March 2010 - mathfun@mailman.xmission.com

[math-fun] Fwd: NYTimes.com: Algebra in Wonderland
by Paul Reiners 09 Mar '10

09 Mar '10

I'm curious what people think of this article: http://www.nytimes.com/2010/03/07/opinion/07bayley.html OPINION | March 07, 2010 Op-Ed Contributor: Algebra in Wonderland By MELANIE BAYLEY The other-worldly events in "Alice's Adventures in Wonderland" can be interpreted as satire on 19th-century advances in mathematics.

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Re: [math-fun] Interesting theorem on graph theory
by Tom Karzes 07 Mar '10

07 Mar '10

Wouldn't a complete graph with k nodes be n-arc symmetric for any n less than k? Tom > I've been recently doing research on highly symmetric > tilings of surfaces, which led to learning of an > interesting theorem in graph theory (Weiss, 1974): > > Define a graph G be as a set V of nodes each pair > of which either is or is not connected by an edge. > We identify the pair of nodes with the edge. > > Let an n-arc on G be defined as a sequence of distinct > nodes v_j, 0 <= j <= n, such that {v_(j-1),v_j} is an > edge for 1 <= j <= n. > > Define a graph to be "n-arc symmetric" if, > for any n-arcs A and B of G, > there is a graph isomorphism G -> G taking A onto B. > > Theorem: If G is n-arc symmetric, then n <= 7. > > Anyone know anything about this theorem? The only > write-up I could find is the original paper, in > German, not a language I'm strong in: > > Weiss, R. M. "Ãber s-regulÃ¤re Graphen." > J. Combin. Th. Ser. B 16, 229-233, 1974. > > (Something I read said this theorem depends on > the classification of the sporadic simple groups. > But since I think that was completed after 1974, > perhaps the universality of this theorem became > known only after the sporadic simple groups > were known?) > > --Dan > > ________________________________________________________________________________________ > "Outside of a dog, a book is man's best friend. Inside of a dog, it's too > dark to read." --Groucho Marx > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > >

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[math-fun] Interesting theorem on graph theory
by Dan Asimov 05 Mar '10

05 Mar '10

I've been recently doing research on highly symmetric tilings of surfaces, which led to learning of an interesting theorem in graph theory (Weiss, 1974): Define a graph G be as a set V of nodes each pair of which either is or is not connected by an edge. We identify the pair of nodes with the edge. Let an n-arc on G be defined as a sequence of distinct nodes v_j, 0 <= j <= n, such that {v_(j-1),v_j} is an edge for 1 <= j <= n. Define a graph to be "n-arc symmetric" if, for any n-arcs A and B of G, there is a graph isomorphism G -> G taking A onto B. Theorem: If G is n-arc symmetric, then n <= 7. Anyone know anything about this theorem? The only write-up I could find is the original paper, in German, not a language I'm strong in: Weiss, R. M. "Über s-reguläre Graphen." J. Combin. Th. Ser. B 16, 229-233, 1974. (Something I read said this theorem depends on the classification of the sporadic simple groups. But since I think that was completed after 1974, perhaps the universality of this theorem became known only after the sporadic simple groups were known?) --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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Re: [math-fun] Correction re Math funnies, lol
by Dan Asimov 05 Mar '10

05 Mar '10

Hilarie wrote: << http://xkcd.com/710/ It's worth the wait! >> Agreed. This guy is great. I've found it rewarding to browse his past comics, which can sometimes get rather edgy and are always endlessly inventive. --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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[math-fun] Math funnies, lol
by Hilarie Orman 05 Mar '10

05 Mar '10

http://imgs.xkcd.com/comics/710 Who is this xkcd guy? Is he fun to hangout with? Hilarie

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[math-fun] Correction re Math funnies, lol
by Hilarie Orman 05 Mar '10

05 Mar '10

Sorry for getting this wrong the first time ... http://xkcd.com/710/ It's worth the wait! Hilarie

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[math-fun] sum 1/fib
by Joerg Arndt 04 Mar '10

04 Mar '10

Surely known, but maybe worthwhile for Eric's mathworld: sum(n=1,infty,1.0/fibonacci(2*n-1)) == sqrt(5)/4*( Th22(be) ) where Th22(q)=4*q * sum( n=0,infty,q^(2*(n^2+n)) )^2 \\ ==Theta2^2(q^2) and be = (sqrt(5)-1)/2; \\ == 0.61803398... sum(n=1,infty,1.0/fibonacci(2*n)) == sqrt(5)*( La22(be) ) where La22(q)=sum(n=1,infty,q^(n*(n+1))/(1-q^(2*n))) \\ Lambert, cf. below and be as before I cannot, however, combine these two into anything neater than just the plain sum (how can one marry Theta with Lambert?). cheers, jj P.S.: some pari/gp code for checking and some additional relations: \\ -------------------------------- default(realprecision,100); default(format,"g.15"); default(echo,1); N=500 \\ sum N terms with linear series S=ceil(sqrt(N)) \\ sum S terms with faster (Theta-type) series Th3(q)=1+2*sum(n=1,S,q^(n^2)) \\ Theta3 Th4(q)=Th3(-q) \\ Theta4 Th2(q)=2*q^(1/4)*sum(n=0,S,q^(n^2+n)) \\ Theta2 \\Th22(q)=(Th3(q)^2-Th4(q)^2)/2 \\ == Theta2^2(q^2) Th22(q)=4*q * sum( n=0,S,q^(2*(n^2+n)) )^2 \\ Theta2^2(q^2) \\La(q)=sum(n=1,N, q^n/(1-q^n)) \\ Lambert series La(q)=sum(n=1,S, q^(n^2)*(1+q^n)/(1-q^n)) \\ Lambert series \\La2(q)=sum(n=1,N,q^(2*n-1)/(1-q^(2*n-1))) \\ Lambert series (odd) \\La2(q)=sum(n=1,N,q^n/(1-q^(2*n))) \\ Lambert series (odd) La2(q)=sum(n=1,S,q^(n*(n+1)/2)/(1-q^n)) \\ Lambert series (odd) La22(q)=sum(n=1,S,q^(n*(n+1))/(1-q^(2*n))) \\ == La2(q^2) T=(1+sqrt(5))/2 \\ == golden == 1.61803398874989... \\vector(10,n,1/sqrt(5)*(T^n-(-1/T)^n)) \\ == Fibs \\be = -(1-sqrt(5))/2; \\ == 1/T == 0.61803398... \\be = 2/(1+sqrt(5)); \\ == 1/T == 0.61803398... be = (sqrt(5)-1)/2; \\ == 1/T == 0.61803398... \\vector(10,n,1/sqrt(5)*(1/be^n-(-be)^n)) \\ == Fibs " ----- EVEN:"; Fe=sum(n=1,N,1.0/fibonacci(2*n)) \\ == 1.53537050883625 Fe - sqrt(5)*sum(n=1,N, T^(2*n)/(T^(4*n)-1) ) \\ == zero Fe - sqrt(5)*( La(T^(-2)) - La(T^(-4)) ) \\ even Fe - sqrt(5)*( La(be^2) - La(be^4) ) \\ == zero Fe - sqrt(5)*( La2(be^2) ) \\ == zero Fe - sqrt(5)*( La22(be) ) \\ == zero " ----- ODD:"; Fo=sum(n=1,N,1.0/fibonacci(2*n-1)) \\ == 1.82451515740692 Fo - sqrt(5)*sum(n=1,N, T^(2*n-1)/(T^(4*n-2)+1) ) \\ == zero Fo - sqrt(5)/4*( Th3(T^(-1))^2 - Th3(T^(-2))^2 ) \\ == zero Fo - sqrt(5)/4*( Th3(be)^2 - Th3(be^2)^2 ) \\ == zero Fo - sqrt(5)/4*( Th2(be^2)^2 ) \\ == zero Fo - sqrt(5)/4*( Th22(be) ) \\ == zero " ----- BOTH:"; F1=sum(n=1,N,1.0/fibonacci(n)) \\ sum 1/Fib == 3.35988566624318... F1 - (Fe+Fo) \\ == zero F1 - sqrt(5)*sum(n=1,N, 1.0/(T^n-(-1/T)^n) ) \\ == zero F1 - sqrt(5)*sum(n=1,N, T^n/(T^(2*n)-(-1)^n) ) \\ == zero F1 - sqrt(5)*(Th22(be)/4 + La22(be)) \\ == zero \\ --------------------------------

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[math-fun] Prime question
by David Wilson 02 Mar '10

02 Mar '10

Can a prime >= 10 have all prime digits but no longer prime substring?

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[math-fun] Dik Winter passed away.
by Warut Roonguthai 01 Mar '10

01 Mar '10

>From Samuel Wagstaff's Page 114 of Update to the Cunningham Project: "Dik Winter passed away on December 28, 2009. He was part of CWI since 1969. As a programmer, he was part of the teams that set records for factoring large integers by the quadratic sieve in the 1980s and by the special and general number field sieves in the 1990s. He was the first to code the APR-CL primality test. He was also known for writing programs to solve Rubik's Cube and to compute pi to many decimal places by the spigot algorithm." Warut

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Re: [math-fun] What is the name of [mod 9 digitsum]
by rcs＠xmission.com 01 Mar '10

01 Mar '10

This is from Paul Payton. -- Rich ------------------------------------------ Date: Thu, 25 Feb 2010 10:43:05 -0700 Subject: RE: What is the name of [mod 9 digitsum] From: "Payton, Paul" <paul.payton(a)lmco.com> I would call it "additive persistence". http://en.wikipedia.org/wiki/Persistence_of_a_number -----Original Message----- From: math-fun-bounces(a)mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Eric Angelini Sent: Thursday, February 25, 2010 9:29 AM To: math-fun Subject: [math-fun] What is the name of [mod 9 digitsum] Hello Math-Fun, is there a name for this operation : - start with 178 for instance; - add the digits: 1+7+8 => 16 - add the digits again: 1+6 => 7 Does the operation going from 178 to 7 have a name (other than "modulus 9 digitsum")? Thank you, Best, É.

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