August 2015 - mathfun@mailman.xmission.com

Re: [math-fun] Here I go again.
by rcs＠xmission.com 08 Aug '15

08 Aug '15

(from Bill Ackerman) ----- Date: Sat, 08 Aug 2015 00:29:18 -0400 Subject: Re: [math-fun] Here I go again. From: Bill Ackerman <wbackerman(a)gmail.com> On 07-Aug-15 22:07, Bill Gosper wrote: [Hide Quoted Text] 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 A long long time ago, when I was interested in writing super-precision math libraries, I had to slog through all the funny notations, including Abramowitz & Stegun and Whittaker & Watson. In order to maintain what was left of my sanity while trying to write routines for E and K, I wrote up the block of comments below. It might possibly be of some help. Or maybe not. But please don't ask me to explain this stuff in further detail, it's long forgotten. /* NOMENCLATURE This is in terms of the "complementary parameter", commonly denoted "m1". This is just 1-m, where m is the parameter. (Using the complement puts the bad behavior at zero rather than one.) Now the parameter is the square of the modulus, and the modulus is the sine of the modular angle, so the parameter is the square of the sine of the modular angle. The complementary parameter is the square of the cosine of the modular angle. It is also the square of the sine of the complementary modular angle, where the complementary modular angle is (pi/2 - modular angle). The normal "complete elliptic integral of the second kind" (which is what this is) is generally denoted "E", and the complementary one (which is what we are actually computing) is denoted "E'" (where the prime is not to be confused with the derivative.) When written in terms of the parameter, normal functional notation is used, as in "E(m)". The complementary function is E'(m), or E(1-m), or E(m1). There is other notation, involving vertical bars and backslashes (I'm not making this up! It's in all the classical textbooks!) for denoting the functions of the modulus and modular angle, respectively. There is also notation involving the use of brackets instead of parentheses to denote composition with the inverse of K, the elliptic integral of the first kind. All in all, the classical notation is incredibly befuddled. */ /* To summarize: parameter = modulus^2 complementary parameter = (complementary modulus)^2 complementary parameter = 1 - parameter complementary modulus = sqrt(1 - modulus^2) elpkcm(x) = elpkc(x^2) elpecm(x) = elpec(x^2) */ // Get pi itself. Argument is ignored. // This is the Brent-Salamin algorithm. const arith arith::getpi(const arith &) { ...... (That last thing is for you, Gene.) <eom>

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[math-fun] treadmill for Slinky toys
by Henry Baker 08 Aug '15

08 Aug '15

Perhaps there's a way to relate this to Somsky Planets? For RWG: http://boingboing.net/2015/08/07/treadmill-for-slinkys.html A treadmill for Slinky toys, for your infinite Slinky-torturing pleasure

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[math-fun] Bridges proceedings (paper) -- Free to good home
by rcs＠xmission.com 07 Aug '15

07 Aug '15

We have an extra. FCFS. Send me a snail address. --Rich

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Re: [math-fun] Fred Lunnon, RWG, and whoever's arctrig confusion
by Fred Lunnon 07 Aug '15

07 Aug '15

Now I look again, I discover that a typo (or fly on my specs) swallowed the 2pi from the RHS from my notes, and it now looks as though WDS was right after all --- the sunset/offset is algebraic! [I think now that I actually realised that while in bed last night, but had forgotten about it again by the time I got near a computer again ... Apologies! WFL On 8/6/15, Fred Lunnon <fred.lunnon(a)gmail.com> wrote: > Warren, > > You don't appear to have seen later messages from myself and > Goucher which addressed this issue. The relevant equation is not > << FINITESUM_j A_j * arccos(B_j(u)) = C*pi >> > but > << ... = C >>> , > which alters the result drastically. > > Fred > > On 8/5/15, Warren D Smith <warren.wds(a)gmail.com> wrote: >> FWL: >> Before this intrusion of ineptitude risks the moral endangerment of any >> further tender young minds, I feel obliged to point out the logical >> inequivalence between the statements >> cos(rational * 2pi) = algebraic ! >> and >> arccos(rational) = algebraic * 2pi ? >> the second of which is in general (to put it mildly) problematic. >> Those remaining in doubt about the matter should reflect upon the >> kindergarten fact >> arccos(1/3) = dihedral angle of regular tetrahedron, >> the expression of which as an algebraic angle ought to be worth a Fields >> at least! >> >> --WDS: >> I'd never claimed these 2 statements equivalent. >> Since FWL seems confused, let me clarify. Or repeat what I'd said >> before. >> Hopefully this also will unconfuse others such as RWG and "tender young >> minds." >> (Although, frankly, I think the problem lies more in cranky old minds.) >> >> FACT: >> Suppose you have an equation of the form >> FINITESUM_j A_j * arccos(B_j(u)) = C*pi >> where C and the A_j are integers, and the functions B_j(u) are >> algebraic functions of u. Then: u is algebraic. >> >> BECAUSE: >> You can rewrite the equation as >> realpart PRODUCT_j Y_j(u)^(A_j) = 0 >> where the Y_j(u) are complex numbers with unit norm >> that are algebraic functions of u, >> having realpart Y_j(u) = B_j(u). >> >> Here by "algebraic function" F(u) >> I mean "if u algebraic then F(u) is, and if y algebraic with F(u)=y, then >> u >> is." >> >> In gear applications, the fact the A_j are integers is related to the >> fact >> gear tooth counts are integers. >> >> Are we clear now? >> This same trick is also how one can prove (or disprove) >> any of my so-called "arccos miracles" in an automated fashion, and note, >> this proof NOWHERE INVOLVES trig or arctrig, which >> was the whole point of the trick -- the arctrig intentionally gets >> removed from the scene, allowing simpler pure-algebraic simplifiers to >> do their thing. >> >> Be happy. >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> >

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[math-fun] "Kepler's Law" for Somsky Planets, and the limited(?) number thereof
by Warren D Smith 07 Aug '15

07 Aug '15

(To Dan Asimov: On the sphere S2, these are bevel gears, and are highly realistic, useful, applicable, etc. On the hyperbolic nonEuclidean plane H2, though, I'm not seeing any realistic use.) Inversive transformations and the stereographic projection all preserve the gear-touch/not combinatorial relations, and preserve rationality of euclidean-radius ratios. Somewhat more interesting is: COSINE RATIONALITY PRESERVATION LEMMA: Doing an inversive map to a set of gear-circles in the plane (to get another such set) can preserve the rationality of cosines. More precisely, if gear A has unit-radius and center (0,0) contacts gears B and C, and the angle on A between the two touch points, has rational cosine, and the AB contact point is (0,1), and the center of inversion has rational coordinates then the cosine will still be rational after the inversive mapping It is conceivable this lemma will enable proving something amazing about Somsky gear existence.... but also conceivable it will lead nowhere. The following problem seems related: POINT SETS WITH INTEGER DISTANCES. It has been conjectured that if N points in the plane exist, with all interpoint distances integer, then either (a) N is upper bounded by some finite constant C (b) the points all lie on the union of a line and a circle As far as I know this conjecture has never been proven, although occasionally somebody raises the record C by one more point. I'm not sure what is the latest greatest record C, but it is at least 6 and I'm pretty sure nobody has managed to reach 10. Update: I think the latest record is C=7, see this 2008 paper: http://arxiv.org/abs/0804.1303v1 Obviously, any two gears in a meshed set have integer center-separations if those two gears mesh. Also by the "Kepler law" Somsky planet centers all lie on an ellipse hence NOT a line U circle if enough of them. So this while not exactly the same problem, it strongly suggests (at least, for those who believe that Conjecture) that is it NOT going to be possible to find an N-planet Somsky planet system in the Euclidean plane with unboundedly great N. There will be some finite upper bound on N. The lack of progress during the last 70 years on the integer-distance problem makes one worry, though, that proving this "Somsky nonexistence conjecture" might also be infeasibly hard. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] "Kepler's Law" for Somsky Planets
by Warren D Smith 06 Aug '15

06 Aug '15

"Kepler's Law" for Somsky Planets =========Warren D Smith===Aug 2015===== Probably everybody knew this, but it is worth saying explicitly. If you have a sun of radius S, and antisun (outer ring) of radius R, and their two centers are separated by some fixed distance D, then the locations of all the planet centers lie on an ELLIPSE with foci being the sun and antisun centers. PROOF: For a planet of radius P, the distance from a planet center to sun center is S+P, the distance from planet center to ring center is R-P, hence this distance sum is S+P+R-P=S+R=constant independent of P. That law describes the ellipse. QED -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Can anyone fix DLMF 15.8.23 ?
by Bill Gosper 06 Aug '15

06 Aug '15

http://dlmf.nist.gov/15.8 It seems to need another term on the right. --rwg

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[math-fun] Rowhammer: Not-so-random memory accesses allow attack
by Henry Baker 05 Aug '15

05 Aug '15

FYI -- This bug reminds me of the old IBM 1403 chain printer hack: printing a not-so-random row of characters would cause all of the hammers to fire simultaneously, thus breaking the printer chain quite spectacularly. https://en.wikipedia.org/wiki/IBM_1403 BTW, it's only a matter of time before DRAM modules incorporate built-in caches/malware that can be triggered by a precise sequence of memory addresses and/or contents. It makes one seriously wonder about the safety and availability of computers, most of whose components come from offshore vendors with uncertain loyalties. The next war might well be fought using mechanical typewriters and slide rules. http://arxiv.org/abs/1507.06955 Rowhammer.js: A Remote Software-Induced Fault Attack in JavaScript As DRAM has been scaling to increase in density, the cells are less isolated from each other. Recent studies have found that repeated accesses to DRAM rows can cause random bit flips in an adjacent row, resulting in the so called Rowhammer bug. This bug has already been exploited to gain root privileges and to evade a sandbox, showing the severity of faulting single bits for security. However, these exploits are written in native code and use special instructions to flush data from the cache. In this paper we present Rowhammer.js, a JavaScript-based implementation of the Rowhammer attack.

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[math-fun] Fred Lunnon, RWG, and whoever's arctrig confusion
by Warren D Smith 05 Aug '15

05 Aug '15

FWL: Before this intrusion of ineptitude risks the moral endangerment of any further tender young minds, I feel obliged to point out the logical inequivalence between the statements cos(rational * 2pi) = algebraic ! and arccos(rational) = algebraic * 2pi ? the second of which is in general (to put it mildly) problematic. Those remaining in doubt about the matter should reflect upon the kindergarten fact arccos(1/3) = dihedral angle of regular tetrahedron, the expression of which as an algebraic angle ought to be worth a Fields at least! --WDS: I'd never claimed these 2 statements equivalent. Since FWL seems confused, let me clarify. Or repeat what I'd said before. Hopefully this also will unconfuse others such as RWG and "tender young minds." (Although, frankly, I think the problem lies more in cranky old minds.) FACT: Suppose you have an equation of the form FINITESUM_j A_j * arccos(B_j(u)) = C*pi where C and the A_j are integers, and the functions B_j(u) are algebraic functions of u. Then: u is algebraic. BECAUSE: You can rewrite the equation as realpart PRODUCT_j Y_j(u)^(A_j) = 0 where the Y_j(u) are complex numbers with unit norm that are algebraic functions of u, having realpart Y_j(u) = B_j(u). Here by "algebraic function" F(u) I mean "if u algebraic then F(u) is, and if y algebraic with F(u)=y, then u is." In gear applications, the fact the A_j are integers is related to the fact gear tooth counts are integers. Are we clear now? This same trick is also how one can prove (or disprove) any of my so-called "arccos miracles" in an automated fashion, and note, this proof NOWHERE INVOLVES trig or arctrig, which was the whole point of the trick -- the arctrig intentionally gets removed from the scene, allowing simpler pure-algebraic simplifiers to do their thing. Be happy.

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[math-fun] DLMF 15.4.26-15.4.33
by Bill Gosper 05 Aug '15

05 Aug '15

I was bugging the DLMF hypergeometric people about having only eight special values (Ira Gessel points out that one of them is redundant), and claimed there should be at least a hundred. Just now I notice http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/ claiming a completely incredible 111271 formulas. The obvious challenge: Find one they missed. Or vastly preferable: an independent contiguous pair. Missing from http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/03/0… Hypergeometric2F1[1/2, 3 a, 1/2 + 2 a, -1/3] == ((27/4)^a Gamma[1 + a] Gamma[1/2 + 2 a])/(Sqrt[\[Pi]] Gamma[1 + 3 a]) and Hypergeometric2F1[1/2, 1 + 3 a, 3/2 + 2 a, -1/3] == 4^-a Sqrt[\[Pi]/3] Gamma[3/2 + 2 a]/Gamma[1 + a]^2 . --rwg

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