January 2016 - mathfun@mailman.xmission.com

[math-fun] Stupid QM question (EPR, B-E statistics)
by Henry Baker 29 Jan '16

29 Jan '16

Bose-Einstein statistics apply to bosons, correct? If I have a number N of these bosons, *all in the "same" state*, then these are all maximally entangled, correct? If I "measure" one of these bosons (is this even possible -- to measure just one?), this measurement necessarily puts *all* of my bosons into the same state ? I.e., the usual EPR experiment works with 2 entangled qubits; measure one and the same measurement on the other will produce the same result. Now, I'd like to have N entangled qubits, so that measuring any one will guarantee the same measurement on *all* of the other qubits. Does this work in theory? Does this work in practise?

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[math-fun] TOTAL SPOILER: A mathemusical sequence
by James Propp 29 Jan '16

29 Jan '16

I wrote: Can any of you mathemusicians spot the significance of the sequence > 10,12,18,21,25,30? > It's the sequence of accented notes in the "Les Augures Printaniers" section of Le Sacre du Printemps by Stravinsky ( https://www.youtube.com/watch?v=XwMjDJPKGig) I suppose there's potential math to be made here, given that Stravinsky clearly wanted the sequence to be devoid of patterns that would enable a listener to predict when the next accented beat would fall. One could construct a model of how listeners predict what a musical piece is going to do, based on conventions of Western music and on patterns laid down by earlier notes in the particular piece being listened to, and then see how well the Stravinsky passage thwarts this kind of extrapolation. Jim Propp

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Re: [math-fun] 2nd intersection
by Henry Baker 29 Jan '16

29 Jan '16

Hi Bill: These look extremely interesting. However, I'm not a Mathematica fan or user. Any chance of showing this in Maxima? BTW, I haven't worked on the complex # stuff for triangle centers for a couple of years, but I still believe that "there's gold in them thar hills" ! In particular, I think that there are complex & quaternion formulae that are analogs of existing scalar formulae. I.e., the scalar formula may be the *absolute value* of the complex and/or quanternion formula. At 02:14 PM 1/28/2016, Bill Gosper wrote: >I hope Henry or someone is collecting our various complex number geometricks e.g., >PolygonArea[L_List] := Total[MapThread[Im[#1\[Conjugate]*#2] &, {L, RotateLeft[L]}]]/2 > >line segment intersect >Interseg[z1_, z2_, z3_, z4_] := Block[{z12 = z1 - z2, z34 = z3 - z4}, >(z34*Im[z2*z1\[Conjugate]] + > z12*Im[z3*z4\[Conjugate]])/Im[z34*z12\[Conjugate]]] > >incenter, circumcenter, inradius, ... . > >For the Clifford circle theorems I needed second intersection: Given centers of >circles though the origin z1, z2, > >secondintersection[z1_, z2_] := (Conjugate[z1] z2 - Conjugate[z2] z1)/Conjugate[z1 - z2] > >Is this obvious to anybody? I had to derive it with 8th grade analytic geometry. --rwg

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[math-fun] 99 Point Hypothesis
by Ed Pegg Jr 29 Jan '16

29 Jan '16

99 Point Hypothesis 99 points of diameter 1 in Euclidean space. 99 points bounding a volume of more than a half. Take one off, move them around (without increasing diameter) You can't bound a volume of 1/2 any more. A set of 99 points bounding a volume of 1/2 <http://math.stackexchange.com/questions/1616138/volume-of-1-2-using-hull-of…> and a picture is shown at math.stackexchange. No two points are further than 1 apart, hence diameter 1. I've been obsessed with the Biggest Little Polyhedron problem for almost a year now. But I've made gigantic amounts of progress since I originally wrote about the problem back in May. http://blog.wolfram.com/2015/05/20/biggest-little-polyhedronnew-solutions-i… In other new, I've had a demoralizing problem with computer crashes for over a year. Every time I worked on something mathematical at home for more than 5 minutes, I tended to have a computer crash. Turns out it was bad RAM, cost $30 to fix. Having a reliable computer at home feels almost weird now, but it's led to a bout of productivity. Ed Pegg Jr

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Re: [math-fun] computer go player kicks human butt
by Warren D Smith 28 Jan '16

28 Jan '16

As far as I can see, mathematical theory of Go is completely uninvolved. The key for them is trained multilayer neural nets; and it strikes me as amazing that said nets were able to be as smart as they apparently are.

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[math-fun] computer go player kicks human butt
by Warren D Smith 28 Jan '16

28 Jan '16

AlphaGo won a Go match versus European 2013, 2014, & 2015 champ Fan Hui (naturalized French citizen, formerly from China, presumably one of the strongest humans outside of China/Korea/Japan) in a London match in October by 5 games to zero. No handicap. At the same time they also played an "informal" 5 game match with shorter time controls. AlphaGo won it by 3-2. Now a match versus Lee Sedol is planned in March in Seoul with $1 million prize to be streamed live on YouTube. Fan Hui is a pro 2-dan. Lee Sedol is a pro 9-dan and one of the top 10 players in the world. DeepMind Technologies, later acquired by Google-Alphabet, wrote the program. It self-learns via self-play as well as play with foreign entities. AlphaGo also played test matches versus other top go programs, namely CrazyStone, Zen, Pachi, Fuego, and GnuGo, as well as playing the first 3 of these with 4 handicap stone disadvantage. Zen and Pachi each are rated 6 amateur dan on KGS. The results were massive wins for AlphaGo in every case. It is about 1000 Elo stronger than the best among those programs on even terms, beating them over 99% of the time; and 350 to 1100 Elo stronger even with 4 handicap stones beating them 77 to 99% of the time. And then on top of that, AlphaGo can run highly parallel, with one configuration using 1202 CPUs and 176 GPUs. This high-parallel version was the one used against Fan Hui and planned to be used versus Lee Sedol. "The Alphabet approach relies on the newest so-called deep learning approach combined with a more traditional type of algorithm known as a Monte Carlo, which is designed to exhaustively explore large numbers of possible combinations of moves. The researchers said they had also trained their program using input from expert human Go players." "Two deep-learning networks were used in AlphaGo: one network learned to predict the next move, and the other learned to predict the outcome from different arrangements on the board. The two networks were combined using a more conventional AI algorithm to look ahead in the game for possible moves." Incredibly, AlphaGo WITHOUT any search, simply using its trained neural network based eval, will beat Pachi using 100000 monte carlo rollouts per move, over 80% of the time. Lee Sedol is a 9-dan pro from Korea and probably really is more like 10 or 11 dan; the rating system stops at 9 but the plain fact is, some 9-dans win a hell of a lot more tournaments than others. Lee Sedol is one of the top 10 players in the world and is 600 Elo above Fan Hui. I would tend to doubt AlphaGo could beat Lee Sedol just based on the fact Lee Sedol would probably have beat Fan Hui 5-0 and 5-0, not 5-0 and 3-2. I think it is an amazing, incredible advance that AlphaGo can kick the butt of a pro 2-dan and even consider trying to take on Lee Sedol. Just a few years ago both seemed way out of reach. 20-author paper: http://www.nature.com/nature/journal/v529/n7587/full/nature16961.html Video: http://www.nature.com/news/google-ai-algorithm-masters-ancient-game-of-go-1… Another piece with a video: http://www.nature.com/news/game-playing-software-holds-lessons-for-neurosci… -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] circles though points
by Bill Gosper 28 Jan '16

28 Jan '16

I think it was Gene Salamin, in an proposed HAKMEM sequel, who described a remarkable infinite sequence of geometry theorems that went something like: Put three arbitrary circles through a given point. They intersect at three other points which (obviously) lie on a 4th circle. Add an arbitrary(?) (5th) point. Put circles through the four triplets of points formed by lumping this point with two of the three intersections. These four circles intersect at one point. Invent a 6th point, and repeat this whole process, to create five quadruple intersections. These all lie on one circle. Iterate, alternately getting n circles through a point, or n points on a circle. Can someone correct and identify this half-remembered theorem? --rwg

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Re: [math-fun] A mathemusical sequence
by Steve Witham 28 Jan '16

28 Jan '16

> Date: Wed, 27 Jan 2016 14:26:45 -0500 > From: James Propp <jamespropp(a)gmail.com> > > Can any of you mathemusicians spot the significance of the sequence > 10,12,18,21,25,30? [edit by Steve: Jim's correction] You could also think of that as the thirty-two bit number (in hex) 00504884. This is a semi-spoiler because you can see what notes on the scale are played: http://www.mac-guyver.com/switham/2016/01/10_12_18_21_25_30/10_12_18_21_25_… --Steve

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[math-fun] POSSIBLE SPOILER: A mathemusical sequence
by James Propp 28 Jan '16

28 Jan '16

Hint: Count to 32 at an even tempo, shouting the seven numbers that are on the list and whispering the other twenty-five. Jim Propp On Wednesday, January 27, 2016, James Propp <jamespropp(a)gmail.com> wrote: > Can any of you mathemusicians spot the significance of the sequence > 1,10,12,18,21,25,30? > > Jim >

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[math-fun] A mathemusical sequence
by James Propp 28 Jan '16

28 Jan '16

Can any of you mathemusicians spot the significance of the sequence 1,10,12,18,21,25,30? Jim

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