August 2012 - mathfun@mailman.xmission.com

[math-fun] Wilf's pi and determinant problems
by Warren Smith 14 Aug '12

14 Aug '12

> Message: 4 > Date: Tue, 14 Aug 2012 02:29:42 -0700 > From: Bill Gosper <billgosper(a)gmail.com> > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] Wilf's pi and determinant problems > Message-ID: > <CAA-4O0H75k=nXijRFLy3drYa0yz2BBk=LJ1nr0x-6HpBO8h1fA(a)mail.gmail.com> > Content-Type: text/plain; charset=UTF-8 > > WDS> > Herb Wilf recently died. > Here are two open problems he posed. > > 1. Is there a series for pi=a0+a1+a2+... > where a(n) is a rational function of n and a(n-1), > which converges faster than geometrically? > Equivalently, is there a hypergeometric entire function F(z) > with F(1)=pi? > --------------- > I think you want a0 algebraic. Also, what's special about z=1? > But I'm still confused. Suppose a0:=1, a(n):=a(n-1)^2/(n+a(n-1)). > What's F? > --rwg > > Do we even have a (presumably AGMish) recurrence a0 = algebraic, > a(n) = algebraicfn(n,a(n-1)), a(?)=?? ---I partially mis-worded the pi problem -- should have said the ratio a(n)/a(n-1) was rational function of n, and a(0)=rational. If none of the pi-experts here can come up with a positive solution, which would be the easy way to solve it, then it is probably a hard problem. One relevant piece of maths is "Siegel E-functions" http://en.wikipedia.org/wiki/E-function Wilf had a web page of "Herb's open problems" here: http://www.math.upenn.edu/~wilf/website/UnsolvedProblems.pdf and the 2 problems I gave were his #1 and #7. --- Re his #7, for which I solved the "is f(n) of superexponential growth?" part (answer: no, I have exponential lower & upperbounds) I can add the following small addendum: lim f(n)^(1/n) exists and is a number C>1, presumably about 3.85. Proof: because f(n) obeys the property f(a+b) >= f(a) * f(b) by partitioning the nXn matrix into an aXa and bXb block along the (wrong way) diagonal, ignoring the two aXb leftover blocks. Essentially, this stops C from decreasing which in view of my upper bound C<=6.75 forces a limit to exist. This also re-proves my lower bound C>=1338076^(1/13)>2.9598. Q.E.D. I have thought of minor improvements which slightly tighten both my lower & upper bounds, but they are hardly worth describing. A pretty large improvement to the upper bound 6.75 ought to be possible. I have no idea how to find the exact limit value of C. I will write the Univ.Penn. maths dept asking them if they have any idea why Wilf posed his #7.

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[math-fun] Wilf's pi and determinant problems
by Bill Gosper 14 Aug '12

14 Aug '12

WDS> Herb Wilf recently died. Here are two open problems he posed. 1. Is there a series for pi=a0+a1+a2+... where a(n) is a rational function of n and a(n-1), which converges faster than geometrically? Equivalently, is there a hypergeometric entire function F(z) with F(1)=pi? --------------- I think you want a0 algebraic. Also, what's special about z=1? But I'm still confused. Suppose a0:=1, a(n):=a(n-1)^2/(n+a(n-1)). What's F? --rwg Do we even have a (presumably AGMish) recurrence a0 = algebraic, a(n) = algebraicfn(n,a(n-1)), a(∞)=π? That MathJax can't be any good. http://www.mathjax.org/demos/mathml-samples/ left the 1/2πi out of Cauchy's integral formula.

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[math-fun] Wilf's pi and determinant problems
by Warren Smith 13 Aug '12

13 Aug '12

Herb Wilf recently died. Here are two open problems he posed. 1. Is there a series for pi=a0+a1+a2+... where a(n) is a rational function of n and a(n-1), which converges faster than geometrically? Equivalently, is there a hypergeometric entire function F(z) with F(1)=pi? 2. Understand the asymptotic behavior of the sequence http://oeis.org/A019447 counting the number of terms in the expansion of an NxN toeplitz-matrix determinant. ----- I think I see a way to solve problem #2 to the extent of proving it grows ultimately exponentially but NOT superexponentially. Argue the number of terms is upper bounded by the number of ways to put N indistinguishable balls into 2N-1 boxes where the Kth box must contain at most min(K, 2N-K) balls. This in turn is upper bounded by the unrestricted #ways to put N indistinguishable balls into 2N-1 boxes, which is (3N-1 choose 2N-2) = (3N-1 choose N+1), which is upper bounded by an exponential function of N, specifically C^N for any C>6.75. You can also prove a lower bound on the #terms by simply chopping the NxN matrix into 13x13 diagonal blocks (ignoring rest of matrix) thus showing C >= 1338076^(1/13) > 2.9598. This (finite upper and lower bounds above 1 on C) was the main qualitative thing Wilf wanted to know. I'm sure it is possible to get tighter upper and lower bounds, though. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] math-fun Digest, Vol 114, Issue 9
by Warren Smith 13 Aug '12

13 Aug '12

On 8/13/12, math-fun-request(a)mailman.xmission.com <math-fun-request(a)mailman.xmission.com> wrote: > Send math-fun mailing list submissions to > math-fun(a)mailman.xmission.com > > To subscribe or unsubscribe via the World Wide Web, visit > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > or, via email, send a message with subject or body 'help' to > math-fun-request(a)mailman.xmission.com > > You can reach the person managing the list at > math-fun-owner(a)mailman.xmission.com > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of math-fun digest..." > > > Today's Topics: > > 1. they have found 5 boxes of notes of Alexandre Grothendieck > (Simon Plouffe) > 2. Re: they have found 5 boxes of notes of Alexandre > Grothendieck (Dan Asimov) > 3. Re: peculiar 2x2 products (Bill Gosper) > 4. MathJax: Beautiful math in all browsers (Henry Baker) > 5. Re: MathJax: Beautiful math in all browsers (Robert Smith) > 6. Re: MathJax: Beautiful math in all browsers (Joerg Arndt) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Sun, 12 Aug 2012 13:12:00 +0200 > From: Simon Plouffe <simon.plouffe(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] they have found 5 boxes of notes of Alexandre > Grothendieck > Message-ID: <50278F80.1070607(a)gmail.com> > Content-Type: text/plain; charset=ISO-8859-1; format=flowed > > > > Hello, > > there is one big news here circulating in France > about Alexandre Grothendieck, they have found 5 > boxes, 20000 pages of documents from him, > written pages about Algebraic Geometry, > > Alexandre Grothendieck is retired (sort of) in > west-south france in the Pyren?es, for about 20 > years, he won fields medal in the 60's and then > retired completely in 68-69, refusing honnors > and money, > > There is a very good account of the life and > mathematics of him here : > http://www.liberation.fr/sciences/2012/07/01/le-tresor-oublie-du-genie-des-… > but this article is in french, I hope that google > translate will do a decent job for those speaking > english only,, > > Bonne lecture, > > Simon Plouffe > > > > > ------------------------------ > > Message: 2 > Date: Sun, 12 Aug 2012 07:44:43 -0700 > From: Dan Asimov <dasimov(a)earthlink.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] they have found 5 boxes of notes of Alexandre > Grothendieck > Message-ID: <70615F75-6EC3-41D1-A548-7358E3796766(a)earthlink.net> > Content-Type: text/plain; charset=iso-8859-1 > > Also, a two-part article about his life & work is at: > > http://www.ams.org/notices/200409/fea-grothendieck-part1.pdf > and > http://www.ams.org/notices/200410/fea-grothendieck-part2.pdf > > --Dan > > > On 2012-08-12, at 4:12 AM, Simon Plouffe wrote: > > << > Hello, > > there is one big news here circulating in France > about Alexandre Grothendieck, they have found 5 > boxes, 20000 pages of documents from him, > written pages about Algebraic Geometry, > > Alexandre Grothendieck is retired (sort of) in > west-south france in the Pyren?es, for about 20 > years, he won fields medal in the 60's and then > retired completely in 68-69, refusing honnors > and money, > > There is a very good account of the life and > mathematics of him here : > http://www.liberation.fr/sciences/2012/07/01/le-tresor-oublie-du-genie-des-… > but this article is in french, I hope that google > translate will do a decent job for those speaking > english only,, > > Bonne lecture, > > Simon Plouffe >>> > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > ------------------------------ > > Message: 3 > Date: Sun, 12 Aug 2012 18:40:33 -0700 > From: Bill Gosper <billgosper(a)gmail.com> > To: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] peculiar 2x2 products > Message-ID: > <CAA-4O0GcziOwe82GSYPJw-LkqjGAOK48X75HT-YH=yQ9kPuvbQ(a)mail.gmail.com> > Content-Type: text/plain; charset=UTF-8 > > On Fri, Aug 10, 2012 at 2:39 AM, Bill Gosper <billgosper(a)gmail.com> wrote: > >> On Mon, Jul 23, 2012 at 12:01 AM, Bill Gosper >> <billgosper(a)gmail.com>wrote: >> >>> We're used to evaluating an infinite continued fraction as a limit of >>> ratios of elements of a running product of 2x2 matrices, >>> while the actual elements of the product blow up. E.g., for 4 arctan(1) >>> >>> In[1153]:= FoldList[Dot, {{0, 4}, {1, 0}}, Table[{{2*k - 1, k^2}, {1, >>> 0}}, {k, 9}]] >>> >>> Out[1153]= {{{0, 4}, {1, 0}}, {{4, 0}, {1, 1}}, {{12, 16}, {4, 4}}, >>> {{76, >>> 108}, {24, 36}}, {{640, 1216}, {204, 384}}, >>> {{6976, 16000}, {2220, 5100}}, {{92736, 251136}, {29520, 79920}}, >>> {{1456704, 4544064}, {463680, 1446480}}, >>> {{26394624, 93229056}, {8401680, 29675520}}, {{541937664, 2137964544}, >>> {172504080, 680536080}}} >>> >>> In[1154]:= N[Divide @@ # & /@ Rest[%]] >>> >>> Out[1154]= {{4., 0.}, {3., 4.}, {3.16667, 3.}, {3.13725, 3.16667}, >>> {3.14234, 3.13725}, >>> {3.14146, 3.14234}, {3.14161, 3.14146}, {3.14159, 3.14161}, {3.14159, >>> 3.14159}} >>> >>> But here's an infinite matrix product where the elements converge >>> individually: >>> >>> Product[MatrixForm[{{(3*a^2 - 3*a*j + j^2)/(j*(-1 - 3*a + b + j)), >>> a^3/(j*(-1 - 3*a + b + j))}, >>> {1/j, 1}}], {j, b, ?}] == >>> MatrixForm[{{"?", (a^3*HypergeometricPFQ[{1 + a, 1 + a, 1 + a}, {1 + >>> b, 1 + b}, 1])/b^2}, >>> {"?", HypergeometricPFQ[{a, a, a}, {b, b}, 1]}}] >>> >>> (Product means matrix product, not Mathematica's product.) The ?s >>> converge, too, but I don't >>> know to what. Note that the determinant ? 1, but approaches 1 in a very >>> particular way. It's not >>> obvious to me how to scale a matrix to make its running product converge >>> elementwise. >>> >>> I actually have an elementwise convergent 2x2 product for the general >>> 3F2[1], but idiot >>> Mathematica 8.04 takes (literally?) forever to convert a simple sum of >>> Pochhammers to a 3F2. >>> --rwg >>> >> >> I gave up waiting and simply guessed the general case, and while >> numerically testing, >> guessed the "?"s: >> >> > I.e., the right column can be proven by path invariance, but the left is > conjectural. > > >> MProd[MatrixForm[{{((-a)*(b + c - j) + (b - j)*(-c + j))/ >> ((1 + a + b + c - d - j)*j), >> -((a*b*c)/((1 + a + b + c - d - j)*j))}, >> {1/j, 1}}], {j, e, ?}] == >> MatrixForm[ >> {{(a*b*c*HypergeometricPFQ[{1 + a, 1 + b, 1 + c}, >> {1 + d, 1 + e}, 1])/((-1 + d)*d*e), >> (a*b*c*HypergeometricPFQ[{1 + a, 1 + b, 1 + c}, >> {1 + d, 1 + e}, 1])/(d*e)}, >> {HypergeometricPFQ[{a, b, c}, {d, e}, 1]/(-1 + d), >> HypergeometricPFQ[{a, b, c}, {d, e}, 1]}}] >> >> I.e., the "?"s were just the right column divided by d-1. So we "only" >> get >> two useful results instead of four. But from a 2x2! Except the >> convergence is lousy. I'm working on it. >> > > Fail. It appears impossible to rapidly compute two pFqs with a 2x2. > They seem always to converge like 1/j. Even 2F1[z]! > > MProd[MatrixForm[{{-(((a + b - j)*z)/(j*(-1 + z))), (a*b*z)/(j - j*z)}, > {1/j, 1}}], {j, e, ?}] == > MatrixForm[{{0, (a*b*z*Hypergeometric2F1[1 + a, 1 + b, 1 + e, z])/e}, > {0, HypergeometricPFQ[{a, b}, {e}, z]}}] > > Other attempts to accelerate convergence were messy, producing, e.g. a > slow octic/cubic CF. > >> >> Here are the two Kochflake >> ring<http://gosper.org/kocharmor-2-8-colored.png> >> pictures <http://gosper.org/kocharmor-2-8-coloredmore.png> Neil withheld >> because he was embarrassed >> about the colors, which were hastily grabbed from a Paint palate for the >> sole purpose >> of visualizing the recurrence relations. >> > > No, he didn't mouthpaint. Duh, I meant palette. > >> >> It looks to me like the boundary dimension is reducing > > > (confirmed by Julian based on > > In[125]:= N[Max[Abs[Eigenvalues[{{0, 0, 0, -2}, {1, 0, 0, 2}, {0, 1, 0, 3}, > {0, 0, 1, 0}}]]], 30] > > Out[125]= 1.87018472683652509239367973786 > ) > >> dramatically--possibly to 1?! >> --rwg >> > > Also, Julian has discovered how pentaflakes > reproduce<http://gosper.org/pentahouse.png>! > Tsk. > --rwg > > >> I think during an earlier discussion here of the necessity of algebra >> instruction I >> mentioned a childhood memory of my great aunt recounting her struggles >> filling >> out a New York State tax form after her carpenter husband neglected to >> charge >> a customer some prescribed sales tax. She was trying to compute by >> successive >> approximations what pre-tax principal he would have to charge so that the >> added >> tax totaled what he actually charged. Ironically, I had already read her >> sister's >> (my grandmother's) highschool algebra text (which I found in a carton), >> and >> could have easily solved her problem. But I said nothing, because I >> could >> tell >> from her tone that she knew I knew. >> >> The book was a plain green hardcover containing zero illustrations. I >> doubt I >> would have learned anything at all from one of those ridiculous damned >> picture books foisted on the current generation. > > > My eavesdropping teacher-cousin calls them hardcover comic books. > > >> It must come as a shock >> when today's college freshmen first open their texts and find only text. >> > > > ------------------------------ > > Message: 4 > Date: Mon, 13 Aug 2012 06:51:20 -0700 > From: Henry Baker <hbaker1(a)pipeline.com> > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] MathJax: Beautiful math in all browsers > Message-ID: <E1T0v4c-0006qV-Iv(a)elasmtp-mealy.atl.sa.earthlink.net> > Content-Type: text/plain; charset="us-ascii" > > I just came across this package, which produces very high quality math on > web pages. It's a bit slow on my cheap laptop w/Firefox, but it is pretty. > > http://www.mathjax.org/demos/ > > > > > ------------------------------ > > Message: 5 > Date: Mon, 13 Aug 2012 10:22:05 -0500 > From: Robert Smith <quadricode(a)gmail.com> > To: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] MathJax: Beautiful math in all browsers > Message-ID: <50291B9D.1070802(a)gmail.com> > Content-Type: text/plain; charset=ISO-8859-1; format=flowed > > Hey: > > This is actually what I use on my website (http://www.symbo1ics.com/blog) > > Be sure to install the STIX font set for high quality renderings using > native fonts as opposed to images! (MathJax might have moved on to using > webfonts, so even that might not be necessary.) > > The fonts are located here: > > http://sourceforge.net/projects/stixfonts/files/Current%20Release/ > > Specifically, the file: STIXv1.1.0.zip > > Cheers! > > -Robert > > On 8/13/2012 8:51 AM, Henry Baker wrote: >> I just came across this package, which produces very high quality math on >> web pages. It's a bit slow on my cheap laptop w/Firefox, but it is >> pretty. >> >> http://www.mathjax.org/demos/ >> >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > > > ------------------------------ > > Message: 6 > Date: Mon, 13 Aug 2012 19:14:43 +0200 > From: Joerg Arndt <arndt(a)jjj.de> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] MathJax: Beautiful math in all browsers > Message-ID: <20120813171443.GA8132(a)jjj.de> > Content-Type: text/plain; charset=us-ascii > > * Henry Baker <hbaker1(a)pipeline.com> [Aug 13. 2012 17:47]: >> I just came across this package, which produces very high quality >> math on web pages. It's a bit slow on my cheap laptop w/Firefox, >> but it is pretty. >> >> http://www.mathjax.org/demos/ >> > > Mathoverflow ( http://mathoverflow.net/ ) is one example, e.g. > (rather random example) > > http://mathoverflow.net/questions/17997/deriving-symmetries-of-a-gauge-theo… > >> [...] > > > > > ------------------------------ > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > End of math-fun Digest, Vol 114, Issue 9 > **************************************** > -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] MathJax: Beautiful math in all browsers
by Henry Baker 13 Aug '12

13 Aug '12

I just came across this package, which produces very high quality math on web pages. It's a bit slow on my cheap laptop w/Firefox, but it is pretty. http://www.mathjax.org/demos/

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[math-fun] they have found 5 boxes of notes of Alexandre Grothendieck
by Simon Plouffe 12 Aug '12

12 Aug '12

Hello, there is one big news here circulating in France about Alexandre Grothendieck, they have found 5 boxes, 20000 pages of documents from him, written pages about Algebraic Geometry, Alexandre Grothendieck is retired (sort of) in west-south france in the PyrenÃ©es, for about 20 years, he won fields medal in the 60's and then retired completely in 68-69, refusing honnors and money, There is a very good account of the life and mathematics of him here : http://www.liberation.fr/sciences/2012/07/01/le-tresor-oublie-du-genie-des-… but this article is in french, I hope that google translate will do a decent job for those speaking english only,, Bonne lecture, Simon Plouffe

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[math-fun] Glider in CA on Penrose tiling (Goucher)
by Warren Smith 07 Aug '12

07 Aug '12

It seems to me the essential features of the Penrose tiling are that (a) there are only a finite set of local patterns (b) the Penrose tiling can be got from a higher dimensional lattice by considering a 2D subspace of the higher space, and projecting stuff that is near to that subspace, orthogonally into the subspace. Due to (a) you can hope to devise a CA and write down only a finite number of rules to define it. Due to (b) there is some notion of "direction" and what physicists call "long-range order." (One way in which that is revealed is the Fourier transform of Penrose, which has delta functions like in diffraction from a crystal. Another is "Ammann bars.") Indeed one way Goucher could have worked (although maybe not the way he actually did it) might be to devise a CA on the higher-D lattice whose rules still worked restricted to the Penrose subset alone. If instead we were to use the Voronoi diagram of Poisson-random points in the plane, then I doubt Goucher (or anyone) could make a CA on THAT with gliders that keep going in some direction... due to properties (a) and (b) now failing...

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Re: [math-fun] Is algebra necessary?
by Ray Tayek 07 Aug '12

07 Aug '12

At 10:03 AM 7/30/2012, you wrote: >For what it is worth, I think it would be absolutely absurd to allow >getting a CS college degree without knowing calculus. i think this stuff http://www.amazon.com/Concrete-Mathematics-Foundation-Computer-Science/dp/0… is way more important and useful than calculus. thanks --- co-chair http://ocjug.org/

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[math-fun] A New Glider Found For Conway's Game of Life
by Ray Tayek 07 Aug '12

07 Aug '12

http://games.slashdot.org/story/12/08/07/1257256/a-new-glider-found-for-con… --- co-chair http://ocjug.org/

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Re: [math-fun] Olympics, tournaments, and intentionally losing
by Phil Carmody 06 Aug '12

06 Aug '12

From: Warren Smith <warren.wds(a)gmail.com> > There was recently a scandal in the Olympics where some > badminton teams were intentionally trying to lose. > > Now with certain tournament designs, such as the one they > used in the Olympics, and also "Swiss system" chess tournaments, > it is possible for situations to arise, where intentionally > losing your game, increases your expected amount of prize-rewards. > But other tournament designs are immune from this problem. ... > PROBLEM: devise other schemes... > > One example would be a "tree of round-robins" where the > players are partitioned into disjoint subsets, each subset > plays a round robin, then the top-K from each subset > enter a "round robin of champions" round. That seems to be very common. Online, when I'm playing abstract strategy games online (at littlegolem), and in real life too (tiddlywinks, don't laugh). Double-elimination seems to work well for smallish numbers of competitors. You can slip up once. The way I've seen such things organised there does seem to be chances for the same people to play each other multiple times. I'm not sure if that can be mostly avoided by clever permutation of how the losers get inserted in the lower tree. It can't be avoided entirely, as if you have 2 entrants who basically never lose to anyone else, the only way they'll be eliminated is if they play each other multiple times. If ~2N is too few matches, can double-elimination be generalised to k-times-elimination for a suitably chosen k? The double-elimination tree is at least (if you don't care about repeated matches) trivial to organise, barely more complex than single elimination. I'm not sure it generalises so easily though. Phil

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