August 2008 - mathfun@mailman.xmission.com

[math-fun] AGM for Gamma(n/12)
by rwg＠sdf.lonestar.org 19 Sep '08

19 Sep '08

In case anybody besides me missed it, Gamma(1/3) = 2^(11/18)*(sqrt(3)+1)^(1/3)*%pi^(2/3)/(3^(1/4)*agm(sqrt(2)*(sqrt(3)+1),1)^(1/3)) 11/18 1/3 2/3 1 2 (sqrt(3) + 1) %pi gamma(-) = ------------------------------------- 3 1/4 1/3 3 AGM (sqrt(2) (sqrt(3) + 1), 1) follows from Borwein&Borwein, pp 15(a) and 28(d), so Gamma(1/3) is nearly as easy as pi. Similarly, Gamma(1/4) = 2^(3/4)*%pi^(3/4)/sqrt(agm(sqrt(2),1)) 3/4 3/4 1 2 %pi Gamma(-) = --------------------- 4 sqrt(AGM(sqrt(2), 1)) finishes off Gamma(n/12). Interesting that, unlike pi, there are no hypergeometric series for these, although the sum(n/(%e^(2*sqrt(3)*%pi*n)-1),n,1,inf) is a degenerate bibasic series. ((1 = AGM(eta(q^2)^10/(eta(q)^4*eta(q^4)^4),eta(q)^4/eta(q^2)^2)) = AGM((16*eta(q^4)^8+eta(q)^8)^(5/12)/eta(q)^(2/3),eta(q)^(10/3)/(16*eta(q^4)^8+eta(q)^8)^(1/12))/eta(q^4)^(2/3)) = AGM(sqrt(64*eta(q^4)^24+eta(q^2)^24)-8*eta(q^4)^12,eta(q^2)^12)/(eta(q^2)^2*eta(q^4)^2*sqrt(sqrt(64*eta(q^4)^24+eta(q^2)^24)-8*eta(q^4)^12)) 10 2 4 eta (q ) eta (q) 1 = AGM(----------------, --------) 4 4 4 2 2 eta (q) eta (q ) eta (q ) 8 4 8 5/12 10/3 (16 eta (q ) + eta (q)) eta (q) AGM(---------------------------, ---------------------------) 2/3 8 4 8 1/12 eta (q) (16 eta (q ) + eta (q)) = ------------------------------------------------------------- 2/3 4 eta (q ) 24 4 24 2 12 4 12 2 AGM(sqrt(64 eta (q ) + eta (q )) - 8 eta (q ), eta (q )) = --------------------------------------------------------------------. 2 2 2 4 24 4 24 2 12 4 eta (q ) eta (q ) sqrt(sqrt(64 eta (q ) + eta (q )) - 8 eta (q )) --rwg Does anyone besides Salamin remember the meaning of the agm error in Peter Samson's (1960s) PDP-1 music compiler?

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[math-fun] Doin' the Hilbert walk
by Fred lunnon 15 Sep '08

15 Sep '08

I've been spending quality time with Hilbert walks (finite approximations to Hilbert curves) in d-space lately (don't tell RWG), my ruminations somewhat inconvenienced by inability to locate a definition of what might actually constitute a Hilbert walk/curve in any dimension d > 2. Previous authors seem to cheerfully ignore the difficulty; or perhaps they just implicitly define a walk as being whatever convoluted Hmaitonian path their proposed algorithm happens to construct. While I have some ideas about an appropriate general definition, it's still not very clear to me exactly what properties --- combinatorial or possibly staistical --- of such a path might be significant in applications --- such as multiple-key database, and equation solving in many variables. Can anybody out there cast light on these matters? Fred Lunnon

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[math-fun] Distributivity of addition over multiplication
by Dan Asimov 07 Sep '08

07 Sep '08

Of course x + yz = (x + y)(x + z) is not true in general for (x,y,z) in R^3. But I don't think I would have guessed which (x,y,z) in R^3 that it does hold for. It only takes a few seconds to figure out what this set is. But can someone give an intuitive reason for why it is what it is? --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Smallest magic squares of powers
by Christian Boyer 01 Sep '08

01 Sep '08

Oct 7th, 2007: the smallest known magic square of 4th powers is now a 44x44 square constructed by Jaroslaw Wroblewski (Poland) and Hugo Pfoertner (Germany). Jaroslaw built the semi-magic square in June, then Hugo needed a very impressive computing job of 3 months, on 57 processors of a SGI computer http://www.sgi.com/products/remarketed/origin3000/, in order to find two magic diagonals in this square! A big progress, because the previous smallest known was a 243x243 square, coming from the 243x243 tetramagic square of Pan Fengchu (China). The best known results, and their first authors, are now: Magic square Smallest known ------------ --------------- Of squares 4x4 (Euler, 1770) Of cubes 9x9 (Boyer, 2006) Of 4th powers 44x44 (Wroblewski-Pfoertner, 2007) Of 5th powers 729x729 (from the pentamagic square of Li Wen, 2003) ON ANY POWER, WE DO NOT KNOW WHAT THE SMALLEST POSSIBLE MAGIC SQUARE IS! For example, as you know, we still do not know if a 3x3 magic square of squares is possible... 4x4, 5x5, 6x6, 7x7, and 8x8 magic squares of cubes are unknown... And the problem of the diagonals is extremely difficult, as proved by the impressive computing search of Hugo. It remains a big gap between the smallest known magic and semi-magic squares (=with or without magic diagonals) of 4th powers: 44x44 versus 4x4... Here is the 4x4 semi-magic square of 4th powers (Morgenstern-Boyer, 2006): 14101^4 938^4 931^4 37762^4 35866^4 20881^4 21038^4 13393^4 24806^4 30191^4 30418^4 9263^4 413^4 32026^4 31787^4 1106^4 Christian.

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Re: [math-fun] Hilbert curve applications
by Steve Witham 31 Aug '08

31 Aug '08

>From: rcs(a)xmission.com > >Steve's calendar lights up a feature that's so disconcerting about >space-filling curves: Our definition of "continuous" is a high- >falutin' way of saying "keeps nearby things nearby". What if we >adopted another requirement, either in addition to, or instead of, >the Nearby rule: How about "Keeps far-apart things far-apart."? >This would address the anomaly in Steve's calendar that Jan 6 and >Aug 7 are adjacent. What sort of "continuity" would we get? Ah! Gray codes also "keep near things near." I've played with what I call "error-minimizing codes" that keep far things far. Not in two dimensions, just with Hamming distance among bit strings. (As I understand it, the more dimensions you have, the more Euclidian distance acts like Hamming distance.) Others have worked at this, too: D.J. Smith, P.H. Stanford, A Random Walk in Hamming Space P.H. Stanford, D.J. Smith, The Multidimensional Scatter Code http://www.santafe.edu/~dsmith/publications.html In scatter codes, you flip some fixed number of randomly-picked bits between adjacent values. This counts on the random walk's sqrt property (not sure how it translates into Hamming distance). I wanted far (value) things to be guaranteed >= ~log2(far) Hamming-wise, because I wanted the result of damaging n bits to be <= 2^n. Here's from something I wrote: >An error-minimizing code is like an error-correcting code except that >instead of insuring perfect recovery of the intended value in the face >of some maximum number of bit errors, it insures that the difference >between the original value and the recovered value is limited to a >function of the number of bit errors. > >In my programs so far, the rule is that > zero errors recover the exact original value > one error => in the range of the original value +/- 1 > two errors => original +/- 2 > 3 => +/- 4 > 4 => +/- 8 > etc. > n>0 => +/- 2^(n-1) > >(Note that if you take real values and round them to integers, the >"original" integer is the real number +/- 2^-1. Seen this way the >above rule is almost consistent.) > >Some people, when I first explain this, ask whether this isn't what >Gray codes do. What's true for Gray codes is that for a given difference >between values, the Hamming distance between their code words is limited: > Hamming distance <= log2( value difference + 1 ) (roughly) > >but for error minimizing codes, for a given Hamming distance, the value >distance is limited: > value difference <= 2^( (Hamming distance) - 1 ) >or > Hamming distance >= log2( value difference + 1 ) > >So the inequality is reversed. Gray codes keep similar values together, >but it's okay if some pairs of dissimilar values are also close. >Error-minimizing codes keep dissimilar values apart, and it's okay if >some pairs of similar values are also far apart. I haven't found a systematic way to generate anti-Gray codes. I wrote various greedy search programs, and for example, Width (bits) Length (# of values encoded) ------------ ---------------------------- 11 17 (my highest exhaustive search) ... ... 31 228 32 258 (narrowest code I found for >= 256 values) ... ... 64 9237 There's an information-theoretic lower bound on the width of the code, buried in my notes somewhere. --Steve

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[math-fun] Fwd: Fermat's last postit note
by rcs＠xmission.com 31 Aug '08

31 Aug '08

Cute. ----- Forwarded message from ken2(a)mirror.org ----- Date: Fri, 29 Aug 2008 09:16:49 -0400 (EDT) From: Ken Roberts <ken2(a)mirror.org> Reply-To: Ken Roberts <ken2(a)mirror.org> Subject: Fermat's last postit note To: Rich Schroeppel <rcs(a)cs.arizona.edu> http://www.irregularwebcomic.net/2041.html ----- End forwarded message -----

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Re: [math-fun] Hilbert curve applications
by Steve Witham 31 Aug '08

31 Aug '08

>From: "Fred lunnon" <fred.lunnon(a)gmail.com> > >I've been spending quality time with Hilbert walks (finite approximations to >Hilbert curves) [...] > >While I have some ideas about an appropriate general definition, it's still >not very clear to me exactly what properties --- combinatorial or possibly >staistical --- of such a path might be significant in applications --- >such as multiple-key database, and equation solving in many variables. Calendars! _ _ _ _ +-----------+-----------+-----------+-----------+ / \ / \ / \ / \| 12 13 14| 17 18 19| 4 5 6| 9 10 11| / | | | | \_/| +---+ + +---+ | +---+ + +---+ | / | | | | / | 11 10| 15 16| 21 20| 3 2| 7 8| 13 12| --- \_/ \_/ / +---+ +-------+ +---+---+ +-------+ +---+ | 8 9| 4 3| 22| 25 26| 1| 20 19| 14 15| | +---+ + | + + +Mar| + +---+ | | 7 6 5| 2| 23 24| 27 28| 21| 18 17 16| |Jan+---+-------+-------+---+ +-------+-------+ +-----------+ | 1 2| 15 16 17 18| 31Feb1| 17 16| 13 12| 22| 25 26 27| +---+ | +---+---+ | +---+ + + + | + +---+ | | 4 3| 14 13| 20 19| 30 29| 18| 15 14| 11| 23 24| 29 28| | +---+---+ | +---+---+ | +---+---+ +-------+ +---+ | 5| 8 9| 12| 21| 24 25| 28| 19 20| 9 10| 5 4| 30 31| | + + + | + + + +---+ | +---+ + +---+Apr| | 6 7| 10 11| 22 23| 26 27| 22 21| 8 7 6| 3 2 1| +-------+-------+-------+-------+ +---+-----------+-----------+ | 7 6| 3 2| 22 21| 18 17| 23 24| 7 8 9| 12 13 14| | + + + | + + + +---+ | +---+ + +---+ | | 8| 5 4| 1| 23| 20 19| 16| 26 25| 6 5| 10 11| 16 15| | +---+---+Aug| +---+---+ | +---+---+ +-------+ +---+ | 9 10| 30 31| 24 25| 14 15| 27| 30May1| 4| 23 22| 17 18| +---+ | +---+---+ | +---+ + + + | + +---+ | | 12 11| 29 28 27 26| 13 12| 28 29| 2 3| 24| 21 20 19| | +---+-------+-------+---+ +-------+-------+ +-----------+ | 13| 16 17 18| 6 7 8| 11| 20 19| 16 15| 25| 28 29 30| | + +---+ | +---+ + | + + + | + +---+ | | 14 15| 20 19| 5 4| 9 10| 21| 18 17| 14| 26 27|1Jun 31| +-------+ +---+---+ +-------+ +---+---+ +-------+ +---+ | 27 26| 21 22| 2 3| 28 27| 22 23| 12 13| 8 7| 2 3| | + +---+ | +---+ + +---+ | +---+ + +---+ | | 28| 25 24 23|1Jul 30 29| 26 25 24| 11 10 9| 6 5 4| | +-----------+-----------+-----------+-----------+-----------+ | 29| 1 2 3| 26 27 28| 31Nov1 2| 15 16 17| 20 21 22| | +Sep+---+ | +---+ + +---+ | +---+ + +---+ | | 30 31| 5 4| 25 24| 29 30| 4 3| 14 13| 18 19| 24 23| +-------+ +---+---+ +-------+ +---+---+ +-------+ +---+ | 12 11| 6 7| 22 23| 18 17| 5| 8 9| 12|1Dec 30| 25 26| | + +---+ | +---+ + | + + + | + +---+ | | 13| 10 9 8| 21 20 19| 16| 6 7| 10 11| 2| 29 28 27| | +---+-------+-------+---+ +-------+-------+ +-----------+ | 14 15| 28 29 30Oct1| 14 15| 29 28| 25 24| 3| 6 7 8| +---+ | +---+---+ | +---+ + + + | + +---+ | | 17 16| 27 26| 3 2| 13 12| 30| 27 26| 23| 4 5| 10 9| | +---+---+ | +---+---+ | +---+---+ +-------+ +---+ | 18| 21 22| 25| 4| 7 8| 11| 31 | 21 22| 17 16| 11 12| | + + + | + + + +---- | +---+ + +---+ | | 19 20| 23 24| 5 6| 9 10| | 20 19 18| 15 14 13| +-------+-------+-------+-------+ +-----------+-----------+ Yeah, I know it's only 2D but I couldn't resist. --Steve

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[math-fun] "An almost random, nonuniform function"
by rwg＠sdf.lonestar.org 31 Aug '08

31 Aug '08

I.e., a squiggle as might be scrawled on a blackboard as an example of an utterly generic continuous function: Plot[Re[DedekindEta[x*(I+5)]],{x,0,.15}] --rwg

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Re: [math-fun] an almost-uniform random variable?
by Eugene Salamin 30 Aug '08

30 Aug '08

----- Original Message ---- From: James Propp <jpropp(a)cs.uml.edu> To: math-fun(a)mailman.xmission.com Sent: Friday, August 29, 2008 5:04:52 PM Subject: Re: [math-fun] an almost-uniform random variable? Dan Asimov is right; I should have written << And here's a fun fact related to Archimedes' insight about the sphere and the cylinder: If X, Y, Z are independent Gaussians of mean 0 and variance 1, X/(X^2+Y^2+Z^2) is uniform on [-1,1]. >> To elaborate on what Dan wrote: Note that if X, Y, and Z are independent Gaussians, then (X,Y,Z) is spherically symmetric in distribution, so if you map to the unit sphere by dividing by sqrt(X^2+Y^2+Z^2) you get a uniform point on the unit sphere. Now proceed as Archimedes did. Jim Propp One more typo: the denominator should be R=sqrt(XX+YY+ZZ) and not R^2. Otherwise there will be values outside [-1,1]. Also, the variances need not be 1, as long as they are equal. Gene

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[math-fun] an almost-uniform random variable?
by James Propp 30 Aug '08

30 Aug '08

Here's a fun fact Oded Schramm just told me: If X, Y, Z are independent and uniform on [0,1], then so is (XY)^Z. And here's a fun fact related to Archimedes' insight about the sphere and the cylinder: If X, Y, Z are independent Gaussians of mean 0 and variance 1, X/(X^2+Y^2+Z^2) is uniform on [0,1]. Since I'm teaching a course on simulation of stochastic processes this coming semester, these could serve as good homework problems of the "do a simulation, make a conjecture, try to prove it" variety. But I'd like a third homework problem where some simple combination of random variables LOOKS like it's uniform but ISN'T, so the students don't come away with the impression that "if it's roughly flat, it must be uniform". So: Does anyone know of a probability distribution that's close to uniform (or close to Gaussian, or something like that), but isn't? Jim Propp

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