December 2015 - mathfun@mailman.xmission.com

[math-fun] Circulant Markov matrices ??
by Henry Baker 19 Dec '15

19 Dec '15

Anyone know anything about these? I was particularly interested, because (1/N) times the matrix of all ones acts like a kind of "zero" in that it wipes out all information about any circulant markov matrix that it multiplies. If Z = matrix([1,1,...,1],[1,1,...,1],...,[1,1,...,1]), and if M is any circulant markov matrix (sum of rows & cols = 1), then Z.M = M.Z = Z Note that |Z.M|=|Z||M|=|M||Z|=|M.Z|=|Z|=0, so this makes sense.

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[math-fun] new particle found?
by Warren D Smith 19 Dec '15

19 Dec '15

CERN allegedly has found a new particle with mass 750 GeV/c^2. The heaviest standard model fundamental particles had been top quark (173) and Higgs boson (125). What is this new particle? Does it really exist? Maybe it is merely some sort of composite "glueball" and nothing beyond the standard model. CERN does not yet consider itself to have the requisite 5sigma confidence level to announce it certainly exists, but presumably less than 1 more year of data would suffice. Various physicists expressed skepticism. http://www.livescience.com/53130-possible-new-particle-found.html -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Mixed numbers?
by Bill Gosper 19 Dec '15

19 Dec '15

A young friend has to convert all his "improper" fractions to mixed numbers. When he repeated my remark that mathematicians, scientists, and engineers shun mixed numbers, he got hauled up before the dean, who angrily insisted that mathematicians actually prefer mixed numbers. Does anybody know a computer algebra system that even offers mixed numbers as an output format? An input format?? --rwg A still younger friend emailed: What?? 3½=3+½?? That's crazy! I was showing this guy rotation matrices, and he seriously asked, "What's a degree? Is that like pi/180?" --rwg

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Re: [math-fun] new particle found?
by Henry Baker 19 Dec '15

19 Dec '15

The new particle might be the missing "Higgs bosoff" antiparticle to the Higgs boson. <bos|on>, <bos|off> (or </bos|on>), in <bra|ket> notation. At 11:04 AM 12/19/2015, Warren D Smith wrote: >CERN allegedly has found a new particle with mass 750 GeV/c^2. >The heaviest standard model fundamental particles had been >top quark (173) and Higgs boson (125). > >What is this new particle? Does it really exist? Maybe it is merely >some sort of composite "glueball" and nothing beyond the standard model. >CERN does not yet consider itself to have the requisite 5sigma >confidence level to announce it certainly exists, but presumably less >than 1 more year of data would suffice. > >Various physicists expressed skepticism. > >http://www.livescience.com/53130-possible-new-particle-found.html

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[math-fun] Golay code question
by Mike Stay 17 Dec '15

17 Dec '15

On wikipedia, it's claimed that this generator matrix for the Golay code is given by the identity matrix and the complement of the vertex adjacency graph of the icosahedron. https://commons.wikimedia.org/wiki/File:BinaryGolayCode.svg However, I haven't been able to figure out a labeling of vertices that actually makes it work. In fact, it looks as though this construction starts with a cyclic 10x10 matrix and then adds two parity bits (which isn't to say it can't be done). I can label the vertices however I want and get an adjacency matrix for that labeling; is there a way to figure out the permutation given the two matrices? -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

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[math-fun] Paul Lockhart?
by Michael Kleber 17 Dec '15

17 Dec '15

Does anyone know Paul Lockhart, author of "A Mathematician's Lament" and more recently "Measurement"? Would like to get in touch. --Michael -- Forewarned is worth an octopus in the bush.

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[math-fun] xkcd
by rcs＠xmission.com 17 Dec '15

17 Dec '15

Today's XKCD cartoon has a nice collection of approximations. http://xkcd.com/1047/ There's the ancient expression 6 pi^5 for the mass ratio of the proton to the electron, and the more accurate (e^8 - 10)/phi. Robert Munafo & REIS are at least partially responsible. Rich

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Re: [math-fun] game spinners & roulette wheels
by Henry Baker 17 Dec '15

17 Dec '15

Good point! I don't know whether the problem can be fixed, but what if we posit that the spinner has to go around at least once; otherwise, the spin doesn't count & we need to try again ? Presumably there is a minimum energy/angular momentum needed to get the spinner to make a complete circle. Similarly, there is a minimum energy/angular momentum needed to get to each of the different marked off sectors. If the "slowing down" function as a function of the initial "kick" is f(k), k is the kick, then the final value is something like f(k) mod 1 (assuming we measure each full turn as being =1). We now need to allocate the interval [0..1) into N pieces s.t. the chance of landing in each piece is the same. I suppose for the sake of at least one solution, we could require exactly M full turns but less than M+1 full turns, where M is a constant decided upon ahead of time. However, we could just as well have started with a smaller energy & angular momentum in the first place. I guess that the "randomness" here comes from the inability to calibrate the initial kick finely enough to distinguish between M and M+1 turns, much less some intermediate sector. At 10:46 AM 12/17/2015, James Propp wrote: >I suspect I've misunderstood the question, but: if the initial impulse is >small, won't the spinner end up pointing at whatever sector it's pointing >at before you start spinning it (leaving aside a small proportion of cases >where it starts close to a boundary between sectors)? > >Jim Propp > >On Thu, Dec 17, 2015 at 11:04 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> I'm sure that everyone has seen and/or used >> one of those little spinners that come with >> many board games. There's usually a piece >> of cardboard with a circle printed on it and >> a metal spinning arrow. In order to produce >> a "random" number, you pluck the spinner arrow >> in such a way that it spins before stopping >> on a color or a number. >> >> Now since the spinner is given a certain >> amount of angular momentum due to an impulse >> from your finger, it has to dissipate that >> angular momentum (and energy) in order to >> stop and indicate the color or number. >> >> There are various models of the physics of >> friction that one might utilize to estimate >> where the arrow will stop. But it should >> be obvious that no matter which model we >> use, these spinner can't possibly produce >> unbiased random numbers -- at least so long >> as each of the colored and/or numbered >> sectors are identical in size. >> >> So... Let's choose a preferred direction >> for the spin -- e.g., *clockwise*. Is there >> a physical model of friction that allows a >> sector size allocation so that the numbers >> chosen *will* be less biased -- at least to >> a first-order approximation? >> >> In particular, is there a model of friction >> that allows a fixed allocation of sector >> sizes *independent* of the absolute size of >> the initial impulse ?

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[math-fun] Trachtenberg arithmetic
by James Propp 17 Dec '15

17 Dec '15

I just got hold of a copy of a book I haven't seen in over forty years: "The Trachtnberg Speed System of Basic Mathematics". Part of the Foreword has me stumped: "... A small girl with beribboned braids was asked to find the solution of 735352314 times 11. She came up with the correct answer --- 8088875454 --- in less time than you can say multiplication table. A thin, studious-looking boy wearing silver-rimmed spectacles was told to multiply 5132437201 times 452736502785. He blitzed through the problem, computing the answer --- 2323641669144374104785 --- in seventy seconds." I know how to do the former trick (the multiplying-by-eleven trick is well known), but I'm baffled by the latter feat. Do any of you have any insights? Jim

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[math-fun] game spinners & roulette wheels
by Henry Baker 17 Dec '15

17 Dec '15

I'm sure that everyone has seen and/or used one of those little spinners that come with many board games. There's usually a piece of cardboard with a circle printed on it and a metal spinning arrow. In order to produce a "random" number, you pluck the spinner arrow in such a way that it spins before stopping on a color or a number. Now since the spinner is given a certain amount of angular momentum due to an impulse from your finger, it has to dissipate that angular momentum (and energy) in order to stop and indicate the color or number. There are various models of the physics of friction that one might utilize to estimate where the arrow will stop. But it should be obvious that no matter which model we use, these spinner can't possibly produce unbiased random numbers -- at least so long as each of the colored and/or numbered sectors are identical in size. So... Let's choose a preferred direction for the spin -- e.g., *clockwise*. Is there a physical model of friction that allows a sector size allocation so that the numbers chosen *will* be less biased -- at least to a first-order approximation? In particular, is there a model of friction that allows a fixed allocation of sector sizes *independent* of the absolute size of the initial impulse ?

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