August 2015 - mathfun@mailman.xmission.com

[math-fun] Forget about runtimes longer than simple exponential
by Warren D Smith 12 Aug '15

12 Aug '15

> APG: Your universe sounds incredibly boring and nihilistic. > I'll continue working in the von Neumann universe instead... --Indeed, if Conway's "life" were the rules of the universe (pseudo-physical laws) then the max amount of time before some N-particle configuration vanished (if it ever vanished) would be, like the "busy beaver number," a function growing faster than any computable function of N. And apparently APG spends a goodly amount of his time in the Conway Life universe...

1 0

[math-fun] Who's Saari now
by Henry Baker 12 Aug '15

12 Aug '15

A delated reaction from an expert in cosmological N-body simulations: "I think [Saari's] is a mathematically fascinating configuration and a nice example that assumptions on smooth/spherical distributions of points can fail catastrophically sometimes. This reminds me of the classic "exploding galaxy" problem, that Salmon & Warren (1994) constructed to demonstrate how a gravitational N-body code based on a tree-code algorithm can lead to unphysical results." "Still, much like we have been using tree codes for simulations without obtaining catastrophic results for decades, I think that the Saari's configuration is very special and not what is realized in galaxies. To support this, we can use observations of disk galaxies at different inclination angles which show the presence of a fairly smooth distribution of stars, typically with a profile exponentially declining with increasing radius. Finally, from the mathematical point of view, it would be interesting to evaluate whether Saari's configuration is stable against perturbations." ----------------- https://www.siam.org/pdf/news/2094.pdf Rethinking ÂStar SoupÂ By Dana Mackenzie One of the greatest mysteries in astrophysics these days is dark matterÂ-an invisible form of matter that has been detected only through its gravitational effects. According to the latest estimates from the European Space AgencyÂs Planck space telescope, dark matter accounts for 26.8% of the matterÂenergy in the universe. That is 5.5 times the amount of ordinary, or baryonic matter. (Both are dwarfed by Âdark energy,Â but thatÂs a subject for another article.) To date, nobody has ever actually seen a ÂlumpÂ of dark matter. All the evidence for its existence depends on mathematical calculationsÂ-something that ought to make mathematicians happy. But Don Saari, a longtime special- ist in celestial mechanics at the University of California at Irvine, is far from convinced. At a standing-room-only lecture at this yearÂs Joint Mathematics Meetings, Saari called into question a classicalÂ-in fact the earliestÂ-method for estimating the amount of dark matter, which uses galactic rotation curves. ÂItÂs a mathematical computation, and that computation is not correct,Â Saari says. http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.407 Mathematics and the ÂDark MatterÂ Puzzle Donald G. Saari Abstract. Surprisingly, aspects of the compelling mystery of Âdark matterÂÂ-the mysterious, undiscovered material that supposedly consumes most of a spiral galaxyÂs massÂ-are math- ematical issues rather than astronomical ones. When analyzed from the perspective of math- ematics, doubt is cast on standard predictions about the existence of huge amounts of this material.

1 0

[math-fun] Forget about runtimes longer than simple exponential
by Warren D Smith 11 Aug '15

11 Aug '15

Forget about runtimes longer than simple exponential ===========Warren D Smith=====Aug 2015============== I saw that computer scientist Bernard Chazelle is now interested in "natural algorithms" which have been "refined by evolution for millions of years," and as a case in point, he in the rather awe-inspiring 106-page paper http://arxiv.org/abs/0905.4241 The Convergence of Bird Flocking (2009) examined the behavior of the "Cucker-Smale model of bird flocking," finding that an N-bird flock would take TIME upper bounded by 2^(2^(2^(...2))) steps to reach its final combinatorial configuration, where the number of 2's in the tower is <=4*lg(N), but, he showed, could under some circumstances exceed lg(N/2). The MODEL is: each bird is a moving point in 3-space, with piecewise-constant velocity vector. Each discrete time step (e.g. each second) the bird readjusts its velocity vector to be a convex combination of its old velocity and the average velocity of itself and that of its "neighbors," i.e. of all other birds within distance 1. He allows the coefficients in the combination (which he calls the "self confidence" factor for that bird) to vary with the bird and with time, in some way he still has not explained after 20 pages (I think he is allowing any such rule and considering the one that yields slowest convergence? His theorem statements and claims in "The Results" section unfortunately ignore this issue), and also you stay my "neighbor" even if your distance from me gets slightly greater than 1 (which Chazelle calls "hysteresis"). What do I have to say about this? I have to say that this model is biophysically wrong. It is not how birds flock. And more importantly, NO ALGORITHM RUNNING IN OUR UNIVERSE CAN TAKE LONGER THAN SIMPLE EXPONENTIAL TIME exp(polynomial(N)) WHERE N IS THE NUMBER OF PARTICLES INVOLVED. Which means that everything Chazelle or anyone else considers that takes, say, double-exponential time, is physically irrelevant. Because of two effects. In the below, let E be the minimum escape energy for any of the N particles (or subsets thereof) in the physical system (E may vary with time). Effect #1: tunneling into a black hole. Consider N particles. For example, the planet Earth consists of about N=1.5*10^52 elementary particles (quarks and electrons). There is a nonzero chance they will quantum-tunnel though the high potential-energy-of-compression barrier to form a black hole. In fact, I claim the expected lifetime of the Earth (or whatever system) before this happens, is upper bounded by exp(C*N^P/E^Q) for some absolute positive constant C and where P and Q also are constants with 0<P<4, 0<Q<4. Effect #2: evaporation. Consider N particles at finite positive temperature T. There at any time is a positive chance one (or a bound subset) of the particles will, just due to thermal fluctuations, attain escape velocity and leave, never to be seen again. This characteristic escape time seems upper bounded by K * N^R * exp(-E/T) where E is the escape energy reckoned as a temperature and K,R are some constants. Note in our universe T is lower bounded by the positive constant "Gibbons Hawking temperature." Combining these 2 effects, we see that no matter what E is, the system will self-destruct in expected simple exponential time at most. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

3 3

[math-fun] FWL more confusion re arctrig
by Warren D Smith 11 Aug '15

11 Aug '15

FWL, you are still confused. Of course nonrigid cases are forbidden.

3 5

[math-fun] "Whig history" treatment of numbers and games?
by James Propp 10 Aug '15

10 Aug '15

I'm re-reading Gardner's article "Conway's Surreal Numbers" (in conjunction with writing my own essay on the subject for my blog), and one passage from Gardner's piece jumped out as me as being, not exactly wrong, but deserving of a certain sort of reply. Gardner quotes Knuth as saying that he wrote "Surreal Numbers" in part "to teach how one might go about developing such a theory". But Knuth's protagonists Alice and Bill, much like Knuth himself, were developing, from the ground up, a theory from someone else's axioms, namely Conway's. It may be worth saying that most math isn't like this, nor is this what Conway himself did. What most math is about is proving new theorems not just from definitions, but from earlier theorems, and using a stockpile of known examples on which one can test out new ideas. But (and here's my main point) what Conway himself did is neither what most mathematicians do (as per the previous paragraph) nor what Alice and Bill and Knuth did (as per the paragraph before that). Conway had to figure out what the right axioms and definitions were, which is an even higher-level mathematical activity than proving theorems. There's a lot that could be said about how mathematical theory-building works, and how it's been practiced by the architects of great theories. One can imagine a book of case-studies that attempts some sort of historical accuracy, in the spirit of David Corfield's notion of bringing philosophy of mathematics into closer alignment with mathematical practice. But that's not exactly what I'm interested in today (though I may have time to be interested in it in the future). What I want to know today is, has anyone written a pedagogically-motivated "Whig history" of the development of the theory of numbers and games, based upon (but not 100% faithful to) what Conway actually did, showing how a Conway *could* have developed the theory, starting from nothing but the observation that some games (such as Go endgames) behave kind of like numbers, and pushing that idea in the directions the idea "wants" to go, and not in directions it doesn't (and having the wisdom to know the difference)? I am thinking of what Lakatos did with the idea of Euler characteristic in "Proofs and Refutations", showing the intense interactions between definitions and theorems and examples (especially but not exclusively counterexamples). Relatedly, has anyone created a "text" on combinatorial game theory that consists basically of a list of problems to be solved, such that working through the problems leads the student to develop the theory on his/her own? If such works exist, I'd like to refer to them in my piece. Jim Propp On Mon, Jul 20, 2015 at 11:03 PM, James Propp <jamespropp(a)gmail.com> wrote: > A smart high school student whom I'm mentoring has expressed interest in > learning about combinatorial game theory. My first thought was to recommend > "Winning Ways", but I think it might be better to start her on something > shorter and more focused, and then suggest "Winning Ways" if the shorter > book whets her appetite for more. Any suggestions? > > Jim Propp >

4 4

[math-fun] Dick Askey always chides me about how easy
by Bill Gosper 10 Aug '15

10 Aug '15

these Gamma identities are: Gamma[-1/12] == (3^(3/8) ((2 + Sqrt[3])/Pi)^(1/4) Gamma[-1/4] Sqrt[-Gamma[-1/6]])/2^(2/3) I guess FullSimplify and I flunk his course. --rwg

2 1

Re: [math-fun] FWL more confusion re arctrig
by Keith F. Lynch 09 Aug '15

09 Aug '15

Fred Lunnon <fred.lunnon(a)gmail.com> wrote: > However, I did slip up again: gamma has not actually been proven > transcendental! I should have employed exp(1), or zeta(3), or > indeed a continuum of other possibilities. Zeta(3) has not actually been proven transcendental either. (It was proven irrational by Apery in 1978.)

2 1

[math-fun] Reminder needed: For what rational a is -8 + 36 a - 27 a^2 square?
by Bill Gosper 09 Aug '15

09 Aug '15

The data are intriguing. Something about 3 and powers of primes 1 mod 6. What subject is this? BQFs? --rwg

4 5

Re: [math-fun] Offset Somsky Gears, coaxial offset
by M. Oskar van Deventer 08 Aug '15

08 Aug '15

Dear Bill, Tom, all, Thanks to Tom's fantastic tool and your explanation, I now understand that any set of Somsky Gears can be offset to produce an new set. I tested this with a random set from Tom's program (26-9-7-5) and I could see that its offset (25-10-6-4) is a Somsky set too. However, those two Somsky sets are not coaxial. I used some crude editing to overlay the two, see the attached figure. It is clear to see that the two are not coaxial. Tom has already provided me with a set of Somsky Gears, which can be coaxially offset into another set of Somsky Gears. However, not all sets of Somsky Gears can be coaxially offset like this. So what is the additional requirement that enables this? Is there a quick way how I can determine whether a Somsky set can be coaxially offset? Best regards, Oskar -----Original Message----- From: wrsomsky(a)gmail.com Sent: Friday, July 31, 2015 6:44 AM >From some thinking I've done, the answer is yes. Ignoring overlap, any set of Somsky Gears can be offset to produce an new set. You can prove it by noting that for a given ring-sun-planet, the strap-length changes by pi (one half a tooth) if you increase the ring and planet radii by one, and decrease the sun radius by one (or vice versa). On 2015-07-21 13:12, M. Oskar van Deventer wrote: > Gentlemen, > > Whiles I am still unsure whether there is agreement about Tom Rokicki's > analysis that all Somsky Gears can be offset, nor can I follow that > various mathematical reasonings, so I did an experiment with regular > 120-degrees planetary gears. The attached sketch shows that a 26-8-10 fits > as well as the 27-9-9 that it was offset from. Of course this does not > prove anything, but at least it is consistent with Tom's analysis. > > Oskar

7 25

[math-fun] Here I go again.
by Bill Gosper 08 Aug '15

08 Aug '15

Both http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html (18) and https://en.wikipedia.org/wiki/Elliptic_integral#Differential_equation give K'[k] == E[k]/k/(1 - k^2) - K[k]/k . Both sources use the k (modulus) notation vs Mathematica's m (parameter) notation. But I can't find *any* interpretation that makes this work. I see no similar claim in DLMF nor Borwein*2. ?? --rwg

3 2