December 2015 - mathfun@mailman.xmission.com

[math-fun] large numbers in ancient India
by James Propp 12 Dec '15

12 Dec '15

Do any of you know about the numbers discussed in the "Ancient India" section of https://en.wikipedia.org/wiki/History_of_large_numbers, or with the book by Georges Ifrah on which it is based? I'm puzzled by *lalitavistarautra *and the succeeding entries, whose definitions refer to "infinities". If the word is being used in its strict mathematical sense, then *lalitavistarautra* et al. cannot be identified with specific positive integers. But perhaps the translator (and/or the creator of this wiki page) erred? Perhaps the word translated as "infinities" is obscure in its meaning but can be assumed from context to have been intended to denote some specific large number? I'd be eager for information on this. Jim Propp

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Re: [math-fun] math-fun Digest, Vol 154, Issue 11
by Warren D Smith 11 Dec '15

11 Dec '15

to Andy Latto, well, consider the context. First I had in mind for 9 7/33 that the / actually was being printed as a horizontal line with 7 above and 33 below it. Second, keeping that in mind, if you saw 2/5 9 7/33 10 8/21 6 2/11 you'd think multiplication, right? Now actually, I am aware of the elementary school vs college conflict so I personally would try to avoid the ambiguity by using TIMES or + or parentheses or something. But if I had made no such effort, what would you think? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] mixed numbers
by Warren D Smith 10 Dec '15

10 Dec '15

They are fine with me, but the problem is 9 7/33 means 9+7/33 or means 9*7/33? In elementary school they mean the former and in college they mean the latter. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] proportional representation
by Warren D Smith 10 Dec '15

10 Dec '15

I wrote 1.7 papers on this topic. Part I: http://rangevoting.org/QualityMulti.html Part II: http://rangevoting.org/NonlinQuality.html The "holy grail" proportional representation voting method is defined http://rangevoting.org/NonlinQuality.html#holygrail and I have been unable to find a method that satisfies those demands. This leads to the conjecture no such voting method is possible. But I have been unable to prove impossibility. The impossibility (if it is impossible) may be topological in nature, so topologists might want to think about it. [People have also asked me about Arrow and Gibbard-Satterthwaite impossibility theorems. Those are not related in the sense they are about single-winner, not PR (multiwinner) voting. But you can learn about them here: http://rangevoting.org/ArrowThm.html http://rangevoting.org/GibbSat.html as well as my own impossibility theorem http://www.rangevoting.org/SimmonsSmithPf.html and there are several more impossibility theorems.] -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] homespun computing conspiracy
by Veit Elser 10 Dec '15

10 Dec '15

Last Sunday an episode of The Simpsons aired where Lisa discovers a loom-based computer. After having it print out a cross-stitched “Hello World”, she inputs 27 x 35 on a type of punched card and the loom responds with 945, the first odd abundant number (https://oeis.org/A005231) Today D-Wave/Google announced a breakthrough in quantum computing, more precisely, “quantum annealing” (http://arxiv.org/pdf/1512.02206v1.pdf) The proof-of-principle demonstration was an optimization problem involving 945 variables. -Veit

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[math-fun] 2015 is palindromic in binary
by Tom Rokicki 09 Dec '15

09 Dec '15

This would have been more appropriate about 11 months ago. 2015 = 11111011111 in base 2. When's the next time that happens? -- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] --

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[math-fun] Floating point matrix product anomalies
by Warren D Smith 09 Dec '15

09 Dec '15

> From: Bill Gosper <billgosper(a)gmail.com> > Subject: [math-fun] Floating point matrix product anomalies --Wow, that is strange. Perhaps related is this: 1. Take N consecutive rightward steps of length 1+noise (where noise is a tiny-variance mean=0 random variable). You will reach N+-sqrt(N)*NoiseAmplitude. 2. Instead take a step, double it, add noise, and keep on doing that for lg(N) doublings. You will reach N+-N*Constant*NoiseAmplitude. 3. If in (1) the noise each step is negatively correlated to the noise the preceding step, then we do even better. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Big numbers
by James Propp 07 Dec '15

07 Dec '15

I seem to recall once seeing a website that described various big numbers, such as the number of "distinguishable" mental states for a human brain, the number of distinguishable mental lives a person could have (where a mental life is defined as a succession of mental states, using some reasonable discretization of time), the number of different states a system the size of the known universe could have, the number of different histories such a universe could have, etc. (Of course there's a certain arbitrariness to the way one calculates these things, but still, the Ackerman-hierarchy complexity of these numbers is fairly insensitive to the details.) I also seem to recall that it was some post on math-fun that steered me toward that website, so can one of you point me in the right direction? Thanks, Jim Propp

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[math-fun] Number of points on convex hull, and number of undominated points
by Warren D Smith 07 Dec '15

07 Dec '15

Let P(X,Y) be any fixed probability density in the plane that is unimodal on lines. For any such density I conjecture these upper bounds: (i) sampling N points from it independently will yield expected #vertices on convex hull growing like O(N^(1/3)) and (ii) expected #undominated points growing like O(N^(1/2)). A point (X,Y) is "undominated" if the infinite closed axis-parallel rectangle with lower left corner (X,Y) is empty of other points. These bounds (if valid) are tight since the uniform density within a circle (zero outside) achieves them. The variances might (more strongly) be conjectured also to be bounded by the same upper bounds. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Floating point matrix product anomalies
by Bill Gosper 05 Dec '15

05 Dec '15

Spin a 14gon RotationMatrix[\[Pi]/7.] // InputForm {{0.9009688679024191, -0.4338837391175581}, {0.4338837391175581, 0.9009688679024191}} one million clicks, less one: Nest[%.# &, IdentityMatrix@2, 999999] // InputForm // tim 0.218347,1 {{-1.000000000000001, -5.551115123125783*^-16}, {5.551115123125783*^-16, -1.000000000000001}} An error this small after a million clicks means the error can not grow, even with a trillion clicks. Couldn't we do even better with just twenty-odd multiplies in a repeated squaring (addition chain) process? MatrixPower[%%, 999999] // InputForm // tim 0.000035,1 {{-1.000000000055938, -5.657918578094723*^-12}, {5.657918578094723*^-12, -1.000000000055938}} Horrific! And 35 musec wasn't enough time for Mathematica to go off and outsmart itself with eigenvalues, e.g. But just to make sure, I rolled my own repeated squarer: mpowr[%%%, 999999] // InputForm // tim 0.000176,1 {{-1.0000000000272093, -3.690142635903726*^-11}, {3.690142635903726*^-11, -1.0000000000272093}} Ten times worse! Here's mpowr: mpowr[M_, 0] := IdentityMatrix[Length[M]] mpowr[M_, n_?EvenQ] := mpowr[M.M, n/2] mpowr[M_, n_] := mpowr[M, n - 1].M Now repeat this whole experiment with 2 pi/7, whose rotation matrix is merely the square of the previous. RotationMatrix[2 \[Pi]/7.] // InputForm {{0.6234898018587336, -0.7818314824680298}, {0.7818314824680298, 0.6234898018587336}} Nest[%.# &, IdentityMatrix@2, 999999] // InputForm // tim 0.215342,1 {{1.0000000000023095, 3.3821279110668456*^-12}, {-3.3821279110668456*^-12, 1.0000000000023095}} This error will eventually balloon exponentially! Addition chains are no help: mpowr[%%, 999999] // InputForm // tim 0.000178,1 {{1.0000000000544191, 7.38031857849819*^-11}, {-7.38031857849819*^-11, 1.0000000000544191}} MatrixPower[%%%, 999999] // InputForm // tim 0.000069,1 {{1.0000000000302447, 5.991646068181922*^-11}, {-5.991646068181922*^-11, 1.0000000000302447}} But a rotation is three skews: In[271]:= Inactive[Dot][{{1,-Tan[t/2]},{0,1}},{{1,0},{Sin[t],1}},{{1,-Tan[t/2]},{0,1}}] Out[271]= {{1,-Tan[t/2]},{0,1}}.{{1,0},{Sin[t],1}}.{{1,-Tan[t/2]},{0,1}} In[272]:= Activate[%] Out[272]= {{1-Sin[t] Tan[t/2],-Tan[t/2]-Tan[t/2] (1-Sin[t] Tan[t/2])},{Sin[t],1-Sin[t] Tan[t/2]}} In[273]:= FullSimplify[%] Out[273]= {{Cos[t],-Sin[t]},{Sin[t],Cos[t]}} And skews are reversible, hence lossless. The three skews: In[448]:= %271/.t->2\[Pi]/7. Out[448]= {{1,-0.481575},{0,1}}.{{1,0},{0.781831,1}}.{{1,-0.481575},{0,1}} exactly equal the 2pi/7 matrix: In[449]:= Activate[%]//InputForm Out[449] {{0.6234898018587336, -0.7818314824680298}, {0.7818314824680298, 0.6234898018587336}} But if we multiply one-at-a-time, In[450]:= Nest[#.%%[[1]].%%[[2]].%%[[3]]&,IdentityMatrix@2 ,999999]//InputForm//tim During evaluation of In[450]:= 15.1245,1 Out[450] {{0.999999999999998, 5.5067062021407764*^-14}, {-5.3512749786932545*^-14, 0.9999999999999998}} No loss! Go for two million. In[452]:= Nest[#.%448[[1]].%448[[2]].%448[[3]]&,IdentityMatrix@2 ,2*999999]//InputForm//tim During evaluation of In[452]:= 20.0325,1 Out[452] {{0.999999999999998, 5.5067062021407764*^-14}, {-5.3512749786932545*^-14, 0.9999999999999998}} No loss. But merely squaring the one million matrix In[454]:= %450.%450//InputForm Out[454] {{0.999999999999996, 1.101341240428154*^-13}, {-1.0702549957386496*^-13, 0.9999999999999996}} doubled the error! Floating point matrix multiplication is *seriously* nonassociative. --rwg

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