February 2016 - mathfun@mailman.xmission.com

[math-fun] solid angle of trihedron, aka area of spherical triangle
by Warren D Smith 26 Feb '16

26 Feb '16

RWG: solidAngle[a_, b_, c_] := { 2 ArcTan[Sqrt[ 1 - Cos[a]^2 - Cos[b]^2 + 2 Cos[a] Cos[b] Cos[c] - Cos[c]^2]/( 1 + Cos[a] + Cos[b] + Cos[c])], ArcCos[-1 + (1 + Cos[a] + Cos[b] + Cos[c])^2/((1 + Cos[a]) (1 + Cos[b]) (1 + Cos[c]))], 2 ArcCos[1/ 4 (1 + Cos[a] + Cos[b] + Cos[c]) Sec[a/2] Sec[b/2] Sec[c/2]], 2 ArcSin[1/(4 Sqrt[2]) Sqrt[-1 - Cos[2 a] - Cos[2 b] + 4 Cos[a] Cos[b] Cos[c] - Cos[2 c]] Sec[a/2] Sec[b/2] Sec[c/2]], 4 ArcTan[\[Sqrt](Tan[1/4 (a + b - c)] Tan[1/4 (a - b + c)] Tan[ 1/4 (-a + b + c)] Tan[1/4 (a + b + c)])]} DanA's would make a sixth, but I can't make sense of it. --WDS; Re Asimov, you appear to be confused about lengths versus angles. Given 3 points on unit sphere, the edge lengths a,b,c are the geodesic distances between them; the (dihredral) angles A,B,C are the angles formed on the sphere at the corners of that triangle with geodesics as edges. Of course these are related. Asimov's formula, also called Albert Girard's formula since apparently it first was known to Thomas Harriot, is area=A+B+C-pi. Of the formulas RWG lists, most were also listed in appendix C of my http://rangevoting.org/BestVrange.html#appC where I listed 7 such formulae due to Cagnoli, Lagrange, Euler, Lhuiler, and Todhunter all before 1860. But RWG's ArcCos[ -1 + (1 + Cos[a] + Cos[b] + Cos[c])^2 / ((1 + Cos[a]) (1 + Cos[b]) (1 + Cos[c])) ] was not previously known to me and might be new. It might be interesting to perform a mildly exhaustive computer search for all such formulae. I programmed such a search, but it may be buggy; if it works it will need to run overnight. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] poker hand probability distribution question
by Ray Tayek 26 Feb '16

26 Feb '16

hi, i've been playing some no limit holdem online. normally holdem is played with 9 or 10 players, but recently 6 handed games have become popular. suppose you are in late position (near the dealer and to his right) and everyone passes to you. since good opening holdem hands usually have two high cards (say Jack or better), this should give you some information about the remaining distribution of cards in the hands of the players still to act after you (not sure what that is mind you, i would think you could argue for a higher probability of high cards in the remaining hands). in one case, say you are the dealer in a 10 handed game and 7 players fold to you. or you are in a 6 handed game and 3 players fold to you. given that you know your hand, in which case do you have /more/ information about the distribution of cards in the remaining two players hands. seems like it would be the 10 handed game? what can you say about the information in these cases? thanks -- 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] Solid angle of a spherical triangle
by Bill Gosper 25 Feb '16

25 Feb '16

On 2016-02-20 11:11, Eugene Salamin via math-fun wrote: > I came across an interesting formula in an optics book for the solid > angle of a spherical triangle. Let a, b, c, be the unit vectors from > the center to the vertices of a spherical triangle on the unit > sphere. Then the solid angle Omega, which equals the area of the > triangle, is given by > > tan(Omega/2) = [a,b,c] / (1 + b.c + a.c + a.b), > > where a.b etc. are dot products, and [a,b,c] is the scalar triple > product. This paper is the cited reference. > > http://www.maa.org/sites/default/files/Eriksson14108673.pdf > > -- Gene > Yow, converting to angles between the vectors, and using the formula 52d8effb9069c8b7454fc03683e6f1ae.png <https://upload.wikimedia.org/math/5/2/d/52d8effb9069c8b7454fc03683e6f1ae.png> makes a *fifth* formula for solid angle in terms of vertex angles: solidAngle[a_, b_, c_] := {2 ArcTan[Sqrt[ 1 - Cos[a]^2 - Cos[b]^2 + 2 Cos[a] Cos[b] Cos[c] - Cos[c]^2]/( 1 + Cos[a] + Cos[b] + Cos[c])], ArcCos[-1 + (1 + Cos[a] + Cos[b] + Cos[c])^2/((1 + Cos[a]) (1 + Cos[b]) (1 + Cos[c]))], 2 ArcCos[1/ 4 (1 + Cos[a] + Cos[b] + Cos[c]) Sec[a/2] Sec[b/2] Sec[c/2]], 2 ArcSin[1/(4 Sqrt[2]) Sqrt[-1 - Cos[2 a] - Cos[2 b] + 4 Cos[a] Cos[b] Cos[c] - Cos[2 c]] Sec[a/2] Sec[b/2] Sec[c/2]], 4 ArcTan[\[Sqrt](Tan[1/4 (a + b - c)] Tan[1/4 (a - b + c)] Tan[ 1/4 (-a + b + c)] Tan[1/4 (a + b + c)])]} DanA's would make a sixth, but I can't make sense of it. --rwg

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[math-fun] Approximate matrix factorization question
by Mike Stay 24 Feb '16

24 Feb '16

Say I've got a mxn matrix that I'd like to factor into an mxk and a kxn matrix. The factorization doesn't have to be exact; I'd just like to minimize some measure of error. What are your favorite algorithms to do this? What assumptions about the matrix and the error measure do they make? There's one blurb in Wikipedia about some nonnegative matrix factorization algorithms: There are several ways in which the W and H may be found: Lee and Seung's multiplicative update rule [10] has been a popular method due to the simplicity of implementation. Since then, a few other algorithmic approaches have been developed. Some successful algorithms are based on alternating non-negative least squares: in each step of such an algorithm, first H is fixed and W found by a non-negative least squares solver, then W is fixed and H is found analogously. The procedures used to solve for W and H may be the same[22] or different, as some NMF variants regularize one of W and H.[17] Specific approaches include the projected gradient descent methods,[22][23]the active set method,[4][24] and the block principal pivoting method[25] among several others. -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

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[math-fun] Approximate matrix factorization
by Warren D Smith 24 Feb '16

24 Feb '16

> Say I've got a mxn matrix that I'd like to factor into an mxk and a > kxn matrix. The factorization doesn't have to be exact; I'd just like > to minimize some measure of error. --Take the SVD, then zero out the smallest singular values until only the k largest are left.

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[math-fun] AlphaGo vs Lee Sedol; Should humankind kiss its ass goodbye?
by Warren D Smith 24 Feb '16

24 Feb '16

$1 million match now set for march 9, 10, 12, 13, 15 in Four Seasons Hotel, Seoul 2016; video will be streamed live on YouTube here: https://www.youtube.com/c/DeepmindAI plus televised all over E.Asia. One game per day starting 1pm Seoul time. This time is 11pm/8pm on the DAY BEFORE for USA eastern/western time zone. Chinese rules with a komi of 7.5, each game expected to take about 5 hours. Match commentators will include Michael Redmond 9 dan, the only professional Western Go player to achieve 9 dan. Redmond will commentate in English, others in other languages. Demis Hassabis said most Go players are giving Sedol the edge over AlphaGo. "They give us a less than 5 percent chance of winning … but what they don’t realize is how much our system has improved," he said. "It's improving while I'm talking with you." I would imagine Hassabis et al can assess how much it improved... they presumably have an objective basis for believing they'll be able to beat Lee Sedol... and Fan Hui, the European Go Champ AlphaGo beat, is now on the AlphaGo team as some kind of advisor. An interview with him is here: https://www.usgo.org/news/2016/01/chinese-professionals-react-to-the-histor… -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] names for some well known facts
by Warren D Smith 24 Feb '16

24 Feb '16

in probability theory: variance + mean^2 = meansquare in minimization: Let F(x1,x2,x3,...,xN) be a concave-U symmetric function of N variables. Then its min necessarily occurs when all the variables are equal. The above are two well known theorems. What I want to know is, do they have names, or cites, or something like that so I do not have to re-prove them myself, I can just name/cite them? There was once a sci-fi book with title "True names" postulating that once I know your true name, I have great power...

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Re: [math-fun] energy flux of gravitational and other fields
by Warren D Smith 24 Feb '16

24 Feb '16

(Trying again, emailer crashed.) E.Salamin: >From this reference http://www.aei.mpg.de/~schutz/download/lectures/AzoresCosmology/Schutz.Azor… the flux (power per unit area) carried by a gravitational wave of frequency f and strain s is F = (pi/4) (c^3/G) f^2 s^2 WDS: 1. this formula cannot be right for large strains, at best it can be right only in the limit of small strains so can use linearized GR. Should be derivable from "stress-energy pseudotensor" https://en.wikipedia.org/wiki/Stress%E2%80%93energy%E2%80%93momentum_pseudo… 2. The contrast with Maxwell is interesting. Maxwell energy flux = ExB where E=electric & B=magnetic field, x=vector cross product note NO frequency dependence. 3. For a scalar field https://en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor https://en.wikipedia.org/wiki/Scalar_field_solution again I see an f^2 frequency dependence. 4. So what is going on? Why for even-spin fields (gravity spin=2, scalar spin=0) do we have f^2 dependence while for odd-spin (Maxwell, photon spin=1) we have no frequency dependence? We get frequency f^n dependence if the stress energy tensor involves DERIVATIVES of the field, which in turn happens if the lagrangian density involves n field derivatives. For spin=0 & 2 we have n=2 (arising as squaring a first derivative). For spin=1 (Maxwell, Proca) we have n=0. The Dirac, Majorana, and Weyl (spin=1/2) fields all have n=1 according to http://arxiv.org/pdf/1006.1718.pdf so for them I would expect f^1 dependence. Rarita-Schwinger, which is one proposal for spin=3/2 fields, has n=1 according to https://en.wikipedia.org/wiki/Rarita%E2%80%93Schwinger_equation so for it also I would expect f^1 dependence. Is there some obvious unified reason these n had to happen in all these cases? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] packings
by Bill Gosper 23 Feb '16

23 Feb '16

Here are the roots of my apparent delusion. (I can't blame senility--this was earlier.) The quantity (d8), ((sqrt(4 * sqrt(3) - 3) + 1) * (n - 1)/2) + 2 claims to be the rightmost extent of the nth triad of gosper.org/pennies1.bmp which was presented via block([equalscale:true],paramplot(''(apply('append,cons([[0,2*t-1],[5*t,-1],[5*t,1]],makelist([[(1+cos(2*%pi*t))/2+''(d8-2),(-1)^n*(1+sin(2*%pi*t))/2],[(3+cos(2*%pi*t))/2+''(d8-2),(-1)^n*(1+sin(2*%pi*t))/2],[(2+cos(2*%pi*t))/2+''(d8-2),(-1)^n*(1-sqrt(3.)+sin(2*%pi*t))/2]],n,1,3)))),t,0,1))$ wherein the "n,1,3" purports to portray triads 1, 2, and 3. Then the command block([equalscale:true],paramplot(''(apply('append,cons([[200,2*t-1],[5*t+195,-1],[5*t+195,1]],makelist([[(1+cos(2*%pi*t))/2+''(d8-2),(-1)^n*(1+sin(2*%pi*t))/2],[(3+cos(2*%pi*t))/2+''(d8-2),(-1)^n*(1+sin(2*%pi*t))/2],[(2+cos(2*%pi*t))/2+''(d8-2),(-1)^n*(1-sqrt(3.)+sin(2*%pi*t))/2]],n,131,133)))),t,0,1))$ purports to portray triads 131, 132, and 133 : gosper.org/pennies2.bmp i.e. 399 pennies with room for two more. Note that the rightmost penny in the bottom row must be 2 mod 3, i.e. #200, not #199. If I'm wrong, at least this was an interesting gotcha! --rwg On 2016-02-22 16:44, Scott Huddleston wrote: > Thanks for the pointer, Richard. > Locally optimizing two pennies at the end improves > 477 pennies in 2x238 > to > 473 pennies in 2x236 > I don't see any further improvements. > > The same idea lets you pack > 400 pennies in 1.964 x 200 > > For Gosper's original 2x200 teaser, I'd be very surprised if 401 were > possible. > > Scott > > > On Fri, Feb 19, 2016 at 10:14 AM, rkg <rkg(a)ucalgary.ca> wrote: > >> We used this as Problem 211 in The Inquisitive Problem Solver. >> I don't know where we got it from. Elwyn Berlekamp and Joe >> Buhler may have used it in their column in Emissary, the >> journal of MSRI. We only gave a packing of 477 balls in a >> 1 x 2 x 238 box. I know that better results are known to >> Ron Graham and to Elwyn Berlekamp. Perhaps either or both >> of them will tell us what they know, and how much has been >> proved. R. >> >> >> On Fri, 19 Feb 2016, Erich Friedman wrote: >> >> I'm a tad suspicious: are the circle packings on Friedman's page proved >>>>> to be maximal, or are they merely the best known so far? There is no >>>>> link >>>>> to any proofs, or key to diameters ... WFL >>>>> >>>> >>>> >>> I had the very same question. But on some related pages, some >>>> packings or coverings are referred to as best known and others >>>> are said to be proven the best . . . so I'm guessing that unless >>>> the word "proved" is used, they are only the best known. >>>> >>>> It's not easy to find the absolute optimum of such things with proof. >>>> >>> >>> this is correct. in general on my packing pages "Trivial" means there is >>> an easy proof not attributed to a given person, "Proved by ..." means it >>> has been proven optimal, and "Found by ..." means it is only the best >>> currently known, with no proof known. >>> >>> much fame awaits those who could find such proofs, but they do tend to be >>> hard. for example, it was only in this century that the best packing of 6 >>> unit squares in a square was confirmed to be a square of side 3. (i proved >>> the cases for 7, 8, 14, and 15 squares a few years earlier.) i suspect 27 >>> unit squares in a square will be the next to be proved. >>> >>> http://www2.stetson.edu/~efriedma/squinsqu/s27.gif >>> >>> One thing a torus-lover like me would like to see added if Erich >>>> decides to do so is additional best-known or proven optima for >>>> the two most symmetrical tori: >>>> >>> >>> when i retire in a few years, my guess is that you will see exactly >>> that. :) >>> >>> erich

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[math-fun] Counterintuitive optimization
by James Propp 23 Feb '16

23 Feb '16

Does anyone have a favorite example of a fake optimum? One class of examples that comes to mind is certain packing problems, where the constraints all obey some sort of symmetry but the optimal symmetrical solution is not as good as the optimal asymmetrical solution. I'm especially interested in optimization problems where a greedy approach seems intuitively optimal but isn't. E.g., try to construct the biggest possible subset of {1,2,...,20} in which no two elements differ by 3 or 5. A greedy construction gives {1,2,3,9,10,11,17,18,19}, but this isn't optimal. (Yes, this is for my blog. My default is to give credit to contributors, so if you prefer anonymity, please let me know!) Jim

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