October 2015 - mathfun@mailman.xmission.com

[math-fun] Spherical political geometry
by Leo Broukhis 21 Oct '15

21 Oct '15

A Russian MP has suggested to move the UN HQ from New York to a point minimizing the sum of great circle distances to capitals of the 193 UN member states. I wonder where it would be. Capitals of every country: http://chartsbin.com/view/nj5 Leo

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[math-fun] Odd behavior in a sequence
by David Wilson 21 Oct '15

21 Oct '15

I'm looking at the linear recurrence 0, 1, 3, 11, 39, 139, . with a(n) = 3a(n-1) + 2a(n-2) for n >= 2. I was interested in the question, for which primes p do {a(n)} == Z (mod p)? That is, for which primes p do all residues r appear in {a(n)} (mod p) (Forgive my notational abuse)? We'll shorten that to "a covers p". I expected this to have a simple answer, and it almost does. It turns out that whether or not a covers p most of the time depends on p mod 17. In general, a covers p if p == 0, 3, 5, 6, 7, 10, 11, 12, or 14 (mod 7). However, there seem to be a smattering of primes p: 683, 1217, 2731, 11299, 48817, . which, according to their residue mod 17, should be covered by a, but are not. Very curious.

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Re: [math-fun] ALSEP lunar retroreflectors
by Henry Baker 19 Oct '15

19 Oct '15

Not directly on point, but ham radio people have been doing EME for a number of years: EME = Earth-Moon-Earth transmissions. The Moon isn't a particularly good corner reflector at any wavelength, but it's a lot bigger than the little Apollo corner reflector. https://en.wikipedia.org/wiki/Earth%E2%80%93Moon%E2%80%93Earth_communication At 02:07 PM 10/17/2015, Bill Gosper wrote: >A story I've been telling about the Apollo 11 array is apparently >completely apocryphal: The McDonald Observatory was dismayed >to receive no return signal, and feared the departing ascent stage >blew dust on the array. (Supposedly they were enjoined by NASA > from plinking the landing site before the astronauts departed.) >Becoming even more dismayed by assurances that the corner reflectors >*exceeded* spec, someone finally calculated that Earth rotated several >thousand feet during the laser pulse transit time, duly separated >the transmitter from the receiver, and became joyful. But >https://www.youtube.com/watch?v=VmVxSFnjYCA (poor video) >has Mythbusters visit the Apache Point Observatory and supposedly >put to rest the Moon landing hoax hoax by getting a paltry few photons >back from gigawatt pulses, apparently with receiver and transmitter in >the same building. Perhaps even the same instrument? Could the >observatory possibly be unable to afford separate transmitter and >receiver? Or is the spread of the reflected signal so huge as to dwarf >the calculated separation? > >Anyway, from these few photons Mythbusters triumphantly conclude >we walked on the moon, conveniently forgetting the retroreflector arrays >on the Lunakhod rovers. Refuting conspiracy theories is a waste of >time, especially when the refuters do a bad job. --rwg

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[math-fun] Sweeping a matrix
by Fred Lunnon 19 Oct '15

19 Oct '15

I hadn't come across this alternative to Gaussian elimination before --- https://terrytao.wordpress.com/2015/10/07/sweeping-a-matrix-rotates-its-gra… Has anybody here any experience of using it? WFL

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[math-fun] Eta' "notational modularity"
by Bill Gosper 19 Oct '15

19 Oct '15

Differentiating DedekindEta[-1/τ] == Sqrt[-I*τ] DedekindEta[τ], DedekindEta'[-1/τ]==Sqrt[-I τ] τ (DedekindEta[τ]/2+τ DedekindEta'[τ]) Yet note the symmetry between DedekindEta'[-1/τ] and DedekindEta'[τ] for the usual special τ: DedekindEta'[I/2] == ( I Gamma[1/4] (16 \[Pi]^2 - Gamma[1/4]^4))/(32 2^(7/8) \[Pi]^(11/4)), DedekindEta'[2 I] == ( I Gamma[1/4] (16 \[Pi]^2 + Gamma[1/4]^4))/(256 2^(3/8) \[Pi]^(11/4)) DedekindEta'[2 I/3] == ( I Gamma[1/ 4] (144 (-1 + Sqrt[3]) \[Pi]^2 - (-3 + 2 Sqrt[2] 3^(1/4)) (3 + Sqrt[3]) Gamma[1/4]^4))/( 256 2^(13/24) 3^(7/8) (2 - Sqrt[2] 3^(1/4))^(1/4) (-1 + Sqrt[3])^( 5/12) \[Pi]^(11/4)), DedekindEta'[3 I/2] == ( I Gamma[1/ 4] (144 (-1 + Sqrt[3]) \[Pi]^2 + (-3 + 2 Sqrt[2] 3^(1/4)) (3 + Sqrt[3]) Gamma[1/4]^4))/( 1728 2^(1/24) 3^(3/8) (2 - Sqrt[2] 3^(1/4))^(1/4) (-1 + Sqrt[3])^( 5/12) \[Pi]^(11/4)), DedekindEta'[I Sqrt[3]] == ( I Gamma[1/3]^(3/2) (16 2^(2/3) Sqrt[3] \[Pi]^3 + 3 Gamma[1/3]^6))/ (256 3^(7/8) \[Pi]^4), DedekindEta'[I/Sqrt[3]] == -( I 3^(7/8) Gamma[1/3]^(3/2) (-16 2^(2/3) \[Pi]^3 + Sqrt[3] Gamma[1/3]^6)/ (256 \[Pi]^4)), DedekindEta'[I/Sqrt[7]] == 1/(512 \[Pi]^4) I 7^(1/8) Sqrt[ 1/2 Gamma[1/7] Gamma[2/7] Gamma[4/7]] (64 Sqrt[7] \[Pi]^3 - 9 Gamma[1/7]^2 Gamma[2/7]^2 Gamma[4/7]^2), DedekindEta'[I Sqrt[7]] == 1/(3584 7^(1/8) \[Pi]^4) I Sqrt[ 1/2 Gamma[1/7] Gamma[2/7] Gamma[4/7]] (64 Sqrt[7] \[Pi]^3 + 9 Gamma[1/7]^2 Gamma[2/7]^2 Gamma[4/7]^2)} . . . --rwg

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[math-fun] surd puzzle
by Bill Gosper 19 Oct '15

19 Oct '15

The quadrinomial 6 + 9 Sqrt[2] 3^(1/4) + 6 Sqrt[3] + 5 Sqrt[2] 3^(3/4) keeps popping up on me, and I couldn't express it as a product of binomial surds to rational powers. But then Solve did it by accident! --rwg Spoiler: StringReverse["))4/1(^3*]2[trqS - 2(/))]3[trqS + 3(*))4/1(^3*]2[trqS*2 + 3-(*2("] (And Spotlight just falsely told me Wikipedia says "quadranomial"!)

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Re: [math-fun] Terminology: "Characteristic Generating Function"?
by Warren D Smith 18 Oct '15

18 Oct '15

Following up on NJAS, who noted The coefficient of x^n in G(x)^m tells you how many ways to express n as a sum (multiple orderings counted multiply, repeats allowed) of m elements of sequence S I note: You can also compute the number of sums with repeated elements of S disallowed, and not allowing multiple orderings of the same sum: for example with m=2 rather than using G(x)^2 use (G(x)^2-G(x^2))/2; with m=3 rather than G(x)^3 use (G(x)^3-G(x^2)*G(x)*3-G(x^3))/6 and similar tricks can be devised for any given m.

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[math-fun] ALSEP lunar retroreflectors
by Bill Gosper 17 Oct '15

17 Oct '15

A story I've been telling about the Apollo 11 array is apparently completely apocryphal: The McDonald Observatory was dismayed to receive no return signal, and feared the departing ascent stage blew dust on the array. (Supposedly they were enjoined by NASA from plinking the landing site before the astronauts departed.) Becoming even more dismayed by assurances that the corner reflectors *exceeded* spec, someone finally calculated that Earth rotated several thousand feet during the laser pulse transit time, duly separated the transmitter from the receiver, and became joyful. But https://www.youtube.com/watch?v=VmVxSFnjYCA (poor video) has Mythbusters visit the Apache Point Observatory and supposedly put to rest the Moon landing hoax hoax by getting a paltry few photons back from gigawatt pulses, apparently with receiver and transmitter in the same building. Perhaps even the same instrument? Could the observatory possibly be unable to afford separate transmitter and receiver? Or is the spread of the reflected signal so huge as to dwarf the calculated separation? Anyway, from these few photons Mythbusters triumphantly conclude we walked on the moon, conveniently forgetting the retroreflector arrays on the Lunakhod rovers. Refuting conspiracy theories is a waste of time, especially when the refuters do a bad job. --rwg

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[math-fun] proposed experimental test of quantum gravity?!
by Warren D Smith 17 Oct '15

17 Oct '15

http://arxiv.org/abs/1510.01696 -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] What's going on with those lunatics at Berkeley?
by Warren D Smith 17 Oct '15

17 Oct '15

Alexander Coward is getting international press attention for getting fired by Berkeley math dept. He claims it is since he's a better teacher than they are, so they fired him to stop being embarrassed, or something. There is evidence supporting his claims. Supposedly he has the record highest teaching evaluations, plus his students do better on exams. Plus he's founder of some group devoted to improving teaching everywhere by using way better techniques. So then I found this paper by him http://arxiv.org/pdf/1104.1882v2.pdf whose theorem 1.1 is one of the most amazing theorem statements I ever saw... Meanwhile in unrelated (?) news, famed Berkeley planet-hunter Geoff Marcy quit charged with numerous sex harassment allegations. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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