- mathfun@mailman.xmission.com - mailman.xmission.com

Re: [math-fun] Compute a Function From its Fourier Series?
by Eugene Salamin 24 Sep '07

24 Sep '07

----- Original Message ---- From: Mike Stay <metaweta(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Sent: Monday, September 24, 2007 9:00:01 AM Subject: Re: [math-fun] Compute a Function From its Fourier Series? On 9/23/07, mcintosh(a)servidor.unam.mx <mcintosh(a)servidor.unam.mx> wrote: > 2) The variance (or better, the standard deviation) of a function > is inverse to the variance of its Fourier Transform. Thus, many > large coefficients are needed to confine a function to a small > region and conversely. Which is a general form of Heisenberg's uncertainty principle. -- Mike Stay metaweta(a)gmail.com http://math.ucr.edu/~mike It is not correct to refer to this as "Heisenberg's" uncertainty principle. Heisenberg's uncertainty relates the variances of position and momentum. In quantum mechanics there is a connection between momentum and spatial frequency, and so the Fourier uncertainty principle can then be applied. The Heisenberg principle is exclusively quantun mechanical in nature. A classical electromagentic pulse has an exactly calculable momentum and an exactly calculable center of mass position. This is true because the energy density (E^2 + B^2) and momentum density (E x H) depend only on the fields, and not on the frequency or wavelength. Gene ____________________________________________________________________________________ Fussy? Opinionated? Impossible to please? Perfect. Join Yahoo!'s user panel and lay it on us. http://surveylink.yahoo.com/gmrs/yahoo_panel_invite.asp?a=7

1 0

[math-fun] Compute a Function From its Fourier Series?
by Robert Baillie 24 Sep '07

24 Sep '07

One of the first examples of Fourier series that a student encounters is something like this: f(x) = x/2, for -Pi < x < Pi. The Fourier series is Sum[ (-1)^(n+1)/n * Sin[n x] ]. A second, more complicated example is: f(x) = -Pi/2, for -Pi < x < 0, f(x) = Pi/2, for 0 < x < Pi. This Fourier series is Sum[ (1 - (-1)^n)/n * Sin[n x] ]. Can one work backwards, from the coefficients to the function? That is: given an expression in n, say, c(n) = (-1)^(n+1)/n, can one find the function that has c(n) as its coefficients? This seems difficult, especially because in the second example, the function consists of two distinct pieces over [-Pi, Pi], and as part of the calculation, you'd have to find the point (x = 0) where the function jumps. One could plot 1000 terms of Sum[ c(n) * Sin[n x] ] for -Pi < x < Pi, then try to guess what the expressions and points of discontinuity of f(x) are. Or, one could compute 1000 terms for many x's and do curve fitting, but guessing would be involved here, too. But is there a systematic method to do this? Bob Baillie

3 2

[math-fun] Problems that led to branches of science ...
by Guy Haworth 24 Sep '07

24 Sep '07

Did the 'Bridges of Koningsberg' lead to topology or just to Graph Theory? I was at a talk on Euler by Prof. Im Hof, and he didn't mention Topology in this context. Guy

1 0

[math-fun] Re: Jaggies
by Steve Witham 22 Sep '07

22 Sep '07

>From: "R. William Gosper" <rwg(a)osots.com> > >http://gosper.org/jaggies.png is a 4x blowup of a detail of a Golly >(Game of Life) >screen, where each pixel is 2^24 x 2^24 cells. The diagonal line >marked by arrows >is a row of equally spaced gliders, (dx,dy) = (14412108,16685960). The gap at >the green arrow is a legitimately missing glider, but every other dot on that >line represents one glider. I.e., Golly darkens a pixel iff there's >anything in >the 2^24 square it represents. Note that dy>dx. >Puzzle: How can the nonmonotonicity at the black arrow be anything but >a display bug? (Does anybody remember BRMs and DDAs?) Both dx and dy are < 2^24. So, going up-right, the next glider can either be on the same row of pixels, or one row up; on the same column, or one column to the right. That means it might be up, up-right, right, or in the same pixel. The arrow points to a place where it goes right (but not up). The places where you don't go up are relatively rare (once every ~184 gliders) because 16685960 is close to 16777216=2^24. Row number and column number as functions of glider number are related to the continued fractions for dy/2^24 and dx/2^24. They will look like fractals until you get to the LCM. The jaggies are like the beating of those two fractals against each other, and resolve to a pattern when you get to the LCM of all three numbers. That is, when the number of gliders in a string is equal to the LCM. For a similarly incoherent explanation of this same integer-division jaggy phenomenon (only with irrational numbers) see the section called "Why the Fractal Patterns?" on this page: http://www.tiac.net/~sw/2005/03/Mandala/index.html Off Google: "BRM -. Binary Rate Multiplier,. and DDA - Digital. Differential. Analyzer" Don't know if a DDA analyzes the way I just did. related to Bresenham's algorithm. Yeah but Bresenham is just an optimized way of drawing the same jaggies. --Steve

1 0

[math-fun] Null polarity group in 3-space
by Fred lunnon 22 Sep '07

22 Sep '07

I have recently been inspecting the Lie group of geometric transformations generated by the "null polarities" in real projective 3-space associated with nonsingular, nondegenerate "linear complexes" [sets of lines subject to single linear constraints on their Pluecker coordinates]. Amongst other things, I have ascertained that: This group is isomorphic to the indefinite orthogonal group O(3,3) [preserving the quadratic form (x1)^2+(x2)^2+(x3)^2-(y1)^2-(y2)^2-(y3)^2] over the real numbers; it has dimension 15. It includes the dimension 6 Euclidean group SE(3) of proper (continuous) isometries; also the dimension 3 "axial dilations" transforming a sphere into an ellipsoid. Are these properties well-known? Are they interesting or useful? What else is known about this group? [I am particularly interested in characterising the continuous component synthetically; as well as establishing its relationship to the other three.] Can anybody direct me to pertinant existing references? Fred Lunnon [23/09/07]

1 0

[math-fun] El Matemático question
by James Buddenhagen 22 Sep '07

22 Sep '07

In 1918 (or 1919?) the acclaimed Mexican artist Diego Rivera painted a painting called "El Matemático" (the mathematician). You can see it here: http://www.abcgallery.com/R/rivera/rivera16.html I am trying the find out who is depicted in that painting, which, by the way, was done in Paris. Can anyone help? Thanks. Jim Buddenhagen

2 2

[math-fun] jaggies
by R. William Gosper 20 Sep '07

20 Sep '07

http://gosper.org/jaggies.png is a 4x blowup of a detail of a Golly (Game of Life) screen, where each pixel is 2^24 x 2^24 cells. The diagonal line marked by arrows is a row of equally spaced gliders, (dx,dy) = (14412108,16685960). The gap at the green arrow is a legitimately missing glider, but every other dot on that line represents one glider. I.e., Golly darkens a pixel iff there's anything in the 2^24 square it represents. Note that dy>dx. Puzzle: How can the nonmonotonicity at the black arrow be anything but a display bug? (Does anybody remember BRMs and DDAs?) --rwg

2 2

Re: RE : [math-fun] First differences spoiler
by Michael Kleber 19 Sep '07

19 Sep '07

Eric Angelini wrote: > O N E N I N E N I N E F I V E F I V E > 1 9 9 5 5 9 9 5 5 9 1 3 13 17 1 3 13 17 --> first (letters rank) diff. is seq. itself Very nice! The existence of "one", with alphabetically adjacent first two letters, is a lucky coincidence -- it's the only place you could start ("two", "five", and "thirteen" are all off by one!). Aha! So the proper French translation of "One, nine, nine, five, five, ..." is "Quatre, vingt, dix-neuf, deux, treize, ..."? But wait! "Neuf, seize, quinze, treize, quatorze,..." works too. Ah, those lucky French. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.

2 1

[math-fun] Puzzle about coloring chess boards
by victor＠idaccr.org 19 Sep '07

19 Sep '07

The following puzzle had been knocking around work (actually it was first given in our place in 1994): Given an n x n chessboard we wish to color the squares black and white with the property that for any subsquare (of size at least 2x2) the four corner squares are not all the same color. These are orthogonal squares (and not tilted ones). The original poser of the puzzle gave a number of 5x5 examples, and asked to either prove that it can be done for all n, or to show that it can't be done for some (and therefore all bigger) n0. I coded up this problem as a satisfiability problem and found a solution for 14x14, but, after 1 1/2 hours of computation the SAT solver (SatEliteGTI) reported that 15x15 is unsatisfiable. Has anyone else seen this puzzle? It would be much more satisfying if there were a proof that 15x15 was impossible by hand. Victor

4 4

[math-fun] Factoring "15" with a Quantum Computer: Quantum Threat to Our Secret Data
by Henry Baker 19 Sep '07

19 Sep '07

FYI -- Only a few more bits to go... Quantum Threat to Our Secret Data New Scientist (09/13/07) Vol. 195, No. 2621, P. 30; Graham-Rowe, Duncan Quantum computing's ability to decrypt the codes that safeguard banking, e-commerce, and business data has taken a step closer to realization with the development of quantum computers that can run Shor's algorithm by two research groups working independently. RSA is an example of a highly common encryption system that can be defeated by Shor's algorithm. RSA involves a widely distributed public key for encrypting messages and a secret private key for decrypting them, and the trick to solving the private key is to work out the large prime numbers that produce the key when they are multiplied together. Shor's algorithm dramatically reduces the time it takes to find the prime factors by searching for telltale patterns in remainders when a key is divided by a prime factor, but quantum computation is essential for performing the massive number of calculations that the algorithm requires to be successful. The first quantum implementation of Shor's algorithm involved the manipulation of nuclear spin, while the more recent experiments--one led by Andrew White at Australia's University of Queensland and the other by Chao-Yang Lu of the University of Science and Technology of China--used quantum photonic computers. Photon pairs were produced with femtosecond lasers and passed through polarizing bismuth borate crystals to generate entangled qubits, which were coaxed by optical devices to run Shor's algorithm to factor the number 15 into its prime components. IT security specialist Bruce Schneier calls the development of techniques to run the algorithm using standard lab optics a significant achievement that could spell trouble for encryption down the road. White contends that cryptography would need to be fundamentally rethought if quantum calculations for much larger numbers could be carried out. "There are paths to a fully scalable quantum computer," he notes.

1 0