August 2014 - mathfun@mailman.xmission.com

[math-fun] Stupid associativity question
by Henry Baker 21 Aug '14

21 Aug '14

I'm interested in binary operations on a finite set S that are associative, period; no other constraints. 1. Are there "fast" methods for checking associativity given the operation table? 2. Can all such associative operations be emulated "efficiently" using an isomorphism in which the operation is matrix multiplication (not just over standard rings, but extended like Knuth & the APL language do) ? (RCS knows why I'm asking this question.)

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[math-fun] A sum
by Warren D Smith 21 Aug '14

21 Aug '14

sum( (D+k-j)! / (k-j)! * (-1)^j * (2*k+1+D-2*j)^(2*k+1) / (j! * (2*k+1+D-j)!), j, 0, k ) = 4^k claims lots of checking. (only checked when D>0 is integer.) But... how to prove it? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Steiner Chain Constraints
by Fred Lunnon 19 Aug '14

19 Aug '14

Readers with a limited attention span are invited to restrict attention to numbered formulae below. At some stage sketched proofs, less obvious sample chains, pictures are envisaged ... WFL %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Steiner Chain Constraints _________________________ "Circles" (more accurately regarded as open discs) shall be oriented, and their curvatures equipped with signs: negative signs indicate circles inside-out (complemented discs). Lines and points, as limiting cases of circles with curvature 0 and oo , will be ignored for the present. A "Steiner chain" comprises a set of circles touching given frontier circles A,B , and indexed by j such that member j touches also j-1 and j+1 . Circles touching in a chain have signs matching just when external to one another, their corresponding discs disjoint: this ensures in particular that the union of chains associated with given frontier constitutes a unique continuous regulus. [Technically such circles are "anti-tangent", their touching not preserved under general transformations of contact geometry.] The distinction may seem frivolous: however, failing to respect signs leads rapidly to horrible confusion. In general j is defined only modulo n , and n and j may be real (elliptic, eg. annular), zero (parabolic, eg. arbelos), or imaginary (hyperbolic with intersecting frontiers). A classical "closed" chain has both n and j natural; or relaxation to n = p/q rational corresponds to a chain of p circles winding q times around some point. An elliptic chain is the (Moebius) inversion of a canonical chain with frontier circles concentric at the origin, and chain members centred on the unit circle: j can be defined by the pre-image of member j having its centre at argument 2pi j/n , for some value of parameter n . [Steiner's porism asserts that n is independent of continuous translations of j , a consequence of rotating the canonical chain with the same value n .] Below, frontier curvatures are denoted kA,kB ; while given some arbitrary value of j , curvatures of a consecutive chain of anti-tangent members j+1, j+2, j+3, ..., are abbreviated to k1, k2, k3, ... . ** Theorem ** Suppose n fixed, and denote t == tan(Pi/n) . Two frontier and any three consecutive member curvatures satisfy constraints kA + kB = b k1 + 2(b-1) k2 + b k3 , (1) where b = (t^2 + 1)/2 ; and (kA - kB)^2 = u k2^2 + v ( k1 k2 + k3 k1 + k2 k3 ) , (2) where u = (t^2 - 3)(t^2 + 1), v = (t^2 + 1)^2 . Proof: Routine using contact (Lie-sphere) transformations, plane pentacyclic coordinates and geometric algebra for inversion --- some other time! QNED These constraints and their integer j-shifts determine all curvatures in terms of n and k1,k2,k3 . Indeed for n = p/q <> 0 they evidently constitute a basis for the polynomial ideal of possible chain constraints: for p >= 2 integer, (2) alone together with p-2 shifts of (1) suffice. Treating (1) as a linear recurrence indexed by j , similar relations may be obtained involving (non-consecutive) subsets of indices. ** Examples ** Basis n = 3 find t^2 = 3 , b = 2 , u, v = 0, 16 ; { - (kA + kB) + 2 (k1 + k2 + k3) , - (kA - kB)^2 + 16 ( k1 k2 + k3 k1 + k2 k3 ) } . Essentially this is Soddy's identity, interpreted as an equation with roots kA,kB . Sample (annular closed) chains: [kA, kB, k1, k2, k3] = [ -1, 3; 2, 3 ] , passim. Basis n = 4 find t^2 = 1 , b = 1 , u, v = -4, 4 ; { - (kA + kB) + k1 + k3 , - (kA + kB) + k2 + k4 , - (kA - kB)^2 - 4 k2^2 + 4 ( k1 k2 + k3 k1 + k2 k3 ) } . Besides the dihedral group D_2n acting on j , and transposing A,B , here any of 3 pairs of disjoint circles may be chosen as bounding, yielding 128 symmetries in all; the constraints thus generated still lie within the theorem ideal. Sample chains: [ -1, 7; 2, 2, 4, 4 ] , [ -2, 16; 3, 8, 11, 6 ] . Basis n = 6 find t^2 = 1/3 , b = 2/3 , u, v = -32/9, 16/9 ; { - (kA + kB) + (2/3) (k1 - k2 + k3) , - (kA + kB) + (2/3) (k2 - k3 + k4) , - (kA + kB) + (2/3) (k3 - k4 + k5) , - (kA + kB) + (2/3) (k4 - k5 + k6) , - (kA - kB)^2 - (32/9) k2^2 + (16/9) ( k1 k2 + k3 k1 + k2 k3 ) } . You get the general idea ... Sample chain: [ -2, 10; 9, 3, 6, 15, 21, 18 ] . Basis n = 5 : find tau == (1+sqrt5)/2 , t^2 = 7 - 4 tau , b = 2/tau^2 , u, v = -tau, 1 ; { - (kA + kB) + (2/tau^2) (k1 + k3) - (2/tau^3) k2 , - (kA + kB) + (2/tau^2) (k2 + k4) - (2/tau^3) k3 , - (kA + kB) + (2/tau^2) (k3 + k5) - (2/tau^3) k4 , - (kA - kB)^2 - tau k2^2 + ( k1 k2 + k3 k1 + k2 k3 ) } . Which might come as a surprise if you casually assumed rational coefficients; no use expecting samples with rational curvatures! ** Corollary ** This elegant " n-free" constraint (k1 - k4)(k4 - k3) = (k2 - k5)(k3 - k2) (3) arises via eliminating b or u/v from consecutive copies of (1) or (2). Now suppose we are given only k1,k2,k3,k4 , with n unspecified. In general t is recovered via d == (k1 - k4)/(k2 - k3) = (3 - t^2)/(1 + t^2) , whence (involutorily!) t^2 = (3 - d)/(1 + d) = 4(k2 - k3)/(k2 - k3 + k1 - k4) - 1 ; and finally n = Pi/arctan(t) , real or imaginary. Expressing b,u,v of the theorem in terms of t^2 , then eliminating t^2 as above, yields these n-free constraints involving kA,kB . ** Corollary ** (kA + kB) (k1 + k2 - k3 - k4) = 2 ( k2 (k2 + k4) - k3 (k3 + k1 ) , (4) (kA - kB)^2 (k1 + k2 - k3 - k4)^2 = 16 (k2 - k3) ( k2^2 (k3 + k4) - k3^2 (k1 + k2) ) . (5) [For a closed chain their integer shifts are plainly still complete; however they no longer constitute a basis, since no linear constraint (1) can lie in this ideal.] Geometric features of a chain, via t^2 > 0 for real n --- -1 < d < 0 or 0 < d < 3 elliptic: A,B disjoint, incl. annular & closed; d = 3 parabolic: Pappus arbelos, A,B anti-tangent; 3 < d < oo or oo < d < -1 hyperbolic: A,B intersecting; d = oo or d = 0 impossible: eg. [1,1,1,2] ; (6) d = 0/0 incomplete: extremum at k_(5/2) , incl. A,B concentric. Fred Lunnon [19/08/14] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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[math-fun] Fwd: See what the Iranian press did to the portrait of a Fields Medalist
by Bill Gosper 18 Aug '14

18 Aug '14

"Bill",via Brent Meeker>Like many, I was very pleased that for the first time, a Fields Medal was awarded to a woman, an Iranian mathematician who teaches at Stanford University. See what an Iranian newspaper did to her portrait here <http://www.juancole.com/2014/08/iranian-genius-officials.html>. ...and she's not the only one. Pretty dumb, huh? < But at least those debauched Shiites let her study advanced math (and lecture with some of her hair showing) in Tehran. Compare Pakistan, where they shoot little schoolgirls in the head, or blind them with acid. The only reason Pakistan has nukes is the apostate http://en.wikipedia.org/wiki/Abdus_Salam, whose reward was to be driven from his homeland for being the wrong kind of Muslim. I'm no fan of feminism, but that may be the Achilles heel that brings down radical Islam before its Final Confrontation with the rest of the world. --rwg Another outstanding Iranian woman: http://en.wikipedia.org/wiki/Jennifer_Tour_Chayes (born here to Iranian immigrants)

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[math-fun] Fwd: See what the Iranian press did to the portrait of a Fields Medalist
by meekerdb 18 Aug '14

18 Aug '14

-------- Original Message -------- Like many, I was very pleased that for the first time, a Fields Medal was awarded to a woman, an Iranian mathematician who teaches at Stanford University. See what an Iranian newspaper did to her portrait here <http://www.juancole.com/2014/08/iranian-genius-officials.html>. ...and she's not the only one. Pretty dumb, huh? Bill

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[math-fun] "clopen" in the wild
by Thane Plambeck 18 Aug '14

18 Aug '14

The mathy definition still ranks highly on Google, but if this popular usage catches on, topologists will have to say goodbye to that. http://sprudge.com/starbucks-makes-barista-history-bans-dreaded-clopen-shif… -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

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Re: [math-fun] Game Theory/Chess strategy question
by Henry Baker 17 Aug '14

17 Aug '14

Just as we can characterize a computer chess player as aggressive or defensive, I characterize evolution in the same way. There may be no "thought" behind either the computer or evolution, but we humans tend to see things this way, no matter what. There is some evidence that evolution becomes more aggressive/attacking as the stress on the species becomes more severe -- e.g., famine, etc. Evolution seems to notice that what was working before isn't working now, and starts broadening its search for new niches. Returning to the original question: In what types of games is an aggressive/attacking strategy optimal, and in what types of games is a defensive strategy optimal? There is obviously a cost in building up defenses, when some of those same resources could be used to attack. In chess, you need to pay close attention to both in order to win. What would be very interesting is a game with a "knob" that you could turn that would force the optimal strategy to smoothly changes from defensive to offensive. I'm just wondering if there's any game theory literature on what types of games might favor the investment in defenses v. the investment in attacks. At 01:58 PM 8/15/2014, Michael Kleber wrote: >Evolution certainly does "try", in the sense that it tries everything! But >I think Henry's question still could make sense: in a game where the >players' strategy is "try everything and do whatever works", do offensive >or defensive things tend to work more? > >In the case of evolution, the best you can hope for in a "defensive" >setting is dominating your niche, i.e. expanding to its carrying capacity. > In the long term there's no up side; species that diversify and expand >their habitat will of course have more ways to grow over time. But at some >point they will count as a different species, so I'm not sure *who* counts >as winning when that happens. > >Well, I suppose you could compete in the defensive niche by expanding your >habitat! And actually beavers do exactly this. But in some sense that's >actually an offensive strategy; I'm not sure the two categories are >well-defined. > >As Dawkins has pointed out eloquently, it makes more sense to think of >*genes*, rather than species, when thinking about the evolution game. I >don't think I know what offense or defense mean there. > >--Michael > >On Fri, Aug 15, 2014 at 1:46 PM, W. Edwin Clark <wclark(a)mail.usf.edu> wrote: > >> Does evolution "attempt", "try", "sacrifice" or "hope"? That's not my >> understanding of evolution. >> >> On Fri, Aug 15, 2014 at 3:51 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> >> > Good question, however, I'm focusing on the game of evolution itself, not >> > on the strategies of individuals of the species. >> > >> > So the real question is whether evolution attempts to protect a particular >> > niche, versus attacking new niches. Many evolutionary biologists would >> > claim that evolution tries to attack new niches with some significant >> > effort, hence the willingness of evolution to sacrifice many failed >> > experiments in genetic mixing in the hope of finding some new successful >> > combinations. >> > >> > At 12:40 PM 8/15/2014, Dan Asimov wrote: >> > >What about rabbits, pill bugs, turtles, porcupines, squirrels? >> > > >> > >--Dan >> > > >> > >On Aug 15, 2014, at 6:44 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> > > >> > >> Some evolutionary biologists have claimed that Darwinian evolution >> > optimizes by using an almost purely offensive strategy.

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[math-fun] work of Maryam Mirzakhani
by Warren D Smith 16 Aug '14

16 Aug '14

http://www.math.harvard.edu/archive/101_spring_00/www/papers/home/text/pape… gives more information, somewhat comprehensible this time, about work of Maryam Mirzakhani. Due to my ignorance I do not appreciate some of the deepest work described, e.g. now that I somewhat understand what is meant by "earthquakes are chaotic," I am not sure what benefit anybody derives from having proved this...

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Re: [math-fun] Game Theory/Chess strategy question
by Bill Gosper 16 Aug '14

16 Aug '14

GHaworth> All chess positions are intrinsically won, drawn or lost for the side to move [Zermelo (?), von Neumann] There may be more than one move which retains the value of the position and this seems to be usually the case. The set of moves SM that progress mate by only one ply may have more than one move in it but usually (so far, about 85% of the time) this appears not to be the case. The set of moves that progress mate to some other objective ('conversion' or 'end of phase') may be different from SM. The set of moves (probably one) that, in a draw situation, maximises the probability of a win against a fallible player may depend on the nature/degree of the opponent's fallibility. I would suggest that 'a strategy' is an algorithm for (successively) filtering down to the chosen move. Papers including the word 'fallible', 'skill', 'skilloscopy' in http://centaur.reading.ac.uk/view/creators/90000763.html may be relevant. A reasonably authority advises me that Lasker was the first to play the opponent as opposed to the ever-present Goddess of Chess, Kaissa. Today, with databases characterising the play of the top players, this is increasingly the case. Tal and Kasparov seem to be examples of players who particularly created 'opportunities to win against fallible players' by not necessarily choosing the best theoretical move. <Guy Another good strategy is not to die right after accepting a draw: http://www.chessgames.com/perl/chessgame?gid=1771703 "[Tournament spokesman Jarle] Heitmann said there was a brief moment of panic as emergency workers rushed to attend to the Seychelles player, as some participants apparently mistook their defibrillator for a gun and fled toward the exits." --rwg

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[math-fun] Mathematica
by Bill Gosper 16 Aug '14

16 Aug '14

A youngster wanted an example of a *really* (but not artificially) long computation, so i let this run (and run): In[166]:= Simplify[ComplexExpand[ mobc[Circle[{x, y}, r], 1, u, u, 1]], -1 < u < 1 && {x, y} \[Element] Reals && r > 0] // tim (*This homographically transforms the general Circle[{x, y}, r] by (z+u)/(u z +1), which preserves the unit circle.*) During evaluation of In[166]:= 3.387416,2 (*This printout was from my timing function tim, meaning 3.4 secs, length 2.*) Out[166]= 2 Circle[{-2 Im[(r (-1 + u ) ( 2/3 2/3 r + (-1) (u + x + I y) 2 (-1 + (-1) ) Abs[-------------------------------] 2/3 r u + (-1) (1 + u (x + I y)) ------------------------------------------------------- + 2/3 (1 + r u + u (x + I y)) (1 + (-1) r u + u (x + I y)) r + u + x + I y 2 I Sqrt[3] Abs[---------------------] 1 + r u + u x + I u y (------------------------------------- - 2/3 1 + (-1) r u + u x + I u y 2/3 1/3 (-1) r + u + x + I y 2 (1 + (-1) ) Abs[-----------------------------] 2/3 1 + (-1) r u + u x + I u y -------------------------------------------------) / 1 + r u + u x + I u y 1/3 (-1 + (-1) r u - u (x + I y)))) / 2 2 2 2 Im[((3 I + Sqrt[3]) r (-1 + u ) ) / 2/3 ((1 + r u + u (x + I y)) (1 + (-1) r u + u (x + I y)) 2/3 Conjugate[r u + (-1) (1 + u (x + I y))] 2/3 (1 + Conjugate[(-1) r u + u (x + I y)]))]], 2/3 2/3 r + (-1) (u + x + I y) 2 (-1 + (-1) ) Abs[-------------------------------] 2/3 2 r u + (-1) (1 + u (x + I y)) 2 Re[(r (-1 + u ) (------------------------------------------------------- + 2/3 (1 + r u + u (x + I y)) (1 + (-1) r u + u (x + I y)) r + u + x + I y 2 I Sqrt[3] Abs[---------------------] 1 + r u + u x + I u y (------------------------------------- - 2/3 1 + (-1) r u + u x + I u y 2/3 1/3 (-1) r + u + x + I y 2 (1 + (-1) ) Abs[-----------------------------] 2/3 1 + (-1) r u + u x + I u y -------------------------------------------------) / 1 + r u + u x + I u y 1/3 (-1 + (-1) r u - u (x + I y)))) / 2 2 2 2 Im[((3 I + Sqrt[3]) r (-1 + u ) ) / 2/3 ((1 + r u + u (x + I y)) (1 + (-1) r u + u (x + I y)) 2/3 Conjugate[r u + (-1) (1 + u (x + I y))] 2/3 (1 + Conjugate[(-1) r u + u (x + I y)]))]]}, 3 2 3 -((6 Sqrt[3] Abs[(r (-1 + u ) ) / 2/3 2 2 ((r u + (-1) (1 + u (x + I y))) (1 + r u + u (x + I y)) 2/3 2 (1 + (-1) r u + u (x + I y)) )]) / 2 2 2 2 Im[((3 I + Sqrt[3]) r (-1 + u ) ) / 2/3 ((1 + r u + u (x + I y)) (1 + (-1) r u + u (x + I y)) 2/3 Conjugate[r u + (-1) (1 + u (x + I y))] 2/3 (1 + Conjugate[(-1) r u + u (x + I y)]))])] (*Method: Circumscribe the transformation of a triangle inscribed in the original circle.*) In[167]:= FullSimplify[ ComplexExpand[List @@ %], -1 < u < 1 && {x, y} \[Element] Reals && r > 0] // tim During evaluation of In[167]:= 184595.531368,2 Out[167]= 2 2 2 x + u (1 - r + x (u + x) + y ) y - u y {{-(-------------------------------), -------------------------------}, 2 2 2 2 2 2 2 2 r u - (1 + u x) - u y 1 + u (2 x + u (-r + x + y )) 2 r (-1 + u ) --------------------------} 2 2 2 2 2 r u - (1 + u x) - u y (*FullSimplify is a heuristic treesearch taking an optional ComplexityFunction whose default is approximately LeafCount.*) (*Attempting to paste the tim printout into a call to Mathematica's somewhat fussy and thuggish units converter resulted in an uneditable expression--it just clunked when I tried modifying most any portion. The "SequenceForm" (??) was completely invisible!*) UnitConvert[SequenceForm[ 184595.53136799999629147350788116455078125`11.286821096829165, ",", 2] (*Trying various alternatives, including a different Mathematica,*) UnitConvert[SequenceForm[ 184595.53136799999629147350788116455078125`11.286821096829165, ",", 2] UnitConvert[184595.531367999996 "seconds", "hours"] Out[169]= UnitConvert[184595.53136800000 "seconds", "hours"] In[172]:= UnitConvert[Quantity[184595.531367999996 , "seconds"], "hours"] Out[172]= Quantity[51.276536491111110, "Hours"] (*In the notebook, this just displays as 51.276536491111110 h !*) In[171]:= UnitConvert[184595.531367999996 "sec", "hr"] Out[171]= UnitConvert[184595.53136800000 "sec", "hr"] --rwg SetAttributes[tim, HoldAll tim[xp_] := Block[{v = AbsoluteTiming[xp]}, Print[v[[1]], ",", Length[v[[2]]]]; If[v[[1]] > 69, Speak[v[[1]]]]; v[[2]]]

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