March 2003 - mathfun@mailman.xmission.com

[math-fun] universe as doughnut
by Shel Kaphan 12 Mar '03

12 Mar '03

Just FYI, a math-fun member is mentioned in an article in yesterday's NY Times, Science Times section: Universe as Doughnut: New Data, New Debate http://www.nytimes.com/2003/03/11/science/space/11COSM.html

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[math-fun] Extension of a theorem on n-gons
by Richard Schroeppel 12 Mar '03

12 Mar '03

Steve Gray posted the message below. It was intercepted (too big), and Steve has offered to send the PDF of his paper to interested parties. His address is "Steve Gray" <stevebg(a)adelphia.net> The paper is about placing similar figures on the edges of a polygon, such as the construction of equilateral triangles on the edges of a starting triangle. Rich rcs(a)cs.arizona.edu ---------------------- Steve Gray> Some of you may be interested in a paper of mine in the March 2003 American Mathematical Monthly, which arrived by mail here today. It's a significant extension of a result that's almost 100 years old, one that has been described as "remarkable" and "beautiful." I enclose a PDF file of the paper in its final format. This is the first of a planned series of papers on further generalizations of this theorem in plane geometry. These theorems apply to n-gons, where n > 2. Proofs are done with elementary matrix theory. For the most part, the results are harder to find than to prove, and while I have two or three more generalizations established and proved, I have been working to characterize and delimit the parameters for two other conjectures for at least six months. I didn't think there was that much left in "ordinary Euclidean" geometry.

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[math-fun] Simple statement equivalent to the Riemann Hypothesis
by asimovd＠aol.com 12 Mar '03

12 Mar '03

The Riemann Hypothesis, often declared to be the number one outstanding unsolved question in mathematics, usually takes some explaining to describe accurately to a layperson. I like the equivalence, recently proved by Jeffrey Lagarias of AT&T Research, of the Riemann Hypothesis to the following Statement: Statement: ------------------------------------------------------------------------------ ------------------------------------------ For any n = 1,2,3,... let sigma(n) = the sum-of-divisors function (so that for example sigma(6) = 12), and let H_n = 1 + 1/2 + 1/3 + ... + 1/n be the nth harmonic number. Let log denote log_e. Then for all n = 1,2,3,..., (*) sigma(n) <= H_n + exp(H_n) * log(H_n) ------------------------------------------------------------------------------ ------------------------------------------. Lagarias adds the words "with equality only for n = 1". I suspect it's easy to prove that for n > 1 the RHS of (*) is never an integer, but maybe a number theorist out there can confirm this. --Dan A.

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Re: [math-fun] Simple statement equivalent to the Riemann Hypothesis
by asimovd＠aol.com 12 Mar '03

12 Mar '03

For those who expressed interest in pursuing this further, here's a reference sent by Neil Sloane: << Note the following sequence %I A057641 %S A057641 0,0,1,0,4,0,7,2,7,5,13,0,17,9,12,8,23,5,27,8,21,20,34,1,33,25, %T A057641 30,17,46,7,50,22,40,37,46,6,62,43,50,19,70,19,74,37,46,55,82,9, %U A057641 79,46,70,47,95,32,83,38,81,74,107,2,112,81,76,56,102,45,125,70 %N A057641 Floor(H(n)+exp(H(n))*log(H(n))) - sigma(n), where H(n) = Sum_{k=1..n} 1/k and sigma(n) (A000203) is the sum of the divisors of n. %C A057641 Showing this is nonnegative is equivalent to proving the Riemann hypothesis. %H A057641 J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, Am. Math. Monthly 109 (#6, 2002), 534-543. <http://arxiv.org/abs/math.NT/0008177> %Y A057641 Cf. A057640, A000203, A076633. %K A057641 nonn,nice,easy %O A057641 1,5 %A A057641 njas, Oct 12 2000 >> --Dan

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[math-fun] Re: Your message to math-fun awaits moderator approval
by Steve Gray 11 Mar '03

11 Mar '03

Rich, I forgot that enclosures are a no-no. If anyone wants the PDF, they can just ask me & I'll send it directly. Thanks, Steve

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[math-fun] Re 'A near vector space'
by Guy Haworth 10 Mar '03

10 Mar '03

My apologies to Fred Helenius: I missed his answer to my question, which was the same as Noam Elkies'. guy

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[math-fun] non-adjacent :: final + insight
by Meeussen Wouter (bkarnd) 10 Mar '03

10 Mar '03

a chessboard shows it all in an instant : (just found it, thanks COS) http://www.theory.csc.uvic.ca/~cos/inf/setp/SetPartitions.html sorry to have bothered you all, Wouter (Blush...) =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached.

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[math-fun] non-adjacent, again ...
by Meeussen Wouter (bkarnd) 10 Mar '03

10 Mar '03

Set Partitions excluding singletons (sets of size one) are counted by N. Sloane and Christian G. Bower (bowerc(a)usa.net) in A000296 (Formerly M3423 and N1387) Sequence : 1, 0, 1, 1, 4, 11, 41, 162, 715, 3425, 17722, 98253 (with Bell[n]=a[n-1]+a[n-2] ) But, Set Partitions excluding neighbours (k excludes k+/-1 in partition) where 1 is taken to neighbour n (making it cyclic) are also counted by A000296. Anyone for a translation from one to the other? no_singletons & cyclic_neighbours_excluded {{{1, 2, 3, 4, 5}} & {{1}, {2, 4}, {3, 5}}}, {{{1, 2}, {3, 4, 5}} & {{1, 4}, {2}, {3, 5}}}, {{{1, 2, 3}, {4, 5}} & {{1, 3}, {2, 4}, {5}}}, {{{1, 4, 5}, {2, 3}} & {{1, 3}, {2, 5}, {4}}}, {{{1, 3}, {2, 4, 5}} & {{1, 4}, {2, 5}, {3}}}, {{{1, 5}, {2, 3, 4}} & {{1}, {2}, {3, 5}, {4}}}, {{{1, 2, 5}, {3, 4}} & {{1}, {2, 4}, {3}, {5}}}, {{{1, 3, 4}, {2, 5}} & {{1}, {2, 5}, {3}, {4}}}, {{{1, 4}, {2, 3, 5}} & {{1, 3}, {2}, {4}, {5}}}, {{{1, 2, 4}, {3, 5}} & {{1, 4}, {2}, {3}, {5}}}, {{{1, 3, 5}, {2, 4}} & {{1}, {2}, {3}, {4}, {5}}} Playfully yours, Wouter =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached.

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[math-fun] A near vector space - example
by Guy Haworth 10 Mar '03

10 Mar '03

To answer my own question, Noam Elkies suggested: K = C(omplex Numbers); one-dimensional V = K Define k * [v] = [Re(k).v], the . being multiplication in K Then all axioms are satisfied except (k1.k2) * v = k1 * (k2 * v) Are there sources on the web of neat examples like this where the independence of the axioms is demonstrated? I know of Gelbaum/Olmstead books I (Analysis) & II. - guy

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Re: [math-fun] Why is e the "best" base? + A symmetrical way to treat digits 0,1,2 base e
by Henry Baker 09 Mar '03

09 Mar '03

Here's the original paper, with corrections: http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html At 09:23 AM 3/7/03 -0800, Marc LeBrun wrote: >I'll try to dig out my copy of Shannon's original paper and see what he said.

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