July 2015 - mathfun@mailman.xmission.com

[math-fun] Fwd: [New post] Gaps between primes
by Fred Lunnon 06 Jul '15

06 Jul '15

I have in the past made such a mess of attempting to paraphrase TT's communications, that it's safer just to forward anything looking remotely like "fun" ... WFL ---------- Forwarded message ---------- From: Whats new <comment-reply(a)wordpress.com> Date: Mon, 6 Jul 2015 17:24:10 +0000 Subject: [New post] Gaps between primes To: fred.lunnon(a)gmail.com Post : Gaps between primes URL : https://terrytao.wordpress.com/2015/07/06/gaps-between-primes/ Posted : 6 July, 2015 at 9:24 am Author : Terence Tao Tags : polymath8 Categories : math.NT, talk Just a short post here to note that the cover story of this month's Notices of the AMS ( http://www.ams.org/notices/201506/rnoti-p660.pdf ) , by John Friedlander, is about the recent work on bounded gaps between primes by Zhang, Maynard, our own Polymath project, and others. I may as well take this opportunity to upload some slides of my own talks on this subject: here are my slides on small and large gaps between the primes ( https://terrytao.files.wordpress.com/2015/07/lat.pdf ) that I gave at the "Latinos in the Mathematical Sciences ( http://www.ipam.ucla.edu/programs/special-events-and-conferences/latinos-in… ) " back in April, and here are my slides on the Polymath project ( https://terrytao.files.wordpress.com/2015/07/polymath.pdf ) for the Schock Prize symposium ( http://www.kva.se/en/Events-List/2014/The-Rolf-Schock-Prize-symposium-in-ma… ) last October. (I also gave an abridged version of the latter talk at an AAAS Symposium ( https://aaas.confex.com/aaas/2015/webprogram/Session9529.html ) in February, and the Breakthrough Symposium ( https://breakthroughprize.org/Symposiums/5 ) from last November.) ____________________________________________________________

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[math-fun] Definition of mathematics
by James Propp 06 Jul '15

06 Jul '15

A couple of weeks ago, in a gathering of mathematicians throwing out candidate definitions of mathematics, I half-seriously ventured the opinion that mathematics is the subset of philosophy consisting of those philosophical questions that actually have answers, along with the answers to those questions. But I don't think this is original. Whom am I quoting (or paraphrasing) here? Jim Propp

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[math-fun] New(?) way to tabulate pFq identities
by Bill Gosper 05 Jul '15

05 Jul '15

By a process not obvious how to automate, I extracted HypergeometricPFQ[{a, b, (2 a)/5 + (6 b)/5 + 1, 2 a + 2 b, a + 3 b}, {(2 a)/5 + (6 b)/5, a + (3 b)/2 + 1/2, a + (3 b)/2 + 1, 2 b + 1}, -(1/4)] == (4^-a Gamma[b + 1/2] Gamma[2 a + 3 b + 1])/( Gamma[a + b + 1/2] Gamma[a + 3 b + 1]) from a summation containing a messy, irreducible cubic factor, analogous to the sort used to discover recurrence relations via "creative telescopy". Punting the cubic cost two degrees of freedom, but there were hundreds of variants. The unpleasant choice seems to be between a useless messy cubic, and a nearly useless identity with only two degrees of freedom. Probably the identity you need was among the hundreds I threw away. Here's an interesting remedy. What you actually tabulate is, in this case, the fully general result in a somewhat intimidating form: ((-b - d) ((1/(-1 + b))(-1 + b - d) (-1 + a + b - d) (-1 + b + c - d) HypergeometricPFQ[{a, -1 + b, c, d, -1 - a + b + d, -1 + b - c + d}, {b/2 + d/2, 1/2 + b/2 + d/2, 1 - a + d, 1 - c + d, -a + b - c + d}, -(1/ 4)] - (19 + 5 b^2 - 7 c + b (-19 + 4 c - 11 d) + a (-7 + 4 b + c - 3 d) + 20 d - 3 c d + 5 d^2) HypergeometricPFQ[{a, b, c, d, -1 - a + b + d, -1 + b - c + d}, {b/2 + d/2, 1/2 + b/2 + d/2, 1 - a + d, 1 - c + d, -a + b - c + d}, -(1/ 4)] - 5 (-3 + b) (-2 + b) HypergeometricPFQ[{a, -1 + b, b, c, d, -1 - a + b + d, -1 + b - c + d}, {-3 + b, b/2 + d/2, 1/2 + b/2 + d/2, 1 - a + d, 1 - c + d, -a + b - c + d}, -(1/ 4)] + 3 (-2 + b) (-8 + a + 3 b + c - 3 d) HypergeometricPFQ[{a, -1 + b, b, c, d, -1 - a + b + d, -1 + b - c + d}, {-2 + b, b/2 + d/2, 1/2 + b/2 + d/2, 1 - a + d, 1 - c + d, -a + b - c + d}, -(1/ 4)]))/(a c (-1 + b + c) (1 - b + c - d) d) == -((Gamma[1 - a + d] Gamma[1 + b + d] Gamma[ 1 - c + d] Gamma[-a + b - c + d])/(a (-1 + b) c (-1 + b + c) Gamma[1 + d] Gamma[-1 - a + b + d] Gamma[ 1 - a - c + d] Gamma[b - c + d])) It's then fairly routine to determine whether your pFq[-1/4] is a special case. It's not pencil and paper, but many people now have access to computers. --rwg Count yourself lucky if you can read "Rauzy fractal" and not remember Them Moose Goosers.

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[math-fun] A neat word I just forgot
by James Propp 05 Jul '15

05 Jul '15

I recently accidentally learned, and then carelessly forgot, a word (based on Greek roots perhaps) that describes prefixes and/or suffixes that negate the word that they're attached to. What are the word(s) you know that have this meaning? Jim Propp

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[math-fun] I'm deriving false identities.
by Bill Gosper 04 Jul '15

04 Jul '15

I don't understand why. By subtracting, dividing, and plotting, I sometimes guess the empirical corrections. They're slightly weird: (-2 h + k) Beta[h, k] HypergeometricPFQ[{-h, 1 - (2 h)/3 + k/3, -k, -2 h + k}, {1/2 - h, -((2 h)/3) + k/3, 1 - h + k}, 1/4] == (1/( h k))(h + k) (2 (h - k) + (-4 h + 3 k) HypergeometricPFQ[{1, -2 h, -h, 1 - (4 h)/5 + (3 k)/5, -k, -2 h + k}, {1/2 - h, 1 + h, -((4 h)/5) + (3 k)/5, 1 + k, 1 - h + k}, -(1/4)]); 2 Beta[h, k] HypergeometricPFQ[{-h, -h - k, 1 - (2 h)/3 - (2 k)/3, -2 k}, {1/2 - h/2 - k, 1 - h/2 - k, -((2 h)/3) - (2 k)/3}, 1/4] == (1/( h k))(-h - 2 k + (3 h + 4 k) HypergeometricPFQ[{1, -h, -h - 2 k, -h - k, 1 - (3 h)/5 - (4 k)/5, -2 k}, {1 + h, 1/2 - h/2 - k, 1 - h/2 - k, -((3 h)/5) - (4 k)/5, 1 + k}, -(1/4)]) (Note +1/4 vs -1/4, and additive terms, and upper-parameter 1) I expected them to come out in Gammas (they may yet), but a trivariate polynomial unexpectedly vanished and I had to take a limit. --rwg

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[math-fun] Definition of mathematics
by Warren D Smith 04 Jul '15

04 Jul '15

Did the ancient Greeks, who used geometric axioms, count as mathematicians under Wilson's "definition" involving formal symbolic manipulation? I guess I would say that mathematics is the study of thinking. And how to be a better thinker. But this is not a mathematical definition :) And I regard CS, physics, and lots of other fields such as biology, chemistry, psychology, economics as all involved and all as fair game for mathematicians. And if not, that is because of a failure of either those fields, or of mathematicians. You know, I once worked at a place called NEC research institute which was supposedly about combining physics, computer science, chemistry, and biology. It self destructed, which is another story. But anyhow, as far as I could tell I was the only one there who tried to do all of them. Mostly, everybody quickly segregated into "their own field" and avoided the others. I also was one of the only ones there (depending on the date) who called himself a mathematician. DId this mean that I was about the sole exemplar of what they wanted? Actually, far as I could tell my approach was not appreciated. More precisely, some appreciated it others not, but the weighted sum was more toward "not." I also think that even within the set of people who call themselves mathematicians, my universalist approach is usually not taken. Usually instead the mathematicians themselves specialize into subfields. This may or may not be a good thing (probably both) but the potential of mathematics as the unifying glue connecting every field, has not been exploited as much as it should. Who are some good universalist mathematicians? List your faves. Von Neumann, Poincare, Feynman spring to mind. Neil Sloane, who sometimes posts on this list, could be argued to be a specialist, but I think he's succeeded in having an impact felt across many fields. I personally would regard Francis Crick as a mathematician, although nobody called him that, including himself. Maybe what I want is what they used to call "natural philosopher" but I think few if any people call themselves that anymore. Newton certainly tried. He also was interested in biology and chemistry, but as far as I know never published in those areas because he never succeeded in his own view in deciphering them. But he tried hard.

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[math-fun] Hartle Hawking "no boundary" proposal
by Warren D Smith 03 Jul '15

03 Jul '15

J. B. Hartle and S. W. Hawking: Wave function of the Universe Phys. Rev. D 28 (1983) 2960- Can anybody explain this idea?

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Re: [math-fun] Haptic hearing
by Henry Baker 03 Jul '15

03 Jul '15

[P.S. We also used an AM radio in the computer room to "hear" how the job was progressing w/o relying on the console lights. It was amazing how quickly the ear could pick up any problems with the computation after "hearing" it a few times. We came to the conclusion that a blind person could easily become gainfully employed as a computer operator; a conclusion that startled most sighted (and blind) people.] At my first computer programming job in high school, the U. of Cincinnati was teaching blind people how to program. These were exceedingly bright people, and we worked out a way to print braille by putting a cushion behind the printer paper so that the printer hammers would dimple the paper. Most of the blind programmers used their fingers, but one of them took off his shirt and pressed the paper against his chest. He couldn't read individual letters that way, but he could quickly "glance" at the page to recognize certain obvious patterns -- e.g., source code, assembly listing, hex/octal dump, etc. At 06:43 AM 7/3/2015, James Propp wrote: >Thanks, Hans! > >Just to make sure my incorrect original question doesn't spread any false >memes: according to Snopes, the physician in question relies on vision and >a compendious knowledge of classical music, NOT touch. > >Could one recruit the sense of touch in the fingertips to replace vision or >hearing? (I believe the nerves in the tongue are being used this way in >some experiments.) > >Jim Propp > >On Friday, July 3, 2015, Hans Havermann <gladhobo(a)teksavvy.com> wrote: > >> http://skepdic.com/vinylvision.html >> http://www.snopes.com/music/media/reader.asp

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Re: [math-fun] Haptic hearing
by Henry Baker 03 Jul '15

03 Jul '15

At my first computer programming job in high school, the U. of Cincinnati was teaching blind people how to program. These were exceedingly bright people, and we worked out a way to print braille by putting a cushion behind the printer paper so that the printer hammers would dimple the paper. Most of the blind programmers used their fingers, but one of them took off his shirt and pressed the paper against his chest. He couldn't read individual letters that way, but he could quickly "glance" at the page to recognize certain obvious patterns -- e.g., source code, assembly listing, hex/octal dump, etc. At 06:43 AM 7/3/2015, James Propp wrote: >Thanks, Hans! > >Just to make sure my incorrect original question doesn't spread any false >memes: according to Snopes, the physician in question relies on vision and >a compendious knowledge of classical music, NOT touch. > >Could one recruit the sense of touch in the fingertips to replace vision or >hearing? (I believe the nerves in the tongue are being used this way in >some experiments.) > >Jim Propp > >On Friday, July 3, 2015, Hans Havermann <gladhobo(a)teksavvy.com> wrote: > >> http://skepdic.com/vinylvision.html >> http://www.snopes.com/music/media/reader.asp

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Re: [math-fun] Polynomials as electric potentials
by Henry Baker 03 Jul '15

03 Jul '15

Also, see Fast Multipole Method for solving such problems: https://en.wikipedia.org/wiki/Fast_multipole_method At 07:19 AM 7/3/2015, Warren D Smith wrote: >Actually, the whole view of a "monic polynomial" P(z) >as P(z)=PRODUCT_j (z-r_j) where r_j are the roots >hence >log P(z) = SUM_j log(z-r_j) >hence >log|P(z)| = SUM_j log|z-r_j| = potential energy of >movable negative unit charge located at z >in the coulomb fields of positive unit charges fixed at the r_j, is often quite useful. > >For one example, suppose you want to make a rootfinding algorithm. >"Newton's iteration" is equivalent to: you assume the whole electric field -P'(z)/P(z) >at z is due to just one root R, and solve for its location; then go there (z:=R) and >do it again. Some other iterations can also be thought of electrostatically. > >One can also create "simultaneous iterations" in this way which attempt to >converge to all N roots of P(z) at once. Place N movable negative unit charges >in the complex plane. They will repel each other. There will also be attractive forces from >the N fixed roots r_j of P(z), which are positive unit charges. The >min-energy locations of the N movable charges are co-located with the N roots. Minimizing energy solves for >the roots. Basically any iteration which keeps decreasing energy in a >sensible way must converge. You can at each guess-location compute the >electric field arising from both the fixed and movable charges, >pretend it is due to just a constant number of them, e.g. to 1 >positive and 1 negative charge, and go to the location of the positive... or take a smaller >step in that direction (keep decreasing step size until cause energy decrease; one >can devise a "safety rule" of that sort to provably assure convergence). >By fast polynomial algorithm methods the simultaneous iteration can be implemented >to run in O(N*(logN)^2) flops per simultaneous-adjust. A famous theorem of >electrostatics assures that there exists no energy local-min for the movable charges aside >from the global min we want. These schemes can be shown to converge >ultimately in a manner such that the number of correct decimals grows >geometrically, if all roots are isolated. Convergence to multiple >roots is slower, geometric, although you can devise variant algorithms >which do not care about multiplicities up to say, 4, and you can >remove all multiple roots ahead of time by solving for the roots not of P(z) but rather of >Q(z)=P(z)/GCD(P(z), P'(z)) which has only single roots. > >I invented and programmed this class of rootfinding methods in the 1980s. >They work pretty well if you use your own floating point arithmetic (not machine's) >with larger exponent field so as to remove worries about numerical overflow >with high-degree polynomials.

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