March 2007 - mathfun@mailman.xmission.com

[math-fun] ACRONYMS : A Common Representation Of Names You Must Shorten
by Eric Angelini 22 Mar '07

22 Mar '07

Hello SeqFans & Math-Fun, If those two seq. are of interest, could someone help in providing more terms? ------------------------ First digit of a(n) is given by the succession of digits in S [a(n+1) is the smallest available integer not yet present in S]: S= 0,1,2,3,4,5,6,7,8,9,11,12,13,21,14,31,22,15,16,41,32,17,23,24,17,51,18,61... ------------------------ First digit of a(n) is given by the succession of digits in S [S is monotonicaly increasing]: S= 0,1,2,3,4,5,6,7,8,9,11,12,13,21,111,311,2111,11111,11112,11113,11114,31111,... ------------------------ Best, É.

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[math-fun] musing with exp(Pi) , Pi and genealized continued fractions.
by Simon Plouffe 22 Mar '07

22 Mar '07

Hello, We all know that Pi and exp(Pi) have some particularities like exp(Pi)-Pi = 19.9990999... apparently found somewhere in 1988 by 3 different people... but that's not the only one, I have found this amusing continued fraction for Pi and exp(Pi), Pi - 3 = [ 1/11,-11/3, -2/12, -11/6, -5/6, -5/3, -2/6, ... ....] in continued fraction expansion. exp(Pi)-23 = [1/11, -11/3, -2/12, -11/24, -23/12, -11/24, -23/14, -13/12, -11/42,...]. There is a striking resemblance, between the two. Another one is Pi-3 = [1/11,-11^2/34, -23^2/172, -149^2/2176, ...] exp(Pi)-23= [1/11,-11^2/34, -23^2/186, -163^2/16042,...] This output comes from an algorithm I developed to <force> prime numbers in the numerator of the gen. continued fraction of some numbers. I computed the expansion of many numbers and these 4 examples have a kind of interesting pattern. Simon Plouffe

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[math-fun] (doubtless) old segment trick
by R. William Gosper 20 Mar '07

20 Mar '07

Given pts A,B in n-space, instead of representing the segment AB by tA + (1-t)B and later checking 0<t<1, just use uA + B, scaling whatever gets added by u+1, and later check u>0. E.g., to see if AB intersects the plane thru P with normal N, solve for u N.(uA+B)=(u+1)N.P, where N.P can be precomputed. The actual intersection is (uA+B)/(u+1). --rwg GREAT PLANES PLANET GEARS PS, apropos Jeopardy!, the TV at Deedees Vegetarian Buffet last nite showed Crorepati, an exact homologue of Philbin's Millionaire show (perhaps minus the lifelines). But winnings were displayed as Rs x,xx,xxxxx instead of $ x,xxx,xxx. (And the quizmaster can break into song.) The math question seemed easy enough: Which of the following is greater than 86? <Four strange words ending in "asi".> Oops.

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[math-fun] Atlas of Finite Groups query
by Fred lunnon 20 Mar '07

20 Mar '07

The Atlas of Finite Groups at http://brauer.maths.qmul.ac.uk/Atlas/ has been the subject of my recent (somewhat apprehensive) investigations. As expected, it proves to make few concessions to the neophyte. In particular, I would be grateful if some kind group-theorist could enlighten me as to the meaning of "19:9" and similar (modulo typo) expressions, occurring frequently therein, but nowhere defined! Fred Lunnon

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[math-fun] E8 : lie group breakthrough
by Simon Plouffe 20 Mar '07

20 Mar '07

Hello, this just came out, http://news.google.com/news?hl=en&q=e8&btnG=Google+Search&ie=UTF-8&oe=UTF-8… it appears in a number of sites already. What is it that we could not do up to now in fact? Simon Plouffe

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[math-fun] Meta analysis spoof
by Henry Baker 19 Mar '07

19 Mar '07

FYI -- If you haven't seen or heard of this paper, it is hilarious... Parachute use to prevent death and major trauma related to gravitational challenge: systematic review of randomised controlled trials http://www.bmj.com/cgi/content/full/327/7429/1459

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Re: [math-fun] bad pun
by Daniel Asimov 18 Mar '07

18 Mar '07

Not sure if I'd ever heard of the Szilassi polyhedron before, but (for those in the same boat) it has 7 (nonconvex) polygonal faces each of meets the other 6, is topologically a torus, and it embeds in 3-space. My favorite torus polyhedron has 7 regular hexagons as faces, any two of which meet each other along an edge.* This embeds isometrically in R^6 (so that its 84 symmetries are all in the orthogonal group O(6)). Does anyone know for sure whether 6 is the lowest possible dimension for this? --Dan _______________________________________________________ * Which is of course the nicest way to 7-color a torus.

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[math-fun] ..and of course
by wouter meeussen 18 Mar '07

18 Mar '07

k=1;NestList[1+1/((k++)#)&,1,12] gives {1,2,5/4,19/15,91/76, ..., A052169(n+1)/A002467(n) } if I got my offsets correct. grin, W.

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[math-fun] Mousetrap and "Kurepa hypothesis for left factorial"
by wouter meeussen 18 Mar '07

18 Mar '07

looking at Sloane's A052169 (Kurepa hypothesis for left factorial) and A002467 (game of Mousetrap), is it self-evident that [[1]] a(n)=1+1/(x (k++) a(n-1)) , a(1)=x, for x->1 produces the same sequence as [[2]] a(2) = 1, a(3) = 2, a(n) = (n-2)*a(n-1) + (n-3)*a(n-2) ? Expanding [[1]], with the usual meaning of k++ as post-increment, we get: k=1;NestList[1+1/((k++)x #)&,x,3] {x, 1 + 1/x^2, 1 + 1/(2*(1 + 1/x^2)*x), 1 + 1/(3*(1 + 1/(2*(1 + 1/x^2)*x))*x)} Taking only the numerators gives: {x, 1 + x^2, 2 + x + 2*x^2, 2 + 6*x + 5*x^2 + 6*x^3, 14 + 27*x + 26*x^2 + 24*x^3, 8 + 94*x + 155*x^2 + 154*x^3 + 120*x^4, 118 + 699*x + 1060*x^2 + 1044*x^3 + 720*x^4, 48 + 1390*x + 5823*x^2 + 8344*x^3 + 8028*x^4 + 5040*x^5, ... set x to 1 and you get A052169 in a different way. As a 'limping' triangular table, the coefficient of the highest power in x stands out as n!, and the definition of A052169 as pairwise sums of A002467 (itself equal to n!-!n) is curious. {0,1}, {1,0,1}, {2,1,2}, {2,6,5,6}, {14,27,26,24}, {8,94,155,154,120}, {118,699,1060,1044,720}, {48,1390,5823,8344,8028,5040}, {1210,16013,54004,74060,69264,40320}, {384,22010,190701,552788,730764,663696,362880}, {14730,364217,2393046,6194420,7931016,6999840,3628800}, {3840,382130,5913397,31851386,75446260,93878136,80627040,39916800}, {208110,8591947,97284270,450355252,992596296,1203535872,1007441280,479001600 } maybe not math-funny to all, but quite amusing to at least one, W.

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[math-fun] counting set systems
by N. J. A. Sloane 17 Mar '07

17 Mar '07

Dear Mathfun, Seqfans: Numbers of set systems of various types --------------------------------------- There are some sequences that may be missing from the OEIS, in case somebody would be interested in calculating them. This was suggested by reading pages 1 and 2 of the book by Heinz Koenig on Measure and Integration (Springer, 1997) We are considering a "set system" S, which is a nonempty collection of subsets A of a set X, where X has n points. There will be two versions of each sequence, depending on whether the points are labeled or unlabeled. We define several properties which S might have: U : A, B in S => A U B in S I : A, B in S => A intersect B in S \ : A, B in S with A contained in B => B\A in S - : A in S => complement of A in X is in S 0 : S contains the empty set 1 : S contains the universe set X We also define a new operation on subsets of X: |P|A|Q| = (P intersect complement of A) union (Q intersect A) [points in P but not in A plus points in both Q and A] Now Koenig defines 4 types of set systems: a lattice : properties U, I an oval : P,A,Q in S => |P|A|Q| in S a ring : properies U, I, \ an algebra : properties U, I, - The question is, how many lattices, ovals, rings, and algebras are there on n labeled/unlabeled points? (In fact one might consider any of the 2^7 combinations of the above 7 properies!) There are of course many sequences like these in the OEIS already, but usually there is no precise definition (this should be corrected!) so it is hard to tell which of these may be new. For example: posets: A000112 (unlabeled), A001035 (labeled) Lattices: A006966 (unlabeled); A055512 (labeled) Boolean lattices: A005493 Set systems of various types, studied recently by Mitch Harris and Don Knuth: A102894-A102897. E.g.: %I A102895 %S A102895 2,2,8,90,4542,2747402,151930948472 %N A102895 Number of ACI algebras or semilattices on n generators, with no identity element. %C A102895 An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation. %C A102895 Or, number of families of subsets of {1, ..., n} that are closed under intersection and contain the empty set. - this seems to count subsets with properties I, 0 in the labeled case %I A102897 %S A102897 2,4,14,122,4960,2771104,151947502948 %N A102897 Number of ACI algebras (or semilattices) on n generators. %C A102897 Also counts Horn functions on n variables, boolean functions whose set of truth assignments are closed under 'and', or equivalently, the boolean functions that can be written as a conjunction of Horn clauses, clauses with at most one negative literal. %C A102897 Also, number of families of subsets of {1,...,n} that are closed under intersection (because we can throw in the universe, or take it out, without affecting anything else). %C A102897 An ACI algebra or semilattice is a system with a single binary, idempotent, commutative and associative operation. - this seems to count subsets with property I in the labeled case ? Likewise A102896 seems to count subsets with properties I, 1 in the labeled case and A102894 seems to count subsets with properties I, 0, 1 in the labeled case (see the full entries for more details) Neil

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