March 2004 - mathfun@mailman.xmission.com

Re: [math-fun] Nonrepeating sequence with 3 digits
by Jeffrey Shallit 12 Mar '04

12 Mar '04

Yes. This is a variant of the Thue sequence. The easiest example is the fixed point of 2 -> 210 1 -> 20 0 -> 1 You can see this in my book _Automatic Sequences_ with Allouche, exercise 33, p. 27. Jeffrey Shallit

1 0

[math-fun] PDF article of ; a search for a mathematical expression using a large database
by Simon Plouffe 12 Mar '04

12 Mar '04

Sorry, some people can't read the fonts ? Here is the document in PDF http://www.lacim.uqam.ca/~plouffe/Search.pdf Simon Plouffe

1 0

[math-fun] Re And you think You have problems ...
by N. J. A. Sloane 12 Mar '04

12 Mar '04

> >> I have a 1Mbit file; I'm willing to add 100K additional bits >> for error correction. What should I do? > >Perhaps you could add six bits of error correcting code for each block >of 64 bits. That provides room for a code of Hamming-distance three, so >you'd get single-error correction in each block. (But double-errors >would be corrected wrong.) > >-- >Don Reble djr(a)nk.ca > We can do a little better than that! If we have length 2^20 = 1048576 and 100000 check bits, we have rate .09536. So by the Gilbert-Varshamov bound there is a binary linear code with minimum distance 12792, which will correct any error patterm with up to 6396 errors. Neil Sloane

4 3

[math-fun] apparent innumeracy in the news
by Marc LeBrun 10 Mar '04

10 Mar '04

Double take dept.: At first sight this looked like a whopper, but there's an interpretation that actually could make sense (although it also could probably have been better expressed!). ======== Berkshire Hathaway Profit Doubles By Philip Klein Sat Mar 6, 9:22 PM ET Reuters ... "In the 1980s, equity holdings were 114 percent of Berkshire's net worth, but that fell to an average of 50 percent in 2000 to 2003, Buffett said." ...

5 5

[math-fun] Calculus of variations ?
by Henry Baker 07 Mar '04

07 Mar '04

This is a problem about optimizing a particular type of human-powered racing sport. A certain type of race occurs over a number of distances and speeds. For any given distance, the conventional wisdom is that the optimum performance is gained by a constant speed over the distance. The best performances of a large number of individuals has been plotted (including a large number of international champions), and for each individual the performances over different distances match very well the formula: 1/speed = a+b*ln(distance) (a,b are positive real constants particular to the individual) i.e., the inverse of the average speed (which is also the constant speed over the course) is equal to a+b*ln(total distance), where a,b are constant parameters specialized for each individual, although the parameter "b" is almost the same for all individuals. Thus, given a particular distance -- e.g., 5000 meters -- and the constants a,b, one can compute the speed down the course as 1/(a+b*ln(5000)). Now comes the question. I'm not convinced that the best racing strategy is a constant speed down the course. My gut feeling is that the optimum strategy involves going faster in the earlier part of the course, and slower later. The problem is: given a fixed distance -- e.g., 5000 meters, and fixed constants a,b -- what is the optimum speed over the entire course, for each different meter ? Do I even have enough information to decide the question ?

1 0

[math-fun] Feeling dumb...
by Michael Kleber 03 Mar '04

03 Mar '04

I thought I'd have no trouble, but my brain seems to be substandard today. Can someone give the easy counting argument that shows the existence of irreducible polynomials of degree n over the finite field F_p? Not the one using splitting fields of x^(p^n)-x, nor the one that gives the explicit formula for the number of irreducibles but requires Mobius inversion. I was sure there was an easy nonzero lower bound on the number of irreducible polynomials, but darned if I can think of it. --Michael Kleber kleber(a)brandeis.edu

1 0

RE: [math-fun] existence of the icosahedron
by Meeussen Wouter (bkarnd) 03 Mar '04

03 Mar '04

try counting the *different* energy levels of 10 (+) and 10 (-) electrostatic charges on the (overall neutral) dodecahedron. Cleaner than permanent magnets. and it's just counting... W. -----Original Message----- From: Ed Pegg Jr [mailto:ed@mathpuzzle.com] Sent: woensdag 3 maart 2004 17:47 To: math-fun Subject: Re: [math-fun] existence of the icosahedron Icosahedra? I don't think they exist. [I've been wanting to say that.] I have lately been playing around with strong spherical magnets. One figure that they can make is a very stable icosahedron (12 spheres). A study of the poles on these spheres reveals a very striking 3-fold rotational symmetry. I've been unsuccessful in making the skew cube out of spherical magnets, so far. The dodecahedron seems to be impossible. Several structures do work, and all of them have unexpected polar qualities, so far. --Ed Pegg Jr _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun =============================== 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.

1 0

Re: [math-fun] existence of the icosahedron
by Daniel Asimov 03 Mar '04

03 Mar '04

On Tuesday, March 2, 2004, at 11:12 AM, Michael Kleber wrote: > Does anyone know the pedigree of the following line of reasoning? > I haven't defined "by continuity," but I think the intent will be clear > from the proof. [...] > There you have it: the polyhedron so constructed clearly has > twenty equilateral triangular faces meeting five at each vertex. Another proof, I believe, can be cobbled from the idea of a cube, each of whose faces has been bisected into two rectangles so that none of 6 new lines meet each other. Now each of these 6 new lines can be symmetrically shrunk toward its midpoint, while allowing the cube to hinge identically along all 6 shrinking lines. By continuity there will be a moment that each of the 12 resulting pentagons is regular, and it can be shown that at this point all dihedral angles are equal. Of course details need to be filled in, but I like this approach because of its symmetry. --Dan Asimov

3 3

Re: [math-fun] Re And you think You have problems ...
by N. J. A. Sloane 03 Mar '04

03 Mar '04

Is it effectively computable? (Revealing how little I know about coding theory...) --Michael Kleber kleber(a)brandeis.edu Me: yes, there are iterative constructions that will get close to that value. even an old-fashioned BCH code with those parameters will correct 100000/log_2(2^20) = 5000 errors. i'll send this off before TKR/Cablevision kills my connection yet again.

1 0

[math-fun] And you think You have problems ...
by Richard Schroeppel 03 Mar '04

03 Mar '04

a few tidbits -- rich ----- From: Balaji Gopalakrishnan <bgopalak(a)ima.umn.edu> To: dmanet(a)zpr.uni-koeln.de Subject: [DMANET] Question on Large-Scale LP's Date: Mon, 1 Mar 2004 21:20:18 -0600 (CST) Hi, I have an LP which has 300 million columns and 200 million rows. The basic structure is like the multicommodity flow problem with about 20 linking constraints (completely dense) and rest are block angular. The blocks are all of the same size (4 rows and 7 columns in each block with about 50 million blocks). The variables are all between 0 and 1. Can someone suggest an approach to such ultra large-scale LP problems ? Any reference to papers/articles that discuss solving such large-scale LPP's will be helpful. I am currently exploring some ideas using potential function methods for obtaining approximate solutions for this problem, but the size of problems solved in literature in relatively small when compared to this one and I am not sure how such methods will perform on such problems. Thanks in advance. -Balaji ----- I recently puzzled over the following problem: I have a 1Mbit file; I'm willing to add 100K additional bits for error correction. What should I do? My best guess is to use random parity bits, with each EC bit based on maybe 100 randomly selected bits from the file. There's some hope that an iterative-improvement decoder would converge if the bit error rate was < 1%. Computing the parity bits is slow. ----- Hilarie has suggested that Feb 28 be called Look Day.

2 1