November 2015 - mathfun@mailman.xmission.com

[math-fun] Quantum mechanical derivation of the Wallis formula for pi
by Hans Havermann 11 Nov '15

11 Nov '15

http://scitation.aip.org/content/aip/journal/jmp/56/11/10.1063/1.4930800 Here's an overview: https://www.aip.org/publishing/journal-highlights/new-derivation-pi-links-q…

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[math-fun] The Uncracked Problem with 33
by Stuart Anderson 10 Nov '15

10 Nov '15

33 is the lowest unsolved problem in "summing three cubes" with Tim Browning. https://www.youtube.com/watch?v=wymmCdLdPvM

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[math-fun] "slick" but not intuitive harmonic series divergence proof
by Warren D Smith 10 Nov '15

10 Nov '15

http://web.williams.edu/Mathematics/lg5/harmonic.pdf seems to me, this proof relies on the principle of the excluded middle, while the term-grouping proof does not. So the latter is better. Intuitionism, baby. I've always felt it would be a somehow desirable thing to go through mathematics determining which parts were provable without relying on excluded middle. That project has never really been done. On the other hand, maybe I'm just nuts.

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[math-fun] Yet another zeta 2 proof
by Bill Gosper 10 Nov '15

10 Nov '15

Young eavesdropper Zack sent me http://mathcircle.berkeley.edu/archivedocs/2009_2010/lectures/0910lecturesp… a weird proof based on (X,Y) = (log(sin(x+y)/sin(x)), log(sin(x+y)/sin(y)) being (to me, remarkably) area-preserving. I think the paper needs a picture of this. gosper.org/passare3.png --rwg Unfortunately, the paper perpetuates the old, unslick proof that Harmonic#(oo) = oo.

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[math-fun] KIC 8462852 (alien megastructures?), saga continues
by Warren D Smith 10 Nov '15

10 Nov '15

"The SETI Institute has trained its Allen Telescope Array on this star for more than two weeks. The Array consists of 42 antennas, each 6 meters in size, and is located approximately 500 km north of San Francisco in the Cascade Mountains... [They found nothing]. Analysis of the Array data show no clear evidence for either type of signal between the frequencies of 1 and 10 GHz. This rules out omnidirectional transmitters of approximately 100 times today's total terrestrial energy usage in the case of the narrow-band signals, and ten million times that usage for broad band emissions." Note, only a small portion of "today's total terrestrial energy usage" is used to power radio stations. So this is not exactly a strong finding of anything.

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[math-fun] Chinese family planning
by James Propp 09 Nov '15

09 Nov '15

Has anyone in the pop math biz tackled the mathematical side of the news story about China's (now abandoned) one-child-per-family policy? Specifically, many families adopted the family planning algorithm "Have kids till you have a son, then stop", which (under idealized assumptions) gives rise to families of average size exactly 2. A non-mathematician friend of mine asked me at dinner last night why the expected size of a family that stops when the first son is born is 2. I began to give him an intuitive argument that doesn't involve calculation, but he ended up preferring the argument that shows that 1/2 + 2/4 + 3/8 + ... = 2 by way of summing the formulas 1/2 + 1/4 + 1/8 + ... = 1 1/4 + 1/8 + ... = 1/2 1/8 + ... = 1/4 ... to obtain 1/2 + 2/4 + 3/8 + ... = 1 + 1/2 + 1/4 + ... = 2 Is there a place where this is explained, and explained well? Jim Propp

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[math-fun] Sim
by James Propp 09 Nov '15

09 Nov '15

Does anyone know of a better video about the graph-game Sim than https://www.youtube.com/watch?v=Y6Lpl9Tq8yY ? I find it somewhat hard to follow; the black lines and red lines look somewhat similar to me, to the point where I could not readily assess whether one of the commenters was correct in asserting that a red K_3 gets inadvertently created at 1:50. Also, does anyone know of a better Sim simulator than http://www.cut-the-knot.org/Curriculum/Combinatorics/ThreeOrThree.shtml ? I can't get it to work, despite my efforts to circumvent Java security. Thanks, Jim Propp

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[math-fun] Kazakhstan "geoglyphs"
by Warren D Smith 08 Nov '15

08 Nov '15

Patterns, such as swastika-like shapes (3-symmetric, though, not the nazi 4-symmetric version), dotted lines, etc were carved into the steppe in Kazakhstan around 8000 years ago out of giant earth mounds. The largest are about 1/4 mile in diameter, the smallest only about 100 meters. There are about 260 in all. They were spotted by Kazakh hobbyist Dimitriy Dey in 2007 using "google earth". Here's some news stories about that including pictures released by NASA. http://www.independent.co.uk/news/science/nasa-puzzled-by-huge-8000-year-ol… http://www.dailymail.co.uk/sciencetech/article-3297282/Mystery-ancient-patt… -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Breakthrough Algorithm Reported For Graph Isomorphsim
by Ray Tayek 07 Nov '15

07 Nov '15

http://science.slashdot.org/story/15/11/06/1651211/breakthrough-algorithm-r… oshuaZ <http://slashdot.org/%7EJoshuaZ> writes: /A major open problem in graph theory is how efficiently one can tell, given two graphs, whether or not they are isomorphic — that is, whether they're the same graph with just the labels changed. This problem is famous, along with factoring integers, as a problem that is potentially in between P and NP in difficulty. Now, Laszlo Babai has reported that he has a quasipolynomial time algorithm which he sketched out at a set of talks at the University of Chicago <http://www.math.uchicago.edu/calendar?calendar=Combinatorics%20and%20Theore…>. Scott Aaronson was one of the first to break the news, and his latest blog entry and its comments contain further discussion of the result <http://www.scottaaronson.com/blog/?p=2521>. The new algorithm places the problem of graph isomorphism as, at most, just barely above P. Babai's result depends on the classification of finite simple groups <https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups>, a deep result in algebra whose proof consists of thousands of pages over hundreds of distinct papers. Unlike the problem of factoring integers, improvements in this algorithm are unlikely to impact cryptography in any direct way, since no cryptographic systems depend on the difficulty of determining when groups are isomorphic./ -- Honesty is a very expensive gift. So, don't expect it from cheap people - Warren Buffett http://tayek.com/

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Re: [math-fun] Chinese Remainders for Audio & Signal Processing ?
by Henry Baker 07 Nov '15

07 Nov '15

Very cool! Thanks, Mike, for the link! My own take on this is that the modp technique is much more useful during processing than during acquisition or reconstruction. I am fond of the *hypberbolic arcsine* companding function for the representation of a signal during transmission to/from sensors & actuators/displays/speakers. This asinh companding function is already used by astronomers for their "magnitude" databases. That having been said, so-called "Class D" (pulse-width modulated binary and/or halftone [for images]) ADC's/amplifiers have elegantly solved the acquisition & reconstruction of analog to digital and back. The MIT modulo camera is a good start at doing an analogous thing for what I might call a "Class P" (modp) acquisition mechanism. http://www.audioholics.com/audio-amplifier/class-d-digital-amplifier At 04:55 PM 11/6/2015, Mike Stay wrote: >Here's a use of a mod p detector for unbounded dynamic range in a camera: >http://web.media.mit.edu/~hangzhao/modulo.html > >On Fri, Nov 6, 2015 at 4:38 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> [Totally different thread from family planning!!] >> >> I sometimes play with MIDI synthesizers to construct & modify music. >> >> The basic operation of a MIDI synthesizer is to *add* sounds to one another to produce chords, etc. >> >> In order for the human ear to later pull apart these synthesized sounds, it is essential for this *addition* to obey *superposition* -- i.e., *linearity*: f(a+b)=f(a)+f(b) and f(c*a)=c*f(a). >> >> The problem with digital sound is that it typically uses too few bits to be able to preserve superposition/linearity; e.g., some systems use 12-16 bits, while others may use fixed precision up to 24 bits. >> >> Newer signal processing systems employ *floating point* arithmetic, which is sometimes a bit better, but still incapable of preserving superposition/linearity 100%. >> >> Has anyone (else) considered a digital sound/signal processing system based upon Chinese Remainders? >> >> The idea is to represent the digital samples (mod 2), (mod 3), (mod 5), etc., (mod p), until we have enough remainders to completely capture the full *dynamic range* of the music or signals. >> >> The wonderful thing about arithmetic (mod p) is that it forms a field, so we can do all of our digital filtering convolutions for each of the primes *separately*, and then reconstruct the final signal for playback using the Chinese Remainder Theorem. Superposition/linearity is faithfully preserved. >> >> Such a signal processing system would be useful in certain types of signal processing where the signal is extremely faint in comparison with another signal (or noise). Such systems include "spread spectrum" techniques, where the signal may be 10-20 dB below the noise floor. Linear system techniques *without roundoff errors* would be essential to performing the necessary computations. >> >> Video is also escaping from the 8-bit world; new cameras can get to 10-bit, 12-bit and even 14-bit dynamic ranges, so Chinese Remainder techniques might be useful for video, as well. >> >> An interesting question is in the final sound/signal *reconstruction*. It may be possible to reconstruct the signal from the Chinese Remainders *out of thin air* -- i.e., by letting the human ear (or eye) put everything back together again. >> >> But I'm not sure how to build a speaker or display which produces (mod p) sounds/brightnesses !. > >-- >Mike Stay - metaweta(a)gmail.com >http://www.cs.auckland.ac.nz/~mike >http://reperiendi.wordpress.com

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