite.verizon.net/cjnelson9/index.htm Your font is too small, too hard for these oldish eyes to read. === === Subject: : Very simple question/paradox about integrals. posting-account=2YcbuwoAAACyq1d4MveawBPtnkVr4m5s Gecko/20080404 Firefox/2.0.0.6 MEGAUPLOAD Toolbar,gzip(gfe),gzip(gfe) The following is true: http://i215.photobucket.com/albums/cc34/Crocodile13/we1.jpg So after applying it to a function we have: http://i215.photobucket.com/albums/cc34/Crocodile13/we2.jpg Something is clearly wrong. There is something, somewhere, that i miss. But where is it? ---------------------- ---------------------- Sorry that you have to copy and paste the images to be shown but i don't know if IMG tags work here. Test(ignore it): [IMG]http://i215.photobucket.com/albums/cc34/Crocodile13/we2.jpg[/IMG] ---------------------- ---------------------- === === Subject: : Re: Very simple question/paradox about integrals. You must change bounds, so if it was -infinity..+infinity, after the change of variables you have +infinity..-infinity which gives another minus. :-) Another way how to do it is to always integrate from lower to upper but substitute g'(x) by |g'(x)|. === === Subject: : Re: Very simple question/paradox about integrals. posting-account=1lThgAoAAAAa_6lVePVy433GcywpX3bx An integral sign without any bound doesn't have any proper sense. Try it with : int_g(a)^g(b) f(u)du=int_a^b f(g(x))g'(x)dx and u=g(x) > The following is any bound doesn't have any proper sensetrue:http://i215.photobucket.com/albums/cc34/Crocodile13/we1.jpg So after applying it to a function we have:http://i215.photobucket.com/albums/cc34/Crocodile13/we2.jpg Something is clearly wrong. There is something, somewhere, that i > miss. But where is it? ---------------------- > ---------------------- > Sorry that you have to copy and paste the images to be shown but i > don't know if IMG tags work here. > Test(ignore it): > [IMG]http://i215.photobucket.com/albums/cc34/Crocodile13/we2.jpg[/IMG] === === Subject: : Re: Why is math important??? > Dave do you think you can send some pics??????? If you're really 13, I think you'll that of the people over 18 who say yes (and especially those who ask you before you ask them), very few have your best interests in mind. In fact, this is probably true if you just say you're 13, even if you aren't. Dave L. Renfro === === Subject: : scale numbers? Assume that: v = 1,...n is a vector containing n real numbers. Is there some procedure to scale all the numbers to lie in the range [a;b]? v = 1 2 3 4 5 6 7 8 9 10 11 12; a = 1; b = 8; max_elm = max(v); s = solve(max_elm/x = b); frac = 1/s; v_scaled = (v.*frac)+1 but the result is: 2 2 3 4 4 5 6 6 7 8 8 9 How is this done correctly and does it have a more formal name? === === Subject: : Re: scale numbers? posting-account=ogMREwkAAAC5xUr8sg7heGtsvzzF18LA Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > Assume that: v = 1,...n is a vector containing n real numbers. Is there some procedure to scale all the numbers to lie in the range > [a;b]? v = 1 2 3 4 5 6 7 8 9 10 11 12; > a = 1; > b = 8; max_elm = max(v); > s = solve(max_elm/x = b); > frac = 1/s; > v_scaled = (v.*frac)+1 but the result is: 2 2 3 4 4 5 > 6 6 7 8 8 9 How is this done correctly and does it have a more formal name? It appears you want to scale a set of integers to keep the same relative values, with a smaller range? In that case, I wouls follow this pattern: wlog, v_1 <= v_2 <= ... <=v_n This is just so I can say the smallest value of the given is v_1, and the largest is v_n. let v' be the scaled vector into [a,b], and v be an intermediate step. v'_1 = a v'_n = b v_k = (v_k - a) * ((b - a) / (v_n - v_1)) + a ; extra parenthesis used to emphasize the scaling factor) v'_k is the approximation used to convert v_k to an integer. Depending on the application, you may want floor, ceiling, rounding, or integer portion. Without any other information, I would use rounding. Thus, for your example (v = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] ; a=1, b=8) v'_1 = 1 v'_12 = 8 v_k = (v_k - 1) * (7 / 11) + 1 --> v = [1, 18/11, 25/11, 32/11, 39/11, 46/11, 53/11, 60/11, 67/11, 74/11, 81/11, 88/11] rounding, we get: v' = [1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8] if you use the integer portion instead (which is the same as the floor, since all values are positive), you get: v' = [1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8] Note that if you don't want to restrict yourself to integers for the scaled vector, then v is exactly scaled, in the sense that the relative differences of the entries are the same in each (i.e. if v_3 - v_2 is five times v_2 - v_1, then v_3 - v_2 will be five times v_2 - v_1) === === Subject: : MURDEROUSMATHS posting-account=4h0TdQoAAAAm-dGoEcWpzOnttI0XiI5o SV1),gzip(gfe),gzip(gfe) See : www.murderousmaths.com === === Subject: : Computing decimals of pi Are there any algorithms out there to compute pi to an arbitrary number of decimal places N, in such a way that intermediate computations do not have to be carried out to precision N? === === Subject: : Re: Computing decimals of pi > Are there any algorithms out there to compute pi to an arbitrary > number of decimal places N, in such a way that intermediate computations > do not have to be carried out to precision N? > If I understand your question correctly, you want to use an iterative algorithm. http://mathworld.wolfram.com/PiIterations.html === === Subject: : Re: Computing decimals of pi > Are there any algorithms out there to compute pi to an arbitrary > number of decimal places N, in such a way that intermediate computations > do not have to be carried out to precision N? > (I'm understanding this question has trying to find the k'th digit of pi without having to generate the (k-1) digits.) In any base of the form 2^n, where n is a positive integer > 0, it is possible to do so. In the decimal base, the answer is There has not been proven to be no algorithm, but we do not know of one. === === Subject: : Re: Computing decimals of pi > Are there any algorithms out there to compute pi to an arbitrary > number of decimal places N, in such a way that intermediate > computations do not have to be carried out to precision N? > (I'm understanding this question has trying to find the k'th digit of pi > without having to generate the (k-1) digits.) Not quite - I am afraid I did not explain myself correctly. My understanding is that when using those methods to produce millions upon millions of decimals of pi, when attempting to compute (say) N decimals one has to carry out all the intermediate computations to a precision of N decimals - which means that one needs huge amounts of memory/disk space when N is large enough. After doing some googling I came across spigot algorithms for pi, which seem to be what I had in mind. === === Subject: : Re: Computing decimals of pi > Are there any algorithms out there to compute pi to an arbitrary > number of decimal places N, in such a way that intermediate > computations do not have to be carried out to precision N? > (I'm understanding this question has trying to find the k'th digit of pi > without having to generate the (k-1) digits.) > Not quite - I am afraid I did not explain myself correctly. > My understanding is that when using those methods to produce > millions upon millions of decimals of pi, when attempting to compute > (say) N decimals one has to carry out all the intermediate computations > to a precision of N decimals - which means that one needs huge amounts of > memory/disk space when N is large enough. If you look at , I think you will find that this is the formula that Joshua Cranmer was describing. It allows for base-16 digits of pi to be computed without computing all the previous digits. > After doing some googling I came across spigot algorithms for pi, > which seem to be what I had in mind. Spigot algorthms have to be arranged ahead of time to compute the desired number of digits. You can't just run a spigot algorithm indefinitely and expect to get arbitrary precision from a single algorithm. Besides, you do have to compute the first n-1 digits in order to get the n-th. === === Subject: : Re: Computing decimals of pi > Are there any algorithms out there to compute pi to an arbitrary > number of decimal places N, in such a way that intermediate > computations do not have to be carried out to precision N? > (I'm understanding this question has trying to find the k'th digit of > pi without having to generate the (k-1) digits.) > Not quite - I am afraid I did not explain myself correctly. > My understanding is that when using those methods to produce > millions upon millions of decimals of pi, when attempting to compute > (say) N decimals one has to carry out all the intermediate computations > to a precision of N decimals - which means that one needs huge amounts > of memory/disk space when N is large enough. If you look at , I think > you will find that this is the formula that Joshua Cranmer was > describing. It allows for base-16 digits of pi to be computed without > computing all the previous digits. > After doing some googling I came across spigot algorithms for pi, > which seem to be what I had in mind. Spigot algorthms have to be arranged ahead of time to compute the > desired number of digits. You can't just run a spigot algorithm > indefinitely and expect to get arbitrary precision from a single > algorithm. Well, the paper below seems to contradict that: http://www.comlab.ox.ac.uk/people/jeremy.gibbons/publications/spigot.pdf > Besides, you do have to compute the first n-1 digits in > order to get the n-th. Right. That was not one of my requirements, which is why the BBP formula is not what I was thinking about. === === Subject: : Re: Computing decimals of pi > Are there any algorithms out there to compute pi to an arbitrary > number of decimal places N, in such a way that intermediate > computations do not have to be carried out to precision N? > (I'm understanding this question has trying to find the k'th digit of > pi without having to generate the (k-1) digits.) > Not quite - I am afraid I did not explain myself correctly. > My understanding is that when using those methods to produce > millions upon millions of decimals of pi, when attempting to compute > (say) N decimals one has to carry out all the intermediate computations > to a precision of N decimals - which means that one needs huge amounts > of memory/disk space when N is large enough. > If you look at , I think > you will find that this is the formula that Joshua Cranmer was > describing. It allows for base-16 digits of pi to be computed without > computing all the previous digits. > After doing some googling I came across spigot algorithms for pi, > which seem to be what I had in mind. > Spigot algorthms have to be arranged ahead of time to compute the > desired number of digits. You can't just run a spigot algorithm > indefinitely and expect to get arbitrary precision from a single > algorithm. > Well, the paper below seems to contradict that: > http://www.comlab.ox.ac.uk/people/jeremy.gibbons/publications/spigot.pdf How, exactly, does it contradict that? I notice the paper begins with a program written by Dik Winter, which I have seen in multiple versions. If I am not mistaken, Dik Winter himself has described in this newsgroup how different versions of the algorithm are used, depending on how many digits are desired. > Besides, you do have to compute the first n-1 digits in > order to get the n-th. > Right. That was not one of my requirements, which is why the BBP > formula is not what I was thinking about. -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === === Subject: : Re: Computing decimals of pi Are there any algorithms out there to compute pi to an arbitrary > number of decimal places N, in such a way that intermediate > computations do not have to be carried out to precision N? > (I'm understanding this question has trying to find the k'th digit > of pi without having to generate the (k-1) digits.) > Not quite - I am afraid I did not explain myself correctly. > My understanding is that when using those methods to produce > millions upon millions of decimals of pi, when attempting to compute > (say) N decimals one has to carry out all the intermediate > computations to a precision of N decimals - which means that one > needs huge amounts of memory/disk space when N is large enough. If you look at , I think > you will find that this is the formula that Joshua Cranmer was > describing. It allows for base-16 digits of pi to be computed without > computing all the previous digits. > After doing some googling I came across spigot algorithms for pi, > which seem to be what I had in mind. Spigot algorthms have to be arranged ahead of time to compute the > desired number of digits. You can't just run a spigot algorithm > indefinitely and expect to get arbitrary precision from a single > algorithm. > Well, the paper below seems to contradict that: > http://www.comlab.ox.ac.uk/people/jeremy.gibbons/publications/ spigot.pdf How, exactly, does it contradict that? I notice the paper begins with a > program written by Dik Winter, which I have seen in multiple versions. > If I am not mistaken, Dik Winter himself has described in this newsgroup > how different versions of the algorithm are used, depending on how many > digits are desired. If I understand it correctly, the paper explicitly says that nothing has to be arranged beforehand as far as the number of decimals that one wants to compute is concerned. The algorithm can run indefintely, churning out more and more decimals. That's why I thought it === === Subject: : Re: Computing decimals of pi > Are there any algorithms out there to compute pi to an arbitrary > number of decimal places N, in such a way that intermediate > computations do not have to be carried out to precision N? > (I'm understanding this question has trying to find the k'th digit > of pi without having to generate the (k-1) digits.) > Not quite - I am afraid I did not explain myself correctly. > My understanding is that when using those methods to produce > millions upon millions of decimals of pi, when attempting to compute > (say) N decimals one has to carry out all the intermediate > computations to a precision of N decimals - which means that one > needs huge amounts of memory/disk space when N is large enough. > If you look at , I think > you will find that this is the formula that Joshua Cranmer was > describing. It allows for base-16 digits of pi to be computed without > computing all the previous digits. > After doing some googling I came across spigot algorithms for pi, > which seem to be what I had in mind. > Spigot algorthms have to be arranged ahead of time to compute the > desired number of digits. You can't just run a spigot algorithm > indefinitely and expect to get arbitrary precision from a single > algorithm. > Well, the paper below seems to contradict that: > http://www.comlab.ox.ac.uk/people/jeremy.gibbons/publications/ > spigot.pdf > How, exactly, does it contradict that? I notice the paper begins with a > program written by Dik Winter, which I have seen in multiple versions. > If I am not mistaken, Dik Winter himself has described in this newsgroup > how different versions of the algorithm are used, depending on how many > digits are desired. > If I understand it correctly, the paper explicitly says that > nothing has to be arranged beforehand as far as the number of decimals > that one wants to compute is concerned. The algorithm can run > indefintely, churning out more and more decimals. That's why I thought it Ok, you're right. I didn't read far enough. -- Dave Seaman Third Circuit ignores precedent in Mumia Abu-Jamal ruling. === === Subject: : Re: Computing decimals of pi posting-account=vI5-YAoAAACpb1I_2s__b0LrjNDZjNTS Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) your intentions. I first thought that you are a PI hunter so planning to go for billions N precision in search of the promised land (hypothetical repetitive series :-) If you need to know N[i] digit of PI and you don't care neither you want to calculate all preceding N then use the Bailey-Borwein-Plouffe formula. It gives any necessary precision for IEEE-754 arithmetic so can be used on an average PC without BigMath addons. The formula and explanations are avaiable at http://crd.lbl.gov/~dhbailey/dhbpapers/digits.pdf David Bailey, Peter Borwein, and Simon Plouffe, 1995 On the rapid computation of various polylogarithmic constants === === Subject: : Re: Computing decimals of pi posting-account=vI5-YAoAAACpb1I_2s__b0LrjNDZjNTS Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > My understanding is that when using those methods to produce > millions upon millions of decimals of pi, when attempting to compute > (say) N decimals one has to carry out all the intermediate computations > to a precision of N decimals - which means that one needs huge amounts of > memory/disk space when N is large enough. No, it is not the case case. While the calculation gets resource expensive for large amount of numbers in the fractional part, the calculation itself is going with integers so protected from otherwise inevitable error accumulation. John Machin's formula normally used for small precisions (several thousands of signs in the fraction part) explained here: http://en.wikipedia.org/wiki/Machin-like_formula A demo Javascript program calculating up to 1000 signs can be found here: http://www.trans4mind.com/personal_development/JavaScript/longnumPiMachin.ht m It takes lesser than 5 sec to get the 1000th digit on 1.9Ghz machine. That could be 10000 or more digits within an acceptable amount of time but it would hit the time limit for a continuously running Javascript program, so if you are targeted for millions of digits then it is better to use C++ === === Subject: : Re: Computing decimals of pi posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 4334.34; Windows NT 5.1; SV1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) spider-mtc-tc04.proxy.aol.com[400C7044] (Prism/1.2.1), HTTP/1.1 cache-mtc-ad05.proxy.aol.com[400C74C7] (Traffic-Server/6.1.5 [uScM]) > ? ? ? ?Are there any algorithms out there to compute pi to an arbitrary > number of decimal places N, in such a way that intermediate > computations do not have to be carried out to precision N? (I'm understanding this question has trying to find the k'th digit of pi > without having to generate the (k-1) digits.) ? ? ? ? Not quite - I am afraid I did not explain myself correctly. ? ? ? ? My understanding is that when using those methods to produce > millions upon millions of decimals of pi, when attempting to compute > (say) N decimals one has to carry out all the intermediate computations > to a precision of N decimals Not necessarily. Pi can be computed to arbitrary precision using an infinite series of arbitrary precision rationals. They start out small and the terms needn't be kept in memory, only the sum does. The decimal places aren't calculated until the end when the rational sum is converted to floating point. > - which means that one needs huge amounts of > memory/disk space when N is large enough. ? ? ? ? After doing some googling I came across spigot algorithms for pi, > which seem to be what I had in mind. === === Subject: : Re: Computing decimals of pi arbitrary number of decimal places N, in such a way that > intermediate computations do not have to be carried out to precision > N? > (I'm understanding this question has trying to find the k'th digit of > pi without having to generate the (k-1) digits.) > ? ? ? ? Not quite - I am afraid I did not explain myself correctly. > ? ? ? ? My understanding is that when using those methods to produce > millions upon millions of decimals of pi, when attempting to compute > (say) N decimals one has to carry out all the intermediate computations > to a precision of N decimals Not necessarily. Pi can be computed to arbitrary precision using an > infinite series of arbitrary precision rationals. They start out small > and the terms needn't be kept in memory, only the sum does. The decimal places aren't calculated until the end when the rational sum > is converted to floating point. Could you please provide specific examples? === === Subject: : Re: Computing decimals of pi posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 4334.34; Windows NT 5.1; SV1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) spider-mtc-th01.proxy.aol.com[400C70E1] (Prism/1.2.1), HTTP/1.1 cache-mtc-ad05.proxy.aol.com[400C74C7] (Traffic-Server/6.1.5 [uScM]) >Not quite - I am afraid I did not explain myself correctly. >My understanding is that when using those methods to produce >millions upon millions of decimals of pi, when attempting to compute >(say) N decimals one has to carry out all the intermediate computations >to a precision of N decimals Not necessarily. Pi can be computed to arbitrary precision using an >infinite series of arbitrary precision rationals. They start out small >and the terms needn't be kept in memory, only the sum does. The decimal places aren't calculated until the end when the rational sum >is converted to floating point. Could you please provide specific examples? Ok, here's the formula I use: pi/2 = 1 + 1/3 + (1*2)/(3*5) + (1*2*3)/(3*5*7) + ... Note that this is for pi/2, so I have to multiply by two to get the actual digits of pi. But also note how each term differs from the previous: numerator: multiply by 1 more 1 then 1*2 then 1*2*3 then 1*2*3*4 etc. denominator: multiply by 2 more 3 then 3*5 then 3*5*7 then 3*5*7*9 etc. For a single term, the rational becomes 2.0, not a very good approximation of pi. So we add a second term 1/3 (times two remember) making the sum 2.666..., still not very good. But if we keep at it long enough current sum: 2.0 next term: 1/3 current sum: 2.66666666666666666663 next term: 2/15 current sum: 2.93333333333333333328 next term: 2/35 current sum: 3.04761904761904761901 next term: 8/315 current sum: 3.09841269841269841261 next term: 8/693 current sum: 3.12150072150072150062 next term: 16/3003 current sum: 3.13215673215673215673 next term: 16/6435 current sum: 3.13712953712953712953 next term: 128/109395 current sum: 3.13946968064615123436 next term: 128/230945 current sum: 3.14057816968033686291 next term: 256/969969 current sum: 3.14110602160137763843 next term: 256/2028117 current sum: 3.1413584725201362703 next term: 1024/16900975 current sum: 3.14147964896114041352 next term: 1024/35102025 current sum: 3.14153799317347574177 next term: 2048/145422675 current sum: 3.14156615934494796917 next term: 2048/300540195 current sum: 3.14157978813759582116 next term: 32768/9917826435 current sum: 3.14158639603706144636 next term: 32768/20419054425 current sum: 3.14158960558823046434 next term: 65536/83945001525 current sum: 3.14159116699150187839 next term: 65536/172308161025 current sum: 3.14159192767514692634 next term: 262144/1412926920405 current sum: 3.1415922987403396326 next term: 262144/2893136075115 current sum: 3.14159247995822444265 next term: 524288/11835556670925 current sum: 3.14159256855363479429 next term: 524288/24185702762325 current sum: 3.1415926119088356046 next term: 4194304/395033145117975 we eventually have pi accurate to 8 digits. We don't need millions of digits at this point. And all these early terms get discarded, only the rational sum remains. Eventually, we'll need some number with millions of digits (bits actually) but a single such number is better that millions of such numbers. To get a 1000 digits of pi takes this program 3317 terms. === === Subject: : Re: Computing decimals of pi My understanding is that when using those methods to produce millions >upon millions of decimals of pi, when attempting to compute (say) N >decimals one has to carry out all the intermediate computations to a >precision of N decimals >Not necessarily. Pi can be computed to arbitrary precision using an >infinite series of arbitrary precision rationals. They start out small >and the terms needn't be kept in memory, only the sum does. >The decimal places aren't calculated until the end when the rational >sum is converted to floating point. >Could you please provide specific examples? Ok, here's the formula I use: pi/2 = 1 + 1/3 + (1*2)/(3*5) + (1*2*3)/(3*5*7) + ... Note that this is for pi/2, so I have to multiply by two to get the > actual digits of pi. But also note how each term differs from the > previous: numerator: multiply by 1 more > 1 then > 1*2 then > 1*2*3 then > 1*2*3*4 etc. > denominator: multiply by 2 more > 3 then > 3*5 then > 3*5*7 then > 3*5*7*9 etc. For a single term, the rational becomes 2.0, not a very good > approximation of pi. So we add a second term 1/3 (times two remember) > making the sum 2.666..., still not very good. But if we keep at it long > enough current sum: 2.0 next term: 1/3 > current sum: 2.66666666666666666663 next term: 2/15 current sum: > 2.93333333333333333328 next term: 2/35 current sum: > 3.04761904761904761901 next term: 8/315 current sum: > 3.09841269841269841261 next term: 8/693 current sum: > 3.12150072150072150062 next term: 16/3003 current sum: > 3.13215673215673215673 next term: 16/6435 current sum: > 3.13712953712953712953 next term: 128/109395 current sum: > 3.13946968064615123436 next term: 128/230945 current sum: > 3.14057816968033686291 next term: 256/969969 current sum: > 3.14110602160137763843 next term: 256/2028117 current sum: > 3.1413584725201362703 next term: 1024/16900975 current sum: > 3.14147964896114041352 next term: 1024/35102025 current sum: > 3.14153799317347574177 next term: 2048/145422675 current sum: > 3.14156615934494796917 next term: 2048/300540195 current sum: > 3.14157978813759582116 next term: 32768/9917826435 current sum: > 3.14158639603706144636 next term: 32768/20419054425 current sum: > 3.14158960558823046434 next term: 65536/83945001525 current sum: > 3.14159116699150187839 next term: 65536/172308161025 current sum: > 3.14159192767514692634 next term: 262144/1412926920405 current sum: > 3.1415922987403396326 next term: 262144/2893136075115 current sum: > 3.14159247995822444265 next term: 524288/11835556670925 current sum: > 3.14159256855363479429 next term: 524288/24185702762325 current sum: > 3.1415926119088356046 next term: 4194304/395033145117975 we eventually have pi accurate to 8 digits. We don't need millions of > digits at this point. And all these early terms get discarded, only the > rational sum remains. Eventually, we'll need some number with millions > of digits (bits actually) but a single such number is better that > millions of such numbers. To get a 1000 digits of pi takes this program 3317 terms. Is it not the case that in order to obtain 8 decimals you need to compute each of those fractions to a precision of at least 8 decimals? In general, if we wanted to compute N decimals of pi we would need to compute each of those fractions to N decimals, and then perform additions to N decimals, right? My question was whether one could obtain N decimals of pi by performing a (possibly humongous) number of operations, each of them to a precision of n decimals, where n < N (possibly << N) === === Subject: : Re: Computing decimals of pi posting-account=OKTeIQkAAAAZk6JK1hK7-grwpoUDNy98 4334.34; Windows NT 5.1; SV1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) spider-mtc-tg03.proxy.aol.com[400C70C3] (Prism/1.2.1), HTTP/1.1 cache-mtc-ad05.proxy.aol.com[400C74C7] (Traffic-Server/6.1.5 [uScM]) >Not quite - I am afraid I did not explain myself correctly. My understanding is that when using those methods to produce millions >upon millions of decimals of pi, when attempting to compute (say) N >decimals one has to carry out all the intermediate computations to a >precision of N decimals >Not necessarily. Pi can be computed to arbitrary precision using an >infinite series of arbitrary precision rationals. They start out small >and the terms needn't be kept in memory, only the sum does. >The decimal places aren't calculated until the end when the rational >sum is converted to floating point. >Could you please provide specific examples? Ok, here's the formula I use: pi/2 = 1 + 1/3 + (1*2)/(3*5) + (1*2*3)/(3*5*7) + ... Note that this is for pi/2, so I have to multiply by two to get the > actual digits of pi. But also note how each term differs from the > previous: numerator: æ multiply by 1 more > 1 æ æ æ then > 1*2 æ æ then > 1*2*3 æ then > 1*2*3*4 etc. > denominator: multiply by 2 more > 3 æ æ æ then > 3*5 æ æ then > 3*5*7 æ then > 3*5*7*9 etc. For a single term, the rational becomes 2.0, not a very good > approximation of pi. So we add a second term 1/3 (times two remember) > making the sum 2.666..., still not very good. But if we keep at it long > enough current sum: 2.0 next term: 1/3 > current sum: 2.66666666666666666663 next term: 2/15 current sum: > 2.93333333333333333328 next term: 2/35 current sum: > 3.04761904761904761901 next term: 8/315 current sum: > 3.09841269841269841261 next term: 8/693 current sum: > 3.12150072150072150062 next term: 16/3003 current sum: > 3.13215673215673215673 next term: 16/6435 current sum: > 3.13712953712953712953 next term: 128/109395 current sum: > 3.13946968064615123436 next term: 128/230945 current sum: > 3.14057816968033686291 next term: 256/969969 current sum: > 3.14110602160137763843 next term: 256/2028117 current sum: > 3.1413584725201362703 next term: 1024/16900975 current sum: > 3.14147964896114041352 next term: 1024/35102025 current sum: > 3.14153799317347574177 next term: 2048/145422675 current sum: > 3.14156615934494796917 next term: 2048/300540195 current sum: > 3.14157978813759582116 next term: 32768/9917826435 current sum: > 3.14158639603706144636 next term: 32768/20419054425 current sum: > 3.14158960558823046434 next term: 65536/83945001525 current sum: > 3.14159116699150187839 next term: 65536/172308161025 current sum: > 3.14159192767514692634 next term: 262144/1412926920405 current sum: > 3.1415922987403396326 next term: 262144/2893136075115 current sum: > 3.14159247995822444265 next term: 524288/11835556670925 current sum: > 3.14159256855363479429 next term: 524288/24185702762325 current sum: > 3.1415926119088356046 next term: 4194304/395033145117975 we eventually have pi accurate to 8 digits. We don't need millions of > digits at this point. And all these early terms get discarded, only the > rational sum remains. Eventually, we'll need some number with millions > of digits (bits actually) but a single such number is better that > millions of such numbers. To get a 1000 digits of pi takes this program 3317 terms. Is it not the case that in order to obtain 8 decimals you need to > compute each of those fractions to a precision of at least 8 > decimals? No. I don't NEED to convert any of the fractions until the end. Now that I know it takes about three times as many terms as decimal places, I could wait until I've done 3000 terms before starting to check the 1000th decimal place for convergence. By doing the calculation entirely with rationals, there are no decimal places involved at all and thus, no rounding errors. Of course, many decimal places of the final rational are discarded, we only keep them to point where they've converged. And even though no decimal places are involed, the integers that make up the numerator and denominator get quite large with up to twice as many digits as we're trying to find decimal places. But we are constantly updating this sum, not creating new ones. That's why the memory use isn't so bad. > In > general, if we wanted to compute N decimals of pi we would need to > compute each of those fractions to N decimals, > and then perform additions to N decimals, right? No, we don't convert the fractions to decimal, they are left as fractions and the math does the calculations as fractions. If you did as you say, you would still have rounding problems even with your millions of decimal places. My question was whether one could obtain N decimals of pi by > performing a (possibly humongous) number of operations, > each of them to a precision of n decimals, where n < N > (possibly << N) Sure, it COULD be done that way. The problem with using floating point with a humungous number of operations is the potential for loss of precision (And having millions of decimal places won't prevent that). The big advantage of rationals is there is NEVER loss of precision no matter how many operations you do (assuming, of course, that the rationals are composed of arbitrary precision integers). When my program converts the rational to decimal, it only checks the desired place for convergence and then discards the floating point number. And all the rounding baggage is discarded along with it, so the errors don't accumulate like they would if we were adding the n decimal places. === === Subject: : Re: Computing decimals of pi posting-account=vI5-YAoAAACpb1I_2s__b0LrjNDZjNTS Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > Are there any algorithms out there to compute pi to an arbitrary > number of decimal places N, in such a way that intermediate computations > do not have to be carried out to precision N? If compute refers to a programming algorithm suitable for an average PC's capabilities then http://en.wikipedia.org/wiki/Pi#Computation_in_the_computer_age === === Subject: : topology posting-account=0oEA4goAAACLSgCzdc1XzJ8bN7P3u9gy Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) can you help about these questions 1 Prove that Every Noetherian topological space is compact. Prove that if X is notherian topological space then every closed subset of X is noetherian === === Subject: : Re: topology posting-account=9QOSvAoAAACEOWJVSDuswW7dB_0wApQO Gecko/2008043010 Fedora/3.0-0.60.beta5.fc9 Firefox/3.0b5,gzip(gfe),gzip(gfe) > can you help about these questions 1 æ Prove that Every Noetherian topological space is compact. > æ æ Prove that if X is notherian topological space then every closed > subset of X is noetherian What have you tried doing? By the way, in the second question there is no need for the subset to be closed for it to inherit a noetherian topology. -- m === === Subject: : Re: topology > can you help about these questions 1 æ Prove that Every Noetherian topological space is compact. > æ æ Prove that if X is notherian topological space then every closed > subset of X is noetherian What have you tried doing? > Assume C is an open cover with no finite subcover. > By the way, in the second question there is no need > for the subset to be closed for it to inherit a noetherian > topology. -- m > === === Subject: : This Week's Finds in Mathematical Physics (Week 264) Also available at http://math.ucr.edu/home/baez/week264.html May 18, 2008 This Week's Finds in Mathematical Physics (Week 264) John Baez Here's a puzzle. Guess the next term of this sequence: 1, 1, 2, 3, 4, 5, 6, ... and then guess the *meaning* of this sequence! I'll give away the answer after telling you about Coleman's videos on quantum field theory and an amazing result on the homotopy groups of spheres. But first... the astronomy picture of the day. The Eaton Collection at UC Riverside may be the world's best library of science fiction: 1) The Eaton Collection of Science Fiction, Fantasy, Horror and Utopian Literature, http://eaton-collection.ucr.edu/ Right now my wife Lisa Raphals is attending a conference there on the role of Mars in SF, called Chronicling Mars. Gregory Benford, Frederik Pohl, Greg Bear, David Brin, Kim Stanley Robinson and even Ray Bradbury are all there! But for some reason I'm staying home working on This Week's Finds. I'd say that shows true devotion - or maybe just stupidity. Anyway, in honor of the occasion, here's an incredible closeup of a crater on Mars' moon Phobos: 2) Astronomy Picture of the Day, Stickney Crater http://apod.nasa.gov/apod/ap080410.html It's another great example of how machines in space now deliver many more thrills per buck than the old-fashioned approach using canned primates. This photo was taken by HiRISE, the High Resolution Imaging Science Experiment - the same satellite that took the stunning photos of Martian dunes which graced week262. Mars has two moons, Phobos and the even tinier Deimos. Their names mean fear and dread in Greek, since in Greek mythology they were sons of Mars (really Ares), the god of war. Interestingly, Kepler predicted that Mars had two moons before they were seen. This sounds impressive, but it was simple interpolation, since Earth has 1 moon and Jupiter has 4. Or at least Galileo saw 4 - now we know there are a lot more. Phobos is only 21 kilometers across, and the big crater you see here - Stickney Crater - is about 9 kilometers across. That's almost half the size of the whole moon! The collision that created it must have almost shattered Phobos. Phobos is so light - just twice the density of water - that people once thought it might be hollow. This now seems unlikely, though it's been the premise of a few SF stories. It's more likely that Phobos is a loosely packed pile of carbonaceous chondrites captured from the asteroid belt. Phobos orbits so close to Mars that it zips around once every 8 hours, faster than Mars itself rotates! Oddly, in 1726 Gulliver's Travels - and he guessed that the inner one orbited Mars every 10 hours. Gravitational tidal forces are dragging Phobos down, so in only 10 million years it'll either crash or - more likely - be shattered by tidal forces and form a ring of debris. So, enjoy it while it lasts. Anyone who's seriously struggled to master quantum field theory is likely to have profited from this book: 3) Sidney Coleman, Aspects of Symmetry: Selected Erice Lectures, Cambridge U. Press, Cambridge, 1988. It's brimming with wisdom and humor. You should have already encountered quantum field theory before trying it: what you'll get are deeper insights. But what if you're just getting started? Sidney Coleman, recently deceased, was one of the best quantum field took a course on quantum field theory from Eddie Farhi, who said he based his class on the notes from Coleman's class at Harvard. So, I've always been curious about these notes. Now they're available online in handwritten form: 4) Sidney Coleman, lecture notes on quantum field theory, http://www.damtp.cam.ac.uk/user/dt281/qft/col1.pdf and http://www.damtp.cam.ac.uk/user/dt281/qft/col2.pdf Someone should LaTeX them up! Even more fun, you can now see *videos* of Coleman teaching quantum field theory: 5) Sidney Coleman, Physics 253: Quantum Field Theory, 50 lectures recorded 1975-1976, http://www.physics.harvard.edu/about/Phys253.html This is a younger, hipper Coleman than I'd ever seen: long-haired, sometimes puffing on a cigarette between sentences. He begins by saying Umm... this is Physics 253, a course in relativistic quantum mechanics. My name is Sidney Coleman. The apparatus you see around you is part of a CIA surveillance project. I wish I'd had access to these when I was a kid! Now for some miraculous math. Daniel Moskovich kindly pointed out a paper that describes all the homotopy groups of the 2-sphere, and I want to summarize the main result. I explained the idea of homotopy groups back in week102. Very roughly, the nth homotopy group of a space X, usually denoted pi_n(X), is the set of ways you can map an n-sphere into that space, where we count two ways as the same if you can continuously deform one to the other. If a space has holes, homotopy groups are one way to detect those holes. Homotopy groups are notoriously hard to compute - so even for so humble a space as the 2-sphere, S^2, there's a sense in which nobody knows all its homotopy groups. People know the first 64, though. Here are a few: pi_1(S^2) = 0 pi_2(S^2) = Z pi_3(S^2) = Z pi_4(S^2) = Z/2 pi_5(S^2) = Z/2 pi_6(S^2) = Z/4 x Z/3 pi_7(S^2) = Z/2 pi_8(S^2) = Z/2 pi_9(S^2) = Z/3 pi_10(S^2) = Z/3 x Z/5 pi_11(S^2) = Z/2 pi_12(S^2) = Z/2 x Z/2 pi_13(S^2) = Z/2 x Z/2 x Z/3 pi_14(S^2) = Z/2 x Z/2 x Z/4 x Z/3 x Z/7 pi_15(S^2) = Z/2 x Z/2 Apart from the fact that they're all finite abelian groups, it's hard to spot any pattern! In fact there's a majestic symphony of patterns in the homotopy groups of spheres, starting from ones that are easy to explain and working on up to those that push the frontiers of mathematics, like elliptic cohomology. But, many of these patterns are too complex for present-day mathematics until we use some tricks to water down or simplify the homotopy groups. So, what people often do first is take the limit of pi_{n+k}(S^n) as n -> infinity, getting what's called the kth stable homotopy group of spheres. It's a wonderful but well-understood fact that these limits really exist. But so far, even these are too complicated to understand until we work at a prime p. This means that we take the kth stable homotopy group of spheres and see which groups of the form Z/p^n show up in it. For example, pi_14(S^2) = Z/2 x Z/2 x Z/4 x Z/3 x Z/7 but if we work at the prime 2 we just see the Z/2 x Z/2 x Z/4. After all this data processing, we get some astounding pictures: 6) Allen Hatcher, Stable homotopy groups of spheres, http://www.math.cornell.edu/~hatcher/stemfigs/stems.html Order teetering on the brink of chaos! If you're brave, you can learn more about this stuff here: 7) Douglas C. Ravenel, Complex Cobordism and Stable Homotopy Groups If you're less brave, I strongly suggest starting here: 8) Wikipedia, Homotopy groups of spheres, http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres But now, I want to talk about an amazing paper that pursues a very different line of attack. It gives a beautiful description of *all* the homotopy groups of S^2, in terms of braids: 9) A. Berrick, F. R. Cohen, Y. L. Wong and J. Wu, Configurations, braids and homotopy groups, J. Amer. Math. Soc., 19 (2006), 265-326. Also available at http://www.math.nus.edu.sg/~matwujie/BCWWfinal.pdf For this you need to realize that for any n, there's a group B_n whose elements are n-strand braids. For example, here's an element of B_3: | | | / | / | / | | / | / | / / | / | / | | / | / | / / | / | / | | / | / | / | | | I actually talked about this specific braid back in week233. But anyway, we count two braids as the same if you can wiggle one around until it looks like the other without moving the ends at the top and bottom - which you can think of as nailed to the ceiling and floor. How do braids become a group? Easy: we multiply them by putting one on top of the other. For example, this braid: | | | / | A = / | / | | | | times this one: | | | | / B = | / | / | | | equals this: | | | / | / | / | | | | AB = | | | | / | / | / | | | and in fact the big one I showed you earlier is (AB)^3. As you let your eye slide from the top to the bottom of a braid, the strands move around. We can visualize their motion as a bunch of points running around the plane, never bumping into each other. This gives an interesting way to generalize the concept of a braid! Instead of points running around the plane, we can have points running around S^2, or some other surface X. So, for any surface X and any number n of strands, we get a surface braid group, called B_n(X). As I hinted in week261, these surface braid groups have cool relationships to Dynkin diagrams. I urged you to read this paper, and I'll urge you again: 10) Daniel Allcock, Braid pictures for Artin groups, available as arXiv:math.GT/9907194. But for now, we just need the spherical braid group B_n(S^2) together with the usual braid group B_n. Let's say a braid is Brunnian if when you remove any one strand, the remaining braid becomes the identity: you can straighten out all the remaining strands to make them vertical. It's a fun little exercise to check that Brunnian braids form a subgroup of all braids. So, we have an n-strand Brunnian braid group BB_n. The same idea works for braids on other surface, like the 2-sphere. So, we also have an n-strand *spherical* Brunnian braid group BB_n(S^2). Now, there's obvious map B_n -> B_n(S^2) Why? An element of B_n describes the motion of a bunch of points running around the plane, but the plane sits inside the 2-sphere: the 2-sphere is just the plane with an extra point tacked on. So, an ordinary braid gives a spherical braid. This map clearly sends Brunnian braids to spherical Brunnian braids, so we get a map f: BB_n -> BB_n(S^2) And now we're ready for the shocking theorem of Berrick, Cohen, Wong and Wu: Theorem: For n > 3, BB_n(S^2) modulo the image of f is the nth homotopy group of S^2. In something more like plain English: when n is big enough, the nth homotopy group of the 2-sphere consists of spherical Brunnian braids modulo ordinary Brunnian braids! Zounds! What do the homotopy groups of S^2 have to do with braids? It's not supposed to be obvious! The proof of this result is long and deep, making use of flows on metric spaces, and also the fact that all the Brunnian braid groups BB_n fit together into a simplicial group whose nth homology is the nth homotopy group of S^2. I'd love to understand all this stuff, but I don't yet. This result doesn't instantly help us compute the homotopy groups of S^2 - at least not in the sense of writing them down as a product of groups like Z/p^n. But, it gives a new view of these homotopy groups, and there's no telling where this might lead. going to tell you about some amazing descriptions of the homotopy groups of the *3-sphere*, due to Wu. However, I later realized - first to my shock, and then my embarrassment for not having known it already - that the nth homotopy group of S^3 is *the same* as the nth homotopy group of S^2, at least for n > 2. Do you see why? Given this, it turns out that Wu's results are predecessors of the theorem just stated, a bit more combinatorial and less geometric. Wu's results appeared here: 11) Jie Wu, On combinatorial descriptions of the homotopy groups of certain spaces, Math. Proc. Camb. Phil. Soc. 130 (2001), 489-513. Also available at http://www.math.nus.edu.sg/~matwujie/newnewpis_3.pdf Jie Wu, A braided simplicial group, Proc. London Math. Soc. 84 (2002), 645-662. Also available at http://www.math.nus.edu.sg/~matwujie/Research2.html and there's a nice summary of these results on his webpage: 12) Jie Wu, 2.1 Homotopy groups and braids, halfway down the page at http://www.math.nus.edu.sg/~matwujie/Research2.html See also this expository paper: 13) Fred R. Cohen and Jie Wu, On braid groups and homotopy groups, Geometry & Topology Monographs 13 (2008), 169-193. Also available at http://www.math.nus.edu.sg/~matwujie/cohen.wu.GT.revised.29.august.2007.pdf Next I want to talk about puzzle mentioned at the start of this Week's Finds... but first I should answer the puzzle I just raised. Why do the homotopy groups of S^2 match those of S^3 after a while? Because of the Hopf fibration! This is a fiber bundle with S^3 as total space, S^2 as base space and S^1 as fiber: S^1 -> S^3 -> S^2 Like any fiber bundle, it gives a long exact sequence of homotopy groups as explained in week151: .. -> pi_n(S^1) -> pi_n(S^3) -> pi_n(S^2) -> pi_{n-1}(S^1) -> ... but the homotopy groups of S^1 vanishes after the first, so we get .. -> 0 -> pi_n(S^3) -> pi_n(S^2) -> 0 -> ... for n > 2, which says that pi_n(S^3) = pi_n(S^2) Okay, now for this mysterious sequence: 1, 1, 2, 3, 4, 5, 6, ... The next term is obviously 7. If you guessed anything else, you were over-analyzing. So the real question is: why the funny hiccup at the beginning? You'll find two explanations of this sequence in Sloane's Online Encyclopedia of Integer Sequences, but neither of them is the reason James Dolan and I ran into it. We were studying theta functions... Say you have a torus. Then the complex line bundles over it are classified by an integer called the first Chern number. In some sense, this integer this measures how twisted the bundle is. For example, you can put any connection on the bundle, compute its curvature 2-form, and integrate it over the torus: up to some constant factor, you'll then get the first Chern number. A torus is a 2-dimensional manifold, but we can also make it into a 1-dimensional *complex* manifold, often called an elliptic curve. In fact we can do this in infinitely many fundamentally different ways, one for each point in the moduli space of elliptic curves. I've explained this repeatedly here - try week125 for a good starting-point - so I won't do so again. The details don't really matter here. Back to line bundles. If we pick an elliptic curve, we can try to classify the *holomorphic* complex line bundles over it - that is, those where the transition functions are holomorphic (or in other words, complex-analytic). Here the classification is subtler! It turns out you need, not just the first Chern number, which is discrete, but another parameter which can vary in a *continuous* way. Interestingly, after you pick a basepoint for your elliptic curve, this other parameter can be thought of as just a point on the elliptic curve! So, the elliptic curve becomes the space that classifies holomorphic line bundles over itself - at least, those with fixed first Chern number. Curiously circular, eh? This is just one of several curiously circular classification theorems that happen in this game... But I'm actually digressing a bit - I'm having trouble resisting the temptation to explain everything I know, since it's so simple and beautiful, and I just learned it. Don't worry - all you need to know is that holomorphic line bundles over an elliptic curve are classified by an integer and some other continuous parameter. The puzzle then arises: how many holomorphic sections do these line bundles have? More precisely: what's the *dimension* of the space of holomorphic sections? Before I answer this, I can't resist adding that these holomorphic sections have a long and illustrious history - they're called theta functions, and you can learn about them here: 14) Jun-ichi Igusa, Theta Functions, Springer, Berlin, 1972. 15) David Mumford, Tata Lectures on Theta, 3 volumes, Birkhauser, Boston, 1983-1991. They're important in geometric quantization, where holomorphic sections of line bundles describe states of quantum systems, and the reciprocal of the first Chern number is proportional to Planck's constant. In fact, I first ran into theta functions years ago, when trying to quantize a black hole - see the end of week112 for more details. But anyway, here's the answer to the puzzle. The dimension turns out not to depend on the continuous parameter labelling our line bundle, but only on its first Chern number. If that number is negative, the dimension is 0. But if it's 0,1,2,3,4,5,6 and so on, the dimension goes like this: 1,1,2,3,4,5,6,... Now, this sequence is fairly weird, because of the extra 1 at the beginning. I hadn't noticed this back when I was quantizing black holes, because the extra 1 happens for first Chern number zero, which would correspond to Planck's constant being *infinite*. But now that I'm just thinking about math, it sticks out like a sore thumb! It's got to be right, since the line bundle with first Chern number zero is the trivial bundle, its sections are just functions, and the only holomorphic functions on a compact complex manifold are constants - so there's a 1-dimensional space of them. But, it's weird. Luckily, Jim figured out the explanation for this sequence. First of all, we can encode it into a power series: 1 + x + 2x^2 + 3x^3 + 4x^4 + ... which we can rewrite as a rational function: (1-x^6) 1 + x + 2x^2 + 3x^3 + 4x^4 + ... = -------------------- (1-x)(1-x^2)(1-x^3) Now, the reason for doing this is that we can pick a line bundle of first Chern number 1, say L, and get a line bundle of any Chern number n by taking the nth tensor power of L - let's call that L^n. We can multiply a section of L^n and a section of L^m to get a section of L^{n+m}. So, all these spaces of sections we're studying fit together to form a commutative graded ring! And, whenever you have a graded ring, it's a good idea to write down a power series that encodes the dimensions of each grade, just as we've done above. This is called a Poincare series. And, when you have a commutative graded ring with one generator of degree 1, one generator of degree 2, one generator of degree 3, one relation of degree 6, and no relations between relations (or syzygies), its Poincare series will be (1-x^6) -------------------- (1-x^1)(1-x^2)(1-x^3) That's how it always works - think about it. So, it's natural to hope that our ring built from holomorphic sections of all the line bundles L^n will have one generator of degree 1, one of degree 2, one of degree 3, and one relation of degree 6. And, this seems to be true! As I mentioned, people usually call these holomorphic sections theta functions. So, it seems we're getting a description of the ring of theta functions in terms of generators and relations. How does it work, exactly? Well, I must admit I'm not quite sure. Jim has some ideas, but it seems I need to do something a bit different to get his story to work for me. Maybe it goes something like this. We can write any elliptic curve as the solutions of this equation: y^2 = x^3 + Bx + C for certain constants B and C that depend on the elliptic curve. (See week13 and week261 for details.) Now, this equation is not homogeneous in the variables y and x, but we can think of it as homogeneous in a sneaky sense if we throw in an extra variable like this: y^2 = x^3 + Bxz^5 + Cz^6 and decree that: y has grade 3 x has grade 2 z has grade 1 Then all the terms in the equation have grade 6. So, we're getting a commutative graded ring with generators of degree 1, 2, and 3 and a relation of grade 6. And, I'm hoping this ring consists of algebraic functions on the total space of some line bundle L* over our elliptic curve. z should be a function that's linear in the fiber directions, hence a section of L. x should be quadratic in the fiber directions, hence a section of L^2. And y should be cubic, hence a section of L^3. If L has first Chern number 1, I think we're in business. If anybody knows about this stuff, I'd appreciate corrections or references. There's a *lot* more to say about this business... because it's all part of a big story about elliptic curves, theta functions and modular forms. But, I want to quit here for now. ----------------------------------------------------------------------- Addenda: I thank David Corfield for pointing out how to get ahold of Wu's papers free online - and earlier, for telling me Wu's combinatorial description of pi_3(S^2). Martin Ouwehand told me that some of Coleman's lecture notes on quantum field theory are available in TeX here: 17) Sidney Coleman, Quantum Field Theory, first 11 lectures notes TeXed by Bryan Gin-ge Chen, available at http://www.physics.upenn.edu/~chb/phys253a/coleman/ 18) Wikipedia, Riemann-Roch theorem, http://en.wikipedia.org/wiki/Riemann-Roch has some very relevant information on the sequence 1, 1, 2, 3, 4, 5, 6, ... though it's phrased not in terms of sections of line bundles, but instead in terms of divisors (secretly another way of talking about the same thing). Let me quote a portion, just to whet your interest: We start with a connected compact Riemann surface of genus g, and a fixed point P on it. We may look at functions having a pole only at P. There is an increasing sequence of vector spaces: functions with no poles (i.e., constant functions), functions allowed at most a simple pole at P, functions allowed at most a double pole at P, a triple pole, ... These spaces are all finite dimensional. In case g = 0 we can see that the sequence of dimensions starts 1, 2, 3, ... This can be read off from the theory of partial fractions. Conversely if this sequence starts 1, 2, ... then g must be zero (the so-called Riemann sphere). In the theory of elliptic functions it is shown that for g = 1 this sequence is 1, 1, 2, 3, 4, 5 ... and this characterises the case g = 1. For g > 2 there is no set initial segment; but we can say what the tail of the sequence must be. We can also see why g = 2 is somewhat special. The reason that the results take the form they do goes back to the formulation (Roch's part) of the [Riemann-Roch] theorem: as a difference of two such dimensions. When one of those can be set to zero, we get an exact formula, which is linear in the genus and the degree (i.e. number of degrees of freedom). Already the examples given allow a reconstruction in the shape dimension - correction = degree - g + 1. For g = 1 the correction is 1 for degree 0; and otherwise 0. The full theorem explains the correction as the dimension associated to a further, 'complementary' space of functions. You can see more discussion of this Week's Finds at the n-Category Cafe: http://golem.ph.utexas.edu/category/2008/05/this_weeks_finds_in_mathematic_2 5.html ----------------------------------------------------------------------- Quote of the Week: The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction. - Sidney Coleman ----------------------------------------------------------------------- mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twfcontents.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html === === Subject: : Re: This Week's Finds in Mathematical Physics (Week 264) posting-account=Rkt6TwoAAACG_SqlrxmgPCl1Ozr0PWSD MathPlayer 2.10b; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30; .NET CLR 1.1.4322; .NET CLR 3.0.04506.648),gzip(gfe),gzip(gfe) On May 24, 3:37 pm, b...@math.removethis.ucr.andthis.edu (John Baez) Mars has two moons, Phobos and the even tinier Deimos. > Their names mean fear and dread in Greek, since in > Greek mythology they were sons of Mars (really Ares), > the god of war. [following is trivia, but I managed to work in a connection with the planet Mars!] The Romans identified their god Mars with the Greek god ares, but Mars was around before that. In the earliest inscriptions, his name appeared as Amartis, and it appears he then had some connection with agriculture as well as war. My guess is Mars/Amartis was originally (as in *way* back, perhaps 800BC or earlier) the god of stirring, associated with spring, when everything, plants in particular, sprang to life and military campaigns and raids resumed, after the inactivity of winter. The word amartis seems similar to the first words one typically learns in Latin amo/amas/amat, meaning love, as in stir or move (and the word move itself also looks etymologically related). In festivals of Mars, and Martial ceremonies, participants had faces painted red. Perhaps that originally represented a sun tan, re-acquired once people started working outside again in the spring. So it's ironic that Mars is called the Red Planet. John Ramsden === === Subject: : What if: we had precise statistics for fertility in the Soviet Union in the 1930's? posting-account=5ayZ-goAAABGZmmwx8zZEwz6gU2OuVSd CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) http://users.erols.com/mwhite28/warstat1.htm There are basically two schools of thought when it comes to the number who died at Stalin's hands. There's the Why doesn't anyone realize that communism is the absolutely worst thing ever to hit the human race, without exception, even worse than both world wars, the slave trade and bubonic plague all put together? school, and there's the Come on, stop exaggerating. The truth is horrifying enough without you pulling numbers out of thin air school. The two schools are generally associated with the right and left wings of the political spectrum, and they often accuse each other of being blinded by prejudice, stubbornly refusing to admit the truth, and maybe even having a hidden agenda. Also, both sides claim that recent access to former Soviet archives has proven that their side is right. Here are a few illustrative estimates from the Big Numbers school: Adler, N., Victims of Soviet Terror, 1993 cites these: Chistyakovoy, V. (Neva, no.10): 20 million killed during the 1930s. Dyadkin, I.G. (Demograficheskaya statistika neyestestvennoy smertnosti v SSSR 1918-1956 ): 56 to 62 million unnatural deaths for the USSR overall, with 34 to 49 million under Stalin. Gold, John.: 50-60 million. Davies, Norman (Europe A History, 1998): c. 50 million killed 1924-53, excluding WW2 war losses. This would divide (more or less) into 33M pre-war and 17M after 1939. Rummel, 1990: 61,911,000 democides in the USSR 1917-87, of which 51,755,000 occurred during the Stalin years. This divides up into: 1923-29: 2,200,000 (plus 1M non-democidal famine deaths) 1929-39: 15,785,000 (plus 2M non-democidal famine) 1939-45: 18,157,000 1946-54: 15,613,000 (plus 333,000 non-democidal famine) TOTAL: 51,755,000 democides and 3,333,000 non-demo. famine William Cockerham, Health and Social Change in Russia and Eastern Europe: 50M+ Wallechinsky: 13M (1930-32) + 7M (1934-38) Cited by Wallechinsky: Medvedev, Roy (Let History Judge): 40 million. Solzhenitsyn, Aleksandr: 60 million. MEDIAN: 51 million for the entire Stalin Era; 20M during the 1930s. And from the Lower Numbers school: Nove, Alec (Victims of Stalinism: How Many? in J. Arch Getty (ed.) Stalinist Terror: New Perspectives, 1993): 9,500,000 surplus deaths during the 1930s. Cited in Nove: Maksudov, S. (Poteri naseleniya SSSR, 1989): 9.8 million abnormal deaths between 1926 and 1937. Tsaplin, V.V. (Statistika zherty naseleniya v 30e gody 1989): 6,600,000 deaths (hunger, camps and prisons) between the 1926 and 1937 censuses. Dugin, A. (Stalinizm: legendy i fakty 1989): 642,980 counterrevolutionaries shot 1921-53. Muskovsky Novosti (4 March 1990): 786,098 state prisoners shot, 1931-53. Gordon, A. (What Happened in That Time?, 1989, cited in Adler, N., Victims of Soviet Terror, 1993): 8-9 million during the 1930s. al., that excess deaths 1926-39 were likely 3.5 million and at most 8 million. MEDIAN: 8.5 Million during the 1930s. As you can see, there's no easy compromise between the two schools. The Big Numbers are so high that picking the midpoint between the two schools would still give us a Big Number. It may appear to be a rather pointless argument -- whether it's fifteen or fifty million, it's still a huge number of killings -- but keep in mind that the population of the Soviet Union was 164 million in 1937, so the upper estimates accuse Stalin of killing nearly 1 out of every 3 of his people, an extremely Polpotian level of savagery. The lower numbers, on the other hand, leave Stalin with plenty of people still alive to fight off the German invasion. ---------------------------------------------------------------------------- --------------- There doesn't seem to be much question that the population of the Soviet Union grew substantially in the 1930's while the population of the United States and Europe stagnated. http://en.wikipedia.org/wiki/Demographics_of_the_Soviet_Union January 1897 (Russia): 125,640,000*** 1911(Russia): 167,003,000** January 1920 : 137,727,000* January 1926 : 148,656,000* January 1937: 162,500,000* January 1939: 168,524,000* June 1941: 196,716,000* January 1946: 170,548,000* January 1951: 182,321,000* January 1959: 209,035,000* January 1970: 241,720,000 1985: 272,000,000 July 1991: 293,047,571 * Andreev, E.M., et al., Naselenie Sovetskogo Soiuza, 1922-1991. Moscow, Nauka, 1993. ISBN 5-02-013479-1 So, the critical question in determining how many people Stalin actually killed in the 1930's seems to be the precise statistics for fertility during this period. The crude birth rate in the USSR throughout its history had been decreasing - from 44.0 per thousand in 1926 to 18.0 in 1974, mostly due to urbanization and rising average age of marriages. The crude death rate had been gradually decreasing as well - from 23.7 per thousand in 1926 to 8.7 in 1974 These figures suggest a 2% per year growth rate. Actual growth appears to have been somewhat less than 1.5%. This difference could be attributed to other deaths. This suggests the possibility that some 10 millions might have been killed in purges or died in famines in the 1930's. Thoughts? === === Subject: : Re: What if: we had precise statistics for fertility in the Soviet ... posting-account=5ayZ-goAAABGZmmwx8zZEwz6gU2OuVSd CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) On May 24, 1:36æpm, Jack Linthicum Is there an executive authority anywhere in time and space that would > permit the collection and exhibition of demographic information that > shows the slaughter of millions? At the very least, the Government du jour would lie about the fatalities > it causes/permits. Alternatively, the title to this thread might be misconstrued to ask: > What if Dr. Alfred Kinsey, not Joseph Stalin, ruled the Soviet Union? . . . . In his most memorable treatise, æ Hereditary Leninist Marxist > Behavior Among Gall Wasps , æGeneral Secretary Kinseyich postulated > that all human behavior could be scientifically and systematically > projected into the future once sufficient data was on record concerning > human sexual behavior. Based on his alleged rational interest in forecasting Soviet society, > Kinseyich instigated and promoted the Great Survey of sexual actvity > within the Soviet Union. This is the act of a monster, said Pope Pius, forbidding communion to > anyone who distributed or answered a sex survey. In England, Ramsey > McDonald's fraction of the Labour Party was known as the No Sex Please > --- We're British Party. æMeanwhile, in America, President Will Hays > declared Clark Gable as Public Enemy Number One for taking his shirt off > his bare torso. . . . . This sort of reminds me of the old intelligence dilemma, we could tell > you the exact number of men in the Soviet Army if we only knew how > many men there were in a squad.- Hide quoted text - - Show quoted text - So, why assume Stalin was a genocidal monster if, effectively, we have no real evidence to that effect? === === Subject: : Re: What if: we had precise statistics for fertility in the Soviet ... posting-account=xtUkLgkAAACOhuoYOJtB4pk1TADupWiX Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > On May 24, 1:36 pm, Jack Linthicum permit the collection and exhibition of demographic information that > shows the slaughter of millions? At the very least, the Government du jour would lie about the fatalities > it causes/permits. Alternatively, the title to this thread might be misconstrued to ask: > What if Dr. Alfred Kinsey, not Joseph Stalin, ruled the Soviet Union? . . . . In his most memorable treatise, _Hereditary Leninist Marxist > Behavior Among Gall Wasps_ , General Secretary Kinseyich postulated > that all human behavior could be scientifically and systematically > projected into the future once sufficient data was on record concerning > human sexual behavior. Based on his alleged rational interest in forecasting Soviet society, > Kinseyich instigated and promoted the Great Survey of sexual actvity > within the Soviet Union. This is the act of a monster, said Pope Pius, forbidding communion to > anyone who distributed or answered a sex survey. In England, Ramsey > McDonald's fraction of the Labour Party was known as the No Sex Please > --- We're British Party. Meanwhile, in America, President Will Hays > declared Clark Gable as Public Enemy Number One for taking his shirt off > his bare torso. . . . . This sort of reminds me of the old intelligence dilemma, we could tell > you the exact number of men in the Soviet Army if we only knew how > many men there were in a squad.- Hide quoted text - - Show quoted text - So, why assume Stalin was a genocidal monster if, effectively, we have > no real evidence to that effect? I don't think I said that. You are making up what you probably feel are valid facts without knowing what the others are. So a better title for your little charade might have been What if Joseph Stalin was really a nice old man whose citizens liked to die? === === Subject: : Re: What if: we had precise statistics for fertility in the Soviet ... posting-account=5ayZ-goAAABGZmmwx8zZEwz6gU2OuVSd CLR 2.0.50727; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) On May 24, 3:23æpm, Jack Linthicum On May 24, 1:36 pm, Jack Linthicum permit the collection and exhibition of demographic information that > shows the slaughter of millions? At the very least, the Government du jour would lie about the fatalities > it causes/permits. Alternatively, the title to this thread might be misconstrued to ask: > What if Dr. Alfred Kinsey, not Joseph Stalin, ruled the Soviet Union? . . . . In his most memorable treatise, æ Hereditary Leninist Marxist > Behavior Among Gall Wasps , æGeneral Secretary Kinseyich postulated > that all human behavior could be scientifically and systematically > projected into the future once sufficient data was on record concerning > human sexual behavior. Based on his alleged rational interest in forecasting Soviet society, > Kinseyich instigated and promoted the Great Survey of sexual actvity > within the Soviet Union. This is the act of a monster, said Pope Pius, forbidding communion to > anyone who distributed or answered a sex survey. In England, Ramsey > McDonald's fraction of the Labour Party was known as the No Sex Please > --- We're British Party. æMeanwhile, in America, President Will Hays > declared Clark Gable as Public Enemy Number One for taking his shirt off > his bare torso. . . . . This sort of reminds me of the old intelligence dilemma, we could tell > you the exact number of men in the Soviet Army if we only knew how > many men there were in a squad.- Hide quoted text - - Show quoted text - So, why assume Stalin was a genocidal monster if, effectively, we have > no real evidence to that effect? I don't think I said that. You are making up what you probably feel > are valid facts without knowing what the others are. So a better > title for your little charade might have been What if Joseph Stalin > was really a nice old man whose citizens liked to die?- Hide quoted text - - Show quoted text - What facts do you feel I'm making up, exactly? Most of what I refer to are statistics from secondary sources. And, quite a wide variety of them, too. I'm just trying to make some sense from them. The range of evaluations of Stalin and the number he killed is very extreme indeed. The fact that you have your own favourite figures is your problem. === === Subject: : Re: What if: we had precise statistics for fertility in the Soviet ... posting-account=CBZzyQkAAADa8LlO-8hxEObjm8IdMLIc > On May 24, 3:23 pm, Jack Linthicum permit the collection and exhibition of demographic information that > shows the slaughter of millions? At the very least, the Government du jour would lie about the fatalities > it causes/permits. Alternatively, the title to this thread might be misconstrued to ask: > What if Dr. Alfred Kinsey, not Joseph Stalin, ruled the Soviet Union? . . . . In his most memorable treatise, _Hereditary Leninist Marxist > Behavior Among Gall Wasps_ , General Secretary Kinseyich postulated > that all human behavior could be scientifically and systematically > projected into the future once sufficient data was on record concerning > human sexual behavior. Based on his alleged rational interest in forecasting Soviet society, > Kinseyich instigated and promoted the Great Survey of sexual actvity > within the Soviet Union. This is the act of a monster, said Pope Pius, forbidding communion to > anyone who distributed or answered a sex survey. In England, Ramsey > McDonald's fraction of the Labour Party was known as the No Sex Please > --- We're British Party. Meanwhile, in America, President Will Hays > declared Clark Gable as Public Enemy Number One for taking his shirt off > his bare torso. . . . . This sort of reminds me of the old intelligence dilemma, we could tell > you the exact number of men in the Soviet Army if we only knew how > many men there were in a squad.- Hide quoted text - - Show quoted text - So, why assume Stalin was a genocidal monster if, effectively, we have > no real evidence to that effect? I don't think I said that. You are making up what you probably feel > are valid facts without knowing what the others are. So a better > title for your little charade might have been What if Joseph Stalin > was really a nice old man whose citizens liked to die?- Hide quoted text - - Show quoted text - What facts do you feel I'm making up, exactly? Most of what I refer > to are statistics from secondary sources. And, quite a wide variety > of them, too. I'm just trying to make some sense from them. The > range of evaluations of Stalin and the number he killed is very > extreme indeed. The fact that you have your own favourite figures is > your problem. Even the people who argue for the smaller figures will generally grant that Stalin was a genocidal monster. If anything, it makes many Marxist feel better if he is portrayed as an anomaly. Of course, Lenin was a genocidal monster also. But two in a row proves nothing, I guess. -- Will in New Haven Good intentions will always be pleaded for every assumption of authority. It is hardly too strong to say that the Constitution was made to guard the people against the dangers of good intentions. There are men in all ages who mean to govern well, but they mean to govern. They promise to be good masters, but they mean to be masters. Daniel Webster === === Subject: : Re: What DARPA Isn't Telling The Rothschilds Re: More evidence proving Apollo Hoax pR9MNF=QZrhF$HA44sG?n(RnC{0!e>5EKJK{?&5QI$HfYWdP*zNGgP79xb%0Sch26o G8|k?UC[~)lu]WnopcGdzzVoCsDY?/{MgR9eDs>Y~X},^2Ay|9n9~~1&UUCoDQ|O| eYokL{N-%%x6C_4lKt2,LOOY#A6rAjZ5~[1-s,It[DTBZa1v_2D7;0Yq1rQ{LcW@J2* )f > The electronics will fail to function when the Morgellons gets >more wide > spread too. > -- Substantiation that you regard yourself as a God to be worhsipped [sic] should be your concern, Deco. -- David Tholen === === Subject: : Re: What DARPA Isn't Telling The Rothschilds Re: More evidence proving Apollo Hoax pR9MNF=QZrhF$HA44sG?n(RnC{0!e>5EKJK{?&5QI$HfYWdP*zNGgP79xb%0Sch26o G8|k?UC[~)lu]WnopcGdzzVoCsDY?/{MgR9eDs>Y~X},^2Ay|9n9~~1&UUCoDQ|O| eYokL{N-%%x6C_4lKt2,LOOY#A6rAjZ5~[1-s,It[DTBZa1v_2D7;0Yq1rQ{LcW@J2* )f > The electronics will fail to function when the Morgellons gets more > wide > spread too. -- Substantiation that you regard yourself as a God to be worhsipped [sic] should be your concern, Deco. -- David Tholen === === Subject: : 3gp jamba videos fuers handy jamba handyvideo converter jamba handy videos downloaden jamba handy videos bearbeiten kostenlose jamba handyvideo jamba handy videos de posting-account=I5cx7QoAAADmrGo-3G-Mnj3ledY0MvJz Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) 3gp jamba videos fuers handy jamba handyvideo converter jamba handy videos downloaden jamba handy videos bearbeiten kostenlose jamba handyvideo jamba handy videos de + + + + + + + + + + + + + + + + + + + + + jamba handy video converter 3gp jamba handyvideo jamba handy videos zum downloaden natascha jamba handy video samsung jamba handy video jamba handy videos format jamba handyvideo 16 j.8ahrige natascha jamba handy video natascha jamba handyvideo mandy jamba handy video jamba handyvideo kostenlos jamba handy videos bearbeiten jamba handy videos kostenlos downloaden jamba handy video auf pc jamba handyvideo akte 07 jamba handyvideo jamba handy video auf pc jamba handyvideo umwandeln jamba handyvideo schule jamba videos fuer handy konvertieren jamba handy video clip jamba videos fuer das handy jamba videos aufs handy laden jamba videos aufs handy jamba handy video programm jamba handy video geburtstag natascha jamba handyvideo jamba handy videos bearbeiten natascha jamba handyvideo jamba handy videos gratis jamba handyvideo umwandeln taff jamba handy video jamba handy video clips natascha jamba handy video jamba handyvideo format jamba videos fuer handy kostenlos jamba video converter fuer handy Input here your backlinks. One in each line. Input here your backlinks. One in each line. Input here your backlinks. One in each line. Input here your backlinks. One in each line. Input here your backlinks. One in each line. Input here your backlinks. One in each line. === === Subject: : Stalk O_x I have a problem showing that the stalk O_x of germs of holomorphic functions at a point O_x is isomorphic to the Ring of convergent power series (coefficients in C) around that point: O_x=~C[[T-x]] Any references or help? === === Subject: : Re: Stalk O_x > I have a problem showing that the stalk O_x of germs of holomorphic functions at a point O_x is isomorphic to > the Ring of convergent power series (coefficients in C) around that point: O_x=~C[[T-x]] Any references or help? Use that (germs of) holomorphic functions and analytic functions are the same === === Subject: : Re: Stalk O_x <69qruhF345vunU1@mid.individual.net Use that (germs of) holomorphic functions and analytic functions are the > same That's right what i cannot show === === Subject: : Re: Stalk O_x <69qruhF345vunU1@mid.individual.net> <48383a3b$0$18158$4fafbaef@reader3.news.tin.it> Use that (germs of) holomorphic functions and analytic functions are the > same That's right what i cannot show It's just Taylor's Theorem or Cauchy's Integral Formula or whatever you want to call it - it will certainly be in any undergraduate complex analysis text. -- email: echo t.adllkhsl@iypzavs.hj.br | tr a-gh-pq-z t-za-ij-s === === Subject: : Re: Stalk O_x I know holomorphic functions and analytic functions are the same. But i can't prove it if we talk about germs of holomorphic functions === === Subject: : Re: Stalk O_x <69qruhF345vunU1@mid.individual.net> <48383a3b$0$18158$4fafbaef@reader3.news.tin.it> <48384755$0$18147$4fafbaef@reader3.news.tin.it I know holomorphic functions and analytic functions are the same. But i > can't prove it if we talk about germs of holomorphic functions If f and g are holomorphic functions defining the same germ at x, then by definition f and g agree on some open disc D(x;r). On the other hand, f and g are given by convergent power series on an open disc D(x;r'). By standard uniqueness results, it follows that the power series coefficients of f and g must be the same. So to each germ at x we can associate a well-defined element of the ring of convergent power series about x. Conversely, if f is analytic at x then it's holomorphic at x, and so defines a germ at x. It's easy to see that this correspondence is a map of rings. -- email: echo t.adllkhsl@iypzavs.hj.br | tr a-gh-pq-z t-za-ij-s === === Subject: : Re: Stalk O_x <69qruhF345vunU1@mid.individual.net> <48383a3b$0$18158$4fafbaef@reader3.news.tin.it> <48384755$0$18147$4fafbaef@reader3.news.tin.it> === === Subject: : Re: Stalk O_x > Use that (germs of) holomorphic functions and analytic functions are the > same That's right what i cannot show Ok ... a text like Cartan or Rudin should have that, essentially it is, that a holomorphic function locally has a convergent (Taylor) series representing it (and it is unique by the Identity theorem) and conversely a convergent power series is holomorphic within its radius of convergence. === === Subject: : Re: 4=3 A maths joke > [...] Tonio Ps. A propo, what about the hoax about induction and all the world > being the same age ? Not only for you, but for the rest of the > participants... anyone interested ? > To recall the hoax : Claim: All the people in the world has exactly the same age. Proof by induction : For a set with one person it is obviously true, So let us assume the claim is true for any set with up > to n people, and let us prove for a set A with n+1 people. [...] => all the elements in A have the same age. Q.E.D. > BTW, this also proves that all integers are equal : all set of integers are composed of equal integers Proof by induction : For any set of 1 integer, it is obviously true. etc... *** But it reminds me about some other hoax : ... If we eat it, we die. ... what is it ? Answer is nothing. (if we eat nothing, we die) *** And then, what about fractional induction ;-) Ok, I get out... -- Philippe C., mail : chephip+news@free.fr site : http://chephip.free.fr/ (recreational mathematics) === === Subject: : Re: 4=3 A maths joke [...] This is getting a little bit tiresome. [...] You asked. How about answering the questions that you excised. > Take n a natural number, then n^2 = n*n = n +...+ n (n times) Is this a valid formula? If yes, are the two sides differentiable functions of n? If yes, then you can take the derivative or both sides wrt n, and you must get the same answer on both sides. -- Michael Press === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=U44YcwkAAAAbGXB70Qr7gA3kornmKE4i > Allow me to continue my defence of WM's point of view. I think this is a bad start, tactically. What you say makes at least >some sense, and doesn't appear to support WM, whose output is >entirlely incoherent. Bad career-wise too, I should think. But >anyway... I thought, still do think (at least I think I still do think), that > WM is on to something. It may be unfortunate that he chose to cast his > argument in the form of a demonstration of ZF's inconsistency (and it does > appear to be important to him that it should be understood in this way). Mainly his arguing serves to show he doesn't understand what mathematicians would mean by a claim that ZF is inconsistent. So when pinned down he admits that all he means is that he personally believes ZF is wrong. Since his supposed arguments supporting this are highly confused, and infinitely feeble, I don't think anyone really cares what he thinks. > But I thought I recognized ideas about the natural numbers similar to my > own, ideas which if accepted do indeed enable one to diiagnose an > equivocation about infinity in set theory. What do you mean by an equivocation? Indeed, what do you mean by infinity? There isn't as far as I know a Thing in set theory called Infinity. > I suppose one could frame this a > justification for or explication of the distinction between potential and > actaul infinity, ... You say a lot that is perhaps poetic, but doesn't seem to have any mathematical content. The distinction between actual infinity (AI) and potential infinity (PI) is not one that has any clear meaning in modern mathematics - though of course it crops up in historical studies. > It means, essentially, that the Naturals, just this familiar > successor structure 1, 2, 3, etc., is fundamental and sui generis. It is > what it is, in a category of its own, an open-ended structure which, by > its nature, aims at infinity but does not possess infinity. Moreover, it, > and it alone, defines for us what 'How Many' means, and it makes no more > sense to ask how many natural numbers there are than it does to ask how > high is 5 metres. The continuation (from 1, 2, 3) etc., or ......, is > unlimited (otherwise we would be limited in what we are able to count), but > not infinite; it is the structural feature of the Naturals to 'point' to > infinity. If by answering how many Xs are there means giving a number-name of >some sort, such that counting the Xs is a process that ends with >reading out the number-name, then indeed it is meaningless to give >such a number-name as an answer to How many naturals are there? I think you also want to capture the intuition - which all bright >children find easy - that counting numbers (naturals) go on and on, >and on and on, and never end. There is never a counting number such >that we cannot add one to find an even bigger counting number. You've called this unlimited - OK. An unlimited set of numbers >thought of as arrayed horizontally has a left end but no right end. >Great. Now get a book on elementary set theory, and go through with a >marker, crossing our every occurrence of the word infinite, >replacing it by unlimited, because that is _all_ that mathematicians >use the i-word to mean. You now have set theory in your grasp. If it were that simple, that would indeed be great, but I am not > sure that everyone even in the set theory camp (if I can put it that way) > would agree with you. You mean that you think that some Cantorians (that's what the cranks call real mathematicians) think that I am _wrong_ in saying that all that infinite set means is an unlimited set, i.e. one for which an attempt at counting never ends? So can you say what else you think that Cantorians mean by infinite set? (And in mathematical terms - I suppose there may be Cantorian poets who claim it's too brutal a simplification... ?) > According to my (counting-based) understanding of the Natural > numbers, the notion of a pure cardinal is a myth, Hmmph. Now let's be a bit honest here: you have done a very common trick - you have gone on a considerable length in a quite pretheoretic way, talking about endless sequences and things, frankly not saying anything we haven't heard before. Now suddenly you announce that the notion of a pure cardinal is a myth. Well what on earth does it mean? Is this some personal intuitive stab at what you feel a cardinal might be? Or are you using the standard set theory meaning of cardinal? You do know what is meant by a cardinal? Or not? I think it would be quickest if you explain what _you_ mean by a cardinal - and then how a pure cardinal is different. >... a misunderstanding. It is > not that ordinal and cardinal number coincide in the finite case, but that > they coincide period. Sorry, you think that ordinals and cardinals are the same? Forgive my way of asking, but do you know what you are talking about (if you want us to understand ordinals and cardinals in the normal mathematical way)? Or do you have private meanings for them, in which case you need to share them. > It is not just a question of not thinking of Aleph or > Omega as being at the end of the chain of finite numbers, Too many nots. In mathematics nothing is at the end of the [chain] of finite numbers, because the [whatever you call it] of finite numbers doesn't have an end. So who do you think thinks this? > ... but a question of > not thinking of Aleph and Omega at all, of believing that the concepts of > Aleph and Omega are ill-motivated. Ah, well, same question. Describe what you understand Aleph to mean. of numbers, according to this conception. Does the existence of a sequence > imply the existence of all its members? If I make a rule for a sequence of > latters as follows: > a, b, c, ......z, aa, bb, cc, ....zz, aaa, bbb, etc. > then does e.g. fffffffff exist in advance of my writing it down? In mathematics, there is no stage of creating as though in an industrial process the objects of discussion. I will agree that if you think mathematics ought to include a creation stage, then you do have something in common with Mueckenheim, who seems to think that numbers (etc) spring into and out of existence as written on a blackboard (or presumably erased). Actually it's an extremely common symptom in crank arguments that at some point there are objects with contradictory properties (such as: they exist, and their existence leads to a contradiction), and the crank attempts to justify this with some sort of dynamic feature, so that when existence is required, the thing exists, when the contradiction would get in the way, we somehow are asked to stop looking. Sorry, I waffle. The short answer is that in a mathematical argument, if we understand a, b, c, ... z to include the whole alphabet, then describing the set of uniliteral strings over the alphabet as all (two-ended, finite) strings consisting of any number of copies of any single letter, then this set exists, and every element of this set exists. Of course it does - that's the whole point of an abstract argument. If I say Any decimal number whose digits sum to a multiple of 9 is itself a multiple of 9, I really mean to include *any* number, including ones nobody has written on the blackboard or ever will. The short answer got almost as long as the long one. Give up. Your question is philosophowaffle, with no mathematical content. Pi = 3.14159265.... > is an an endless decimal expansion. which approximates, not to a number, except derivatively, but to a > position on the spatial line, or equivalently a ratio of spatial > magnitudes. No. 3.14159265.... refers to the endless sequence of digits, which obviously, being endless cannot be written on a piece of paper, but any digit of which can be calculated, so there is no doubt as to what it is. This endless sequence of digits is the decimal expansion of pi, which is precisely the number pi, and not an approximation to anything. Brian Chandler http://imaginatorium.org === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=vI5-YAoAAACpb1I_2s__b0LrjNDZjNTS Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > A standard 32-bit PC you are probably using right now to read this > thread, without any extra memory or libraries is gladly calculate for > you much bigger numbers than 10^50 IT IS NOT THE NUMBER 10^50. It is a number that requires 10^50 bits to > be written or stored! In what schema? IEEE-754-64bit FP-DP (Floating Point - Double Precision) currently used in your computer doesn't use bit-per-bit representation. It stores a number in four separate parts: the number sign (+/-), the exponent (all non-zero digits), mantissa (the number to count left or right to mark the integer part), mantissa sign (gives the direction to count for the integer part position). This way all numbers, except integers <= 4 294 967 295 involved in bitwise operations, are not stored as numbers but merely as special type of internal functions returning the number itself on request. I don't know of any non-positional storage schema where say decadic 125 would be represented by exactly 125 bits. AFAIK such schema was last time used in Sumer at pre-dynastic period and on Crete at Linear B period: 5 fish shown as 5 fish drawings, 8 jugs of wine shown as 8 jugs drawings etc. If you insist that the math operates only with bit- per-bit runtime physically existing exact representations of numbers then the math is passed away circa since the pre-dynastic Sumer times and whatever humanity is doing ever since is not a math :-) > That would not help. What *counts* is only that matter which really > can be used to store bits and to process information - be it in brains > or in computers. Why exactly bits? Dyadic system currently used in the computers is a forced compromise to accommodate numbers to the physical medium used to store them, namely to the medium what is able to have only two distinct states at the time: on (0) and off (1). Quantum computer prototypes allow to have three and four distinct states for each memory cell: with the target to eventually have 10 distinct states so making decadic calculations native to PCs > See the limits I posted above. If you like I'll make a web-page for > you using Javascript to say allocate all hydrogen atoms in the 10^100 > ly area having the Earth as the center. For that purpose you would have to write 10^80 numbers (if you only > gave the distance). How would you store these numbers? On a sheet of > paper? On a screen? They would exists in potential :-) just like say decadic 2 or dyadic 10 you see in this line of text: these are not numbers but formulas to extract numbers if needed. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) A standard 32-bit PC you are probably using right now to read this > thread, without any extra memory or libraries is gladly calculate for > you much bigger numbers than 10^50 IT IS NOT THE NUMBER 10^50. It is a number that requires 10^50 bits to > be written or stored! In what schema? Unimportant. It must be a schema that allows to infer what number is meant. > I don't know of any non-positional storage schema where say decadic > 125 would be represented by exactly 125 bits. That would be unary. No. In order to store a number you need some bits. If there is a simple rule, like 1,232323..., then you need only few bits. If the period is longer, then you need more bits, and if there is no period, and no other rule, then you need infinitely many bits. > See the limits I posted above. If you like I'll make a web-page for > you using Javascript to say allocate all hydrogen atoms in the 10^100 > ly area having the Earth as the center. For that purpose you would have to write 10^80 numbers (if you only > gave the distance). How would you store these numbers? On a sheet of > paper? On a screen? They would exists in potential :-) just like say decadic 2 or dyadic > 10 you see in this line of text: these are not numbers but formulas > to extract numbers if needed. But formulas can only be applied to a countable set of numbers because the set of formulas is countable. So most of the real numbers are not covered by formulas. And in case of your special example, a formula would not do the job because a formula to infer the position of every Atom would need at least as many bits as are required for storing the positions separately --- unless there is a simple law ruling the distribution of all the atoms, which I doubt. And if it were, it must be known, before using it. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=vI5-YAoAAACpb1I_2s__b0LrjNDZjNTS Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > on (0) and off (1). A typo of course: please read instead on (1) and off (0). === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor > Yes, if s(n) is the set of naturals (not initial segments) that preceed n > and {} is not assigned the value 0 (von Neumann) then that is right. But > the null set has no predecessor. > That is precisely why the set of predecessors of 0 is the empty set. Note that s(n) is defined as the ***set*** of predecessors of n, so that > s(n) is necessarily a set. If it is not the empty set, which set is it? A limit ordinal, zero, the von Neumann ordinal, has no predecessor, for which the successor operation upon it evaluates to zero. Similarly omega and all the infinite limit ordinals through the cumulative limit hierarchy, each have no predecessors. Naturally within the ring of integers where succession is interpreted as increment the predecessor of zero is negative one. In physics, nature, just below absolute zero: is positive infinity. In a general consideration of clock arithmetic of the natural integers, which is not an alien concept to the sophisticate, zero goes to infinity and back again, through infinity. Similarly iota increments count through integers. As fluxions to fluents as it were, the fluxions have their fluxions as do theirs etcetera (Cantor's bacilli). Cantor would count backwards from infinity. Negative one is generally established to exist in the integers for some thousands of years. As a natural extension of the natural integers, the Zahlen or integers positive, negative, and zero, contain a value which incremented equals zero, in closure of addition. Where there are only the natural integers upon which to lay those objects in their representation, there is in closure something to represent that item, and in a particular and simple abstraction, infinity equals negative one. ZF's universe <-> Russell set. Ross F. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor > ... > Absolutely, {} itself is not a predecessor of 0. The problem with the > claim > that s(0)={} is that it DOES require {} itself to be a predecessor of > 0. > Note, the definition is: -------------------------------------------------- > Let (A, <=) be a well ordered set. Then the set: > s(k) = {a in A | a < k}, for some k in A > is called an initial segment of A. > -------------------------------------------------- The definition says that the set s(k) = {a in A | a < k} *itself* is an > initial segment of A. If s(0)={} were true then {} itself would be an > initial segment of 0 and hence {} itself would be a predecessor to 0 Why? Why do you think an initial segment is a predecessor? 0 = {} > 1 = {0} > 2 = {0,1} > If A=2={0,1} then the intial segments of A, for k=1, are: According to your definition quoted above, s(1) = {0}, not 0. > Check your definition. s({0}) = s(1) = the set of k with k < 1 = {0} = 1. Or in common language, within the ordinal numbers the set of predecessors > of a number is just that number. You are right, my mistake. (So much for posting at midnight after a long day's work!) It's the predecessor of {} that does not exist in N, not an initial segment. Apology to virgil on that point since he was initially right about it. k === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor > ... > Absolutely, {} itself is not a predecessor of 0. The problem with the > claim > that s(0)={} is that it DOES require {} itself to be a predecessor of > 0. > Note, the definition is: -------------------------------------------------- > Let (A, <=) be a well ordered set. Then the set: > s(k) = {a in A | a < k}, for some k in A > is called an initial segment of A. > -------------------------------------------------- The definition says that the set s(k) = {a in A | a < k} *itself* is an > initial segment of A. If s(0)={} were true then {} itself would be an > initial segment of 0 and hence {} itself would be a predecessor to 0 Why? Why do you think an initial segment is a predecessor? 0 = {} > 1 = {0} > 2 = {0,1} > If A=2={0,1} then the intial segments of A, for k=1, are: According to your definition quoted above, s(1) = {0}, not 0. > Check your definition. s({0}) = s(1) = the set of k with k < 1 = {0} = 1. Or in common language, within the ordinal numbers the set of predecessors > of a number is just that number. You are right, my mistake. (So much for posting at midnight after a long > day's work!) It's the predecessor of {} that does not exist in N, not an > initial segment. Apology to virgil on that point since he was initially > right about it. k No problem! === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > æ> When claiming: A n E m : m > n, > æ> but ~ E m An : m > n > æ> then set theory uses potential infinity. Ok, it uses potential infinity. (Assuming m and n are natural numbers.) æ> When claiming there are all n, or omega is larger than every n, then > æ> it uses actual infinity. Ok, it uses actual infinity. æ(Again assuming n is a natural number, which > omega is not.) What is the problem? æI do not see a conflict between the two statements. In reality there can only be form of infinity, because both exclude each other. Either we can never come to an end, or we can. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor Nntp-Posting-Host: hera.cwi.nl > > > When claiming: A n E m : m > n, > > but ~ E m An : m > n > > then set theory uses potential infinity. > > Ok, it uses potential infinity. (Assuming m and n are natural numbers.) > > > When claiming there are all n, or omega is larger than every n, then > > it uses actual infinity. > > Ok, it uses actual infinity. (Again assuming n is a natural number, which > omega is not.) > > What is the problem? I do not see a conflict between the two statements. > > In reality there can only be form of infinity, because both exclude > each other. Either we can never come to an end, or we can. You are still confused. Going step by step we never get to an end. On the other hand, due to the axiom of infinity the set of all does exist. Still I only detect a conflict between your views and ZF. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > æ æ> æ> When claiming: A n E m : m > n, > æ> æ> but ~ E m An : m > n > æ> æ> then set theory uses potential infinity. > æ æ> Ok, it uses potential infinity. (Assuming m and n are natural numbers.) > æ æ> æ> When claiming there are all n, or omega is larger than every n, then > æ> æ> it uses actual infinity. > æ æ> Ok, it uses actual infinity. æ(Again assuming n is a natural number, which > æ> omega is not.) > æ æ> What is the problem? æI do not see a conflict between the two statements. > æ æ> In reality there can only be form of infinity, because both exclude > æ> each other. Either we can never come to an end, or we can. You are still confused. æGoing step by step we never get to an end. æOn > the other hand, due to the axiom of infinity the set of all does exist. The set of all N is !countable. Do you know what it means to count? It is a procedure going step by step. > Still I only detect a conflict between your views and ZF. The countable and actually infinite are contradictions. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor æ> When claiming: A n E m : m > n, > æ> but ~ E m An : m > n > æ> then set theory uses potential infinity. Ok, it uses potential infinity. (Assuming m and n are natural numbers.) æ> When claiming there are all n, or omega is larger than every n, then > æ> it uses actual infinity. Ok, it uses actual infinity. æ(Again assuming n is a natural number, which > omega is not.) What is the problem? æI do not see a conflict between the two statements. In reality there can only be form of infinity, because both exclude > each other. Either we can never come to an end, or we can. Fortunately, the form of reality that WM insists upon occurs only in WM's mytheology, and is irrelevant elsewhere, such as in the world of mathematics. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > æ> In unary representation only o's are usable for enumerating purposes. Sorry, above you stated that the FISONs enumerated themselves. æIs a FISON > an o nowadays? The first FISON is an o. The second is two o's, and so on. Therefore there are only finitely many o's available for enumerating purposes. You can not remove omega, because omega is not one of the sets united. > Clearly you are using remove with two different meanings. > (1) remove from the set of sets that are being united. > (2) remove from the result of the union. > When you remove a FISON you are clearly using meaning (1), but when you > remove omega you are clearly meaning (2). æConsider using unambiguous > language. I mean omega FISON is not empty. > æ> Why is there a need to distinguish more FISONs than there are? > æ æ> Didn't you say that there are infinitely many FISONs? Yes. So to distinguish infinitely many FISONs we need infinitely many > FISONs, because (as you did state) each FISON is used to enumerate itself. And that is tantamount with infinitely many o's because the number of o's distinguishes the FISONs. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor Nntp-Posting-Host: hera.cwi.nl ... > > In unary representation only o's are usable for enumerating purposes. > > Sorry, above you stated that the FISONs enumerated themselves. Is a FISON > an o nowadays? > > The first FISON is an o. The second is two o's, and so on. Therefore > there are only finitely many o's available for enumerating purposes. I do not understand. Are FISONs sets of natural numbers or sequences of os? > You can not remove omega, because omega is not one of the sets united. > Clearly you are using remove with two different meanings. > (1) remove from the set of sets that are being united. > (2) remove from the result of the union. > When you remove a FISON you are clearly using meaning (1), but when you > remove omega you are clearly meaning (2). Consider using unambiguous > language. > > I mean omega FISON is not empty. So you remove from the result. But in that case, omega FISON(1) != omega. And so the result changes when you remove. On the other hand you claimed that the result did not change when you did remove. So what is it? > > Why is there a need to distinguish more FISONs than there are? > > > > Didn't you say that there are infinitely many FISONs? > > Yes. So to distinguish infinitely many FISONs we need infinitely many > FISONs, because (as you did state) each FISON is used to enumerate itself. > > And that is tantamount with infinitely many o's because the number of > o's distinguishes the FISONs. And does not imply that there are infinitely many o's in a single line. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > ... > æ> æ> In unary representation only o's are usable for enumerating purposes. > æ æ> Sorry, above you stated that the FISONs enumerated themselves. æIs a FISON > æ> an o nowadays? > æ æ> The first FISON is an o. The second is two o's, and so on. Therefore > æ> there are only finitely many o's available for enumerating purposes. I do not understand. æAre FISONs sets of natural numbers or sequences of > os? FISONs are both, sequences of numbers and numbers, expressed by o's. The sequence ooo contains the sequences o and oo. o, oo, and ooo are also numbers, namely 1, 2, and 3. It is this covering of both meanings, sequences or sets and numbers, which shows that the infinite set of finite numbers is a self-contradiction. æ> You can not remove omega, because omega is not one of the sets united. > æ> Clearly you are using remove with two different meanings. > æ> (1) remove from the set of sets that are being united. > æ> (2) remove from the result of the union. > æ> When you remove a FISON you are clearly using meaning (1), but when you > æ> remove omega you are clearly meaning (2). æConsider using unambiguous > æ> language. > æ æ> I mean omega FISON is not empty. So you remove from the result. æBut in that case, omega FISON(1) != omega. > And so the result changes when you remove. æOn the other hand you claimed > that the result did not change when you did remove. æSo what is it? I just explained it in a recent posting. æ> æ> Why is there a need to distinguish more FISONs than there are? > æ> æ æ> æ> Didn't you say that there are infinitely many FISONs? > æ æ> Yes. So to distinguish infinitely many FISONs we need infinitely many > æ> FISONs, because (as you did state) each FISON is used to enumerate itself. > æ æ> And that is tantamount with infinitely many o's because the number of > æ> o's distinguishes æthe FISONs. And does not imply that there are infinitely many o's in a single line. Unless there are infinitely many o's in a line there are not infinitely many FISONs. That is because FISONs and numbers are identical in unary representation. It's just the clue of my proof. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor æ> In unary representation only o's are usable for enumerating purposes. Sorry, above you stated that the FISONs enumerated themselves. æIs a FISON > an o nowadays? The first FISON is an o. The second is two o's, and so on. Therefore > there are only finitely many o's available for enumerating purposes. WM's o's are at most only names of fisons, but are not themselves fisons. Wm still cannot distinguish between a name and the thing named. A very poor performance for one who claims such infallibility. You can not remove omega, because omega is not one of the sets united. > Clearly you are using remove with two different meanings. > (1) remove from the set of sets that are being united. > (2) remove from the result of the union. > When you remove a FISON you are clearly using meaning (1), but when you > remove omega you are clearly meaning (2). æConsider using unambiguous > language. I mean omega FISON is not empty. AS that is irrelevant to the issue, it is another poor performance for one who claims such infallibility. æ> Why is there a need to distinguish more FISONs than there are? > æ æ> Didn't you say that there are infinitely many FISONs? Yes. So to distinguish infinitely many FISONs we need infinitely many > FISONs, because (as you did state) each FISON is used to enumerate itself. And that is tantamount with infinitely many o's because the number of > o's distinguishes the FISONs. One need not name a thing in order to have it exist, even in the physical world. So that WM's o's are irrelevant. > === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor æ> In unary representation only o's are usable for enumerating purposes. Sorry, above you stated that the FISONs enumerated themselves. æIs a FISON > an o nowadays? The first FISON is an o. The second is two o's, and so on. Therefore > there are only finitely many o's available for enumerating purposes. WM's o's are at most only names of fisons, but are not themselves fisons. Wm still cannot distinguish between a name and the thing named. A very poor performance for one who claims such infallibility. You can not remove omega, because omega is not one of the sets united. > Clearly you are using remove with two different meanings. > (1) remove from the set of sets that are being united. > (2) remove from the result of the union. > When you remove a FISON you are clearly using meaning (1), but when you > remove omega you are clearly meaning (2). æConsider using unambiguous > language. I mean omega FISON is not empty. AS that is irrelevant to the issue, it is another poor performance for one who claims such infallibility. æ> Why is there a need to distinguish more FISONs than there are? > æ æ> Didn't you say that there are infinitely many FISONs? Yes. So to distinguish infinitely many FISONs we need infinitely many > FISONs, because (as you did state) each FISON is used to enumerate itself. And that is tantamount with infinitely many o's because the number of > o's distinguishes the FISONs. One need not name a thing in order to have it exist, even in the physical world. So that WM's o's are irrelevant. > === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > æ> In unary representation only o's are usable for enumerating purposes. Sorry, above you stated that the FISONs enumerated themselves. æIs a FISON > an o nowadays? The first FISON is an o. The second is two o's, and so on. Therefore > there are only finitely many o's available for enumerating purposes. WM's o's are at most only names of fisons, but are not themselves fisons. If you are unable to complete the indices of the o's, you are invited to write 1,2,3 instead of ooo. Wm still cannot distinguish between a name and the thing named. Not if the thing is only a name like numbers are, > A very poor performance for one who claims such infallibility. I never claimed infallibility. But I would not deliberately use a wrong concept. Numbers do not exist without their names. You can not remove omega, because omega is not one of the sets united. > Clearly you are using remove with two different meanings. > (1) remove from the set of sets that are being united. > (2) remove from the result of the union. > When you remove a FISON you are clearly using meaning (1), but when you > remove omega you are clearly meaning (2). æConsider using unambiguous > language. I mean omega FISON is not empty. AS that is irrelevant to the issue, only innocent outsiders. æ> Why is there a need to distinguish more FISONs than there are? > æ æ> Didn't you say that there are infinitely many FISONs? Yes. So to distinguish infinitely many FISONs we need infinitely many > FISONs, because (as you did state) each FISON is used to enumerate itself. And that is tantamount with infinitely many o's because the number of > o's distinguishes æthe FISONs. One need not name a thing in order to have it exist, even in the > physical world. Unless that thing is only a name like numbers are. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > ... > > > Of course all FISONs can be removed. Which one should remain? > > And again, sloppy. You can remove them all, indeed, but the result is > > different. So the result you get when you remove a collection of FISONs > > one by one can not be used as a result when you remove them all. > > That is not acceptable. But if it is accepted, then also Cantor's > > diagonal proof is invalid. Nope. You are still confused. > FISONs: > each element of the set of FISONs you can remove, individually. > Cantor: > each element of the list is different from the diagonal, individually. Your conclusion so the set of all FISONs can be removed is equivalent > (in the diagonal case) the set of all elements of the list is different > from the diagonal. > You are still confused. I do not claim that the set of FISONs can be removed or that the diagonal is a list. I claim that *all* FISONs without exception can be removed as Cantor claims that all lines are different from the diagonal. > The first need proof, as need the second. The proof of the second is > trivial: the set of all elements of the list is a set of reals, the > diagonal is a single real, so they are trivially different. > > And about this there is no disagreement. Why do you think there is > > > disagreement about this? > > > > Because you seem to believe there is a difference between removing > > > every FISON and all FISONs. > > That is *not* what I state. You are again imprecise. > > You said above: So the result you get when you remove a collection of > > FISONs one by one can not be used as a result when you remove them > > all. Yes, what I state is that removing one by one is not the same as removing > all at once. So you agree that I did not state what you said. I think that you are confused about the difference of all and the set of all. I do not claim to remove a set of FISONs. I remove only all FISONs at once. You can also say I remove every FISON at once. > But perhaps here is what you state: > > Consider the set A of all FISONs. > > Here is a unary representation. We need no representations. If you want to swindle that will best be detected by a solid example. > When removing all FISONs, the set is empty. Right. > Consider the set B that has all FISONs and omega as an element. > > Here is a unary representation. There is no unary representation of omega. No? You deny the existence of a unary representation of omega? Do you also deny the existence of a unary representation of (the set of) all natural numbers? > When removing all FISONs, the element omega remains. Indeed. > Consider the union of A. When removing all FISONs, the union is empty. Right. union(n in N) FISON(n) = N. When we remove a single FiSON, let's > say the first we get union(n in N, > 1) FISON(n) = N. But when we remove > all FISONS we have: union(n in {}) FISON(n) = {}. > Consider the union of B. When removing all FISONs, the union is empty. Wrong. Again: N U union(n in N) FISON(n) = N. > N U union(n in {}) FISON(n) = N. Why do you say wrong??? You say union A = N. All FISONs removed gives { }. But B is nothing than the union of N U N = N. Why should the result of a removal be different? > That means, after the union is done B has dwindled. Nope. If you remove one of the FiSONs from the set of FISONs united the > result is *still* the same. But contrary to A set B had omega as an element before the union was done. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor Nntp-Posting-Host: hera.cwi.nl ... > > That is not acceptable. But if it is accepted, then also Cantor's > > diagonal proof is invalid. > > Nope. You are still confused. > FISONs: > each element of the set of FISONs you can remove, individually. > Cantor: > each element of the list is different from the diagonal, individually. > > Your conclusion so the set of all FISONs can be removed is equivalent > (in the diagonal case) the set of all elements of the list is different > from the diagonal. > > You are still confused. I do not claim that the set of FISONs can be > removed or that the diagonal is a list. > I claim that *all* FISONs without exception can be removed as Cantor > claims that all lines are different from the diagonal. And again you are conflating two different meanings of all. In the first you mean the complete collection, in the second you do *not* mean that because the complete collection would be the complete list, but the complete list is not an entry of the list. The distinction is quite fundamental. But you refuse to make that distinction. But that is why logic uses specific quantifiers. That is: A (f is a FISON) f can be removed A (l is an element of the list) l is different from the diagonal. It does *not* mean: {f | f is a FISON} can be removed {l | l is an element of the list} is different from the diagonal. > > > Because you seem to believe there is a difference between removing > > > every FISON and all FISONs. > > > > That is *not* what I state. You are again imprecise. > > > > You said above: So the result you get when you remove a collection of > > FISONs one by one can not be used as a result when you remove them > > all. > > Yes, what I state is that removing one by one is not the same as removing > all at once. So you agree that I did not state what you said. > > I think that you are confused about the difference of all and the > set of all. I do not claim to remove a set of FISONs. I remove only > all FISONs at once. And so you remove a set of FISONs. > > But perhaps here is what you state: > > Consider the set A of all FISONs. > > Here is a unary representation. > > We need no representations. > > If you want to swindle that will best be detected by a solid example. I have already discussed this example with you some time ago. You are just repeating yourself. I simply do not wat to repeat that discussion. > > Consider the set B that has all FISONs and omega as an element. > > Here is a unary representation. > > There is no unary representation of omega. > > No? You deny the existence of a unary representation of omega? Yes. > Do you > also deny the existence of a unary representation of (the set of) all > natural numbers? Yes. Representations are finite. So to represent the set of all natural numbers you can not use unary representation. There are of course non-unary representations, like 111..., N, and we can go on. > > Consider the union of B. When removing all FISONs, the union is empty. > > Wrong. Again: N U union(n in N) FISON(n) = N. > N U union(n in {}) FISON(n) = N. > > Why do you say wrong??? You say union A = N. All FISONs removed > gives { }. > But B is nothing than the union of N U N = N. Why should the result of > a removal be different? Where do you remove? You remove from the set of things united. If I remove from the left hand side one N, I get N = N. On the other hand, if you remove from the right hand side, after removing FISON(1) you can not remove any other FISON, because there is no FISON that is any longer a subset of the result. > > That means, after the union is done B has dwindled. > > Nope. If you remove one of the FiSONs from the set of FISONs united the > result is *still* the same. > > But contrary to A set B had omega as an element before the union was > done. A pretty confused terminology. A = union FISON(n). The elements of A are *not* FISONs. The elements of A are natural numbers. And B is N U union FISON(n). In the same way B has as elements the natural numbers. Omega is not an element of A or B. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > ... > æ> æ> That is not acceptable. But if it is accepted, then also Cantor's > æ> æ> diagonal proof is invalid. > æ æ> Nope. æYou are still confused. > æ> FISONs: > æ> æ æ each element of the set of FISONs you can remove, individually. > æ> Cantor: > æ> æ æ each element of the list is different from the diagonal, individually. > æ æ> Your conclusion so the set of all FISONs can be removed is equivalent > æ> (in the diagonal case) the set of all elements of the list is different > æ> from the diagonal. > æ æ> You are still confused. I do not claim that the set of FISONs can be > æ> removed or that the diagonal is a list. > æ> I claim that *all* FISONs without exception can be removed as Cantor > æ> claims that all lines are different from the diagonal. And again you are conflating two different meanings of all. æIn the first > you mean the complete collection, in the second you do *not* mean that > because the complete collection would be the complete list, but the > complete list is not an entry of the list. Is it so hard to comprehend? In both cases I means, what is true in fact, that we can compare omega (or diagonal number, respectively) with every FISON (or line entry, respectively). We can remove every non fitting FISON (line entry) from the set of possibly fitting items . We can conclude that all can be removed. All means all. The distinction is quite fundamental. æBut you refuse to make that > distinction. æBut that is why logic uses specific quantifiers. And that is why I use linear sets which are not subject to such silly arguments as the a man who dances with every woman. EWvery set of FISONs can be replaced by one FISON. > æ> æ> But perhaps here is what you state: > æ> æ> Consider the set A of all FISONs. > æ> æ> Here is a unary representation. > æ æ> We need no representations. > æ æ> If you want to swindle that will best be detected by a solid example. I have already discussed this example with you some time ago. But you have not found any argument against it. (Simply because there is no argument - other than ZF is not suitable for unary numbers.) > æ> æ> Consider the set B that has all FISONs and omega as an element. > æ> æ> Here is a unary representation. > æ æ> There is no unary representation of omega. > æ æ> No? You deny the existence of a unary representation of omega? Yes. That is hard. So current number theory is not possible woth unaries. What about binaries? æ> æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æDo you > æ> also deny the existence of a unary representation of (the set of) all > æ> natural numbers? Yes. æRepresentations are finite. æ 111... ? > So to represent the set of all natural > numbers you can not use unary representation. æThere are of course > non-unary representations, like 111..., N, and we can go on. N is non unary, 111... is unary. Otherwise 1,2,3,... is not decimal. æ> æ> Consider the union of B. When removing all FISONs, the union is empty. > æ æ> Wrong. æAgain: N U union(n in N) FISON(n) = N. > æ> N U union(n in {}) FISON(n) = N. > æ æ> Why do you say wrong??? You say union A = N. All FISONs removed > æ> gives { }. > æ> But B is nothing than the union of N U N = N. Why should the result of > æ> a removal be different? Where do you remove? æYou remove from the set of things united. æIf I > remove from the left hand side one N, I get N = N. æOn the other hand, > if you remove from the right hand side, after removing FISON(1) you > can not remove any other FISON, because there is no FISON that is any > longer a subset of the result. Here I will explain again what I understand by to remove a FISON from omega. I compare a FISON and omega by omega FISON = ? If the result is not empty, then I take that FISON and remove it from the set of FISONs that are possible useful to be united in order to obtain omega. By this procedure I can remove every FISON form the set of possibly useful FISONs without ever removing omega. (The latter would occur if a FISON was omega.) æ> æ> That means, after the union is done B has dwindled. > æ æ> Nope. æIf you remove one of the FiSONs from the set of FISONs united the > æ> result is *still* the same. > æ æ> But contrary to A set B had omega as an element before the union was > æ> done. A pretty confused terminology. æA = union FISON(n). æThe elements of A > are *not* FISONs. æThe elements of A are natural numbers. A union of FISONs is a FISON. Consider the set ofnFISONs o oo ooo The union of this set is the FISON ooo. It is obtained by removing the FISONs o and oo from the set. The union of all FISONs is, according to my observation, a FISON, according to set theory it is omega. > æAnd > B is N U union FISON(n). æIn the same way B has as elements the > natural numbers. æOmega is not an element of A or B. Omega is A. The union of Omega and omega is also omega. nSo Omega is B. remove every FISON, omega remains. But from the unions of A or B we can remove, by the procedure outlined above, every FISON, and, according to set theory, nothing remains. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor > ... > > > Of course all FISONs can be removed. Which one should remain? > > And again, sloppy. You can remove them all, indeed, but the result is > > different. So the result you get when you remove a collection of > > FISONs > > one by one can not be used as a result when you remove them all. > > That is not acceptable. But if it is accepted, then also Cantor's > > diagonal proof is invalid. Nope. You are still confused. > FISONs: > each element of the set of FISONs you can remove, individually. > Cantor: > each element of the list is different from the diagonal, individually. Your conclusion so the set of all FISONs can be removed is equivalent > (in the diagonal case) the set of all elements of the list is different > from the diagonal. You are still confused. I do not claim that the set of FISONs can be > removed or that the diagonal is a list. Wm has quite frequently claimed that all fisons can be removed from the set of all fison and still have a union equalling omega. More sensible people see that when one takes everything away, nothing is left. > I claim that *all* FISONs without exception can be removed We do to, but it is what is left afterwards on which we differ from WM. WM claims everything is left while we claim nothing is left. > as Cantor > claims that all lines are different from the diagonal. The first need proof, as need the second. The proof of the second is > trivial: the set of all elements of the list is a set of reals, the > diagonal is a single real, so they are trivially different. > > And about this there is no disagreement. Why do you think there > > > is > > > disagreement about this? > > > > Because you seem to believe there is a difference between removing > > > every FISON and all FISONs. > > That is *not* what I state. You are again imprecise. > > You said above: So the result you get when you remove a collection of > > FISONs one by one can not be used as a result when you remove them > > all. Yes, what I state is that removing one by one is not the same as removing > all at once. So you agree that I did not state what you said. I think that you are confused about the difference of all and the > set of all. I do not claim to remove a set of FISONs. I remove only > all FISONs at once. You can also say I remove every FISON at once. WM claims to remove all and yet have something left unreemoved, which is nonsense everywhere except WM's mytheology. > But perhaps here is what you state: > > Consider the set A of all FISONs. > > Here is a unary representation. We need no representations. If you want to swindle that will best be detected by a solid example. > When removing all FISONs, the set is empty. Right. > Consider the set B that has all FISONs and omega as an element. > > Here is a unary representation. There is no unary representation of omega. No? You deny the existence of a unary representation of omega? Do you > also deny the existence of a unary representation of (the set of) all > natural numbers? If WM thinks he has a complete unary representation (without ellipis) let him not post again until he can post it. > When removing all FISONs, the element omega remains. Indeed. > Consider the union of A. When removing all FISONs, the union is empty. Right. union(n in N) FISON(n) = N. When we remove a single FiSON, let's > say the first we get union(n in N, > 1) FISON(n) = N. But when we remove > all FISONS we have: union(n in {}) FISON(n) = {}. > Consider the union of B. When removing all FISONs, the union is empty. Wrong. Again: N U union(n in N) FISON(n) = N. > N U union(n in {}) FISON(n) = N. Why do you say wrong??? You say union A = N. All FISONs removed > gives { }. > But B is nothing than the union of N U N = N. Why should the result of > a removal be different? A = the set of all fisons B = union ({A, {omega}}), whose members are every fison and omega. A - A = {} B - A = {omega}, and thus union ({omega}) = omega > That means, after the union is done B has dwindled. Nope. If you remove one of the FiSONs from the set of FISONs united the > result is *still* the same. But contrary to A set B had omega as an element before the union was > done. After removing any finite set of fisons from A or B, the resultant unions are both omega. After removing any set of fisons which leaves infinitely many fisons in A and B,the resultant unions are still both omega. After removing all but finitely many fisons from both A and B, only then will the resultant unions differ. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Your conclusion so the set of all FISONs can be removed is equivalent > (in the diagonal case) the set of all elements of the list is different > from the diagonal. You are still confused. I do not claim that the set of FISONs can be > removed or that the diagonal is a list. Wm has quite frequently claimed that all fisons can be removed from the > set of all fison and still have a union equalling omega. Yes, of course that is true if omega is actually larger than all FISONs. But the set of all seems to induce some magic feelings. More sensible people see that when one takes everything away, nothing is > left. Not if linear sequences are concerned and if that from which everything is taken away is larger than everything. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Well what you consider fundamental is different from what other > people consider fundamental. Perhaps, but many find it convincing. Here is the summary: For every FISON X and all smaller FISONs we know that there is a FISON larger than X. This for every and all smaller is tantamount to all FISONs. It is not necessary to identify the border between all FISONs and omega in order to know of its existence. Rather it is a property of mathematics and logic to use indirect conclusions. So between all FISONs and omega there is an unsurpassable border. Omega is not an actually infinite set. ________________________________________________________ 1) Every FISON is a finite union of FISONs. Omega, if being not a finite union of FISONs, must contain more than every FISON. 2) An actually infinite number of distinct FISONs cannot exist because the finite information present in any FISON (and therefore all FISONs in unary representation) is not sufficient to distinguish infinitely many FISONs. 3) It can be shown for every FISON that it is not required for the asserted infinite union covering omega. ___________________________________________________ Every FISON is finite. Every FISON is a union of a set of FISONs. The set of all FISONs contains only finite FISONs. The set of all FISONs contains only finite unions. The set of all FISONs contains all finite sets of FISONs as subsets. The union of the set of all FISONs is formed by removing many copies of FISONs from the set of all finite sets of FISONs. But there is nothing that could be added, because all FISONs are already there in the set of all FISONs. ________________________________________________________________ Consider the set A of all FISONs. Here is a unary representation. 1 11 111 ... When removing all FISONs, the set is empty. Consider the set B that has all FISONs and omega as elements. Here is a unary representation. 1 11 111 ... 111... When removing all FISONs, the element omega remains. Consider the union of A. When removing all FISONs, the union is empty. Consider the union of B. When removing all FISONs, the union is empty. That means, after the union is done B has dwindled. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor Well what you consider fundamental is different from what other > people consider fundamental. Perhaps, but many find it convincing. Here is the summary: For every FISON X and all smaller FISONs we know that there is > a FISON larger than X. This for every and all smaller is tantamount > to all FISONs. Thus for every fison there is a larger fison. What else is new? > It is not necessary to identify the border between > all FISONs and omega in order to know of its existence. The only such border is that a fison is necessarily finite set and an omega, while a set, is necessarily not a finite set. > Rather it is > a > property of mathematics and logic to use indirect conclusions. Indirect conclusions which are opposed by conflicting provable direct conclusions lose out in mathemtics, thugh the seem to thrive in WM's mytheology. > between all FISONs and omega there is an unsurpassable border. Fisons are all finite, omega is not. > ________________________________________________________ > (1) Every FISON is a finite union of FISONs. > Omega, if being not a > finite union of FISONs, must contain more than every FISON. More than any fison. Omega conatians, as subsets, ALL fisons, which no fison can do. > 2) An actually infinite number of distinct FISONs cannot exist Except that for ZF and related systems, where set and exist do not mean what WM takes them to mean, WM's constraints do not apply. > 3) It can be shown for every FISON that it is not required for the > asserted infinite union covering omega. That does not man that the union of all of them is any less than omega, nor does it mean that the empty set covers omega. ___________________________________________________ > Every FISON is finite. > Every FISON is a union of a set of FISONs. > The set of all FISONs contains only finite FISONs. > The set of all FISONs contains only finite unions. > The set of all FISONs contains all finite sets of FISONs as subsets. > The union of the set of all FISONs is formed by removing many copies > of FISONs from the set of all finite sets of FISONs. Unions are not formed by *removing* anything from the set of sets being unioned.. __________________________ > Consider the set A of all FISONs. > Here is a unary representation. > 1 > 11 > 111 > ... > Then o, oo, ooo,..., is no longer the unary representation? Besides which, WM's alleged representation does not represent any fisons, only naturals, unless fisons are naturals, in which case WM's argument collapses. Consider the set B that has all FISONs and omega as elements. > Here is a unary representation. 1 > 11 > 111 > ... > 111... > This also has no fisons as elements, unless fisons are naturals, in which case WM's argument collapses. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > It is not necessary to identify the border between > all FISONs and omega in order to know of its existence. The only such border is that a fison is necessarily finite set and an > omega, while a set, is necessarily not a finite set. That's enough. Rather it is > a > property of mathematics and logic to use indirect conclusions. Indirect conclusions which are opposed by conflicting provable direct > conclusions lose out in mathemtics, So the indirect conclusion of uncountably many real numbers loses out by the fact that there are only countably many numbers which can be individually identified. between all FISONs and omega there is an unsurpassable border. Fisons are all finite, omega is not. So it is. Therefore no FISON and no heap of FISONs will yield omega. 2) An actually infinite number of distinct FISONs cannot exist Except that for ZF and related systems, where set and exist do not > mean that such individuals really exist in a way that they can be identified. 3) It can be shown for every FISON that it is not required for the > asserted infinite union covering omega. That does not man that the union of all of them is any less than omega, > nor does it mean that the empty set covers omega. We can remove every FISON from omega in the following sense, which I intriduced some time ago but which seems to have been forgotten: We can take a FISON, look whether omega FISON is empty, and if not, put on the pile of FISONs not needed. By this removing process we can remove every FISON from omega, i.e., we can remove all FISONs from omega. ___________________________________________________ > Every FISON is finite. > Every FISON is a union of a set of FISONs. > The set of all FISONs contains only finite FISONs. > The set of all FISONs contains only finite unions. > The set of all FISONs contains all finite sets of FISONs as subsets. > The union of the set of all FISONs is formed by removing many copies > of FISONs from the set of all finite sets of FISONs. Unions are not formed by *removing* anything from the set of sets being > unioned.. That depends on the set. The union of the set (deliberately I avoid magic set theoretic symbols) o oo ooo is formed by removing the elements o and oo. __________________________> Consider the set A of all FISONs. > Here is a unary representation. > 1 > 11 > 111 > ... Then o, oo, ooo,..., is no longer the unary representation? It is the same whether you use ooo or 111 or VVV. Besides which, WM's alleged representation does not represent any > fisons, only naturals, unless fisons are naturals, Of course FISONs can be identified with naturals. In unary every FISON contains all its predecessors as YYY contains YY and Y. It is just this identification of sets and numbers that reveals the contradiction in the infinite set of finite numbers. > in which case WM's > argument collapses. Consider the set B that has all FISONs and omega as elements. > Here is a unary representation. 1 > 11 > 111 > ... > 111... This also has no fisons as elements, unless fisons are naturals, in > which case WM's argument collapses. 111 is a FISON, because it contains 1 and 11 and 111. But the argument will not be different when you use 1,2,3 instead. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor WM says... > Well what you consider fundamental is different from what other > people consider fundamental. Perhaps, but many find it convincing. I don't think so. -- Daryl McCullough Ithaca, NY === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > WM says... > Well what you consider fundamental is different from what other > people consider fundamental. Perhaps, but many find it convincing. I don't think so. No contradiction. You are not many. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor > WM says... > Well what you consider fundamental is different from what other > people consider fundamental. Perhaps, but many find it convincing. I don't think so. No contradiction. You are not many. > Based on posting here, not many agree with WM, but many disagree. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 CLR 1.1.4322; InfoPath.1),gzip(gfe),gzip(gfe) Well what you consider fundamental is different from what other > people consider fundamental. Perhaps, but many find it convincing. Here is the summary: For every FISON X and all smaller FISONs we know that there is > a FISON larger than X. This for every and all smaller is tantamount > to all FISONs. It is not necessary to identify the border between > all FISONs and omega in order to know of its existence. Rather it is > a > property of mathematics and logic to use indirect conclusions. So > between all FISONs and omega there is an unsurpassable border. Omega > is not an actually infinite set. Again, another route WM takes starting with a couple of sensible points and then ending with bizarre logic at an untenable conclusion. > 1) Every FISON is a finite union of FISONs. Yes. > Omega, if being not a > finite union of FISONs, must contain more than every FISON. (1) Yes, omega has 0 as a member. 0 is not a member of any fison. > 2) An actually infinite number of distinct FISONs cannot exist > because > the finite information present in any FISON (and therefore all FISONs > in unary representation) is not sufficient to distinguish infinitely > many FISONs. In your ontology. But that is not something you can use toward a contradiction in set theory, since set theory has no predicate amount of information in in the way you are using it. > 3) It can be shown for every FISON that it is not required for the > asserted infinite union covering omega. Given any PARTICULAR fison, it is not required. But SOME infinite set of fisons is required. No contradiction. > Every FISON is finite. Yes. > Every FISON is a union of a set of FISONs. Yes. > The set of all FISONs contains only finite FISONs. Yes. 'finite fison' is redundant. > The set of all FISONs contains only finite unions. If 'contains' means 'has as members', then yes. > The set of all FISONs contains all finite sets of FISONs as subsets. Yes. > The union of the set of all FISONs is formed by removing many copies > of FISONs from the set of all finite sets of FISONs. There's no requirement about removing copies. Simply: U{f | f is a fison} = {n | Ef(f is a fison & nef)}. > But there is > nothing that could be added, because all FISONs are already there in > the set of all FISONs. Whatever you mean by removing copies and adding, we still know that the union of the set of all fisons is: U{f | f is a fison} = {n | Ef(f is a fison & nef)}. > Consider the set A of all FISONs. > Here is a unary representation. > 1 > 11 > 111 > ... When removing all FISONs, the set is empty. Okay. > Consider the set B that has all FISONs and omega as elements. > Here is a unary representation. 1 > 11 > 111 > ... > 111... When removing all FISONs, the element omega remains. Okay. > Consider the union of A. When removing all FISONs, the union is empty. Do you mean: Start with the set of all fisons, then remove all fisons, then take the union. If so, yes, the result is the empty set. But if you mean: Start with the set of all fisons, take the union, then remove all fisons. Then, no, the result is not the empty set. Indeed NO member of said union is itself a fison. So there ARE NO fisons to remove from said union. > Consider the union of B. When removing all FISONs, the union is empty. Do you mean: Start with B, then remove all fisons, then take the union. If so, then the result is not empty, since omega (or even the set of positive integers, since you keep conflating omega with the set of all positive integers) is not a fison, thus not removed, and U{omega} is not empty. And if you mean: Start with B, take the union, then remove all fisons. Then still the result is not empty, since the union is just omega, of which no member is a fison. > That means, after the union is done B has dwindled. Nope. No dwindling. No dwindling, spindling, or kindling. Your arguments are a form of swindling, though. MoeBlee === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) >There is no such function already, if only we use Q in its normal >ordering by size. That means that cardinality depends on the ordering. Your statement is what is sometimes called a non sequitur. Obviously the possibility of a bijection between a set Q and N depends on the ordering. It is a non sequitur to assume without any reasonable reason that that the cardinal number is independent of the ordering. Of course Q in its normal order cannot be countable if R is uncountable. > It > has nothing to do with what I said. I didn't say anything about > ordering! That is one of the main errors of set theory. Try reading it again: There is a function f from Q to N, and a function g from N > to Q such that for any rational q, g(f(q)) = q, and for any > natural n, f(g(n)) = n. That's just a fact. That's not a fact and it is not justified by axioms. It is assumed without any reason like most errors of set theory. > Ordering has nothing to do with it. Then show such a function for the set Q in its order by size. It is also a fact that if we replace Q by R (the set of reals) > then there are no such functions f and g. There is no pair of > functions such that f is a function from R to N > and g is a function from N to R, and for any real number r, > g(f(r)) = r, and for any natural number n, f(g(n)) = n. Do you agree, or disagree? Look here for an answer: A severe inconsistency of transfinite set theory, http://arxiv.org/pdf/math.GM/0408089 === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor WM says... >There is no such function already, if only we use Q in its normal >ordering by size. That means that cardinality depends on the ordering. > Your statement is what is sometimes called a non sequitur. Obviously the possibility of a bijection between a set Q and N depends >on the ordering. No, it certainly does not. Let me go through an argument for the existence of a bijection from Q to N. If q is any rational number, define the simplest denominator of q to be the smallest positive integer d such that q*d is an integer. Define the simplest numerator of q to be q*d, where d is the lowest denominator. Finally, if q has simplest denominator d and simplest numerator n, then q = n/d, where n and d have no common factors. Define the level of a rational q to be |n|+d where n is its simplest numerator, and d is its simplest denominator (and where |n| means the absolute value of n). So 0/1 is the only rational of level 1. -1/1 and 1/1 are the two rationals of level 2. -2/1, -1/2, 1/2, and 2/1 are the rationals of level 3. Etc. Now, define the following function next(q) from Q to Q: If q is a nonnegative integer, then next(q) = -(q+1). Otherwise, next(q) = the smallest rational number p such that p > q and the level of p = the level of q. Let's start at q = 0/1 and repeatedly apply the next operator: next(0/1) = -1/1 next(-1/1) = 1/1 next(1/1) = -2/1 next(-2/1) = -1/2 next(-1/2) = 1/2 next(1/2) = 2/1 etc. Now, if n is a natural number, then let g(n) = the nth item in the sequence: 0/1, next(0/1), next(next(0/1)), etc. If q is a rational number, then let f(q) = the number of distinct rationals p such that either level(p) < level(q), or level(p) = level(q) and p < q. (By distinct, I mean ignoring equivalent rationals, such as 1/2 and 2/4). My claim is that for any rational q, g(f(q)) = q, and for any natural number n, f(g(n)) = n. If you think otherwise, then give an example of some rational q or some natural n that violate the claim. >Try reading it again: > There is a function f from Q to N, and a function g from N > to Q such that for any rational q, g(f(q)) = q, and for any > natural n, f(g(n)) = n. > That's just a fact. That's not a fact and it is not justified by axioms. I have no idea why you would say something so completely false. Perhaps you can come up with an example of an n or q that you think fails? -- Daryl McCullough Ithaca, NY === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > Let's start at q = 0/1 and repeatedly apply the next > operator: Let's start at FISON F(1) and repeatedly apply the next operator. In this case it is very simple nect(F(n)) is F(n+1). === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor WM says... > Let's start at q = 0/1 and repeatedly apply the next > operator: Let's start at FISON F(1) and repeatedly apply the next operator. In >this case it is very simple nect(F(n)) is F(n+1). Very good! The correct conclusion is The set of all FISONs is countable which means There is a bijection between the set of all FISONs and omega. So maybe it is possible for you to learn something. -- Daryl McCullough Ithaca, NY === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > WM says... >There is no such function already, if only we use Q in its normal >ordering by size. That means that cardinality depends on the ordering. > Your statement is what is sometimes called a non sequitur. Obviously the possibility of a bijection between a set Q and N depends >on the ordering. No, it certainly does not. Let me go through an argument for > the existence of a bijection from Q to N. I know of many possible bijections between N and Q. But there is none preserving the natural ordering by size of Q (while all respect the natural ordering of N or some of its subsets. That is because countable means you can count on and on and need not stop). I have no idea why you would say something so completely > false. It is simply true by the fact that there are not any two real numbers without a rational between them. But the argument is the same as with the binary tree or the simple fact that all irrational numbers are not more than infinite sequences of rational numbers. And these sequences all belong to a countable set. There is no chance for the irrationals to get uncountable. > Perhaps you can come up with an example of an n > or q that you think fails? Can you come up with an example of a FISON that is required for the set of FISONs covering omega? === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor WM says... >I know of many possible bijections between N and Q. Then why would you say that there is no such function? You are a bizarre person. > I have no idea why you would say something so completely > false. It is simply true by the fact that there are not any two real numbers >without a rational between them. That is not good enough for there to be a bijection. To say that between any two reals, there is a rational simply says that there is a function (*NOT* a bijection) from RxR (the set of pairs of reals) to Q. A bijection is not simply a function. It is a function with a unique *inverse*. Specifically, for a function f from RxR to Q to be a bijection, you need to have an inverse function g from Q to RxR, and the two functions must satisfy the properties: 1. For any pair of reals , g(f()) = 2. For any rational number q, f(g(q)) = q. Yes, you can certainly define a function f from RxR to Q. And you can define a function g from Q to RxR. But what you can't do is find such functions where f and g are inverses. > Perhaps you can come up with an example of an n > or q that you think fails? Can you come up with an example of a FISON that is required for the >set of FISONs covering omega? You are very bizarre person. Nobody said that there is a FISON that is required. Everyone has said exactly the opposite. What people are denying is the implication: No single FISON is required to cover omega --> No set of FISONs can cover omega. That implication is what is provably false. -- Daryl McCullough Ithaca, NY === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor Nntp-Posting-Host: hera.cwi.nl >There is no such function already, if only we use Q in its normal >ordering by size. That means that cardinality depends on the ordering. > > Your statement is what is sometimes called a non sequitur. > > Obviously the possibility of a bijection between a set Q and N depends > on the ordering. Obviously this is false. There *are* explicit bijections between the rationals and the naturals. And they are independent of ordering. Have a look at: . You have been pointed to this but apparently never have looked at it. (I agree that it is a mapping between the non-negative rationals and the non-negative integers, but it is easily adapted to a mapping between all the rationals and the natural numbers.) Why you think that a bijection depends on ordering escapes me. > Of course Q in its normal order cannot be countable if R is > uncountable. And that is a non-sequitur. > There is a function f from Q to N, and a function g from N > to Q such that for any rational q, g(f(q)) = q, and for any > natural n, f(g(n)) = n. > > That's just a fact. > > That's not a fact and it is not justified by axioms. It is assumed > without any reason like most errors of set theory. It is a fact, look at the reference I gave above, which I gave earlier but which you wish to ignore. > Ordering has nothing to do with it. > > Then show such a function for the set Q in its order by size. See in the reference given above where the mapping is given independent on ordering. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=X9VdBgoAAAA0ZF8HT8BN_JvL2DEZQ6_G CLR 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) > æ>There is no such function already, if only we use Q in its normal > æ>ordering by size. That means that cardinality depends on the ordering. > æ æ> Your statement is what is sometimes called a non sequitur. > æ æ> Obviously the possibility of a bijection between a set Q and N depends > æ> on the ordering. Obviously this is false. æThere *are* explicit bijections between the > rationals and the naturals. æAnd they are independent of ordering. > Have a look at: > . Why do you think that there is no ordering? In particular why do you think that this mapping has anything to do with the set of rationals in the natural order? You have been pointed to this but apparently never have looked at it. > (I agree that it is a mapping between the non-negative rationals and > the non-negative integers, but it is easily adapted to a mapping between > all the rationals and the natural numbers.) Why you think that a bijection depends on ordering escapes me. It is obvious that a bijection between Q and N in the natural order of Q cannot be given. That seems to be reason enough. æ> Of course Q in its normal order cannot be countable if R is > æ> uncountable. And that is a non-sequitur. No, it is not. For every pair ofreals we can find a rational between them. If there were more reals than rationals, then there must be at least two reals without a rational between them. æ> There is a function f from Q to N, and a function g from N > æ> to Q such that for any rational q, g(f(q)) = q, and for any > æ> natural n, f(g(n)) = n. > æ æ> That's just a fact. > æ æ> That's not a fact and it is not justified by axioms. It is assumed > æ> without any reason like most errors of set theory. It is a fact, look at the reference I gave above, which I gave earlier > but which you wish to ignore. I did not see it earlier. Nevertheless this reference does not give a bijection between N and Q in natural order. I did not object that there are many, in fact very many, bijections between N and Q. æ> Ordering has nothing to do with it. > æ æ> Then show such a function for the set Q in its order by size. See in the reference given above where the mapping is given independent > on ordering. It is not. You have always 2n -> 2n+1 etc. === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor WM says... > =A0>There is no such function already, if only we use Q in its normal > =A0>ordering by size. That means that cardinality depends on the order= >ing. > =A0> =A0> Your statement is what is sometimes called a non sequitur. > =A0> =A0> Obviously the possibility of a bijection between a set Q and N depend= >s > =A0> on the ordering. > Obviously this is false. There *are* explicit bijections between the > rationals and the naturals. And they are independent of ordering. > Have a look at: > . Why do you think that there is no ordering? ordering. >In particular why do you think that this mapping has anything to do >with the set of rationals in the natural order? It DOESN'T have anything to do with rationals in the natural order. That's the whole point. To say there is a bijection between the rationals and the naturals has *nothing* to do with any natural order. Let's try some examples. Let S be the set { apple, grape, lemon } Let T be the set { David, Henry, John }. I don't have to say anything about a natural order in orde > Why you think that a bijection depends on ordering escapes me. It is obvious that a bijection between Q and N in the natural order of >Q cannot be given. That seems to be reason enough. Nobody said anything about natural order of Q. What we said was There is a bijection between Q and N. And there certainly is. -- Daryl McCullough Ithaca, NY === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor There is no such function already, if only we use Q in its normal >ordering by size. That means that cardinality depends on the ordering. Your statement is what is sometimes called a non sequitur. Obviously the possibility of a bijection between a set Q and N depends > on the ordering. On the contrary, the possiblity of a bijection is no way dependent on the given ordering, because existence of bijections is quite independent of the given orderings of the sets being compared. > It is a non sequitur to assume without any reasonable > reason that that the cardinal number is independent of the ordering. The definitions of ordering and of cardinality, at least in any sane set theory, requires that cardinality be independent of ordering. > Of course Q in its normal order cannot be countable if R is > uncountable. Another of WM's weird mytheological theorems which are true nowhere outside that mytheology. Q in any order is just as countable as in any other order, as it is not the order that a set may come with but the orders which can be imposed on it which determines its cardinality. It > has nothing to do with what I said. I didn't say anything about > ordering! That is one of the main errors of set theory. That Daryl said nothing about ordering is hardly an error of set theory. WM is off in his never never land of delusions again. Try reading it again: There is a function f from Q to N, and a function g from N > to Q such that for any rational q, g(f(q)) = q, and for any > natural n, f(g(n)) = n. That's just a fact. That's not a fact and it is not justified by axioms. It is assumed > without any reason like most errors of set theory. On the contrary, many such pairs of functions have actually been constructed. I have constructed a couple myself. And it is even more easy to prove that such pairs of functions must exist in ZF and its like. Ordering has nothing to do with it. Then show such a function for the set Q in its order by size. One can show quite easily that there cannot be any more rationals than naturals by injecting the rationals into the naturals. Each non-zero rational has a unique representation as i/n, where i is a non-zero integer and n is a non-zero natural. So let f map Q to N by f(0) = 1, for i < 0, let f(i/n) = 2^|i| * 3^n, for i > 0, let f(1/n) = 5^i * 7^n. then every rational maps to a different natural, so that the cardinality of Q is no greater than that of N. It is also a fact that if we replace Q by R (the set of reals) > then there are no such functions f and g. There is no pair of > functions such that f is a function from R to N > and g is a function from N to R, and for any real number r, > g(f(r)) = r, and for any natural number n, f(g(n)) = n. Do you agree, or disagree? Look here for an answer: > A severe inconsistency of transfinite set theory, > http://arxiv.org/pdf/math.GM/0408089 Why look into that nesting of errors and idiocies for truth? === === Subject: : Re: A consideration concerning the diagonal argument of G. Cantor posting-account=EL3hgwoAAABtyRFrR2z7EBO1tnJeMiO7 CLR 1.1.4322; InfoPath.1),gzip(gfe),gzip(gfe) There is no such function already, if only we use Q in its normal >ordering by size. That means that cardinality depends on the ordering. Your statement is what is sometimes called a non sequitur. Obviously the possibility of a bijection between a set Q and N depends > on the ordering. WRONG WRONG WRONG. That remark by WM again shows his real lack of understanding of the basic concepts. > It is a non sequitur to assume without any reasonable > reason that that the cardinal number is independent of the ordering. We don't ASSUME it. Rather, we simply prove the existence of bijections, and we may do so without mentioning any particular ordering. > Of course Q in its normal order cannot be countable if R is > uncountable. Things aren't countable or uncountable in an ordering. Things are either countable or uncountable without that being relative to orderings. > There is a function f from Q to N, and a function g from N > to Q such that for any rational q, g(f(q)) = q, and for any > natural n, f(g(n)) = n. That's just a fact. That's not a fact and it is not justified by axioms. It is assumed > without any reason like most errors of set theory. WRONG. It is not ASSUMED. It is PROVEN. > Ordering has nothing to do with it. Then show such a function for the set Q in its order by size. What does show a function for the set Q in its order by size mean? Are you asking for an order preserving bijection from Q onto N based on their standard orderings? There is none. So what? There still exists a bijection from Q onto N and its inverse is a bijection from N onto Q. > It is also a fact that if we replace Q by R (the set of reals) > then there are no such functions f and g. There is no pair of > functions such that f is a function from R to N > and g is a function from N to R, and for any real number r, > g(f(r)) = r, and for any natural number n, f(g(n)) = n. Do you agree, or disagree? Look here for an answer: > A severe inconsistency of transfinite set theory,http://arxiv.org/pdf/math.GM/0408089 Where exactly in that swamp of crank nonsense do we look for your answer to the simple question just asked of you? Why can't you just answer the simple question? MoeBlee === === Subject: : singular value of a matrix posting-account=1vQ5xwoAAADIDQUVBSMlqBb6NsFD508y CLR 1.1.4322),gzip(gfe),gzip(gfe) Consider any arbitrary matrix A, not necessarily square. let sing be the largest singular value of A. Is it true that sing is atleast as large as the magnitude of A's largest entry? === === Subject: : Re: singular value of a matrix > Consider any arbitrary matrix A, not necessarily square. let sing be > the largest singular value of A. Is it true that sing is atleast as > large as the magnitude of A's largest entry? Yes. sing is equal to the maximal value of x^t A y over all appropriately dimensioned vectors x and y whose Euclidean norm is 1. === === Subject: : Re: Mobius Transformation / S Commutes with T if they have same fixed point posting-account=06BQLAoAAADoC7Y4z9FWcUwGvMa7xMG9 7.4),gzip(gfe),gzip(gfe) > So can we say that any two Mobius transformation can > commutate if both of them at the form of translation, > dilation and inversion. æ >Can any two MT commutate with each other even if one > of them at the form of translation and the other one > dilation? > When you ask this question it seems clear that either > you're not ready > to be studying this sort of material or you haven't > even tried to > figure out anything for yourself. If T(z) = z + A and > S(z) = Bz > you should really be able to determine on your own > whether > TS = ST - if you can't do that there's not much > point in > helping you with this. >Mehmet > David C. Ullrich You are right David, I have just started learning this. I am always imagining a group action when we write Tz =z+A and Sz=Bz. There is a group action here. By the way, TS =Bz +A æ and ST =Bz +BA ... so they will not commutate... That wasn't hard, was it? Sure enough, as soon as you tried to find > the answer to your question you had the answer. You should > think about questions before asking other peopler about them... > David C. Ullrich- Masquer le texte des messages pr.8ec.8edents - - Afficher le texte des messages pr.8ec.8edents - Bonsoir, As said by Robert Israel there are two cases : One fixed point x1 , 1/(f(x)-x1)= 1/(x -x1) + c two fixed points x1,x2 , (f(x)-x1)/(f(x)-x2)= a*(x-x1)/(x-x2) Since Commuting functions just differ only by their constant c or a . We might also consider in this case commuting functions as powers : h^[a1] o h^[a2] = h^[a1+a2] , Alain === === Subject: : Re: The quaternion group as a product <9303181.1211518435631.JavaMail.jakarta@nitrogen.mathforum.org>, > I do not intend to stop posting, as I enjoy this forum, and I do not think that many people are experiencing the same problems as you. I re-iterate, however, that you can feel free to just ignore my posts, and I will not be offended. As it happens others people are experiencing the same problems. I am, for instance. Had you read carefully all the replies you received you would know this is the case, and would not have made this false assertion. If it were possible to ignore you, I would; but you insist upon putting yourself forward in a manner that cannot be ignored. -- Michael Press === === Subject: : Re: The quaternion group as a product > mathforum.org does not maintain their own forums: the > site's > discussion area is entirely an interface for certain > usenet groups and > mailing lists. As a user of the mathforum discussion > site, you are > automatically also user of those groups and mailing > lists and you are > therefore responsible for following established > community conventions > within those groups. No, I think mathforum.org is responsible for this, not me. I have posted to mathforum.org. I intend my message to be read by other users of mathforum.org. If mathforum.org decides to transmit my message via Usenet, then it is responsible for the consequences, not me. Analogy: if I tell my friend something in private, and they decide to spread it around the whole world, then I am not really responsible if others find my words inappropriate. I am, however, prepared to follow the community guidelines of mathforum.org. This will be a subset of those of Usenet, but will not include everything, and in particular not limiting the length of my lines. === === Subject: : Re: The quaternion group as a product days. My association with the Department is that of an alumnus. > mathforum.org does not maintain their own forums: the > site's > discussion area is entirely an interface for certain > usenet groups and > mailing lists. As a user of the mathforum discussion > site, you are > automatically also user of those groups and mailing > lists and you are > therefore responsible for following established > community conventions > within those groups. No, I think mathforum.org is responsible for this, not me. I have >posted to mathforum.org. I intend my message to be read by other users >of mathforum.org. Your intentions are immaterial. You are simply unaware of the nature of mathforum. The section you are in is INTENDED to be a portal to Usenet. > If mathforum.org decides to transmit my message via > Usenet, then it is responsible for the consequences, not me. Mathforum did not decide to transmit your messages to Usenet. YOU instruct mathforum to post your messages to usenet by your choice of the section of the Matforum that you are using. The section on Discussions, sci.math* that you are in is expressly meant to be a portal to usenet. >Analogy: if I tell my friend something in private, and they decide to >spread it around the whole world, then I am not really responsible if >others find my words inappropriate. The analogy is flawed because you are not in fact telling your friend something in private. You are sending a letter to the editor in the mistaken impression that you are telling something to your friend in private. The mistake is yours, in misunderstanding the nature of the forum in which you are participating. >I am, however, prepared to follow the community guidelines of > mathforum.org. So long as you can do so while pretending that nothing is your fault, at any rate. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === === Subject: : Re: The quaternion group as a product I am a little confused by the fact that there are two almost identical replies to my last post, but I will try to reply to both in one. >I am looking at: >http://en.wikipedia.org/wiki/Internet_forum and it mentions >...forums perform a function similar to that of > Usenet newsgroups ...that were common from the late > 1970s to the 1990s. >Precursor systems like Usenet have been archived as > far back as 1981 by Google Groups Sigh. If you call the tail of a cat a leg, then how many > legs does the cat > have? Answer: Four. Calling the tail a leg does not make it > one. > I would say five, but I think you would then say I was being contrary. > Your intentions do not control the nature of the > medium you are using. > No, unfortunately I have no control over said medium. > It's not just a river in Egypt, dear, and it's about > time you realize > that you are quite simply wrong and mistaken about > the nature of the > messages you have been sending. > Again, very patronising. > Drexel's Math Forum is not an Internet forum within > the meaning of > the wikipedia page you are reading, despite its name Drexel's Math > Forum is a portal that administers a number of > mailing lists, AND a > portal to the usenet groups [...] > and you will see right at the top that it reads: sci.math.* Usenet newsgroups about mathematics. > ^^^^^^ Yes, indeed my posts have been sent to usenet newsgroups by mathforum.org. I am quite prepared to accept that mathforum.org is, technically, an interface to Usenet. But I choose to ignore it. This is not a new concept. When I _finally_ beat that last race in the video game, I choose to ignore that really all that is happening is a lot of zeros and ones buzzing about inside my games console. When I watch that emotional scene on my DVD player for the umpteenth time, I choose to ignore that really it is a tiny laser shining onto a load of bumps and troughs on the back of a disc. The interface is an abstraction of the boring technical stuff underneath. It is _meant_ to make you ignore what is really going on. > Clearly, you were not aware that you were posting to > sci.math and to a > usenet group. But when you were informed of this, > your reply was to > blame me I did not blame you. If by asking you to check if you were running the latest version of Firefox, you thought I was blaming you, then you misunderstand me. I was merely offering a suggestion to attempt to solve the problem. I was trying to help. > then blame mysterious others who had been > copying your > message in places you did not intend, etc. The reason why these others are mysterious is because they do not exist. What is happening is that the message is being transmitted automatically over Usenet. But it is mathforum.org who have chosen to set up their system this way. >It provides appropriate end-of-line characters to > me, as shown by the link I sent you. Sigh. Are you really this dense, or just contrary? > I do not believe I am either. > Math Forum does not in fact provide end-of-line > characters. What you > are seeing is a rendering of the message you are > sending, local to > Math Forum's display. Most portals to Usenet (of which, whether you like it > or not, Drexel's > Math Forum is one) format their messages in some way. > Most include > either soft or hard carriage-return characters for > formatting purposes > ->in<- the text. Drexel's Math Forum does not. The > interface of the Math > Forum handles this (as does the Google interface). > But this is not > part of the MESSAGE, it is part of the interface. I agree. Math Forum provides appropriate end-of-line characters to me, through its interface. >Please do not patronise me. Then perhaps you should not try patronizing me, > telling me about > checking my browser or learning all about the Maths > Forum if I am > interested in Math. I could say the same, Mr If you are interested in > maths you should > check out the forum, nee Perhaps you should update > your browser. > You have interpreted my comments as patronising, that was not how I intended them. I think that mathforum.org is a great site, obviously you don't, but I was just suggesting maybe you could give it a second chance. This is not patronising, I am not assuming that you are stupid. I just thought that perhaps you had not realised the full scope of what was offered by mathforum.org, and perhaps I was mistaken. >I have a 2:1 Masters degree in Maths from a major > university >(graduated last July) and I really do not appreciate > it when you use >phrases like saying those big words you don't > understand and >calling me son like I am still in high school. Congratulations. Then perhaps you can stop calling me > Doctor or > Professor and stop patronizing me about visiting > the Math Forum if I > am interested in maths. Point is, you just don't know what you are talking > about but you > pretend you do. That's why you are being patronized. > Son. Congratulations! I have a Ph.D. from, let's say a > reasonably good > university, and have had it for a bit longer than 11 > months. > Well congratulations back then. I think you have misunderstood me as perhaps boasting or claiming superiority. I was merely asserting that I am not still in high school and I found the tone of your messages to be a little patronising. I was _not_ demanding respect, or anything of the sort. Certainly the rank of professor does demand some respect. However, you continue to patronise me. > You are demonstrating that you enjoy giving advise > from a position of > ignorance, and blaming mysterious others for doing > what you yourself > did (in ignorance). That suggests how you ought ot be > treated. Yes. However, we are all ignorant. There is no one who knows everything there is to know. All of us are still learning and that is the prerogative of any academic or further, any individual. However, we still give advice, that is human nature, we are attempting to help our fellow man, especially when he is in trouble. > and I do not think that many people are > experiencing the same > problems as you. How would you know? It is pretty clear you don't > really have much of a > clue as to what you are doing or what is going on. > I didn't attest to know, I simply stated that I _think_ this to be the case. This assumption comes from the fact that in the last 5 years I have not encountered anyone else with this problem. Suppose I am a prolific author (I'm not by the way) and I write books in English which are distributed worldwide. And everyone who understands them, reads them, but those who don't just ignore them. It is just a normal, status quo. However then one day, someone comes along from Japan and says to me that they don't understand my books because they don't speak English. And could I, from now on, translate all my published books into Japanese, so that they can understand them. I might consider it, but eventually refuse, because my target audience is those who speak English, and I do not feel that I am obligated to appease those who speak a different language. It is the same here: I am writing posts intended for people who use mathforum.org, but they are being read worldwide by people who use other Usenet interfaces. And some of those are not compatible with the messages that I am sending. However, I am fine with that, because I expect such people to just ignore my posts. I know that anyone who uses mathforum.org, or any other interface which wraps my lines, can understand and reply to my messages, and this is a large enough target audience. I know previously I suggested that you just ignore my posts from now on. Well perhaps I should go a little further and request that you ignore them. We are at a complete textit{impasse} here, and I do not think there is anything to be gained from continuing this discussion, as we have conflicting viewpoints. I will just get more offended, you will get more angry, and anyone else reading this thread will get more bored! I will try, from now on, not to reply directly to any of your posts, and I will not expect you to reply to any of mine. There is nothing stopping you from doing so, though, and so you have the last word in this discussion. Once again, despite the fact that you believe it to be an empty phrase, I am truly sorry for the problems I have caused you. === === Subject: : Re: The quaternion group as a product days. My association with the Department is that of an alumnus. Just for what it may be worth... It is the same here: I am writing posts intended for people who use >mathforum.org, but they are being read worldwide by people who use >other Usenet interfaces. have generated 184 replies; of these, no more than 62 have come from the Math Forum, and 40 of these were in 2002; of more recent posts, the ratio is much worse. For example, the recent thread Principles of induction in non-well-founded set theories generated 36 replies, none of which came from the MathForum. You might want to rethink the notion that you only intend to reach those who use mathforum. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === === Subject: : Re: The quaternion group as a product days. My association with the Department is that of an alumnus. >I am a little confused by the fact that there are two almost > identical replies to my last post, but I will try to reply to both in >one. accident. Yet another of the drawbacks of Drexel's Math Forum, that despite it being a portal to Usenet it does not acknowledge a number > If you call the tail of a cat a leg, then how many > legs does the cat > have? > Answer: Four. Calling the tail a leg does not make it > one. > I would say five, but I think you would then say I was being > contrary. No. I would simply say that you missed the point. And that I was not surprised at that. [...] >The reason why these others are mysterious is because they do not >exist. What is happening is that the message is being transmitted >automatically over Usenet. But it is mathforum.org who have chosen to >set up their system this way. What is actually happening is that YOU are sending your messages TO Usenet. They are not being automatically transmitted. They are MEANT for Usenet, because that is what mathforum is meant to do. Your comments are like complaining that your telephone automatically transmits what you say in your house to the other end of the line, and that this is beyond your intentions even though you are the one who picked it up and dialed. That's what you are doing each time you are posting: you are picking up the phone, and dialing sci.math. It's just that your local telephone central happens to be Drexel's Math Forum. >I agree. Math Forum provides appropriate end-of-line characters to >me, through its interface. Math Forum DISPLAYS end-of-line characters in its graphic interface. Click on the plain text button next time to see how Math Forum does not in fact provide end-of-line characters. There is a difference between providing and displaying. [...] >You have interpreted my comments as patronising, that was not how I >intended them. I think that mathforum.org is a great site, obviously >you don't, Mathforum.org is a pretty lousy interface, as such interfaces go. I speak with some experience both in using and in programming user interfaces in general. The CONTENT of the site, on the other hand, is not mathforum's. I am familiar not only with most of the content of mathforum, but with a lot more that Mathforum does not provide because of its limited scope. I was commenting on the INTERFACE, but again, since you were apparently utterly confused about the nature of the beast, you misinterpreted it and then proceded to make what was, in fact, a rather patronizing remark whether you intended it be such or not. > However, you continue to patronise me. You continue to behave as if your intentions and your misinterpretation (and ignorance) about mathforum and usenet should somehow control everything. That's why you are being treated with a certain modicum of contempt. >I didn't attest to know, I simply stated that I _think_ this to be >the case. This assumption comes from the fact that in the last 5 >years I have not encountered anyone else with this problem. Many people don't bother making the suggestion. >Suppose I am a prolific author (I'm not by the way) and I write books >in English which are distributed worldwide. And everyone who >understands them, reads them, but those who don't just ignore >them. It is just a normal, status quo. However then one day, someone >comes along from Japan and says to me that they don't understand my >books because they don't speak English. And could I, from now on, >translate all my published books into Japanese, so that they can >understand them. I might consider it, but eventually refuse, because >my target audience is those who speak English, and I do not feel that >I am obligated to appease those who speak a different language. You really do love to make utterly ridiculous and misguided analogies. I did not ask you to start translating your messages in Japanese or anything close to that. Simply suggesting pressing the carriage return key at the end of the line is hardly a good parallel to that. Apparently, however, it is indeed too much work. >It is the same here: No, it is not. Here is a better analogy: speaking clearly and loudly is a good way to communicate. One can still communicate while mumbling, or while saying er and like every other word; however, this makes it more difficult for your words to reach the audience. I was not asking you to learn a new language, I was asking you take a breath, speak clearly and not mumble, when you speak. > I am writing posts intended for people who use mathforum.org, Which is prima facie nonsense, and betrays deep misunderstandings about the nature of the beast. Misunderstandings that have been explained, but which you prefer to adhere to because it is too much work to try to figure out what is really happening or the real nature of what you are doing. -- It's not denial. I'm just very selective about what I accept as reality. --- Calvin (Calvin and Hobbes by Bill Watterson) Arturo Magidin magidin-at-member-ams-org === === Subject: : How many types of moves are there in Chess? posting-account=fWSjlAoAAABhdvLz9gHpERiqVNX5yD3B 98),gzip(gfe),gzip(gfe) How many types of moves are there in Chess? To formulate the question mor precisly, consider the following: Moves in Chess can be divided in 6 classes. 1. Simple moves. 2. Simple captures. 3. passant. 4. Simple pawn-tranformation. 5. Capture and pawn-transformation. 6. Castling, short or short long or long long. As simply can be seen from this, is that there are never more than 3 pieces involved in a chess-move. Definition. A type of chess-move is a class of chess moves, under the following equal-relation. Chess move A and B are equal if you can't se any difference on the moves when they are performed on the chess-board without looking on a piece not involved in the chess-move.' The board is pressumed to have the letters A-H and the numbers 1-8 written on the sides. CALCULATION OF THE NUMBER OF TYPES OF MOVES IN CHESS: I only write the formulas, you have to think out what it means yourself. White King: 4*(3+3*4+2) + (6+6).(5+5*4+ 3) +4. (5+5*4+3) +(6+6).(8+8*4+5)+(8*4). (8+8*4+8)=2592 control: 4+(6+6)+4+(6+6)+(8*4)=64 Note: I have not counted the castling, I will do that later. Black king: The same: 2592 White Rook: (8+8).(14+14*4 +6) + (8*6).(14+14*4+7+5) = 5152 control: (8+8)+(8*6)=64 Note: No castling yet. Black rook: The same: 5152 White bishop: 4.(7+7*4+6) + (6+6).(7+7*4+7)+ (6+6).(7+7*4+5)+ 4*(9+9*4+6)+8*(9+9*4+7)+(4+4)*(9+9*4+7) + 4.(11+11*4+8) + 8.(11+11*4+9)+4.(13+13*4+10) = 3188 control: 4+(6+6)+(6+6)+4+8+(4+4)+4+8+4=64 Black bishop: the same: 3188 White Queen Rook +bishop= 5152 + 3188= 8340 Black queen: The same: 8340 White Night: 4.(2+2*4+2)+ 4*(3+3*4+3) + (4+4).(4+4*4+4)+4.(3+3*4+2)+4.(4+4*4+3) + (4+4).(6+6*4+4)+ + 4.(4+4*4+3) + 4.(6+6*4+4) + (4+4).(8+8*4+6) + 4.(4+4*4+4) + 4. (6+6*4+4)+ 8.(8+8*4+4)= =1924 control: 4+4+(4+4)+4+4+(4+4)+4+4+(4+4)+4+4+8=64 Black Night: The same: 1924 White Pawn: (8+8).0+2.(2+1*4+1)+ 6.(2+2*4+2) + (3+3).(1+1*4+1) + (6+6+6) (1+2*4+2) + (1+1).(1+1*4+1+1)+ + 6.(1+2*4+2+2)+ 2.(1*4+1*4*4)+ 6.(1*4 +2*4*4) = 770 control: (8+8)+2+6+(3+3)+(6+6+6)+(1+1)+6+2+6=64 Black pawn: The same: 770 Rocckad: short, short long, long long =3 NUMBER OF MOVES= 2592+2592+770+770+3+3+5152+5152+3188+3188+8340+8340+1924+1924= 2232 ------------ 5184 1540 6 6376 16680 3848 ---------------- 33634 Note that this number might be wrong, I did the same calculation som time ago and I did get another number. === === Subject: : Re: How many types of moves are there in Chess? lundslaktare@yahoo.com a .8ecrit : > How many types of moves are there in Chess? > ... > 6. Castling, short or short long or long long. ... > Rocckad: short, short long, long long =3 > What's that ???? Short is Ke1-g1 and Rh1-f1 for White (or Ke8-g8 and Rh8-f8 for Black Long is Ke1-c1 and Ra1-d1 for White (Ke8-c8 and Ra8-d8 for Black) That's all. Other kind of Rocckad are illegal moves. -- Philippe C., mail : chephip+news@free.fr site : http://chephip.free.fr/ (recreational mathematics) === === Subject: : diagonal posting-account=bC1KEQoAAADQT48JtbxGptwByVrNqgqY 1.1.4322; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) http://www.meb.gov.tr/baglantilar/okullar/yonlendir.asp?KOD=711531 === === Subject: : ? displacement operator with vector argument Hi: A displacement operator for a scalar function with a real scalar argument is defined as follow. f(x+b) = exp( b*d/dx )*f(x) = sum( b^k*D(f(x), k)/k!, k = 0 to Inf ), where D(f, k) denotes the kth order derivative, e.g., D(f(x), 1) = df(x)/dx. Now I have question on extending this operator to deal with the case when its argument is a real vector. As the 1st thought, it seems to be straightforward. f(x+b) = exp( b'*d/dx )*f(x) = sum( b'^k*D(f(x)/k!, k), k = 0 to Inf ) Here the symbol ' denotes a transpose operation. My problem arose when I tried to write this expression term-by-term explicitly. f(x+b) = f(x)+b'*D(f(x), 1)/1!+b'^2*D(f(x), 2)/2!+b'^3*D(f(x), 3)/3!+... The 1st and the 2nd term in the RHS are easy. But how about the rest? I guess the 3rd term should be: b'*d^2f(x)/dx/dx'*b/2! Something like a quadratic form so that it remains a scalar since f(x+b) is a scalar function. But how about the 4th term and so on? Besides, I got the 3rd term by guessing or trial-and-error. Are there more systematic or more intuitive way to derive the 3rd term from the 2nd term? by Cheng Cosine May/24/2k8 NC === === Subject: : Re: ? displacement operator with vector argument posting-account=K5WE3woAAAAXArsybjkbN6LjMxWdHtbX Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > Hi: A displacement operator for a scalar function with a real scalar argument is defined as follow. f(x+b) = exp( b*d/dx )*f(x) = sum( b^k*D(f(x), k)/k!, k = 0 to Inf ), where D(f, k) denotes the kth order derivative, e.g., D(f(x), 1) = df(x)/dx. Now I have question on extending this operator to deal with the case when its argument is a real vector. As the 1st thought, it seems to be straightforward. f(x+b) = exp( b'*d/dx )*f(x) = sum( b'^k*D(f(x)/k!, k), k = 0 to Inf ) Here the symbol ' denotes a transpose operation. My problem arose when I tried to write this expression term-by-term explicitly. f(x+b) = f(x)+b'*D(f(x), 1)/1!+b'^2*D(f(x), 2)/2!+b'^3*D(f(x), 3)/3!+... The 1st and the 2nd term in the RHS are easy. But how about the rest? I guess the 3rd term should be: b'*d^2f(x)/dx/dx'*b/2! Something like a quadratic form so that it remains a scalar since f(x+b) is > a scalar function. But how about the 4th term and so on? Besides, I got the 3rd term by guessing or trial-and-error. Are there more > systematic or more intuitive way to derive the 3rd term from the 2nd term? by Cheng Cosine > May/24/2k8 NC Look at the two-variable case (the case of more than two variables is an easy generalization). What is the Taylor expansion for f(x1+b1,x2+b2)? Let F(t) = f(x1+t*b1, x2+t*b2), where t is a scalar. We want F(1). The Maclaurin expansion of F is F(t) = F(0)+sum{t^n*F^(n) (0)/n!, n=1..infinity}, where F^(n)(t) = nth derivative of F(t) with respect to t. We have F'(t) = b1*D1f + b2*D2f, where D1 and D2 are the partial derivatives in the x1 and x2 directions and the argument of f is x + t*b =(x1 + t*b1,x2+t*b2). Thus, F'(0) = b1*D1f(x) + b2*D2f*x) = [b1*D1 + b2*D2]f(x). We have F^(2)(t) = [b1*D1 + b2*D2]F'(t) = [b1*D1+b2*D2]^2 f(x+b*t), F^(3)(t) = [b1*D1 + b2*D2]F^(2)(t) = [b1*D1 + b2*D2]^3 f(x+b*t), etc. Thus, F(1) = sum{(1/n!)*(b1*D1 + b2*D2)^n f(x),n=0..infinity) = exp(b1*D1 + b2*D2)f(x). R.G. Vickson === === Subject: : Re: ? displacement operator with vector argument Look at the two-variable case (the case of more than two variables is > an easy generalization). What is the Taylor expansion for > f(x1+b1,x2+b2)? Let F(t) = f(x1+t*b1, x2+t*b2), where t is a scalar. > We want F(1). The Maclaurin expansion of F is F(t) = F(0)+sum{t^n*F^(n) > (0)/n!, n=1..infinity}, where F^(n)(t) = nth derivative of F(t) with > respect to t. We have F'(t) = b1*D1f + b2*D2f, where D1 and D2 are the > partial derivatives in the x1 and x2 directions and the argument of f > is x + t*b =(x1 + t*b1,x2+t*b2). Thus, F'(0) = b1*D1f(x) + b2*D2f*x) = > [b1*D1 + b2*D2]f(x). We have F^(2)(t) = [b1*D1 + b2*D2]F'(t) = > [b1*D1+b2*D2]^2 f(x+b*t), F^(3)(t) = [b1*D1 + b2*D2]F^(2)(t) = [b1*D1 > + b2*D2]^3 f(x+b*t), etc. Thus, F(1) = sum{(1/n!)*(b1*D1 + b2*D2)^n > f(x),n=0..infinity) = exp(b1*D1 + b2*D2)f(x). > Ah, I see! Your expression is easier to extend to higher dimension. :) But how about if the function f is not just a scalar-valued function but also a vector? by Cheng Cosine May/25/2k8 NC === === Subject: : Re: ? displacement operator with vector argument > Look at the two-variable case (the case of more than two variables is > an easy generalization). What is the Taylor expansion for > f(x1+b1,x2+b2)? Let F(t) = f(x1+t*b1, x2+t*b2), where t is a scalar. > We want F(1). The Maclaurin expansion of F is F(t) = F(0)+sum{t^n*F^(n) > (0)/n!, n=1..infinity}, where F^(n)(t) = nth derivative of F(t) with > respect to t. We have F'(t) = b1*D1f + b2*D2f, where D1 and D2 are the > partial derivatives in the x1 and x2 directions and the argument of f > is x + t*b =(x1 + t*b1,x2+t*b2). Thus, F'(0) = b1*D1f(x) + b2*D2f*x) = > [b1*D1 + b2*D2]f(x). We have F^(2)(t) = [b1*D1 + b2*D2]F'(t) = > [b1*D1+b2*D2]^2 f(x+b*t), F^(3)(t) = [b1*D1 + b2*D2]F^(2)(t) = [b1*D1 > + b2*D2]^3 f(x+b*t), etc. Thus, F(1) = sum{(1/n!)*(b1*D1 + b2*D2)^n > f(x),n=0..infinity) = exp(b1*D1 + b2*D2)f(x). Ah, I see! Your expression is easier to extend to higher dimension. :) But how about if the function f is not just a scalar-valued function but > also a vector? > Hold on. Vector function has nothing but more components, so I only need to treat it component-by-component. by Cheng Cosine May/25/2k8 NC === === Subject: : Re: DVT risk and air travel > The only study that I can find that measures the effect of class of > travel on the chance of getting a DVT is this on (all the other .be > The BEST study--a prospective study to compare business class versus > economy class air travel as a cause of thrombosis. .be .be[NonBreakingSpace]¢ > Only 434 subjects had a full venous duplex scan performed. None had > ultrasonic evidence of venous thrombosis. Nine passengers tested at > departure had elevated D-dimer levels and these volunteers were > excluded from further study. Seventy-four of the 899 passengers had > raised D-dimers on arrival. Twenty-two of 180 business class > passengers (12%) developed elevated D-dimers compared with 52 of 719 > economy class passengers (7%). There was no significant association > between elevation of D-dimers and the class flown (odds ratio (OR) > 0.61, p = 0.109). .be[NonBreakingSpace]¢ .be > I'd imagine that a huge amount of money, potentially, rests on this > evidence, so I'm surprised that this is the only study. I haven't got > Dr Jacobson's e-mail address at Wits, nor the whole text of the study > yet, but I'll follow these up to understand more detail and if there > was any special funding. .be > I find something curious about the above, though, maybe somebody with > some knowledge of statistics can help explain it. .be > 1. If all air passengers were the same, and there was no bias caused > by the class of travel, then you'd expect, I'd have thought, to find > the same raised D-dimers (the proxy for potential DVT in the study) in > both populations. To me, finding 14% in on population and 7% in the > other would suggest that the first population was twice as likely to > suffer the effect. .be > 2. Clearly the size of study is important. So, though the whole study > includes nearly 900 passengers, the study only examines 180 business > class passengers. So, these are less likely to be representative than > those not in business class. .be > 3. Isn't it also likely that those flying in Business class will have > other characteristics that differ that might be significant in their > risk of DVT? Shouldn't these factors be exluded before a comparison is > made? .be > 4. How, then, do they come to the conclusion that there is no effect? Their conclusion is There was no significant association between elevation of D-dimers and the class flown (odds ratio (OR)> 0.61, p = 0.109), which means that, according to their calculations, the probability of the apparent association being due to nothing more than chance is 10.9%; by convention, a result is only regarded as statistically significant if this probability is less than 5%. It is not clear to me how they reached this conclusion. According to the figures given, the odds for business class passengers to have raised D-dimers on arrival were found to be 22/158 and for economy class passengers 52/667. The ratio between these is 1.79 or 0.56, depending on which way round you take them, and calculating the 95% confidence intervals as described, for example, here http://www.bmj.com/cgi/content/full/320/7247/1468 , gives intervals of 1.05-3.03 and 0.33-0.95 respectively. Since these intervals do not include the value 1, that seems to indicate p < 0.05. Does the full text of the study give any more information on how they arrived at their stated figures for the odds ratio and p? === === Subject: : Re: DVT risk and air travel posting-account=p-xPhAkAAADjHQWEO7sFME2XBdF1P_2H AppleWebKit/525.18 (KHTML, like Gecko) Version/3.1.1 Safari/525.18,gzip(gfe),gzip(gfe) > The only study that I can find that measures the effect of class of > travel on the chance of getting a DVT is this on (all the other The BEST study--a prospective study to compare business class versus > economy class air travel as a cause of thrombosis. > Only 434 subjects had a full venous duplex scan performed. None had > ultrasonic evidence of venous thrombosis. Nine passengers tested at > departure had elevated D-dimer levels and these volunteers were > excluded from further study. Seventy-four of the 899 passengers had > raised D-dimers on arrival. Twenty-two of 180 business class > passengers (12%) developed elevated D-dimers compared with 52 of 719 > economy class passengers (7%). There was no significant association > between elevation of D-dimers and the class flown (odds ratio (OR) > 0.61, p = 0.109). > I'd imagine that a huge amount of money, potentially, rests on this > evidence, so I'm surprised that this is the only study. I haven't got > Dr Jacobson's e-mail address at Wits, nor the whole text of the study > yet, but I'll follow these up to understand more detail and if there > was any special funding. I find something curious about the above, though, maybe somebody with > some knowledge of statistics can help explain it. 1. If all air passengers were the same, and there was no bias caused > by the class of travel, then you'd expect, I'd have thought, to find > the same raised D-dimers (the proxy for potential DVT in the study) in > both populations. To me, finding 14% in on population and 7% in the > other would suggest that the first population was twice as likely to > suffer the effect. 2. Clearly the size of study is important. So, though the whole study > includes nearly 900 passengers, the study only examines 180 business > class passengers. So, these are less likely to be representative than > those not in business class. 3. Isn't it also likely that those flying in Business class will have > other characteristics that differ that might be significant in their > risk of DVT? Shouldn't these factors be exluded before a comparison is > made? 4. How, then, do they come to the conclusion that there is no effect? Their conclusion is There was no significant association between elevation > of D-dimers and the class flown (odds ratio (OR)> 0.61, p = 0.109), which > means that, according to their calculations, the probability of the apparent > association being due to nothing more than chance is 10.9%; by convention, a > result is only regarded as statistically significant if this probability is > less than 5%. It is not clear to me how they reached this conclusion. æAccording to the > figures given, the odds for business class passengers to have raised > D-dimers on arrival were found to be 22/158 and for economy class passengers > 52/667. æThe ratio between these is 1.79 or 0.56, depending on which way > round you take them, and æcalculating the 95% confidence intervals as > described, for example, herehttp://www.bmj.com/cgi/content/full/320/7247/1468, gives intervals of > 1.05-3.03 and 0.33-0.95 respectively. æSince these intervals do not include > the value 1, that seems to indicate p < 0.05. æDoes the full text of the > study give any more information on how they arrived at their stated figures > for the odds ratio and p? > No, it doesn't, it simply declares that it is not significant. === === Subject: : Half Tetration? posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) Hi. I was wondering about the following approach to Tetration given the recent discussion. I recently saw what is called Fa.88 di Bruno's Formula on a formal power series: http://en.wikipedia.org/wiki/Fa%C3%A0 di Bruno's formula#Formal power series version Now, I was wondering: could these be used to obtain coefficients for a power series such that, when summed, would yield a function f(x) such that f(f(x)) = exp(x), by using the formula for the composite then equating to the terms of exp(x)? === === Subject: : Re: Half Tetration? > Hi. I was wondering about the following approach to Tetration given the > recent discussion. I recently saw what is called Fa[Thorn] di Bruno's Formula on a > formal power series: > http://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno's_formula#Formal_power_series _version Now, I was wondering: could these be used to obtain coefficients > for a power series such that, when summed, would yield a function > f(x) such that f(f(x)) = exp(x), by using the formula for the > composite > then equating to the terms of exp(x)? I think (but I am not absolutely certain) that Daniel Geisler has used the Fa di Bruno formula to obtain an extension to tetration. His results (for sqrt(2)) are given in terms of Conway's Chained Arrow notation in this page: http://www.tetration.org/Ackermann/index.html As far as I know he's the only one who has explicitly utilized this formula for tetration, but I could well be wrong. -- I.N. Galidakis === === Subject: : Re: Half Tetration? <1211687909.940067@athprx04> posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx (KHTML, like Gecko) Safari/412,gzip(gfe),gzip(gfe) > Hi. I was wondering about the following approach to Tetration given the > recent discussion. I recently saw what is called Fa.88 di Bruno's Formula on a > formal power series: http://en.wikipedia.org/wiki/Fa%C3%A0 di Bruno's formula#Formal power... Now, I was wondering: could these be used to obtain coefficients > for a power series such that, when summed, would yield a function > f(x) such that f(f(x)) = exp(x), by using the formula for the > composite > then equating to the terms of exp(x)? I think (but I am not absolutely certain) that Daniel Geisler has used the Fa di > Bruno formula to obtain an extension to tetration. His results (for sqrt(2)) are > given in terms of Conway's Chained Arrow notation in this page: http://www.tetration.org/Ackermann/index.html As far as I know he's the only one who has explicitly utilized this formula for > tetration, but I could well be wrong. Well I suppose I could see where this all goes. The rub though is that the formula given on the page I referenced has series starting at n = 1 instead of n = 0... In addition to that, though, we also run into the fact that those Bell polynomials, being polynomials after all, will have multiple ROOTs. That means we need to figure out which ROOTs will give the most natural possible tetration. === === Subject: : Here you can see the amazing pictures posting-account=OYA7QwoAAADCTrcqfGDBau5ErGuHYCfE SV1),gzip(gfe),gzip(gfe) Here you can see the amazing pictures http://www.flixya.com/photos/u/superphoto === === Subject: : solutions manual to Elements of engineering electromagnetics (6/e) by N.N.RAO posting-account=AIT25goAAAD4PInVOqQYW2U7xf3SSqUF Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) i have the solution manual to Elements of engineering electromagnetics (6/e) by N.N.RAO Introduction to Mathematical Statistics 6/E Robert V. Hogg.. if you want to get it,please email to trustsolution(at)hotmail.com. === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity <483122e6$0$30464$afc38c87@news.optusnet.com.au> posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) A manifestation of Einstein's genius was that he fixed mechanics > to conform to Lonrentz Invariance. No. I believe that the credit for that belongs to Henri Poincar.8e. > Almost everyone else would have tried to trick up > electrodynamics to conform to Newtonian Mechanics. What are you talking about? Einstein spent about 10 years of his life trying to do exactly that. Shubee http://www.everythingimportant.org/relativity/directory.htm === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity <483122e6$0$30464$afc38c87@news.optusnet.com.au> A manifestation of Einstein's genius was that he fixed mechanics to > conform to Lonrentz Invariance. No. I believe that the credit for that belongs to Henri Poincar.8e. Indeed as Lorentz recognized (blockquote For certain of the physical magnitudes which enter in the formulae I have not indicated the transformation which suits best. This has been done by Poincar.8e, and later by Einstein and Minkowski. [...] I have not established the principle of relativity as rigorously and universally true. Poincar.8e, on the other hand, has obtained a perfect invariance of the electro-magnetic equations, and he has formulated 'the postulate of relativity', terms which he was the first to employ. [...] Poincar.8e remarks [..] that if one considers x,y,z, and t sqrt{-1} as the coordinates of a space of four dimensions, the transformations of relativity are reduced to rotations in that space. ) For other aspects about history of relativity http://www.canonicalscience.org/en/researchzone/history.html -- Center for CANONICAL |SCIENCE) http://canonicalscience.org === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity <483122e6$0$30464$afc38c87@news.optusnet.com.au> posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) On May 25, 8:42 am, Juan R. Gonz.87lez-.8dlvarez > A manifestation of Einstein's genius was that he fixed mechanics to > conform to Lonrentz Invariance. No. I believe that the credit for that belongs to Henri Poincar.8e. Indeed as Lorentz recognized (blockquote > For certain of the physical magnitudes which enter in the formulae I > have not indicated the transformation which suits best. This has been > done by Poincar.8e, and later by Einstein and Minkowski. > [...] I have not established the principle of relativity as rigorously > and universally true. Poincar.8e, on the other hand, has obtained a > perfect invariance of the electro-magnetic equations, and he has > formulated 'the postulate of relativity', terms which he was the first > to employ. [...] Poincar.8e remarks [..] that if one considers x,y,z, and > t sqrt{-1} as the coordinates of a space of four dimensions, the > transformations of relativity are reduced to rotations in that space. > ) Juan, > For other aspects about history of relativity http://www.canonicalscience.org/en/researchzone/history.html This page of yours is excellent. For that reason I added it to the Everything Important Directory: http://www.everythingimportant.org/relativity/directory.htm Shubee > Center for CANONICAL |SCIENCE)http://canonicalscience.org === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity <48322397$0$30460$afc38c87@news.optusnet.com.au> posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) On May 19, 8:04 pm, Peter Webb There is a more fundamental theory than special relativity. One > consequence is that Einstein's theory could be false and our theory > could be true. However, if Einstein's theory is true, then our theory > is also true. Our fundamental axioms are: 1. Newton's First law of motion. > 2. There is a simple definition of clock time for each point in an > inertial frame of reference. (i) This time is mathematically well- > defined. (ii) Time is defined the same way in all inertial frames of > reference. http://www.everythingimportant.org/relativity/special.pdf Shubee These axioms are fine, from a mathematical point of view (ie they are not > self-contradictory). Indeed, they are a subset of Newton's laws of motion. Unfortunately, they do not correspond with observation. There is no > universal time. My fundamental axiom 2 (ii) states: ñTime is defined the same way in > all inertial frames of reference.î I think youÍve stretched the phrase, ñthe same wayî to presuppose that > I believe in the existence of a cosmic everywhere present ñnow. You > are mistaken. ********************** > Well, what does your axiom mean exactly, if it is not implying the > existence of universal time. The exact meaning is represented perfectly in the simplest spacetime imaginable. That explanation is contained in the section titled, My First Toy Universe. http://www.everythingimportant.org/relativity/special.pdf > Does your theory predict the same time dilation > as predicted by SR? Yes. > If so, how does it differ from SR? My theory removes the unnecessary restrictions in Einstein's theory and thus allows for motion faster than light. > Identically constructed clocks measure time the same way in all > inertial frames of reference. Does that concept trouble you? ********************** > Yes, because it appears to be meaningless. What is it supposed to mean? That > physics behaves the same in all inertial frames? I'm saying that all the laws of physics may be divided into two categories. There are physical laws that are true in all inertial frames of reference (such as my definition of time) and there may be other physical laws that depend on an absolute frame of reference. Shubee http://www.everythingimportant.org/relativity/special.pdf > If so, this axiom was > used Einstein, so its hard to see how your theory could differ from SR > without some other axioms which conflict with SR being used. Is it the > same as SR? If not, what additional axioms that you use cause the > difference in results? Shubee http://www.everythingimportant.org/relativity/special.pdf === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity I'm saying that all the laws of physics may be divided into two > categories. There are physical laws that are true in all inertial > frames of reference (such as my definition of time) and there may be > other physical laws that depend on an absolute frame of reference. Has nature provided us an absolute frame of referrence? If so what is it, and how has it been observed (or detected) and measured. What empirical evidence can you assert indicating that such a frame exists and if it exists what is it? I am happy to entertain any theory, hypothesis, model that explains or is consistent with all phehomena that have been observed up to this point in time. As long as it makes testable predictions and can encompass doable methods of observation and measurement. Is your stuff well grounded empirically? Bob Kolker === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) I'm saying that all the laws of physics may be divided into two > categories. There are physical laws that are true in all inertial > frames of reference (such as my definition of time) and there may be > other physical laws that depend on an absolute frame of reference. Has nature provided us an absolute frame of referrence? Perhaps nature doesn't have a choice. I consider it a distinct possibility that, in the Hilbert atlas of all conceivable universes, the only spacetime without an absolute frame of reference is the Newtonian one defined by Galilean transformations. Conceivably, spacetime according to Einstein's 1905 formulation could be a mathematical fantasy * without any possibility of real existence. * By mathematical fantasy, I mean something mathematically and empirically consistent but ultimately false, such as the continuum hypothesis or its negation in Zermelo[CapitalEth]Fraenkel set theory. > If so what is it, and how has it been observed (or detected) > and measured. I will be updating http://www.everythingimportant.org/relativity/special.pdf with a very clear derivation of superluminal coordinate transformations. I will explain the consequences and give experimentalists the physical laws that govern superluminality. > What empirical evidence can you assert indicating that such a > frame exists and if it exists what is it? Just as Einstein predicted without any direct empirical evidence that motion faster than light will never be observed, I think that I also have the intellectual freedom to believe and conjecture that superluminal motion will be observed and to teach the consequences. Isn't it obvious that superluminality requires an absolute frame of reference? > I am happy to entertain any theory, hypothesis, model that explains or > is consistent with all phehomena that have been observed up to this > point in time. Presupposing an absolute frame is perfectly consistent with all empirical evidence. > As long as it makes testable predictions and can encompass > doable methods of observation and measurement. Is your stuff > well grounded empirically? I consider my theory as a restoration and improvement of the relativity theory of Henri Poincar.8e, which predates Einstein's theory. Shubee http://www.everythingimportant.org/relativity/special.pdf === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=IwEXfQoAAADCjdz2TIptLhugu5MWo47W Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) I'm saying that all the laws of physics may be divided into two > categories. There are physical laws that are true in all inertial > frames of reference (such as my definition of time) and there may be > other physical laws that depend on an absolute frame of reference. Has nature provided us an absolute frame of referrence? If so what is > it, and how has it been observed (or detected) and measured. What > empirical evidence can you assert indicating that such a frame exists > and if it exists what is it? > CMB http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation That's the preferred frame of reference > I am happy to entertain any theory, hypothesis, model that explains or > is consistent with all phehomena that have been observed up to this > point in time. As long as it makes testable predictions and can > encompass doable methods of observation and measurement. Is your stuff > well grounded empirically? Bob Kolker === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity CMB > http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation > That's the preferred frame of reference There is nothing intrinsically better about one frame over another... It's all relative. One frame may be preferred over another depending on the application. === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=IwEXfQoAAADCjdz2TIptLhugu5MWo47W Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) CMB >http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation > That's the preferred frame of reference There is nothing intrinsically better about one frame over another... > It's all relative. One frame may be preferred over another depending > on the application. Then, try to find another frame of reference where CMB would vanish. === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=IwEXfQoAAADCjdz2TIptLhugu5MWo47W Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > CMB >http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation > That's the preferred frame of reference There is nothing intrinsically better about one frame over another... > It's all relative. One frame may be preferred over another depending > on the application. Then, try to find another frame of reference where CMB > would vanish. IOW, try to find a frame of reference where CMB's anisotropy would look as isotropy. === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=vma-PgoAAABrctSmMdefNKZ-c5S8buvP CMBhttp://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation > That's the preferred frame of reference No it isn't. Good try, though :-) === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=rIfu6QoAAAD5nXG3h9QEE0J3dZn1U45R Gecko/20080518 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > On May 19, 8:04 pm, Peter Webb > There is a more fundamental theory than special relativity. One > consequence is that Einstein's theory could be false and our theory > could be true. However, if Einstein's theory is true, then our theory > is also true. Our fundamental axioms are: 1. Newton's First law of motion. > 2. There is a simple definition of clock time for each point in an > inertial frame of reference. (i) This time is mathematically well- > defined. (ii) Time is defined the same way in all inertial frames of > reference. http://www.everythingimportant.org/relativity/special.pdf Shubee These axioms are fine, from a mathematical point of view (ie they are not > self-contradictory). Indeed, they are a subset of Newton's laws of motion. Unfortunately, they do not correspond with observation. There is no > universal time. My fundamental axiom 2 (ii) states: ñTime is defined the same way in > all inertial frames of reference.î I think youÍve stretched the phrase, ñthe same wayî to presuppose that > I believe in the existence of a cosmic everywhere present ñnow. You > are mistaken. ********************** > Well, what does your axiom mean exactly, if it is not implying the > existence of universal time. The exact meaning is represented perfectly in the simplest spacetime > imaginable. That explanation is contained in the section titled, My > First Toy Universe.http://www.everythingimportant.org/relativity/special.pdf Does your theory predict the same time dilation > as predicted by SR? Yes. If so, how does it differ from SR? My theory removes the unnecessary restrictions in Einstein's theory > and thus allows for motion faster than light. Why not show us something more important? Derive E^2 = [mc^2]^2 + [pc]^2, or its' analog, from your theory. Identically constructed clocks measure time the same way in all > inertial frames of reference. Does that concept trouble you? ********************** > Yes, because it appears to be meaningless. What is it supposed to mean? That > physics behaves the same in all inertial frames? I'm saying that all the laws of physics may be divided into two > categories. There are physical laws that are true in all inertial > frames of reference (such as my definition of time) and there may be > other physical laws that depend on an absolute frame of reference. Shubeehttp://www.everythingimportant.org/relativity/special.pdf If so, this axiom was > used Einstein, so its hard to see how your theory could differ from SR > without some other axioms which conflict with SR being used. Is it the > same as SR? If not, what additional axioms that you use cause the > difference in results? Shubeehttp://www.everythingimportant.org/relativity/special.pdf === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Why not show us something more important? Derive E^2 = [mc^2]^2 + > [pc]^2, or its' analog, from your theory. WouldnÍt it be more fun to solve longstanding problems that were never solved before? You really do need to get a life. Shubee http://www.everythingimportant.org/relativity/special.pdf === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=vma-PgoAAABrctSmMdefNKZ-c5S8buvP Why not show us something more important? Derive E^2 = [mc^2]^2 + > [pc]^2, or its' analog, from your theory. WouldnÍt it be more fun to solve longstanding problems that were never > solved before? You really do need to get a life. ...meaning that you have no clue how to do it, Bert... === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=rIfu6QoAAAD5nXG3h9QEE0J3dZn1U45R Gecko/20080518 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Why not show us something more important? Derive E^2 = [mc^2]^2 + > [pc]^2, or its' analog, from your theory. WouldnÍt it be more fun to solve longstanding problems that were never > solved before? You really do need to get a life. There is much more to special relativity than the Lorentz transformations - which you only obtained after many special case assumptions. If you cannot derive the most basic yet important features of special relativity, why should we care about what you have to say on the matter? If you want people to care about your vastly more complicated and physically un-intuitive version of relativity, there had better be a corresponding tradeoff. Shubeehttp://www.everythingimportant.org/relativity/special.pdf === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > Why not show us something more important? Derive E^2 = [mc^2]^2 + > [pc]^2, or its' analog, from your theory. WouldnÍt it be more fun to solve longstanding problems that were never > solved before? You really do need to get a life. There is much more to special relativity than the Lorentz > transformations - which you only obtained after many special case > assumptions. If you cannot derive the most basic yet important features of special > relativity, why should we care about what you have to say on the > matter? If you want people to care about your vastly more complicated > and physically un-intuitive version of relativity, there had better be > a corresponding tradeoff. The most absolutely beautiful and general way to derive the equations of relativistic mass and momentum conservation is with pretensors, discovered by me while in graduate school. I certainly donÍt feel a great need to publish my discovery because I believe I can apply it to yet unsolved problems. If pretensors were only good to derive relativistic mechanics, then they wouldn't be very important. IÍm following the advice given me by Wolfgang Rindler. ñNever publish anything that can be developed further because if you donÍt fully develop and generalize your own work, someone else surely will.î Shubee > Shubee http://www.everythingimportant.org/relativity/special.pdf === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity > Why not show us something more important? Derive E^2 = > [mc^2]^2 + [pc]^2, or its' analog, from your theory. > WouldnÍt it be more fun to solve longstanding problems that > were never solved before? You really do need to get a life. > There is _much more_ to special relativity than the Lorentz > transformations - which you only obtained after many special case > assumptions. > If you cannot derive the most basic yet important features of > special relativity, why should we care about what you have to say > on the matter? If you want people to care about your vastly more > complicated and physically un-intuitive version of relativity, > there had better be a corresponding tradeoff. The most absolutely beautiful and general way to derive the > equations of relativistic mass and momentum conservation is with > pretensors, discovered by me while in graduate school. I certainly > donÍt feel a great need to publish my discovery because I believe > I can apply it to yet unsolved problems. If pretensors were only > good to derive relativistic mechanics, then they wouldn't be very > important. IÍm following the advice given me by Wolfgang Rindler. > ñNever publish anything that can be developed further because if > you donÍt fully develop and generalize your own work, someone else > surely will.î In your case, Shooby, I wouldn't worry about anybody jumping in and fully developing your work. No need to thank me for those kind words. So, how goes your fight against the physicists? Was the strategy I suggested helpful? Do you know what they call a person who constantly makes false claims? A pretensor. :) (Know any?) === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=lBRURwoAAAB_-Q_b04pGziaymfr5yRFx Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > Why not show us something more important? Derive E^2 = > [mc^2]^2 + [pc]^2, or its' analog, from your theory. > WouldnÍt it be more fun to solve longstanding problems that > were never solved before? You really do need to get a life. > There is much more to special relativity than the Lorentz > transformations - which you only obtained after many special case > assumptions. > If you cannot derive the most basic yet important features of > special relativity, why should we care about what you have to say > on the matter? If you want people to care about your vastly more > complicated and physically un-intuitive version of relativity, > there had better be a corresponding tradeoff. The most absolutely beautiful and general way to derive the > equations of relativistic mass and momentum conservation is with > pretensors, discovered by me while in graduate school. I certainly > donÍt feel a great need to publish my discovery because I believe > I can apply it to yet unsolved problems. If pretensors were only > good to derive relativistic mechanics, then they wouldn't be very > important. IÍm following the advice given me by Wolfgang Rindler. > ñNever publish anything that can be developed further because if > you donÍt fully develop and generalize your own work, someone else > surely will.î So, how goes your fight against the physicists? Just try to argue against http://www.everythingimportant.org/relativity/special.pdf mathematically and perhaps a few mathematicians will chime in and defend the mathematics that you fail to understand. Shubee === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=vma-PgoAAABrctSmMdefNKZ-c5S8buvP > I'm saying that all the laws of physics may be divided into two > categories. There are physical laws that are true in all inertial > frames of reference (such as my definition of time) and there may be > other physical laws that depend on an absolute frame of reference. > This one is precious, Bert === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity <%M4Yj.170028$yE1.118703@attbi_s21> posting-account=G-TjQAkAAADYg6rno3bWQPnIwKFBrf1t 1.1.4322; .NET CLR 2.0.50727; .NET CLR 3.0.04506.30),gzip(gfe),gzip(gfe) > æ æShubee--You do remember that there has never been a prediction > æ æof GTR or SR that was contradicted by an observation. Of courst there has been. All classical theories make predictions that are false; namely the prediction of the absence of coherent superpositions. In a classical theory, states may only combine incoherently. These combinations are in accord with the Bell inequality (among other things). This inequality is known to be violated. Hence: an observation that contradicts prediction. === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=rIfu6QoAAAD5nXG3h9QEE0J3dZn1U45R Gecko/2008051206 Firefox/3.0,gzip(gfe),gzip(gfe) æ æShubee--You do remember that there has never been a prediction > æ æof GTR or SR that was contradicted by an observation. Of courst there has been. All classical theories make predictions that > are false; namely the prediction of the absence of coherent > superpositions. In a classical theory, states may only combine > incoherently. These combinations are in accord with the Bell > inequality (among other things). This inequality is known to be > violated. Hence: an observation that contradicts prediction. Bell's inequality isn't an observation - it is a condition derived from quantum mechanics. Naturally if you derive something from one theory and apply it to a different theory it will be wrong - which surprises nobody. The point is that neither special or general relativity have been contradicted by observation in a repeatable manner. Special relativity is especially safe since it forms the foundation of quantum field theory as well as being the local limit of general relativity. === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity <%M4Yj.170028$yE1.118703@attbi_s21 The point is that neither special or general relativity have been > contradicted by observation in a repeatable manner. Special relativity > is especially safe since it forms the foundation of quantum field theory > as well as being the local limit of general relativity. Well that is your _belief_ -- Center for CANONICAL |SCIENCE) http://canonicalscience.org === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=rIfu6QoAAAD5nXG3h9QEE0J3dZn1U45R Gecko/2008051206 Firefox/3.0,gzip(gfe),gzip(gfe) The point is that neither special or general relativity have been > contradicted by observation in a repeatable manner. Special relativity > is especially safe since it forms the foundation of quantum field theory > as well as being the local limit of general relativity. > Well that is your belief If I didn't believe it I wouldn't have just said it. -- > Center for CANONICAL |SCIENCE) > http://canonicalscience.org === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity <%M4Yj.170028$yE1.118703@attbi_s21> The point is that neither special or general relativity have been > contradicted by observation in a repeatable manner. Special > relativity is especially safe since it forms the foundation of > quantum field theory as well as being the local limit of general > relativity. > Well that is your _belief_ If I didn't believe it I wouldn't have just said it. If you had said I believe that neither special or general relativity have been contradicted by observation in a repeatable manner. I would not object to :-) -- Center for CANONICAL |SCIENCE) http://canonicalscience.org === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=rIfu6QoAAAD5nXG3h9QEE0J3dZn1U45R Gecko/2008051206 Firefox/3.0,gzip(gfe),gzip(gfe) > The point is that neither special or general relativity have been > contradicted by observation in a repeatable manner. Special > relativity is especially safe since it forms the foundation of > quantum field theory as well as being the local limit of general > relativity. > Well that is your belief If I didn't believe it I wouldn't have just said it. If you had said I believe that neither special or general relativity > have been contradicted by observation in a repeatable manner. I would not object to :-) > Do you disagree that quantum field theory is explicitly Lorentz invariant? Do you disagree that SR is the local limit of GR? -- > Center for CANONICAL |SCIENCE) > http://canonicalscience.org === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity <%M4Yj.170028$yE1.118703@attbi_s21> If you had said I believe that neither special or general relativity > have been contradicted by observation in a repeatable manner. > I would not object to :-) > Do you disagree that quantum field theory is explicitly Lorentz > invariant? Do you disagree that SR is the local limit of GR? > Do you disagree that your name is Eric Gisse? Do you disagree you are not a physicist? :-) -- Center for CANONICAL |SCIENCE) http://canonicalscience.org === === Subject: : Re: There Is A More Fundamental Theory Than Special Relativity posting-account=vma-PgoAAABrctSmMdefNKZ-c5S8buvP On May 24, 6:42 am, Juano R. Gonz.87lez-.8dlvarez http://www.helinium.nl/trolltech.gif === === Subject: : Inequality with max I want to understand posting-account=U_S7ZQoAAABl0eAVoTKiBqWajBkK438l 1.1.4322; .NET CLR 2.0.50727; InfoPath.2; .NET CLR 3.0.04506.590; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) I was reading a paper and hit an inequality I have never seen before: (a + b) / (c + d) <= max (a/c, b/d) In the paper I have the additional constraints that c > 0, d > 0, a >= 0, b >= 0. I am interested in where this comes from and other examples. If there are books or other resources that contain stuff similar to this I would like to know. I have no idea how I might search for something like this online for example. I haven't sat down yet for an extended period to try and prove this. I don't really have an idea of how to try and tackle it either. Mathematica can find counter examples with negative variables but not with the additional constraints. Neill. === === Subject: : Re: Inequality with max I want to understand I was reading a paper and hit an inequality I have never seen before: (a + b) / (c + d) <= max (a/c, b/d) In the paper I have the additional constraints that c > 0, d > 0, a >= > 0, b >= 0. > I am interested in where this comes from and other examples. If there > are > books or other resources that contain stuff similar to this I would > like to know. > I have no idea how I might search for something like this online for > example. > I haven't sat down yet for an extended period to try and prove this. > I don't really have an idea of how to try and tackle it either. > Mathematica can find counter examples with negative variables but not > with the additional constraints. Neill. Another way to prove it is to write (a + b) / (c + d) as a convex combination of a/c and b/d ... Here is where you need to know certain things are nonnegative. -- G. A. Edgar http://www.math.ohio-state.edu/~edgar/ === === Subject: : Re: Inequality with max I want to understand <250520080450125841%edgar@math.ohio-state.edu.invalid> posting-account=U_S7ZQoAAABl0eAVoTKiBqWajBkK438l 1.1.4322; .NET CLR 2.0.50727; InfoPath.2; .NET CLR 3.0.04506.590; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022),gzip(gfe),gzip(gfe) whatÍs going on now. A general question for anyone still reading. When I meet something like this inequality that I don't understand in a paper should I try to prove it myself or try to find some reference material that might have it described? A few times I have spent many days on single lines like this trying to understand where what the author did came from. For example in Knuth volume 2 3rd edition 19th printing I was looking at the answer to question 27 in section 4.6.3. The first line makes an assumption that a particular function (c(r)) is monotonically increasing. ThatÍs not proved or even stated in the text. I spent quite a bit of time trying to understand why this was true. It also makes an assumption that another sequence is monotonically increasing. I asked one mathematician about this and he said he thought it was obvious. Not to me though! For those interested c(r) is built on a sequence of values from a function l(n) with the following properties: l(1) = 0, l(n+1) <= l(n) + 1, l(n) >= 0, n >= 1. c(r) is defined as the first n with l(n) = r. So l(c(r)) = r. So with this I was able to convince myself (prove even) that c(r) < c(r +1) and the same property for the other sequence. It became obvious to me then by thinking of l(n) as walking up stairs were you can only go up one stair at a time but you can slip down multiple stairs. So I started thinking that there are likely books that contain things like this and maybe I should be reading them rather than trying to fill in the gaps myself. This particular case payed off as there are two problems in the answer to this question the Knuth says he will write checks for (the brackets are wrong and the text in the question is now out of date with the tables in the book). Neill. === === Subject: : Re: Inequality with max I want to understand I was reading a paper and hit an inequality I have never seen before: (a + b) / (c + d) <= max (a/c, b/d) In the paper I have the additional constraints that c > 0, d > 0, a >= > 0, b >= 0. > I am interested in where this comes from and other examples. If there > are > books or other resources that contain stuff similar to this I would > like to know. > I have no idea how I might search for something like this online for > example. > I haven't sat down yet for an extended period to try and prove this. > I don't really have an idea of how to try and tackle it either. > Mathematica can find counter examples with negative variables but not > with the additional constraints. Since we can multiply the numerators or denominators by a positive constant without changing the inequality, we may assume wlog that a+b = c+d = 1. Thus the inequality says 1 <= max(a/c, (1-a)/(1-c)) for 0 < a < 1, 0 < c < 1. Let the right side be R. There are three cases: 1) for a = c we have R = 1; 2) for a > c we have R >= a/c > 1; 3) for a < c we have 1-a > 1-c, and R >= (1-a)/(1-c) > 1. So the inequality is true. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === === Subject: : Re: Inequality with max I want to understand I was reading a paper and hit an inequality I have never seen before: (a + b) / (c + d) <= max (a/c, b/d) In the paper I have the additional constraints that c > 0, d > 0, a >= > 0, b >= 0. Oh let's see. Some old fashion proportions come to mind. What were they? . . a/c = b/d ==> a/c = (a + b)/(c + d) = b/d Let's see. . . a(c + d) = c(a + b) . . b(c + d) = d(a + b) Ok, it's correct. Hm. . . a/c <= b/d ==> a/c <= (a + b)/(c + d) <= b/d Gee whiz, I'm beginning to understand what the inequality is about. Interpolation of fractions. Interesting. > I am interested in where this comes from and other examples. If there > are books or other resources that contain stuff similar to this I would > like to know. How's what I've conjured up? Execise. For all a,b,c,d > 0, . . min( a/c, b/d) <= (a + b) / (c + d) <= max (a/c, b/d) For example, . . min( n,m) <= (n + m)/2 <= max( n,m ) > I have no idea how I might search for something like this online for > example. Interpolation of fractions? > I haven't sat down yet for an extended period to try and prove this. > I don't really have an idea of how to try and tackle it either. Use the basic inequalities for addition, multiplication and division. . . r <= s ==> a + r <= a + s . . 0 <= a, r <= s ==> ar <= as . . 0 < a ==> 0 < 1/a > Mathematica can find counter examples with negative variables but not > with the additional constraints. > Using computers for math is the faster way to dupe yourself into thinking you're learning some math. > Neill. > === === Subject: : True random number generator module for Python posting-account=2ubq6QkAAAAiC35lIjVuIdMNqgA0rCki Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) truerandom.py is a Python module that generates true random numbers obtained from www.random.org. Use with the form: mylist=truerandom.getnum(min,max,amount) mylist will be a list containing the true random numbers. If for some reason the numbers cannot be generated, the list will contain -1. You can download it here: === === Subject: : Everyone must know that the important information posting-account=OYA7QwoAAADCTrcqfGDBau5ErGuHYCfE CLR 2.0.50727),gzip(gfe),gzip(gfe) Everyone must know that the important information http://www.popotj.cn/ === === Subject: : Re: Everyone must know that the important information posting-account=oDMc0AoAAADvG78Zlak-Ht17zhW-b40r Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > Everyone must know that the important information > http://www.popotj.cn/ It's a good website includeing a great deal of useful information! === === Subject: : Re: yeah sure !!! posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > Does anything in any of those papers define fractional > iteration for arbitrary functions? > (Meaning function in the modern sense, including > f(x) = sqrt(abs) x if x is rational, sin(x) if x is irrational. > Is there a definition of the result of applying that function > f pi times?) No, they don't. The authors of those papers focus only on the case where f is an exponential function. The problem is that if we have a solution to Abel's functional equation, a function F such that F^-1(f(x)) = F^-1(x) + 1 then the solution to Ullrich's question would simply be F(F^-1(x)+pi), and we'd be done. (Notice that tommy1729's equation is simply Abel's.) The problem, of course, is that finding such an F is in general a nontrivial task. === === Subject: : Re: yeah sure !!! Am 25.05.2008 10:04 schrieb lwalke3@lausd.net: > Does anything in any of those papers define fractional > iteration for arbitrary functions? > (Meaning function in the modern sense, including > f(x) = sqrt(abs) x if x is rational, sin(x) if x is irrational. > Is there a definition of the result of applying that function > f pi times?) No, they don't. The authors of those papers focus only > on the case where f is an exponential function. Well, my tractate is not really a paper, but I've given at least some other examples of functions and explained the process of finding of the powerseries for a fractional iterate I've linked to this texts several times here - but see always the same questions. Is the text in someway disgusting? If so - please let me know (and possibly give some help) so that I either can improve the Gottfried Helms for the basics of the method http://go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf === === Subject: : Re: yeah sure !!! Gottfried Helms a .8ecrit : > Am 25.05.2008 10:04 schrieb lwalke3@lausd.net: > Does anything in any of those papers define fractional > iteration for arbitrary functions? > (Meaning function in the modern sense, including > f(x) = sqrt(abs) x if x is rational, sin(x) if x is irrational. > Is there a definition of the result of applying that function > f pi times?) > No, they don't. The authors of those papers focus only > on the case where f is an exponential function. Well, my tractate is not really a paper, but I've > given at least some other examples of functions and > explained the process of finding of the powerseries > for a fractional iterate > I've linked to this texts several times here - but > see always the same questions. Is the text in someway > disgusting? If so - please let me know (and possibly > give some help) so that I either can improve the Gottfried Helms for the basics of the method > http://go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf At first sight, it looks okay, but there is here a serious lack of discussion of motivation : at least, I would explain that one intend to extend f^{on} =fofo...of (n times) in such a way that (for instance) f^{ox}of^{oy)=f^o{(x+y)}, plus some conditions on f and f^{ox} (like analytic in some neighbourhood of 0, say). And an important part of your paper(is it here ?I didn't sse it) would then be to *prove* that the constructions you give (matrix approach, etc.) satisfy indeed the conditions. Else, you have only a plausible definition of fractional iteration, but (especially if thereiis many solutions) no real way to go farther... === === Subject: : Re: yeah sure !!! Am 25.05.2008 12:59 schrieb Denis Feldmann: > Gottfried Helms a .8ecrit : > Am 25.05.2008 10:04 schrieb lwalke3@lausd.net: > Does anything in any of those papers define fractional > iteration for arbitrary functions? > (Meaning function in the modern sense, including > f(x) = sqrt(abs) x if x is rational, sin(x) if x is irrational. > Is there a definition of the result of applying that function > f pi times?) > No, they don't. The authors of those papers focus only > on the case where f is an exponential function. > Well, my tractate is not really a paper, but I've > given at least some other examples of functions and > explained the process of finding of the powerseries > for a fractional iterate > I've linked to this texts several times here - but > see always the same questions. Is the text in someway > disgusting? If so - please let me know (and possibly > give some help) so that I either can improve the > Gottfried Helms > for the basics of the method > http://go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf At first sight, it looks okay, but there is here a serious lack of > discussion of motivation : at least, I would explain that one intend to > extend f^{on} =fofo...of (n times) in such a way that (for instance) > f^{ox}of^{oy)=f^o{(x+y)}, plus some conditions on f and f^{ox} (like > analytic in some neighbourhood of 0, say). And an important part of your > paper(is it here ?I didn't sse it) would then be to *prove* that the > constructions you give (matrix approach, etc.) satisfy indeed the > conditions. Else, you have only a plausible definition of fractional > iteration, but (especially if thereiis many solutions) no real way to go > farther... Denis - pain with my teeth since yesterday, so I don't want to come back to this tomorrow. Any substantiated feedback helps me to make steps from my primarily heuristic situation to achieve a better communicable level. Sorry, this msg is so short - but it is too diffcult to concentrate currently... Gottfried === === Subject: : Here you can see the most lovely photos,contain the motion I have. Love animals,fruit,pepole,homeland.... posting-account=7Cwr5woAAAAiM0R25b0xkpHQJwLOZuOm CLR 2.0.50727),gzip(gfe),gzip(gfe) Here you can see the most lovely photos,contain the motion I have. Love animals,fruit,pepole,homeland.... http://www.popotj.cn/ === === Subject: : set-squares posting-account=fl4D2woAAAC4QBFmZeykoadHa2UXfAKY Gecko/20060731 Ubuntu/dapper-security Firefox/1.5.0.5,gzip(gfe),gzip(gfe) Hi all, what exactly is the purpose of set-squares ? AFAIK they are used for drawing parallel lines and for making angles. is there any other purpose for it ...??? === === Subject: : data entry home posting-account=xzg2_AoAAAD-pPKMjpzw8rRD3J-wBQ5t Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; InfoPath.2),gzip(gfe),gzip(gfe) make money online home based jobs entry and email jobs easy earn home based online world first company list and more income daily and no risk join today home easy make money online jobs list top 3 company home jobs Learn how to create successful Internet campaigns. 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If you have any quary related Data entry Work or Data Processing service, we have solutions for that. vist www.max7cab.com === === Subject: : Re: the tetration difference-differential equation. posting-account=euF15goAAACbw3KIqEWxZHCIPUc2KPmU .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506),gzip(gfe),gzip(gfe) > let T(0) = e > for all positive real numbers x : > T(x - 1) = ln( T(x) ) > taking derivates on both sides : Interesting idea. I've actually thought about this before. > ( note that we might need to add a +C since we took a derivate ) No, we don't. We only need a constant when we integrate , not when we differentiate . > T ' (x - 1) = æT ' (x) / T(x) > that is the tetration difference-differential equation mentioned in the title. Yes, I obtained this equation too. But I was stumped as to what to do next. > since we avoided poles by the restrictions on x , we can get series expansions solutions for T(x). Hmmm. Let's see that part again: > when i was very young i came up with this : > tetration(x) = T(x) > exp(A) - A = 0 > it is obvious that ln(ln(ln(... converges to A since > exp(A) - A = ln(A) - A or exp(A) = ln(A) > ( fixed point should nowadays be the correct term i guess ) > because of the above : T(-oo) = A. > and -oo is the only value where T(z) gives A. > ( unless T(z) = A for all z , but then we are dealing with the wrong tetration because we cannot compute the semi-exponential of lets say 3 .. this resembles louivilles theorem (where T(-oo + z) is considered ) but T(z) is not entire as i will say below ) But is it possible to have T(x+ir) = A for all real x (with a fixed, nonzero real r)? Then we may be still able to compute the semi-exponential of 3 since T takes on values other than A off the line z = x+ir, but now we may longer claim that the function 1/(T(z)-A) is entire. Is that possible? I admit that it's been a while since I've had complex analysis so I may be wrong. Still, this is a promising idea... === === Subject: : Re: the tetration difference-differential equation. posting-account=06BQLAoAAADoC7Y4z9FWcUwGvMa7xMG9 7.4),gzip(gfe),gzip(gfe) let T(0) = e > for all positive real numbers x : > T(x - 1) = ln( T(x) ) > taking derivates on both sides : Interesting idea. I've actually thought about this before. ( note that we might need to add a +C since we took a derivate ) No, we don't. We only need a constant when we > integrate , not when we differentiate . T ' (x - 1) = æT ' (x) / T(x) > that is the tetration difference-differential equation mentioned in the title. Yes, I obtained this equation too. But I was stumped as > to what to do next. since we avoided poles by the restrictions on x , we can get series expansions solutions for T(x). Hmmm. Let's see that part again: when i was very young i came up with this : > tetration(x) = T(x) > exp(A) - A = 0 > it is obvious that ln(ln(ln(... converges to A since > exp(A) - A = ln(A) - A or exp(A) = ln(A) > ( fixed point should nowadays be the correct term i guess ) > because of the above : T(-oo) = A. > and -oo is the only value where T(z) gives A. > ( unless T(z) = A for all z , but then we are dealing with the wrong tetration because we cannot compute the semi-exponential of lets say 3 .. this resembles louivilles theorem (where T(-oo + z) is considered ) but T(z) is not entire as i will say below ) But is it possible to have T(x+ir) = A for all real x (with a fixed, > nonzero real r)? Then we may be still able to compute the > semi-exponential of 3 since T takes on values other than A > off the line z = x+ir, but now we may longer claim that the > function 1/(T(z)-A) is entire. Is that possible? I admit that it's been a while since I've had > complex analysis so I may be wrong. Still, this is a promising idea... Bonjour, if fractionnal iterate exp^[a](L) has got a meaning then T(u) =exp^[u](L) , u in [1,infinity[ is a solution to T(x - 1) = ln( T(x) ) Alain === === Subject: : acne posting-account=xzg2_AoAAAD-pPKMjpzw8rRD3J-wBQ5t Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; InfoPath.2),gzip(gfe),gzip(gfe) Clear Your Acne - Naturally! Most people who suffer from acne go out and spend good money on common over-the-counter acne treatment products. The truth is, most of these products are full of chemicals that can actually slow down the healing of acne, and irritate your skin, causing farther breakouts. When an acne product has 20 ingredients listed on it, it can be hard to know if one of those ingredients is actually stopping you from having success. It can be even harder when you can't pronounce the ingredient, much less know what it is and how it is going to affect your skin. Most mainstream acne treatments are not only much more expensive than common household items, but they also don't always work as well either. With that in mind, I will list some basic household items that can work wonders on clearing up your skin. http://www.max7cab.com/acne/ === === Subject: : Re: hahaha semi exponential , tetration <26277512.1211492765497.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx (KHTML, like Gecko) Safari/412,gzip(gfe),gzip(gfe) >( and lied ) > ................................................ > So a proponent oftetrationmay be considered > a crank (or an eccentric) by someone like > Tonio, for example. But if someone thinks > that tommy1729 is a crank only because he > usestetration,.............................. >http://en.wikipedia.org/wiki/Tetration > But tommy1729 is just as bad here......... > But usingtetrationis not this same as > denying Cantor's proof or trying to prove > that ZFC is inconsistent, so one shouldn't > puttetrationon the same level. One is > trying to find a way to extend to the > standard reals, i.e., good ol' R. > The reason that some people, including Tonio, > may associatetetrationwith crankdom is that > many people who work withtetrationare not > adherents of ZFC............... > *************************************************** > 1) No, I don't consider Tommy a crank only because > he uses >tetration. In fact, and as somebody else already > pointed out, tommy > himself didn't know abouttetration, either. LIAR !! i consideredtetrationmore than 15 years ago , what the hell do you know about that !? He was > just quoting from > wikipedia or something like that, cheater , when i say stuff different from wikipedia or a book , people say i got it wrong. when i say stuff equivalent to wikipedia or some book, they say i stole it. thats dirty cheating ! but unlike others, > he just found it > suitable to laugh at somebody (I think it was David) > and even barked > never heard of that hmm. some mathematician you are > ! hahahaha yes , and with good reason , david ullrich does not believe in fractional iterations > Not really - they give things analogous to fractional iteration, > not actual fractional iteration. But never mind that. How do we define fractional iteration for > arbitrary functions? > What exactly do you mean by actual fractional iteration? You mean actually carrying through the operation of half an addition? Actually one should mind that since how can we talk about it even for NON-arbitrary functions if we don't even know what it means? === === Subject: : Re: hahaha semi exponential , tetration mike3 a .8ecrit : > ( and lied ) > ................................................ > So a proponent oftetrationmay be considered > a crank (or an eccentric) by someone like > Tonio, for example. But if someone thinks > that tommy1729 is a crank only because he > usestetration,.............................. > http://en.wikipedia.org/wiki/Tetration > But tommy1729 is just as bad here......... > But usingtetrationis _not_ this same as > denying Cantor's proof or trying to prove > that ZFC is inconsistent, so one shouldn't > puttetrationon the same level. One is > trying to find a way to extend to the > _standard_ reals, i.e., good ol' R. > The reason that some people, including Tonio, > may associatetetrationwith crankdom is that > many people who work withtetrationare not > adherents of ZFC............... > *************************************************** > 1) No, I don't consider Tommy a crank only because > he uses > tetration. In fact, and as somebody else already > pointed out, tommy > himself didn't know abouttetration, either. > LIAR !! > i consideredtetrationmore than 15 years ago , what the hell do you know about that !? > He was > just quoting from > wikipedia or something like that, > cheater , when i say stuff different from wikipedia or a book , people say i got it wrong. > when i say stuff equivalent to wikipedia or some book, they say i stole it. > thats dirty cheating ! > but unlike others, > he just found it > suitable to laugh at somebody (I think it was David) > and even barked > never heard of that hmm. some mathematician you are > ! hahahaha > yes , and with good reason , david ullrich does not believe in fractional iterations > Not really - they give things analogous to fractional iteration, > not actual fractional iteration. > But never mind that. How do we define fractional iteration for > _arbitrary_ functions? > What exactly do you mean by actual fractional > iteration? No , he asked first :-) You mean actually carrying through > the operation of half an addition? No, he ask what this is supposed to mean for general enough functions. Look ; iterate 3 times f means (by *definition* of iteration) calculate f(f(f(x))) (or more rigorously fofof), and it is reasonable to say that iterate 1 times means f, iterate 0 times means identity, and even iterate -1 times means take the inverse bijection of f (at least locally, if f is locally one-to-one). But what is supposed to mean iterate 1/2 times, and even more iterate i times ? There is obviously not enough properties of iteration (in general) to allow for such a broad generalisation ... Actually one should mind that since how can > we talk about it even for NON-arbitrary functions > if we don't even know what it means? > Exactly > === === Subject: : Re: hahaha semi exponential , tetration <26277512.1211492765497.JavaMail.jakarta@nitrogen.mathforum.org> posting-account=Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) On May 25, 1:42 am, Denis Feldmann ( and lied ) > ................................................ > So a proponent oftetrationmay be considered > a crank (or an eccentric) by someone like > Tonio, for example. But if someone thinks > that tommy1729 is a crank only because he > usestetration,.............................. >http://en.wikipedia.org/wiki/Tetration > But tommy1729 is just as bad here......... > But usingtetrationis not this same as > denying Cantor's proof or trying to prove > that ZFC is inconsistent, so one shouldn't > puttetrationon the same level. One is > trying to find a way to extend to the > standard reals, i.e., good ol' R. > The reason that some people, including Tonio, > may associatetetrationwith crankdom is that > many people who work withtetrationare not > adherents of ZFC............... > *************************************************** > 1) No, I don't consider Tommy a crank only because > he uses > tetration. In fact, and as somebody else already > pointed out, tommy > himself didn't know abouttetration, either. > LIAR !! > i consideredtetrationmore than 15 years ago , what the hell do you know about that !? > He was > just quoting from > wikipedia or something like that, > cheater , when i say stuff different from wikipedia or a book , people say i got it wrong. > when i say stuff equivalent to wikipedia or some book, they say i stole it. > thats dirty cheating ! > but unlike others, > he just found it > suitable to laugh at somebody (I think it was David) > and even barked > never heard of that hmm. some mathematician you are > ! hahahaha > yes , and with good reason , david ullrich does not believe in fractional iterations > Not really - they give things analogous to fractional iteration, > not actual fractional iteration. > But never mind that. How do we define fractional iteration for > arbitrary functions? What exactly do you mean by actual fractional > iteration? No , he asked first :-) You mean actually carrying through the operation of half an addition? No, he ask what this is supposed to mean for general enough functions. > Look ; iterate 3 times f means (by *definition* of iteration) calculate > f(f(f(x))) (or more rigorously fofof), and it is reasonable to say that > iterate 1 times means f, iterate 0 times means identity, and even > iterate -1 times means take the inverse bijection of f (at least > locally, if f is locally one-to-one). But what is supposed to mean > iterate 1/2 times, and even more iterate i times ? There is obviously > not enough properties of iteration (in general) to allow for such a > broad generalisation ... ? f#n := f(f(..(f(x))..) --- n compositions (f#n) o (f#m) = f#(n+m) = (f#m) o (f#n) as with exponentation we could use this appropriately and assume (f#(1/2)) o (f#(1/2)) = f for fractions q it seems the meaning of iteration q times has at least a functional property that defines it whether any function obeys this relation or none do or many do will depend on the function considered but certainly there is meaning here -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === === Subject: : Re: hahaha semi exponential , tetration posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > Last post about that subject, whether you > understand > or not. you dont understand it. > Philippe 92 : > Same bad answer as amy666. > seems you dont understand that n times is > indeed > multiplication by n. > a multiplication is an iteration of a sum. for > you to *iterate* > a non integer number of times ? guess you dont know about iterates. Giggle. If you know what it means to iterate > something 1/2 times you're the only one. You > should explain what it means. hahaha !! tetration semi-exponential never heard of that hmm. some mathematician you are ! hahahaha hint : semiexponential is 1/2 iterate of exponential. indeed i might be the only one here who knows about that. thus most are cranks and im not. *********************************************************** Oh, hollie mollie! After a welcome hiatus, when many of us were > convinced you had begun to think (go figure..!) and had decided to > stop posting in this thread because of the unbelievably ammount of > nonsense you've posted....you once again attack common sense, logic > and decency and crank around again! How did he attack logic in that post? As far as > I can tell, tetration is a real mathematical > operation. Although I've never heard the term > semiexponential either.- *********************************************************** How, you ask? Hint: explain me, or ask him to explain you, what does > hint : semiexponential is 1/2 iterate of exponential. mean in any > context, in general, and in mathematics, in particular, and even more > in particular within the context of the problem we've been dealing > with from the beginning of this thread (have you read it?). Beware though! It may be that if you ask him that, he'll write back > something like æsome mathematician you are!, hahahaha!, and other > very robust and intelligent arguments. > Then I'll might ask him. However what I do find telling is his use of this term semiexponential. Googling semiexponential function did not give anything that looked like it had to do with a half iterate of the exponential function. === === Subject: : Re: hahaha semi exponential , tetration posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > ................................................ So a proponent of tetration may be considered > a crank (or an eccentric) by someone like > Tonio, for example. But if someone thinks > that tommy1729 is a crank only because he > uses tetration,.............................. http://en.wikipedia.org/wiki/Tetration But tommy1729 is just as bad here......... But using tetration is not this same as > denying Cantor's proof or trying to prove > that ZFC is inconsistent, so one shouldn't > put tetration on the same level. One is > trying to find a way to extend to the > standard reals, i.e., good ol' R. The reason that some people, including Tonio, > may associate tetration with crankdom is that > many people who work with tetration are not > adherents of ZFC............... *************************************************** 1) No, I don't consider Tommy a crank only because he uses > tetration. In fact, and as somebody else already pointed out, tommy > himself didn't know about tetration, either. He was just quoting from > wikipedia or something like that, but unlike others, he just found it > suitable to laugh at somebody (I think it was David) and even barked > never heard of that hmm. some mathematician you are ! hahahaha > because, in his opinion, if somebody hasn't heard of tetration that > that somebody isn't a mathematician. > There are pleny of other reasons to consider such a nasty character as > a crank (read the lines above, for one), as anyone willing to read the > trash Tommy's been posting the last few months can realize. 2) Beyond ZFC, the most important thing we have to see here after is > context: in the present thread's context, neither Tommy nor anyone > else willing to insist on that have yet explained how can one take a > sum of an item, call it, x times, when x is not a natural number. > There can though be a misunderstanding here: in english, > multiplication usually is orally denoted by means of the word times, > so the product 9*Sqrt(2) is read in english nine times the square > root of two. It could be that some people here is misunderstanding > this with the use of the written expression > 9*sqrt(2) = Sqrt(2) + Sqrt(2)+...+ Sqrt(2) æ( 9 times ) > The latter is just an abridged form or writing the expression Sqrt(2)+ > 9 times in a row. Now, the above is quite simple when one of the factors is, as already > said, an integer. What happens though with, for example, an expression > like Pi*Sqrt(2) -- with Pi = the quotient of any circle's perimeter to > twice its radius -- ? An english speaker would read sucn an expression > as Pi times the square root of 2, but this would NOT mean that you > HAVE to sum Pi with itself a Square-of-two numnber of times > Now the only thing that both Tommy and Michael have been asked about > is: how would YOU define the iteration IN A SUM of such a thing as the > above? How would you iterate IN A SUM (and this is what we've been > talking about all the time since I first posted that hoax) any > number a non-integer number of times? > Tetration does not answer that question, at least not so far, since > nobody has yet answered that question, in spite of the mocking and > HAHAHAHA!! remarks of the prince of charismatic cranks = Tommy. 3) Finally: no, I don't tie together tetration with Crank. I do > tie together Tommy with nasty crank, though. Of course, it says > lots that he has to go to the edges of wikipedia searches to find > something well out of mathematical consensus, to say the least, that > he thinks (figure of speech) can serve him well to win in a debate, > which by the way hasn't been a debate at all, at least from his stand. > Ah, so then his attack on logic in the post I mentioned is because he said 1/2 iterate of exponential which sounds like do the exponential function half a time? But the fractional iteration of functions is something that legitimate mathematical exploration has been done on. Is it because usually one calls it that, instead of saying stuff like half a time, which doesn't seem to make any sense, even if we can define fractional multiplication like 3/5 * 7/9 -- but those definitions move away from the multiplication- as- repeated-addition idea, whereas he would be trying to preserve said idea by talking about adding 3/5 to zero 7/9ths of a time? PS. did you rate my question post with 1 star? Or am I just too paranoid??? (hope the latter) Well, to whoever did it I thought it was a legitimate question to help increase my understanding. === === Subject: : Re: hahaha semi exponential , tetration posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e InfoPath.1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) ................................................ So a proponent of tetration may be considered > a crank (or an eccentric) by someone like > Tonio, for example. But if someone thinks > that tommy1729 is a crank only because he > uses tetration,.............................. http://en.wikipedia.org/wiki/Tetration But tommy1729 is just as bad here......... But using tetration is not this same as > denying Cantor's proof or trying to prove > that ZFC is inconsistent, so one shouldn't > put tetration on the same level. One is > trying to find a way to extend to the > standard reals, i.e., good ol' R. The reason that some people, including Tonio, > may associate tetration with crankdom is that > many people who work with tetration are not > adherents of ZFC............... *************************************************** 1) No, I don't consider Tommy a crank only because he uses > tetration. In fact, and as somebody else already pointed out, tommy > himself didn't know about tetration, either. He was just quoting from > wikipedia or something like that, but unlike others, he just found it > suitable to laugh at somebody (I think it was David) and even barked > never heard of that hmm. some mathematician you are ! hahahaha > because, in his opinion, if somebody hasn't heard of tetration that > that somebody isn't a mathematician. > There are pleny of other reasons to consider such a nasty character as > a crank (read the lines above, for one), as anyone willing to read the > trash Tommy's been posting the last few months can realize. 2) Beyond ZFC, the most important thing we have to see here after is > context: in the present thread's context, neither Tommy nor anyone > else willing to insist on that have yet explained how can one take a > sum of an item, call it, x times, when x is not a natural number. > There can though be a misunderstanding here: in english, > multiplication usually is orally denoted by means of the word times, > so the product 9*Sqrt(2) is read in english nine times the square > root of two. It could be that some people here is misunderstanding > this with the use of the written expression > 9*sqrt(2) = Sqrt(2) + Sqrt(2)+...+ Sqrt(2) æ( 9 times ) > The latter is just an abridged form or writing the expression Sqrt(2)+ > 9 times in a row. Now, the above is quite simple when one of the factors is, as already > said, an integer. What happens though with, for example, an expression > like Pi*Sqrt(2) -- with Pi = the quotient of any circle's perimeter to > twice its radius -- ? An english speaker would read sucn an expression > as Pi times the square root of 2, but this would NOT mean that you > HAVE to sum Pi with itself a Square-of-two numnber of times > Now the only thing that both Tommy and Michael have been asked about > is: how would YOU define the iteration IN A SUM of such a thing as the > above? How would you iterate IN A SUM (and this is what we've been > talking about all the time since I first posted that hoax) any > number a non-integer number of times? > Tetration does not answer that question, at least not so far, since > nobody has yet answered that question, in spite of the mocking and > HAHAHAHA!! remarks of the prince of charismatic cranks = Tommy. 3) Finally: no, I don't tie together tetration with Crank. I do > tie together Tommy with nasty crank, though. Of course, it says > lots that he has to go to the edges of wikipedia searches to find > something well out of mathematical consensus, to say the least, that > he thinks (figure of speech) can serve him well to win in a debate, > which by the way hasn't been a debate at all, at least from his stand. Ah, so then his attack on logic in the post I mentioned is because > he said 1/2 iterate of exponential which sounds like do the > exponential > function half a time? But the fractional iteration of functions > is something that legitimate mathematical exploration has been > done on. Is it because usually one calls it that, instead of > saying stuff like half a time, which doesn't seem to make any > sense, even if we can define fractional multiplication like > 3/5 * 7/9 -- but those definitions move away from the multiplication- > as- > repeated-addition idea, whereas he would be trying to > preserve said idea by talking about adding 3/5 to zero 7/9ths > of a time? PS. did you rate my question post with 1 star? Or am I just too > paranoid??? (hope the latter) Well, to whoever did it I thought it > was a legitimate question to help increase my understanding.- *************************************************************** I don't usually do stars; it seems a little pointless. Tonio === === Subject: : Re: hahaha semi exponential , tetration posting-account=fwSgtAkAAACFnX70ssKwbvm9_oCZVHrx 2.0.50727; .NET CLR 3.0.04506.648; .NET CLR 3.5.21022; .NET CLR 1.1.4322),gzip(gfe),gzip(gfe) > ................................................ So a proponent of tetration may be considered > a crank (or an eccentric) by someone like > Tonio, for example. But if someone thinks > that tommy1729 is a crank only because he > uses tetration,.............................. http://en.wikipedia.org/wiki/Tetration But tommy1729 is just as bad here......... But using tetration is not this same as > denying Cantor's proof or trying to prove > that ZFC is inconsistent, so one shouldn't > put tetration on the same level. One is > trying to find a way to extend to the > standard reals, i.e., good ol' R. The reason that some people, including Tonio, > may associate tetration with crankdom is that > many people who work with tetration are not > adherents of ZFC............... *************************************************** 1) No, I don't consider Tommy a crank only because he uses > tetration. In fact, and as somebody else already pointed out, tommy > himself didn't know about tetration, either. He was just quoting from > wikipedia or something like that, but unlike others, he just found it > suitable to laugh at somebody (I think it was David) and even barked > never heard of that hmm. some mathematician you are ! hahahaha > because, in his opinion, if somebody hasn't heard of tetration that > that somebody isn't a mathematician. > There are pleny of other reasons to consider such a nasty character as > a crank (read the lines above, for one), as anyone willing to read the > trash Tommy's been posting the last few months can realize. 2) Beyond ZFC, the most important thing we have to see here after is > context: in the present thread's context, neither Tommy nor anyone > else willing to insist on that have yet explained how can one take a > sum of an item, call it, x times, when x is not a natural number. > There can though be a misunderstanding here: in english, > multiplication usually is orally denoted by means of the word times, > so the product 9*Sqrt(2) is read in english nine times the square > root of two. It could be that some people here is misunderstanding > this with the use of the written expression > 9*sqrt(2) = Sqrt(2) + Sqrt(2)+...+ Sqrt(2) æ( 9 times ) > The latter is just an abridged form or writing the expression Sqrt(2)+ > 9 times in a row. Now, the above is quite simple when one of the factors is, as already > said, an integer. What happens though with, for example, an expression > like Pi*Sqrt(2) -- with Pi = the quotient of any circle's perimeter to > twice its radius -- ? An english speaker would read sucn an expression > as Pi times the square root of 2, but this would NOT mean that you > HAVE to sum Pi with itself a Square-of-two numnber of times > Now the only thing that both Tommy and Michael have been asked about > is: how would YOU define the iteration IN A SUM of such a thing as the > above? How would you iterate IN A SUM (and this is what we've been > talking about all the time since I first posted that hoax) any > number a non-integer number of times? > Tetration does not answer that question, at least not so far, since > nobody has yet answered that question, in spite of the mocking and > HAHAHAHA!! remarks of the prince of charismatic cranks = Tommy. 3) Finally: no, I don't tie together tetration with Crank. I do > tie together Tommy with nasty crank, though. Of course, it says > lots that he has to go to the edges of wikipedia searches to find > something well out of mathematical consensus, to say the least, that > he thinks (figure of speech) can serve him well to win in a debate, > which by the way hasn't been a debate at all, at least from his stand. Ah, so then his attack on logic in the post I mentioned is because > he said 1/2 iterate of exponential which sounds like do the > exponential > function half a time? But the fractional iteration of functions > is something that legitimate mathematical exploration has been > done on. Is it because usually one calls it that, instead of > saying stuff like half a time, which doesn't seem to make any > sense, even if we can define fractional multiplication like > 3/5 * 7/9 -- but those definitions move away from the multiplication- > as- > repeated-addition idea, whereas he would be trying to > preserve said idea by talking about adding 3/5 to zero 7/9ths > of a time? PS. did you rate my question post with 1 star? Or am I just too > paranoid??? (hope the latter) Well, to whoever did it I thought it > was a legitimate question to help increase my understanding.- *************************************************************** I don't usually do stars; it seems a little pointless. > Ah. === === Subject: : Acne Information and Acne Treatment posting-account=xzg2_AoAAAD-pPKMjpzw8rRD3J-wBQ5t Mozilla/4.0 (compatible; MSIE 6.0; Windows NT 5.1; SV1) ; InfoPath.2),gzip(gfe),gzip(gfe) The Truth About Acne Skin Care by: John Lenaghan Let's face it, you are engaged in an ongoing battle when you are afflicted with acne. While acne treatments for a mild case of acne are usually successful, moderate acne is a greater problem. Even dermatologists cannot cure a severe case of acne but can only provide an acne treatment regimen. However, you can support your acne treatment efforts with proper acne skin care in order to reduce the recurrence of acne flare-ups and not aggravate existing acne. Knowing What You're Doing One of the first pieces of advice is to make sure that you understand and follow the directions for any acne remedy that you are using. If you are taking a prescription acne medicine, read the patient leaflet that accompanies the medication carefully. Typically for moderate to severe acne a dermatologist may prescribe several acne medicines, each with its own treatment protocol and directions. http://www.max7cab.com/acne/The_Truth_About_Acne_Skin_Care.html === === Subject: : Fully inflated air pillow posting-account=33KaEgkAAAA9tz8WICNABjrkyMKXFbGS Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) It is a fourth order surface where sections parallel x- , y - axes are ellipses of variable eccentricity,and a constant major axis. The foci move from center of square (case of circle) to corners (narrow digon line ellipse) along diagonals of square .The x =y sections are parabolas It looks like a fully inflated square pillow whose surface equation is a^2 z^2 = ( a^2 - x^2) (a^2 - y^2) and the image written in Mathematica is: a = 1; ContourPlot3D[ z^2 == a^2 - (x^2 + y^2) + ( x y /a)^2 , {x, - a, a}, {y, -a, a}, {z, -a, a}] May be known already but hope it is interesting. Narasimham === === Subject: : Re: Fully inflated air pillow posting-account=33KaEgkAAAA9tz8WICNABjrkyMKXFbGS Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Such a shape can be represented by a a fourth order surface where sections parallel x- , y - axes are ellipses of variable eccentricity,and a constant major axis. The foci move from center of square (case of circle) to corners (narrow digon line ellipse) along diagonals of square .The x =y sections are parabolas It looks like a fully inflated square pillow whose surface equation is a^2 z^2 = ( a^2 - x^2) (a^2 - y^2) and the image can be got from Mathematica as: a = 1; ContourPlot3D[ z^2 == a^2 - (x^2 + y^2) + ( x y /a)^2 , {x, - a, a}, {y, -a, a}, {z, -a, a}] May be it is known already, but hope is interesting. Narasimham === === Subject: : Re: Fully inflated air pillow posting-account=33KaEgkAAAA9tz8WICNABjrkyMKXFbGS Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) > It is a fourth order surface where sections parallel x- , y - axes are > ellipses of variable eccentricity,and a constant major axis. The foci > move from center of square (case of circle) to corners (narrow digon > line ellipse) along diagonals of square .The x =y sections are > parabolas It looks like a fully inflated square pillow whose surface > equation is a^2 z^2 = ( a^2 - x^2) (a^2 - y^2) and the image written > in Mathematica is: a = 1; ContourPlot3D[ z^2 == a^2 - (x^2 + y^2) + ( x y /a)^2 , {x, - > a, a}, {y, -a, a}, {z, -a, a}] May be known already but hope it is interesting. Narasimham === === Subject: : Re: A Curious Question > A Curious Question > ~v~~ > Is there anything which is not predicated of not? > ~v~~ > Any object you care to mention is not predicated of not. > Would you care to mention some and explain to us exactly how they're > not predicated of not? 'Not' is not predicated of any object, It's not? ~v~~ === === Subject: : Conway's group Co0 - elements of orders 72, 80, 120? A question related to J.H.Conway's groups Co0 and Co1 ----------------------------------------------------- The group Co0 is the group of automorphisms of the Leech lattice which leave one lattice point invariant. Is is a finite subgroup of the 24D rotation group SO(24). It is commonly denoted by .0 (by its discoverer John Horton Conway) or by Co0. Its centre Z2 consists of the non-rotation and the central inversion. The factor group Co0/Z2 is a simple group, the first of Conway's sporadic simple groups. It is commonly denoted by .1 or by Co1. Co1 contains elements of orders 1 through 16, 18, 20, 21 through 24, 26, 28, 30, 33, 35, 36, 39, 40, 42, 60. (Reference: J.H.Conway et al.: ATLAS of Finite Groups) Some time ago I explored the group Co0 by means of a specially designed exploration and visualisation program (*). I found 24D rotations of all the orders mentioned above and furthermore of orders 46, 52, 56, 66, 70, 78, 84. Cyclic subgroups of Co0 of these orders necessarily contain the central inversion; otherwise they would be mapped onto cyclic subgroups of Co1 of the same orders when descending from Co0 to Co1. One could also expect rotations of orders 72, 80, 120, but I never observed these orders. My question: how to prove or disprove that Co0 contains elements of orders 72, 80, 120. The answer might be found in Nicholas James Patterson's thesis On Conway's group .0 and some subgroups (University of Cambridge, UK, 1973), but I hope that there is more readily accessible literature on the fine structure of Co0. Up to now I have only one speculation: if the Monster group contains a subgroup isomorphic to Co0, then Co0 does not contain elements of order 120. This is because the highest order found in the Monster is 119. (Reference: ATLAS) (*) ---------------------------------------------------------------------------- -------------- http://www.xs4all.nl/~jemebius/Ab4help.htm and http://www.xs4all.nl/~jemebius/AB4-2007.ZIP reachable from my home page at http://www.xs4all.nl/~jemebius/index.html , item downloads === === Subject: : Conway's group Co0 - Orders 72, 80, 120? - URLs corrected A question related to J.H.Conway's groups Co0 and Co1 ----------------------------------------------------- The group Co0 is the group of automorphisms of the Leech lattice which leave one lattice point invariant. Is is a finite subgroup of the 24D rotation group SO(24). It is commonly denoted by .0 (by its discoverer John Horton Conway) or by Co0. Its centre Z2 consists of the non-rotation and the central inversion. The factor group Co0/Z2 is a simple group, the first of Conway's sporadic simple groups. It is commonly denoted by .1 or by Co1. Co1 contains elements of orders 1 through 16, 18, 20, 21 through 24, 26, 28, 30, 33, 35, 36, 39, 40, 42, 60. (Reference: J.H.Conway et al.: ATLAS of Finite Groups) Some time ago I explored the group Co0 by means of a specially designed exploration and visualisation program (*). I found 24D rotations of all the orders mentioned above and furthermore of orders 46, 52, 56, 66, 70, 78, 84. Cyclic subgroups of Co0 of these orders necessarily contain the central inversion; otherwise they would be mapped onto cyclic subgroups of Co1 of the same orders when descending from Co0 to Co1. One could also expect rotations of orders 72, 80, 120, but I never observed these orders. My question: how to prove or disprove that Co0 contains elements of orders 72, 80, 120. The answer might be found in Nicholas James Patterson's thesis On Conway's group .0 and some subgroups (University of Cambridge, UK, 1973), but I hope that there is more readily accessible literature on the fine structure of Co0. Up to now I have only one speculation: if the Monster group contains a subgroup isomorphic to Co0, then Co0 does not contain elements of order 120. This is because the highest order found in the Monster is 119. (Reference: ATLAS) (*) ---------------------------------------------------------------------------- -------------- http://www.xs4all.nl/~jemebius/Abconwayhelp.htm and http://www.xs4all.nl/~jemebius/ABCONWAY.ZIP reachable from my home page at http://www.xs4all.nl/~jemebius/index.html , item downloads === === Subject: : Earth radius posting-account=fl4D2woAAAC4QBFmZeykoadHa2UXfAKY Gecko/20060731 Ubuntu/dapper-security Firefox/1.5.0.5,gzip(gfe),gzip(gfe) Hi all, I recently read in a book that the volume of earth is (4/3) x (22/7) x (6380 x 6380 x 6380) cu km. Does this mean that radius of earth is 6380 km , how the radius of earth is measured ..??? === === Subject: : Re: Earth radius posting-account=EFrT1AoAAABqVlHo1bRMHSNKdfktVxk- SV1),gzip(gfe),gzip(gfe) > Hi all, > I recently read in a book that the volume of earth is > æ (4/3) x (22/7) x (6380 x 6380 x 6380) cu km. Does this mean that radius of earth is 6380 km , how the radius of > earth is measured ..??? dear radius can easily be calculated from V=4/3(3.142*Re^3) as u know the volume another method is Re^2=G*Me/g where re is the radius of the earth G is the gravitational constant Me is the mass of the earth g is the acceleration due to gravity === === Subject: : Re: Earth radius- > Hi all, > I recently read in a book that the volume of earth is > (4/3) x (22/7) x (6380 x 6380 x 6380) cu km. Does this mean that radius of earth is 6380 km , how the radius of > earth is measured ..??? http://en.wikipedia.org/wiki/Earth_radius http://en.wikipedia.org/wiki/Eratosthenes The volume of an ellipsoid: V = 4pi.ABC / 3, where A, B, C are the half-axes. If A = B = C one has a ball. If A = B and C < B one has an oblate spheroid, like the Earth. 22/7 is a common not too crude approximation to pi. Happy studies: Johan E. Mebius === === Subject: : math puzzles posting-account=AABsHgoAAAAbDpvVc5qQ5QBQtWqga9ZN .NET CLR 2.0.50727; Media Center PC 5.0; .NET CLR 3.0.04506; Tablet PC 2.0),gzip(gfe),gzip(gfe) The vanishing Leprechaun This puzzle consists of 15 leprechauns and is rectangular in shape. The puzzle is cut into three pieces as follows: First it is cut in half lengthwise and then the top half is cut into two pieces. The top two pieces are then switched. The left half becomes the right half and the right half becomes the left half. When the switch is made and the leprechauns are counted, one is missing. Where did it go? The puzzle was first produced by W.A. Elliott Co., 212 Adelaide Street West, Toronto,, Ontario, Canada, M5H 1W7 === === Subject: : Re: math puzzles lacher07@sympatico.ca a .8ecrit : > The vanishing Leprechaun This puzzle consists of 15 leprechauns and is rectangular in shape. > The puzzle is cut into three pieces as follows: First it is cut in > half lengthwise and then the top half is cut into two pieces. The top > two pieces are then switched. The left half becomes the right half and > the right half becomes the left half. When the switch is made and the > leprechauns are counted, one is missing. Where did it go? The puzzle was first produced by W.A. Elliott Co., 212 Adelaide Street > West, Toronto,, Ontario, Canada, M5H 1W7 At least, give a link, like : http://www.planetperplex.com/en/img.php?id=26 === === Subject: : Love in Islam posting-account=oBvL6AoAAADRozlt7Mf0pYS_ixDs1x6C Gecko/20080404 Firefox/2.0.0.14,gzip(gfe),gzip(gfe) Love - Woman and her Husband Love - Your Mother But, what kind of a Friend? FlowersLove is one of the greatest blessings Allah has bestowed on humanity. Allah has created human nature in such a way that a person will take pleasure from loving and being loved, from friendship and from intimacy. Being with people who live by the moral values of the Qur'an, enjoying love and friendship with them, brings a believer greater pleasure than many other blessings. Therefore, the Paradise that Allah has promised to His servants whom He loves and with whom He is well pleased is a place of extraordinary beauty in which true love, friendship and intimacy will be experienced with enormous joy for ever. The information Allah provides about the life of Paradise in the Qur'an, always refers to joy, friendship, love, conversation, pleasing words and peace. Everything that might prove an obstacle to love and friendship has been kept far removed from the people of Paradise. In one verse, for example, Allah has revealed that He will strip away any rancour in the hearts of the companions of Paradise (Surat al-A'raf, 43). All vices that might represent an obstacle to love and friendship, such as jealousy, enmity, rivalry, anger, discord and touchiness, will be excluded from Paradise. PS. Christian Arabs call God Allah. Allah is the word in Arabic translation of God, but there is no plural to this unique word (no Gods, but single God) One of the main characteristics of believers, who have been given the glad tidings of Paradise, is that they love all believers who abide by Allah's messengers, prophets and approval, in this world too. This bond of love and friendship between believers is revealed in these terms in the Qur'an: Your friend is only Allah and His Messenger and those who believe: those who perform prayer and give the alms, and bow. (Surat al-Ma'ida, 55) The faithful feel a closeness to all devout believers who strive to earn Allah's approval, and adopt them as their friends and guardians. They take great pleasure from being with them under all conditions. They are bound to all Muslims with great loyalty and devotion. This love is not based on any interest, or lineage or race. Money, rank, culture or material values are of no importance to them. They approve of the people of whom they expect Allah to approve; devout believers on the path of Allah are some of His most beloved servants. For that reason, believers also have great love and consideration for one another. Many verses of the Qur'an refer to the love, attachment, compassion and consideration that believers have for one another. Some of these verses read: Hold fast to the rope of Allah all together, and do not separate. Remember Allah's blessing to you when you were enemies and He joined your hearts together so that you became brothers by His blessing. You were on the very brink of a pit of the Fire and He rescued you from it. In this way Allah makes His Signs clear to you, so that hopefully you will be guided. (Surah al 'Imran, 103) The believers are brothers, so make peace between your brothers and heed Allah so that hopefully you will gain mercy. (Surat al-Hujurat, 10) Allah loves those who fight in His Way in ranks like well-built walls. (Surat as-Saff, 4) In another verse Allah reveals this love among believers thus: Those who were already settled in the abode [Madina], and in faith, before they came, love those who have migrated to them and do not find in their hearts any need for what they have been given and prefer them to themselves even if they themselves are needy. It is the people who are safe-guarded from the avarice of their own selves who are successful. (Surat al-Hashr, 9) Believers regard all the faithful as their brothers. They do not hesitate to make any sacrifice for the good of another believer or to ensure his or her comfort. This conception of love possessed by believers can only be acquired through faith and living by the moral values of the Qur'an. Allah reveals in one verse that He will bestow love on those who believe and do devout works: As for those who believe and do right actions, the All-Merciful will bestow His love on them. (Surah Maryam, 96) Our Prophet (may Allah bless him and grant him peace) has expressed the importance of love and the superior nature of believers who live by true love in one of his hadiths: Allah (subhanahu wa ta`ala) said: 'Those who love one another for My glory, will have minbars of light, and the Prophets and martyrs will wish that they had the same. (Reported by Tirmidhi, 4/24, Bab ma ja'a fi al-hubb fi-Allah) The true source of this love that believers have for one another is their deep love of Allah. Believers whose purpose in the life of this world is to attain the approval, mercy and Paradise of Allah spend their entire lives for Him. As is revealed in the verse Say: 'My prayer and my rites, my living and my dying, are for Allah alone, the Lord of all the worlds,' (Surat al-An'am, 162) they aim at attaining His approval in their every deed, in all their behaviour. The love of believers, who devote all they have to attaining the good pleasure of Allah, is again for Allah alone. The love of Allah of a believer who knows Allah by all His names, who witnesses His might and greatness at every moment, who feels His mercy, love and compassion throughout his life, is incomparably greater than any other love. Believers' love for other believers is similarly powerful and deep because it is based on the love of Allah, and they love in order to attain the approval of Allah. Another reason why this love is strong and permanent is because they know that the friendship with the faithful will last for ever in the Hereafter. In the Qur'an Allah has given the moral values of the Prophet Yahya (peace be upon him) as an example of this sensitivity of believers regarding love: [We said:] Yahya, take hold of the Book with vigour. We gave him judgement while still a child, and affection and purity from Us-he guarded against evil... (Surah Maryam, 12-13) In the hadiths, our Prophet (may Allah bless him and grant him peace) has recalled that the love believers feel for one another is for the approval of Allah: Whoever loves for Allah's sake, hates for Allah's sake, gives for Allah's sake and withholds for God's sake has a perfect faith. (Abu Dawud) There are three qualities whosoever has them will taste the sweetness of Iman: loving Allah and His Messenger above all else, loving someone solely for the sake of Allah, and hating to return to disbelief after Allah has rescued him from it, as much as he would hate being thrown into Hellfire. (Sahih Bukhari). Believers who have great love of Allah, fear Allah, and make sincere endeavours so that He might approve of them, are auspicious individuals who bring beauty to the world. Due to these superior moral values, they also love the things that Allah has created, feel affection and compassion for them, and wish to protect them and bring them goodness and beauty. Allah has given the glad tidings that in return for this pleasing love stemming from the faith in believers' hearts and their fear of Allah, and for their sincere loyalty to Him, He will reward them with Paradise, the most beautiful sphere of love and faithfulness. It is an important religious observance for the faithful to remind one another of the importance of loving Allah and believers. Every believer must strive to live by the love to be experienced in Paradise while still in this world, and they must bind themselves to Allah, our only friend and guardian, and to other believers, with love and devotion. THE SPIRITUAL STRENGTH THAT COMES FROM BELIEVERS' LOVE AND SOLIDARITY What lies behind the attachment of people who do not believe in Allah and the Hereafter is generally the importance imputed to worldly values and expectations directed towards worldly interests. By coming together, these people in one sense make a mutual agreement of interests; the parties support one another and thus seek to achieve mutual advantages. The people in that alliance know that their togetherness is not based on trust or friendship, and that, even though this is not openly stated, the alliance is dependent on a number of conditions. When the element of either party that provides the other with these advantages is eliminated, the alliance also falls apart. Whether this leaves the other person in difficulties or needing support is of no interest to other. That is because the alliance formed has stemmed solely from a joining of forces and an expectation of benefits. For that reason, when these expectations are removed, it is perfectly natural that the alliance should also fall apart. As revealed by Allah in the verse Their hostility towards each other is intense. You consider them united but their hearts are scattered wide. That is because they are people who do not use their intellect, (Surat al-Hashr, 14) no matter how much those who do not believe may appear to enjoy unity and solidarity, fundamentally they cannot actually be united. The only force on Earth that can establish true friendship and alliance among human beings is faith. With their friendships, believers who fear the Day of Judgement have laid the foundations of a sound alliance that begins in this world and that will continue for ever in the Hereafter. They love one another with no expectation of benefit or advantage, out of pure intentions and solely for Allah's approval, and are one another's friends for that approval. It is impossible for this bond, based on love of Allah and fear of Allah, to be loosed unless Allah so wills. WORDS OF OUR PROPHET (MAY ALLAH BLESS HIM AND GRANT HIM PEACE) RECOMMENDING LOVE When a man loves his brother for sake of Allah, he should tell him that he loves him. (Abu Dawud) Give gifts to each other, as this will make you love one another. (Sahih Muslim) Give one another gifts and love one another. Give one another food. This will produce breadth in your daily bread.(Al Hafiz ibn al-Dayba al-Shaybani, Taysir al-'usul ilaJami al-'usul, vol. 16, p. 239) The Prophet (may Allah bless him and grant him peace) said:One who is the best of you in good conduct is nearest to me. A believer loves and is loved. There is no good in one who does not love and is not loved. (Imam Ghazzali, vol. 2 , p. 95) Two brother are like two hands one of which clears the dust of the other. (Imam Ghazzali, vol.2, p.95) Do not be angry with each other and do not envy each other and do not turn away from each other, and be slaves of Allah, brothers. (Muwatta, narrated by Anas ibn Malik) Allah's Messenger (may Allah bless him and grant him peace) said, A Muslim is a brother of another Muslim, so he should not oppress him, nor should he hand him over to an oppressor. Whoever fulfilled the needs of his brother, Allah will fulfill his needs; whoever brought his (Muslim) brother out of a discomfort, Allah will bring him out of the discomforts of the Day of Resurrection, and whoever screened a Muslim, Allah will screen him on the Day of Resurrection. (Narrated by Abdullah bin Umar, Vol 3: #622) The faithful constitute a great spiritual force with the strength their love for one another for Allah's approval gives them. As revealed in the words of one verse, But those who were sure that they were going to meet Allah said, 'How many a small force has triumphed over a much greater one by Allah's permission! Allah is with the steadfast, (Surat al-Baqara, 249), even if they are few in number, with the faith in their hearts they acquire great enthusiasm and will with which to overcome terrible difficulties and troubles. They obtain the assistance and support of Allah because of the moral values they display. As Allah has revealed in the verse, You shall be uppermost if you are believers, (Surah Al 'Imran, 139), they constitute such a spiritual force that nobody can turn them against one another, and that nobody can break. Since they sincerely seek Allah's approval, they never engage in any confusion, disagreement or dispute among themselves. That is because the word of Allah is one; the verses of the Qur'an are clear. Since all believers abide unconditionally by the Qur'an and always act with a view to gaining as much approval from Allah as possible, a great harmony and order ensues. All matters can be easily resolved within a harmonious order. A powerful solidarity is formed because they behave in the light of the moral values of the Qur'an and the interests of believers, even when they conflict with their own interests, and hold their brothers' desires above their own. Since believers intend to be one another's eternal friends in the Hereafter they are bound to one another with a deep love, respect and loyalty. Therefore, they know no rivalry, disagreement or dispute. Due to their fear of and sincere faith in Allah, no matter what difficulties or troubles they may encounter they never fall into defeatism, moral relativism or lack of will. If there is a flaw in one of them, the others will support him with proper moral values and encourage him towards goodness. Since they constantly command one another to perform what is good and to avoid evil, their faith and strength constantly grow. This spiritual strength possessed by believers, whose objectives, endeavours and prayers are always the same, which stems from faith and love, has been described by Bediuzzaman Said Nursi with the following example: For just as one of man's hands cannot compete with the other, neither can one of his eyes criticize the other, nor his tongue object to his ear, nor his heart see his spirit's faults. Each of his members completes the deficiencies of the others, veils their faults, assists their needs, and helps them out in their duties. Otherwise man's life would be extinguished, his spirit flee, and his body be dispersed. Similarly, the components of machinery in a factory cannot compete with one another in rivalry, take precedence over each other, or dominate each other. They cannot spy out one another's faults and criticize each other, destroy the other's eagerness for work, and cause them to become idle. They rather assist each other's motions with all their capacity in order to achieve the common goal; they march towards the aim of their creation in true solidarity and unity. Should even the slightest aggression or desire to dominate interfere, it would throw the factory into confusion, causing it to be without product or result. Then the factory's owner would demolish the factory entirely. (Bediuzzaman Said Nursi, Risale-i Nur Collection, TheTwenty- First Flash) This example given by Bediuzzaman is of great importance with regard to being able to comprehend the union and unity stemming from the love among believers. On account of the sincere love and devotion that stem from their faith, in the same way that the machinery in a factory comes together to constitute a great force, so they acquire an unshakable spiritual strength with their mutual love and devotion. Your friend is only Allah and His Messenger and those who believe: those who perform prayer and give the alms, and bow. (Surat al Ma'ida, 55) Allah loves those who fight in His way in ranks like well-built walls. (Surat as-Saff, 4) === === Subject: : Re: continuous iteration posting-account=Yn5cwwoAAADntcMuRwk-EwLg-DMZ_hXN Gecko/20070509 Camino/1.5,gzip(gfe),gzip(gfe) > sometimes > as in the case of gamma > you need to add conditions for uniqueness > (like log convexity does with bohr-mollerup) > (and it seems like in your last message Tetration seems to be a pretty edgy subject, > and since it apparently uses the well known > Gamma function, poor tommy thinks (so to speak) > that if someone doesn't know about tetration > then he also doesn't know about the Gamma > function...:) you may not be noticing gamma itself is a continuous iterate) there has never been anything edgy about tetration it has a long history ****************************************************** I don't think so: I think it's hugely edgy, but it doesn't really > matter: you may as well not consider it so. > About gamma being continuous iterate: do you mean that it fulfills G(s+1) = s*G(s), or what? I never heard it called that way, but that > jsut could be my bad. I hear id called the factorial function, we > even proved in an advanced calculus course that G is the unique real > function which fulfills: > 1) G a log-convex function; > 2) G(1) = 1 > 3) G(s+1) = sG(s) the factorial of n can be seen as multiply the n numbers from 1 to n together 1.2.3...n which can be written in the rising pochhammer (1) n now much as i demonstrated for changing sum the n numbers f(.)... there is a way to make sense of continuous products again much as with sums the idea is to extend the iteration infinitely but some extra care to avoid infinities is necessary write (1) (1) (1+n) x+n n x (1) = ------ = ---------- n (1+x) (1+x) n n now the leading term of (1+n) x x is n so the limit as n->oo of the above is the same limit as (1) x n n lim -------- n->oo (1+x) n which is defined for all x and is one way euler defined __ | ' | (1 + x) .. so now i've shown you ways of making continuous iterative processes for sums and products do you see how iterative exponentiation can also be so generalised? several different ways have been proposed but there is one approach that follows this same idea can you guess it now? -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar === === Subject: : Re: continuous iteration posting-account=suWj4AkAAADE1IvGmj55Nmq3f98qb17e InfoPath.1; .NET CLR 2.0.50727),gzip(gfe),gzip(gfe) sometimes > æ as in the case of gamma > you need to add conditions for uniqueness > (like log convexity does with bohr-mollerup) > (and it seems like in your last message Tetration seems to be a pretty edgy subject, > and since it apparently uses the well known > Gamma function, poor tommy thinks (so to speak) > that if someone doesn't know about tetration > then he also doesn't know about the Gamma > function...:) æyou may not be noticing gamma itself is a continuous iterate) there has never been anything edgy about tetration it has a long history ****************************************************** I don't think so: I think it's hugely edgy, but it doesn't really > matter: you may as well not consider it so. > About gamma being continuous iterate: do you mean that it fulfills G(s+1) = s*G(s), or what? I never heard it called that way, but that > jsut could be my bad. I hear id called the factorial function, we > even proved in an advanced calculus course that G is the unique real > function which fulfills: > 1) G a log-convex function; > 2) G(1) = 1 > 3) G(s+1) = sG(s) the factorial of n can be seen as > multiply the n numbers from 1 to n together 1.2.3...n which can be written in the rising pochhammer (1) > æ æn now > æ much as i demonstrated for changing > æ sum the n numbers f(.)... > there is a way to make sense of continuous products again > æ much as with sums > the idea is to extend the iteration infinitely > æ but some extra care to avoid infinities is necessary write æ æ æ æ(1) æ æ æ(1) (1+n) > æ æ æ æ æ x+n æ æ æn æ æ x > (1) æ= ------ = ---------- > æ æn æ (1+x) æ æ æ(1+x) > æ æ æ æ æ æ n æ æ æ æ æn now > the leading term of (1+n) > æ æ æx æ æ x > is n so the limit as n->oo of the above > æ is the same limit as æ æ æ æ(1) æ æx > æ æ æ æ æ n æn > ælim æ -------- > n->oo æ (1+x) > æ æ æ æ æ æ æn which is defined for all x > and is one way euler defined > | ' > | æ(1 + x) æ æ æ æ æ æ.. so > now i've shown you ways of making continuous > æ iterative processes for sums and products do you see how iterative exponentiation > æ can also be so generalised? several different ways have been proposed > but there is one approach that follows this same idea can you guess it now? > ******************************************************** I sincerely thank you for trying to make some sense for me out of all the above, but I still can't see it: you began by saying that n! = (1)~ n -- meaning (1) is kind of power to the left of n --. Then you write (1)~(x + n), which I can udnerstand as being (x + n)! generalized, most like the infinite product in (a+b)^1/2 as series, trying to generalize Newton. But then you write (1+x)~ n, and I just can't understand what this mean: factorial from 1+x to n or what..?? Not that iteration or whatever has anything to do with the solution of that darn old hoax, but it would nevertheless be nice to understand something new in maths. Tonio === === Subject: : What did you know about The Holy Quran? posting-account=_DBhBAoAAAAm99vuHauWW0fIeBO1VZfW .NET CLR 2.0.50727; .NET CLR 3.0.04506; InfoPath.1),gzip(gfe),gzip(gfe) Let the scripture talk... http://www.harunyahya.com/Quran_translation/Quran_translation_index.php === === Subject: : This Week's Finds in Mathematical Physics (Week 264) Originator: bergv@math.uiuc.edu (Maarten Bergvelt) Also available at http://math.ucr.edu/home/baez/week264.html May 18, 2008 This Week's Finds in Mathematical Physics (Week 264) John Baez Here's a puzzle. Guess the next term of this sequence: 1, 1, 2, 3, 4, 5, 6, ... and then guess the *meaning* of this sequence! I'll give away the answer after telling you about Coleman's videos on quantum field theory and an amazing result on the homotopy groups of spheres. But first... the astronomy picture of the day. The Eaton Collection at UC Riverside may be the world's best library of science fiction: 1) The Eaton Collection of Science Fiction, Fantasy, Horror and Utopian Literature, http://eaton-collection.ucr.edu/ Right now my wife Lisa Raphals is attending a conference there on the role of Mars in SF, called Chronicling Mars. Gregory Benford, Frederik Pohl, Greg Bear, David Brin, Kim Stanley Robinson and even Ray Bradbury are all there! But for some reason I'm staying home working on This Week's Finds. I'd say that shows true devotion - or maybe just stupidity. Anyway, in honor of the occasion, here's an incredible closeup of a crater on Mars' moon Phobos: 2) Astronomy Picture of the Day, Stickney Crater http://apod.nasa.gov/apod/ap080410.html It's another great example of how machines in space now deliver many more thrills per buck than the old-fashioned approach using canned primates. This photo was taken by HiRISE, the High Resolution Imaging Science Experiment - the same satellite that took the stunning photos of Martian dunes which graced week262. Mars has two moons, Phobos and the even tinier Deimos. Their names mean fear and dread in Greek, since in Greek mythology they were sons of Mars (really Ares), the god of war. Interestingly, Kepler predicted that Mars had two moons before they were seen. This sounds impressive, but it was simple interpolation, since Earth has 1 moon and Jupiter has 4. Or at least Galileo saw 4 - now we know there are a lot more. Phobos is only 21 kilometers across, and the big crater you see here - Stickney Crater - is about 9 kilometers across. That's almost half the size of the whole moon! The collision that created it must have almost shattered Phobos. Phobos is so light - just twice the density of water - that people once thought it might be hollow. This now seems unlikely, though it's been the premise of a few SF stories. It's more likely that Phobos is a loosely packed pile of carbonaceous chondrites captured from the asteroid belt. Phobos orbits so close to Mars that it zips around once every 8 hours, faster than Mars itself rotates! Oddly, in 1726 Gulliver's Travels - and he guessed that the inner one orbited Mars every 10 hours. Gravitational tidal forces are dragging Phobos down, so in only 10 million years it'll either crash or - more likely - be shattered by tidal forces and form a ring of debris. So, enjoy it while it lasts. Anyone who's seriously struggled to master quantum field theory is likely to have profited from this book: 3) Sidney Coleman, Aspects of Symmetry: Selected Erice Lectures, Cambridge U. Press, Cambridge, 1988. It's brimming with wisdom and humor. You should have already encountered quantum field theory before trying it: what you'll get are deeper insights. But what if you're just getting started? Sidney Coleman, recently deceased, was one of the best quantum field took a course on quantum field theory from Eddie Farhi, who said he based his class on the notes from Coleman's class at Harvard. So, I've always been curious about these notes. Now they're available online in handwritten form: 4) Sidney Coleman, lecture notes on quantum field theory, http://www.damtp.cam.ac.uk/user/dt281/qft/col1.pdf and http://www.damtp.cam.ac.uk/user/dt281/qft/col2.pdf Someone should LaTeX them up! Even more fun, you can now see *videos* of Coleman teaching quantum field theory: 5) Sidney Coleman, Physics 253: Quantum Field Theory, 50 lectures recorded 1975-1976, http://www.physics.harvard.edu/about/Phys253.html This is a younger, hipper Coleman than I'd ever seen: long-haired, sometimes puffing on a cigarette between sentences. He begins by saying Umm... this is Physics 253, a course in relativistic quantum mechanics. My name is Sidney Coleman. The apparatus you see around you is part of a CIA surveillance project. I wish I'd had access to these when I was a kid! Now for some miraculous math. Daniel Moskovich kindly pointed out a paper that describes all the homotopy groups of the 2-sphere, and I want to summarize the main result. I explained the idea of homotopy groups back in week102. Very roughly, the nth homotopy group of a space X, usually denoted pi_n(X), is the set of ways you can map an n-sphere into that space, where we count two ways as the same if you can continuously deform one to the other. If a space has holes, homotopy groups are one way to detect those holes. Homotopy groups are notoriously hard to compute - so even for so humble a space as the 2-sphere, S^2, there's a sense in which nobody knows all its homotopy groups. People know the first 64, though. Here are a few: pi_1(S^2) = 0 pi_2(S^2) = Z pi_3(S^2) = Z pi_4(S^2) = Z/2 pi_5(S^2) = Z/2 pi_6(S^2) = Z/4 x Z/3 pi_7(S^2) = Z/2 pi_8(S^2) = Z/2 pi_9(S^2) = Z/3 pi_10(S^2) = Z/3 x Z/5 pi_11(S^2) = Z/2 pi_12(S^2) = Z/2 x Z/2 pi_13(S^2) = Z/2 x Z/2 x Z/3 pi_14(S^2) = Z/2 x Z/2 x Z/4 x Z/3 x Z/7 pi_15(S^2) = Z/2 x Z/2 Apart from the fact that they're all finite abelian groups, it's hard to spot any pattern! In fact there's a majestic symphony of patterns in the homotopy groups of spheres, starting from ones that are easy to explain and working on up to those that push the frontiers of mathematics, like elliptic cohomology. But, many of these patterns are too complex for present-day mathematics until we use some tricks to water down or simplify the homotopy groups. So, what people often do first is take the limit of pi_{n+k}(S^n) as n -> infinity, getting what's called the kth stable homotopy group of spheres. It's a wonderful but well-understood fact that these limits really exist. But so far, even these are too complicated to understand until we work at a prime p. This means that we take the kth stable homotopy group of spheres and see which groups of the form Z/p^n show up in it. For example, pi_14(S^2) = Z/2 x Z/2 x Z/4 x Z/3 x Z/7 but if we work at the prime 2 we just see the Z/2 x Z/2 x Z/4. After all this data processing, we get some astounding pictures: 6) Allen Hatcher, Stable homotopy groups of spheres, http://www.math.cornell.edu/~hatcher/stemfigs/stems.html Order teetering on the brink of chaos! If you're brave, you can learn more about this stuff here: 7) Douglas C. Ravenel, Complex Cobordism and Stable Homotopy Groups If you're less brave, I strongly suggest starting here: 8) Wikipedia, Homotopy groups of spheres, http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres But now, I want to talk about an amazing paper that pursues a very different line of attack. It gives a beautiful description of *all* the homotopy groups of S^2, in terms of braids: 9) A. Berrick, F. R. Cohen, Y. L. Wong and J. Wu, Configurations, braids and homotopy groups, J. Amer. Math. Soc., 19 (2006), 265-326. Also available at http://www.math.nus.edu.sg/~matwujie/BCWWfinal.pdf For this you need to realize that for any n, there's a group B_n whose elements are n-strand braids. For example, here's an element of B_3: | | | / | / | / | | / | / | / / | / | / | | / | / | / / | / | / | | / | / | / | | | I actually talked about this specific braid back in week233. But anyway, we count two braids as the same if you can wiggle one around until it looks like the other without moving the ends at the top and bottom - which you can think of as nailed to the ceiling and floor. How do braids become a group? Easy: we multiply them by putting one on top of the other. For example, this braid: | | | / | A = / | / | | | | times this one: | | | | / B = | / | / | | | equals this: | | | / | / | / | | | | AB = | | | | / | / | / | | | and in fact the big one I showed you earlier is (AB)^3. As you let your eye slide from the top to the bottom of a braid, the strands move around. We can visualize their motion as a bunch of points running around the plane, never bumping into each other. This gives an interesting way to generalize the concept of a braid! Instead of points running around the plane, we can have points running around S^2, or some other surface X. So, for any surface X and any number n of strands, we get a surface braid group, called B_n(X). As I hinted in week261, these surface braid groups have cool relationships to Dynkin diagrams. I urged you to read this paper, and I'll urge you again: 10) Daniel Allcock, Braid pictures for Artin groups, available as arXiv:math.GT/9907194. But for now, we just need the spherical braid group B_n(S^2) together with the usual braid group B_n. Let's say a braid is Brunnian if when you remove any one strand, the remaining braid becomes the identity: you can straighten out all the remaining strands to make them vertical. It's a fun little exercise to check that Brunnian braids form a subgroup of all braids. So, we have an n-strand Brunnian braid group BB_n. The same idea works for braids on other surface, like the 2-sphere. So, we also have an n-strand *spherical* Brunnian braid group BB_n(S^2). Now, there's obvious map B_n -> B_n(S^2) Why? An element of B_n describes the motion of a bunch of points running around the plane, but the plane sits inside the 2-sphere: the 2-sphere is just the plane with an extra point tacked on. So, an ordinary braid gives a spherical braid. This map clearly sends Brunnian braids to spherical Brunnian braids, so we get a map f: BB_n -> BB_n(S^2) And now we're ready for the shocking theorem of Berrick, Cohen, Wong and Wu: Theorem: For n > 3, BB_n(S^2) modulo the image of f is the nth homotopy group of S^2. In something more like plain English: when n is big enough, the nth homotopy group of the 2-sphere consists of spherical Brunnian braids modulo ordinary Brunnian braids! Zounds! What do the homotopy groups of S^2 have to do with braids? It's not supposed to be obvious! The proof of this result is long and deep, making use of flows on metric spaces, and also the fact that all the Brunnian braid groups BB_n fit together into a simplicial group whose nth homology is the nth homotopy group of S^2. I'd love to understand all this stuff, but I don't yet. This result doesn't instantly help us compute the homotopy groups of S^2 - at least not in the sense of writing them down as a product of groups like Z/p^n. But, it gives a new view of these homotopy groups, and there's no telling where this might lead. going to tell you about some amazing descriptions of the homotopy groups of the *3-sphere*, due to Wu. However, I later realized - first to my shock, and then my embarrassment for not having known it already - that the nth homotopy group of S^3 is *the same* as the nth homotopy group of S^2, at least for n > 2. Do you see why? Given this, it turns out that Wu's results are predecessors of the theorem just stated, a bit more combinatorial and less geometric. Wu's results appeared here: 11) Jie Wu, On combinatorial descriptions of the homotopy groups of certain spaces, Math. Proc. Camb. Phil. Soc. 130 (2001), 489-513. Also available at http://www.math.nus.edu.sg/~matwujie/newnewpis_3.pdf Jie Wu, A braided simplicial group, Proc. London Math. Soc. 84 (2002), 645-662. Also available at http://www.math.nus.edu.sg/~matwujie/Research2.html and there's a nice summary of these results on his webpage: 12) Jie Wu, 2.1 Homotopy groups and braids, halfway down the page at http://www.math.nus.edu.sg/~matwujie/Research2.html See also this expository paper: 13) Fred R. Cohen and Jie Wu, On braid groups and homotopy groups, Geometry & Topology Monographs 13 (2008), 169-193. Also available at http://www.math.nus.edu.sg/~matwujie/cohen.wu.GT.revised.29.august.2007.pdf Next I want to talk about puzzle mentioned at the start of this Week's Finds... but first I should answer the puzzle I just raised. Why do the homotopy groups of S^2 match those of S^3 after a while? Because of the Hopf fibration! This is a fiber bundle with S^3 as total space, S^2 as base space and S^1 as fiber: S^1 -> S^3 -> S^2 Like any fiber bundle, it gives a long exact sequence of homotopy groups as explained in week151: ... -> pi_n(S^1) -> pi_n(S^3) -> pi_n(S^2) -> pi_{n-1}(S^1) -> ... but the homotopy groups of S^1 vanishes after the first, so we get ... -> 0 -> pi_n(S^3) -> pi_n(S^2) -> 0 -> ... for n > 2, which says that pi_n(S^3) = pi_n(S^2) Okay, now for this mysterious sequence: 1, 1, 2, 3, 4, 5, 6, ... The next term is obviously 7. If you guessed anything else, you were over-analyzing. So the real question is: why the funny hiccup at the beginning? You'll find two explanations of this sequence in Sloane's Online Encyclopedia of Integer Sequences, but neither of them is the reason James Dolan and I ran into it. We were studying theta functions... Say you have a torus. Then the complex line bundles over it are classified by an integer called the first Chern number. In some sense, this integer this measures how twisted the bundle is. For example, you can put any connection on the bundle, compute its curvature 2-form, and integrate it over the torus: up to some constant factor, you'll then get the first Chern number. A torus is a 2-dimensional manifold, but we can also make it into a 1-dimensional *complex* manifold, often called an elliptic curve. In fact we can do this in infinitely many fundamentally different ways, one for each point in the moduli space of elliptic curves. I've explained this repeatedly here - try week125 for a good starting-point - so I won't do so again. The details don't really matter here. Back to line bundles. If we pick an elliptic curve, we can try to classify the *holomorphic* complex line bundles over it - that is, those where the transition functions are holomorphic (or in other words, complex-analytic). Here the classification is subtler! It turns out you need, not just the first Chern number, which is discrete, but another parameter which can vary in a *continuous* way. Interestingly, after you pick a basepoint for your elliptic curve, this other parameter can be thought of as just a point on the elliptic curve! So, the elliptic curve becomes the space that classifies holomorphic line bundles over itself - at least, those with fixed first Chern number. Curiously circular, eh? This is just one of several curiously circular classification theorems that happen in this game... But I'm actually digressing a bit - I'm having trouble resisting the temptation to explain everything I know, since it's so simple and beautiful, and I just learned it. Don't worry - all you need to know is that holomorphic line bundles over an elliptic curve are classified by an integer and some other continuous parameter. The puzzle then arises: how many holomorphic sections do these line bundles have? More precisely: what's the *dimension* of the space of holomorphic sections? Before I answer this, I can't resist adding that these holomorphic sections have a long and illustrious history - they're called theta functions, and you can learn about them here: 14) Jun-ichi Igusa, Theta Functions, Springer, Berlin, 1972. 15) David Mumford, Tata Lectures on Theta, 3 volumes, Birkhauser, Boston, 1983-1991. They're important in geometric quantization, where holomorphic sections of line bundles describe states of quantum systems, and the reciprocal of the first Chern number is proportional to Planck's constant. In fact, I first ran into theta functions years ago, when trying to quantize a black hole - see the end of week112 for more details. But anyway, here's the answer to the puzzle. The dimension turns out not to depend on the continuous parameter labelling our line bundle, but only on its first Chern number. If that number is negative, the dimension is 0. But if it's 0,1,2,3,4,5,6 and so on, the dimension goes like this: 1,1,2,3,4,5,6,... Now, this sequence is fairly weird, because of the extra 1 at the beginning. I hadn't noticed this back when I was quantizing black holes, because the extra 1 happens for first Chern number zero, which would correspond to Planck's constant being *infinite*. But now that I'm just thinking about math, it sticks out like a sore thumb! It's got to be right, since the line bundle with first Chern number zero is the trivial bundle, its sections are just functions, and the only holomorphic functions on a compact complex manifold are constants - so there's a 1-dimensional space of them. But, it's weird. Luckily, Jim figured out the explanation for this sequence. First of all, we can encode it into a power series: 1 + x + 2x^2 + 3x^3 + 4x^4 + ... which we can rewrite as a rational function: (1-x^6) 1 + x + 2x^2 + 3x^3 + 4x^4 + ... = -------------------- (1-x)(1-x^2)(1-x^3) Now, the reason for doing this is that we can pick a line bundle of first Chern number 1, say L, and get a line bundle of any Chern number n by taking the nth tensor power of L - let's call that L^n. We can multiply a section of L^n and a section of L^m to get a section of L^{n+m}. So, all these spaces of sections we're studying fit together to form a commutative graded ring! And, whenever you have a graded ring, it's a good idea to write down a power series that encodes the dimensions of each grade, just as we've done above. This is called a Poincare series. And, when you have a commutative graded ring with one generator of degree 1, one generator of degree 2, one generator of degree 3, one relation of degree 6, and no relations between relations (or syzygies), its Poincare series will be (1-x^6) -------------------- (1-x^1)(1-x^2)(1-x^3) That's how it always works - think about it. So, it's natural to hope that our ring built from holomorphic sections of all the line bundles L^n will have one generator of degree 1, one of degree 2, one of degree 3, and one relation of degree 6. And, this seems to be true! As I mentioned, people usually call these holomorphic sections theta functions. So, it seems we're getting a description of the ring of theta functions in terms of generators and relations. How does it work, exactly? Well, I must admit I'm not quite sure. Jim has some ideas, but it seems I need to do something a bit different to get his story to work for me. Maybe it goes something like this. We can write any elliptic curve as the solutions of this equation: y^2 = x^3 + Bx + C for certain constants B and C that depend on the elliptic curve. (See week13 and week261 for details.) Now, this equation is not homogeneous in the variables y and x, but we can think of it as homogeneous in a sneaky sense if we throw in an extra variable like this: y^2 = x^3 + Bxz^5 + Cz^6 and decree that: y has grade 3 x has grade 2 z has grade 1 Then all the terms in the equation have grade 6. So, we're getting a commutative graded ring with generators of degree 1, 2, and 3 and a relation of grade 6. And, I'm hoping this ring consists of algebraic functions on the total space of some line bundle L* over our elliptic curve. z should be a function that's linear in the fiber directions, hence a section of L. x should be quadratic in the fiber directions, hence a section of L^2. And y should be cubic, hence a section of L^3. If L has first Chern number 1, I think we're in business. If anybody knows about this stuff, I'd appreciate corrections or references. There's a *lot* more to say about this business... because it's all part of a big story about elliptic curves, theta functions and modular forms. But, I want to quit here for now. ----------------------------------------------------------------------- Addenda: I thank David Corfield for pointing out how to get ahold of Wu's papers free online - and earlier, for telling me Wu's combinatorial description of pi_3(S^2). Martin Ouwehand told me that some of Coleman's lecture notes on quantum field theory are available in TeX here: 17) Sidney Coleman, Quantum Field Theory, first 11 lectures notes TeXed by Bryan Gin-ge Chen, available at http://www.physics.upenn.edu/~chb/phys253a/coleman/ 18) Wikipedia, Riemann-Roch theorem, http://en.wikipedia.org/wiki/Riemann-Roch has some very relevant information on the sequence 1, 1, 2, 3, 4, 5, 6, ... though it's phrased not in terms of sections of line bundles, but instead in terms of divisors (secretly another way of talking about the same thing). Let me quote a portion, just to whet your interest: We start with a connected compact Riemann surface of genus g, and a fixed point P on it. We may look at functions having a pole only at P. There is an increasing sequence of vector spaces: functions with no poles (i.e., constant functions), functions allowed at most a simple pole at P, functions allowed at most a double pole at P, a triple pole, ... These spaces are all finite dimensional. In case g = 0 we can see that the sequence of dimensions starts 1, 2, 3, ... This can be read off from the theory of partial fractions. Conversely if this sequence starts 1, 2, ... then g must be zero (the so-called Riemann sphere). In the theory of elliptic functions it is shown that for g = 1 this sequence is 1, 1, 2, 3, 4, 5 ... and this characterises the case g = 1. For g > 2 there is no set initial segment; but we can say what the tail of the sequence must be. We can also see why g = 2 is somewhat special. The reason that the results take the form they do goes back to the formulation (Roch's part) of the [Riemann-Roch] theorem: as a difference of two such dimensions. When one of those can be set to zero, we get an exact formula, which is linear in the genus and the degree (i.e. number of degrees of freedom). Already the examples given allow a reconstruction in the shape dimension - correction = degree - g + 1. For g = 1 the correction is 1 for degree 0; and otherwise 0. The full theorem explains the correction as the dimension associated to a further, 'complementary' space of functions. You can see more discussion of this Week's Finds at the n-Category Cafe: http://golem.ph.utexas.edu/category/2008/05/this_weeks_finds_in_mathematic_2 5.html ----------------------------------------------------------------------- Quote of the Week: The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction. - Sidney Coleman ----------------------------------------------------------------------- mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twfcontents.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html === === Subject: : leibniz's theorem on differentiation under the integral sign hello, im an amateur mathematician who needs help with this theorem. I don't understand the proof. can anyone help me? raul === === Subject: : Re: leibniz's theorem on differentiation under the integral sign <17577040.1211578034117.JavaMail.jakarta@nitrogen.mathforum.org>, raul > hello, im an amateur mathematician who needs help with this theorem. I don't > understand the proof. can anyone help me? > raul See ; it is easier than you think. -- Paul Sperry Columbia, SC (USA) === === Subject: : Re: leibniz's theorem on differentiation under the integral sign === === Subject: : Need solution manuals of physics books posting-account=RiP9KwoAAAA1lewGbY3vgVccW5jzW017 SV1),gzip(gfe),gzip(gfe) My name is Sam and I am preparing to write my GRE physics subject test.Can you please send me the ebooks of the following list? 1) concepts of modern physics by Arthur Beiser 2)Introduction to Solid state physics by Kittel 3) Introduction to Quantum mechanics by Griffiths 4) Introduction to Electromagnetism by Grifiths 5) Fundamentals of physics by Resnic and Haliday 6th edition. If you have any books on GRE physics, please send.It will be reaaly very helpful to me. Sam === === Subject: : A lot of Solution Manuals in Electronic (PDF)Format! posting-account=AIT25goAAAD4PInVOqQYW2U7xf3SSqUF 5.1),gzip(gfe),gzip(gfe) A lot of Solution Manuals in Electronic (PDF)Format! A lot of Solutions Manuals in Electronic (PDF)Format! Just contact with trustsolution (at) hotmail.com (my email address), these are parts of our solutions, if the solution you want isn.81ft on the list, don.81ft give up,please email to me. Solutions manual list: Fundamentals of Electric Circuits(6th) By Charles Alexander, Matthew Sadiku Introduction to VLSI Circuits and Systems By John P. Uyemura Linear Algebra with Applications( 6th )edition by Leon Mechanics of Materials(-3rd) ,by Ferdinand P. Beer, E. Russell Johnston Jr., John T. DeWolf Physics for Scientists and Engineers with Modern Physics (3rd Edition )by Douglas C. 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Freedman, use with Fundamentals of Corporate Finance, 4th Edition By Bruce Swenson vector analysis (1961) By Lewis Richard Shorter Vector Mechanics for Engineers: Dynamics, 7th By Ferdinand P. Beer(selected chapters) E. R. Johnston Vector Mechanics for Engineers: Statics, 7th By Ferdinand P. Beer(selected chapters) Vector Mechanics for Engineers: Statics, 7th Edition ,By Ferdinand P. Beer, E. Russell Johnston Jr., ElliotR Visual C++ How to Program, (3rd Edition) ,by Harvey & Paul Deitel & Associates Wireless communication and networks 2th by willian stallings Zill's a First Course in Differential Equations with boundary value problem 5th Zill's a First Course in Differential Equations with Modeling Applicants 7/e.81isolitions manual.81j http://solutionsmanual.spaces.live.com I have many solution manual, http://solutionsmanual.spaces.live.com === === Subject: : Re:Need solution manuals of physics books Hello everybody, My name is Sam and I am preparing to write my GRE physics subject test.Can you please send me the ebooks of the following list? 1) concepts of modern physics by Arthur Beiser 2)Introduction to Solid state physics by Kittel 3) Introduction to Quantum mechanics by Griffiths 4) Introduction to Electromagnetism by Grifiths 5) Fundamentals of physics by Resnic and Haliday 6th edition. If you have any books on GRE physics, please send.It will be reaaly very helpful to me. email: sammystrings@yahoo.com Sam