May 2016 - mathfun@mailman.xmission.com

[math-fun] Wilson a^2+b^2-c^2 Conjecture
by Warren D Smith 18 May '16

18 May '16

computer output if anybody wants to look at it: https://dl.dropboxusercontent.com/u/3507527/WilsonConj.out (do text search for the letter e) By the way, if anybody cared the runtime could probably be reduced from order N^2 to order N roughly, by doing a less-brute search for primes, i.e. only trying primes that might plausibly work to fill in holes in the current table of represented N. If that were done, you might be able to confirm Wilson out to N=10^10. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] horses finally finding water
by Warren D Smith 18 May '16

18 May '16

to FWL, perhaps your working by columns not rows is since you transposed the matrices versus my convention? In any case, doesn't really matter. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Wilson a^2+b^2-c^2 question
by Warren D Smith 18 May '16

18 May '16

Apparently he intended to ask a question equivalent to the following: Is there a way to express every integer N>=0 in the form a^2 + b^2 - c^2 where a,b,c integers and c^2 <= a^2+b^2 < (c+1)^2 ? An integer S is expressible as a^2+b^2 if and only if its prime divisors of the form 4m+3 have even powers inside S's prime factorization. In particular, every number whose prime factors are 4m+1 primes and the prime 2, only, is thus-expressible. So for example Wilson's question would have a "yes" answer if CONJECTURE: every number N>0 is expressible as 2^k * P - c^2 where P is a prime of form 4m+1, k is in {0,1,2}, and c is integer, and c^2 < 2^k * P < (c+1)^2. Well, what can I say. This conjecture seems extremely likely to be true, in fact I would think every N>0 is thus-expressible in an infinite number of ways. If there were any N only thus-expressible in a finite number of ways, that would be an astonishing find that would win you a Fields medal immediately. But I do not see a way to prove it. Perhaps of some relevance: The ternary form a^2 + b^2 - c^2 is known to represent all integers [Leonard Eugene Dickson: Modern Elementary Theory of Numbers, University of Chicago Press, Chicago, 1939, theorem 109? A.Oppenheim: The determination of all universal ternary quadratic forms, Quart. J. Math. 1 (1930) 179-185] -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] solar cell efficiency record
by Henry Baker 18 May '16

18 May '16

I don't know about this particular solar cell, but: 1. different wavelengths can propagate further below the surface of the receiving chip; this effect has been used for some color camera chips. 2. if one can spread the Ge and In extremely thinly -- which is probably required by #1 -- then the amount of such material required is quite small. Ultimately, solar cells will probably do what plants have done for >1 billion years: they "downconvert" certain wavelengths for handling by molecules that can only process longer wavelengths. At 10:34 AM 5/18/2016, Warren D Smith wrote: >35.4% energy conversion (sun->electric) efficiency >achieved. The light was split into 4 spectral bands >each handled with a different semiconductor: > Ge: 0.67eV > Si: 1.1eV > GaInAs: 1.41eV > GaInP: 1.88eV >I don't understand their splitting scheme. Can it be used >with scattered sunlight, i.e. which is not coming from just one direction? >In any event, it seems to me this whole approach >probably is totally uneconomical because Ge and In >are rare. > http://newsroom.unsw.edu.au/news/science-tech/milestone-solar-cell-efficien… > >(Even greater efficiency has been achieved by the same team >but using concentrated sunlight.) > >-- >Warren D. Smith >http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Question
by David Wilson 18 May '16

18 May '16

Is every nonnegative integer of the form a^2 + b^2 - [sqrt(a^2 + b^2)]^2 ?

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[math-fun] Orthogonal group generators
by Fred Lunnon 16 May '16

16 May '16

The following questions arose as a result of an enquiry raised on the list Geometric_Algebra <geometric_algebra(a)googlegroups.com> --- SO(n) denotes the proper isometries of an (n-1)-sphere, embedded in standard fashion in a Cartesian coordinate basis (x_1,...,x_n); R_i(t) denotes rotation through angle t about axis the coline meet of coordinate hyperplanes x_(i-1), x_i . [ Ambiguity in rotation sense need not concern us just now. ] Question (A) : Does the set G = { R_2, ..., R_n } generate SO(n) (irredundantly)? Question (B) : Given an arbitrary isometry, is its minimum length as a word over G necessarily at most n_C_2 = n(n-1)/2 (sharply)? Question (C) : Is there a practical algorithm expressing an isometry, given in the form (say) of a numerical matrix, as a word over G ? Fred Lunnon

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[math-fun] "swarm intelligence"
by Warren D Smith 15 May '16

15 May '16

Is there anything to this, or is it, er, horseshit? A "swarm intelligence" AI "platform" working with expert humans supposedly brilliantly predicted the Kentucky Derby horserace to win a 550:1 odds bet...: http://m.csmonitor.com/Technology/2016/0512/Smarter-as-a-group-How-swarm-in…

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Re: [math-fun] Simple finite group problem
by Keith F. Lynch 15 May '16

15 May '16

Dan Asimov <dasimov(a)earthlink.net> wrote: > P.S. Now I'd like to know the same thing for the Gaussian integers > Z[i] and the Eisenstein integers Z[w] where w = exp(2pi*i/3). These > apparently have some complicated factor rings. I'm not sure what you mean by modular arithmetic on Gaussian integers. * Modular in both real and imaginary, meaning that the complex plane is tessellated with rectangles of some integer side lengths, and the rectangles are all identified with each other? Equivalently, there's just one rectangle, and its opposite edges are connected, making a torus? * Modular just in the reals, with the imaginaries unbounded? * Modular just in the imaginaries, with the reals unbounded? I think modular in both real and imaginary sounds most interesting. Especially since if you think of it as a tessellation of rectangles, you can generalize to offset rectangles, like bricks in a wall. And if you think of it as a torus, you can generalize to a Klein bottle. (I wonder what the Mandelbrot set looks like in the modular complex plane.) Long ago I noticed that modulo a number that's one more than a square, negative one has a real square root. For instance mod 10, i = 3 and -i = 7. (Or vice versa if you prefer.) All Gaussian integers have consistent real integer values mod 10. For instance 2i is 6, 2i-5 is 1, etc. But this doesn't seem to get you anything new and interesting. I played around a lot with Gaussian integers. I calculated and plotted, by hand, the first hundred or so Gaussian primes. I also calculated and plotted Gaussian squares, and noticed that they formed interlocking parabolas. I never played with Eisenstein integers. Would you use a triangular tessellation for them when going modular? Are those the only two interesting ways of generalizing integers to the complex plane? What about the vertices of a regular hexagonal tessellation?

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Re: [math-fun] 473-gene "artificial bactierium"
by Henry Baker 14 May '16

14 May '16

We've been running this evolutionary experiment with Unix/Linux/OpenWRT/... for the past 50 years, and the outcome is clear: it is nearly impossible to come up with a mechanism that enforces small DNA/RNA/kernel size. Or to flip this around, the marginal cost of adding additional "functionality" -- especially functionality that is executed only occasionally (i.e., small *dynamic* cost) -- is almost free. Nature has apparently come to the same conclusion: functionality that hasn't been extensively used for 1,000 generations still remains more- or-less functional in humans today. Thus, it is likely that bacteria & viruses trapped in the Siberian permafrost for 50,000 to 100,000 years will still find humans reasonably resistant. Good example: the indigenous Americans were cut off from Europeans for at least 10,000 and possibly as much as 20,000 years. Yes, Columbus brought over diseases that likely wiped out 60-80% of indigenous Americans. Nevertheless, these diseases didn't wipe them *all* out. (I don't recall specifically, but I think the indigenous Americans gave Columbus & the other explorers some diseases that Europeans hadn't seen for quite a while, as well.) <shields up for incoming flack> At 10:38 AM 5/12/2016, Allan Wechsler wrote: >A very late reply to this interesting thread. I am curious to know whether >any biologists have tried letting evolution do this work: that is, breed >some well-understood model organism (almost certainly E. coli K12 or >something similar) and select for small genome. I think genome size can be >easily calculated by electrophoresis. This ought to be easy to mechanize, >and with a generation time of 17 minutes I suspect that evolution would >start whittling away at the 4.6 Mbp genome pretty quickly. Leave it running >for a few months and see how close it can get to this 531 Kbp goal line. I >have a sneaking suspicion that it would surprise us. > >2016-03-27 13:02 GMT-04:00 Warren D Smith <warren.wds(a)gmail.com>: > >> http://www.livescience.com/54165-artificial-bacterium-has-smallest-genome.h… >> >> 531K base pairs.

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Re: [math-fun] Differential equations. See something, say something.
by Keith F. Lynch 13 May '16

13 May '16

James Propp <jamespropp(a)gmail.com> wrote: > Some math lingo does smack of terrorism: when we talk about Killing, An SF story by Charles Sheffield made use of this. A scientist hostage said "no Killing vector," which the space pirate interpreted as meaning nobody would be killed, when what the scientist meant is that all spin should be removed from the mini-black-hole so that the spaceship wouldn't do what the space pirate wanted. > ... or Lie theory ("A big lie can be easier to sell than a littler", > said Hitler). Unfortunately, it's pronounced "lee." (This was perhaps the most obscure factoid that I found useful during my short career as an audio proofreader, in which I made sure transcripts agreed with the audio they were intended to be the transcript of. A good technical vocabulary was a plus. Too bad they were looking for speed, not accuracy. They're out of business now, mostly for that reason.)

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