October 2003 - mathfun@mailman.xmission.com

Re: [math-fun] Matrix Arithmetic Compendium
by Henry Baker 31 Oct '03

31 Oct '03

Knuth has lots of cool stuff. There's a large 3 volume (?) set on determinants (?) from 1880 or so, which tries to be encyclopedic. There are some cool matrix algorithms to find & characterize roots of polynomials. At 10:22 PM 10/28/03 -0700, Schroeppel, Richard wrote: >Does anyone know of a compendium of matrix arithmetic? >I'm imagining a collection of numerical & simple algebraic >facts about matrices-- things like the determinant of a >Vandermonde matrix, or the inverse of a finite section >of Pascal's Triangle. > >Rich rcs(a)cs.arizona.edu

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[math-fun] Glider
by Richard Schroeppel 30 Oct '03

30 Oct '03

Ed Pegg Jr <ed(a)mathpuzzle.com> reports Eric Raymond is advocating that open source computer programmers (hackers) use the Game of Life Glider as their emblem. http://www.catb.org/~esr/hacker-emblem/ This opens up all sorts of possibilities. Eric is claiming only the shape .o. ..o ooo There are 15 others available for other purposes or organizations. I've always been partial to ooo ..o .o. which seems more upward looking. Covert hackers might use oo. .oo o.. a suitably obscure glider. NASA could adopt .ooo o..o ...o ...o ..o. and the State of Utah might use .o. o.o o.o .o. Rich rcs(a)cs.arizona.edu

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RE: [math-fun] Simple EF
by Dan Hoey 30 Oct '03

30 Oct '03

> > > This may be completely dumb, but is it always possible to > > > solve x/y = a/b + c/d, where all algebraic quantities are > > > positive integers, for all x and y? How can we make this not completely dumb? Given that x,y are positive and relatively prime, can you give formulas for a,b,c,d that guarantee that all are positive and that (a,b)=(c,d)=1? Hmm, still pretty dumb. Suppose we also require that a,b,c,d are four different positive integers? I've gotten b and d different, but I haven't managed all four. In case that's still dumb, can all of a,b,c,d be made relatively prime in pairs? Dan

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[math-fun] MathWorld
by Jon Perry 30 Oct '03

30 Oct '03

at http://mathworld.wolfram.com/ makes the strange boast of having 0 animated GIF's and 0 live Java applets. Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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[math-fun] archimedian kaleidoscope
by Shel Kaphan 29 Oct '03

29 Oct '03

This is a nice Java applet. I've always wanted something like this: http://www.csd.uwo.ca/~morey/CogEng/AK.html

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Re: [math-fun] cyclic quadrilaterals
by reid＠math.arizona.edu 29 Oct '03

29 Oct '03

dan hoey: > By considering the point at infinity, I understand this to mean that > if we have points 01234567 in cocircular quartets 0123, 4567, 0145, > 2367, and 0246, then quartet 1357 must also be cocircular. I didn't in fact this is easy to see directly. the three angles at each vertex, suitably interpreted, add to 2 pi . use this relation at vertices 0 , 3 , 5 and 6 , along with the fact that opposite angles of a cyclic quadrilateral add to pi to see that if five of those quadrilaterals are cyclic, then so is the sixth. alternatively, use the cross ratio, which keeps track of lengths as well as angles. if k(a, b, c, d) = (a - b)(c - d) / ((c - b)(a - d)) is the usual cross ratio, then for eight distinct complex numbers, we have k(a,b,c,d) k(a,e,f,b) k(a,d,h,e) k(g,f,e,h) k(g,c,b,f) k(g,h,d,c) = 1 (the cross ratio of four distinct complex numbers is real iff the four points are concyclic.) the cross ratio is nice in that it automatically handles the "suitable interpretation". > Are all such patterns known? Say, given a mapping of a hypergraph to > the plane, where vertices are mapped to points, and the images of the > vertices of each hyperedge are cocircular, can we tell what other > subsets of vertices are forced to map to cocircular points? interesting question! i might have speculated that any such theorem (that a particular pattern holds) could be deduced by looking at the angles. but i don't see that conway's configuration (see below) can be. maybe i just didn't see it. the question of determining all such patterns seems ambitious to me. perhaps first we should try to find a collection of patterns from which all others can be built up. i'll give the first of these: if two of the quadruples ABCD , BCDE , CDEA , DEAB , EABC are concyclic, then so are the remaining three. (one needs to assume that the five points are distinct.) these extra conditions to avoid degenerate cases are annoying, and perhaps their complexity is not bounded over all such configurations. the condition above, that the five points be distinct, is not unreasonable. but conway's configuration has a different kind of restriction. john conway: ) For example, if all but one of the quadrilaterals ) AaBb,AaCc,AaDd,BbCc,BbDd,CcDd are cyclic, then so is ) that last one. there must be an extra condition required here. the condition that five of the six quadrilaterals are cyclic, and their circumcircles are all distinct, is sufficient. perhaps there's a more elegant restriction, but that seems like the most natural. (one could say that at least four of the five circumcircles are distinct, but that would imply that all five are.) mike

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[math-fun] More on EF's
by Jon Perry 29 Oct '03

29 Oct '03

A certain R. Guy sent me a chapter form his book on EF's today (for which I wish to say thanks). One of the unsolved conjectures was that if x/y is rational, that x/y = sum{i=1,n,1/a_i}, where n is finite and each a_i is minimal. It struck me that the rule could be altered, e.g. say a_1 is minimal, a_2 is minimal+1, etc... Or any other variation of the rules.... Does this alter the conjecture?, and can someone mathematically phrase my new conjecture? Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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Re: [[math-fun] Simple EF]
by Christian G. Bower 29 Oct '03

29 Oct '03

"Jon Perry" <perry(a)globalnet.co.uk> wrote: > This may be completely dumb, but is it always possible to solve x/y = a/b + > c/d, where all algebraic quantities are positive integers, for all x and y? Maybe I'm missing something here, but how about: x/y = x/(2*y) + x/(2*y) ?

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Re: [math-fun] Matrix Arithmetic Compendium
by N. J. A. Sloane 29 Oct '03

29 Oct '03

There is 30-page booklet by Marcus that has a really excellent summary of matrix theory. From MathSciNet: [4] MR0109824 (22 #709) Marcus, Marvin Basic theorems in matrix theory. Nat. Bur. Standards Appl. Math. Ser. 57 1960 iv+27 pp. (Reviewer: A. S. Householder) 15.00 For anyone with even a passing interest in the subject, this is certainly the best book bargain one could imagine! It is a compact collection of definitions, identities, inequalities, and basic theorems, grouped into six chapters, as follows: General definitions and elementary properties; Canonical forms, invariance, congruence, commutativity, orthogonalization; Eigenvalues, determinants, submatrices, rank; Determinant and rank inequalities; Linear equations $Ax=b$; Inversion of $A$; Some particular matrices and their condition numbers. Each of these is well subdivided, so that there is no great difficulty in finding what is there even though the index is perhaps less complete than one might like. While the collection is to be recommended enthusiastically, it is, nevertheless, not beyond criticism. Statements are often referenced, but by no means always. The list of 47 references is quite meager, and the selection seems a bit haphazard. Localization theorems of the Perron-Gersch-gorin-Brauer type are given, but there is no mention of the Weinstein theorem. The Givens method for computing eigenvalues of a hermitian matrix is stated after Jacobi's method, and is called "the Givens' modification", whereas the two methods have only superficial resemblance. The condition for convergence of the Jacobi iteration for solving a system of equations is incorrectly stated (top of p. 18). It is to be hoped, however, that the author will consider subsequent expansions, with appropriate revision. Reviewed by A. S. Householder NJAS

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Re: [math-fun] Matrix Arithmetic Compendium
by asimovd＠aol.com 29 Oct '03

29 Oct '03

Rich asks: << Does anyone know of a compendium of matrix arithmetic? I'm imagining a collection of numerical & simple algebraic facts about matrices-- things like the determinant of a Vandermonde matrix, or the inverse of a finite section of Pascal's Triangle. >> This is probably more advanced than you're looking for, but a wonderful book that's jam-packed with matrix facts but whose emphasis is on good robust algorithms for various matrix computations is by Golub & Van Loan, now in its nth edition where n >= 3. --Dan

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