July 2009 - mathfun@mailman.xmission.com

[math-fun] Horizonward and Z-ward with the Arts
by Hilarie Orman 12 Jul '09

12 Jul '09

Did anyone happen to see this opera last month? Hypermusic Prologue: A Projective Opera in Seven Planes by Hector Parra & Lisa Randall Pompidou Centre, Paris. 14-15 June 2009 Hilarie

1 0

Re: [math-fun] Fractal puzzles
by rwg＠sdf.lonestar.org 11 Jul '09

11 Jul '09

> Thanks, Alan. > > So, does this make it the same as starting with a plus sign of 5 adjacent squares, > and taking 5 congruent copies snuggled together and normalized -- then iterate to > the limit? (Actually the boundary of this limit.) If you do it right, you can start with a square for Q[0] and get the "+" sign for Q[1]. See http://gosper.org/erezd.PNG . These aren't quite the same but they have the same closure and boundary dimension. A toy analogy is the approximations to the unit interval you get by varying Q[0] in the iteration Q[n+1] = (Q[n] U (1+Q[n]))/2. Allan's method, Q[0] := {0}, produces the dyadic rationals. But Q[0] := (0,1] gives the complement of the dyadic rationals. Same closure, utterly unequal measures. You can't zoom down to a self contact because of self similarity. For greater rigor, you can construct sausage links that will remain demonstrably disjoint. You can also write a function analogous to that obscure Peano Mathematica function that will exactly and continuously map the rationals in the unit interval onto one quarter of the boundary in question. There are also open and closed loop spacefills of this region, which can also be computed exactly for rationals. I like D=log_5(9) . Any takers on 3D revisitation? --rwg > If so, it's a fractal I was studying just a couple of months ago. > I'll go review what I had about it. > > =-Dan > > << > I'm pretty sure I understand the intended construction. We'll construct a sequence of sets of > complex numbers; the limit of this sequence will be (well-defined and) the intended fractal. > > Let Q[0] contain only 0. Then for any nonnegative integer i, let Q[i+1] = (Q[i] + {0, 1, -1, > i, -i}) / (2i + 1). Here, if A and B are sets, the set A+B is intended to mean {a+b | a in A > and b in B}. > > Each Q is roughly cross-shaped; RWG observes (very tersely) that dividing by 2i+1 rotates and > shrinks each such cross by just enough that five crosses can snuggle together to make a > meta-cross. > > I think this fractal is in Mandelbrot; I cannot dig up my copy at the moment. >>> > > _____________________________________________________________________ > "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >

3 11

[math-fun] [Fwd: Re: Addition formula paper]
by rwg＠sdf.lonestar.org 10 Jul '09

10 Jul '09

Oleg has asked me to forward this: -------------------------------------- Original Message -------------------------------------- Subject: Re: [math-fun] Addition formula paper From: "Marichev Oleg" <oleg(a)wolfram.com> Date: Thu, July 2, 2009 11:52 am To: rwg(a)sdf.lonestar.org Cc: "math-fun" <math-fun(a)mailman.xmission.com> ---------------------------------------------------------------------------------------------- My collection for function s site includes all formulas for all arctrigs with structure a ArcTrig[z]+b OtherArctrig[z]== c ArcTrig4 +Pi d. They are correct for all values of arguments. We have corresponding program for such operations. But it was not used yet in Mathematica deeply and was not shown to public in Internet yet. For example: In[20]:= Chop[Table[ a*ArcCsc[x] + b*ArcTanh[y] - (-2*I*Pi*(Floor[(-Arg[1/((Sqrt[1 - 1/x^2] + I/x)^(I*a)*(1 - y)^(b/2))] - Arg[(y + 1)^(b/2)] + Pi)/(2*Pi)] + Floor[(Pi - Im[Log[1/((Sqrt[1 - 1/x^2] + I/x)^(I*a)*(1 - y)^(b/2))]])/ (2*Pi)] + Floor[(Pi - (1/2)*Im[ b*Log[y + 1]])/(2*Pi)]) - 2*I*Pi*(Floor[(-Arg[(Sqrt[1 - 1/x^2] + I/x)^ ((-I)*a)] - Arg[(1 - y)^(-(b/2))] + Pi)/(2*Pi)] + Floor[ ((1/2)*Im[b*Log[1 - y]] + Pi)/(2*Pi)] + Floor[(Re[a*Log[Sqrt[1 - 1/x^2] + I/x]] + Pi)/(2*Pi)]) + I*Pi* (1 - (-1)^(Floor[-(Arg[(y + 1)^(b/2)/ ((1 - y)^(b/ 2)*(Sqrt[1 - 1/x^2] + I/x)^(I*a)) + 1]/(2*Pi))] - Floor[-(Arg[(y + 1)^(b/2)/ ((Sqrt[1 - 1/x^2] + I/x)^(I*a)* (1 - y)^(b/2))]/(2*Pi))])) + I*(-1)^(Floor[-(Arg[(y + 1)^(b/2)/ ((Sqrt[1 - 1/x^2] + I/x)^(I*a)* (1 - y)^(b/2))]/Pi)] + Floor[Arg[(y + 1)^(b/2)/((Sqrt[1 - 1/x^2] + I/x)^(I*a)*(1 - y)^(b/ 2)) - 1]/(2*Pi) - Arg[(y + 1)^(b/2)/ ((1 - y)^(b/2)*(Sqrt[1 - 1/x^2] + I/x)^(I*a)) + 1]/(2*Pi) + 1/2])* (ArcCsc[(2*(y + 1)^(b/2))/ ((Sqrt[1 - 1/x^2] + I/x)^(I*a)* (1 - y)^(b/2)*((y + 1)^b/((1 - y)^b* (Sqrt[1 - 1/x^2] + I/x)^(2*I* a)) + 1))] - Pi/2)) /. {a -> Random[Complex]} /. {b -> Random[Complex]} /. {x -> (j + Random[])*Exp[Pi*I*(k/4)]} /. {y -> (i + Random[])*Exp[Pi*I*(n/4)]}, {k, 0, 7}, {j, 0, 1}, {n, 0, 7}, {i, 0, 1}]] Out[20]= {{{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}, {{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}, {{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}, {{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}, {{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}, {{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}, {{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}, {{{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}, {{0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}, {0, 0}}}} Oleg rwg(a)sdf.lonestar.org wrote: >> I'm looking for a paper I read once (more than six years ago) on >> addition formulas. It had the tangent addition formula as a special >> case: >> atan(x) + atan(y) = atan((x+y)/(1-xy)) >> as well as log >> log(x) + log(y) = log(xy) >> and various other ones. The idea was to find a function p and an >> operator @ such that >> p(x) + p(y) = p(x@y), >> and they showed how to come up with a large class of these. It also, >> as I recall, had something to do with intersecting chords of a conic. >> >> Does anyone know what paper this was, or could anyone suggest keywords >> to google for it? >> -- >> Mike Stay - metaweta(a)gmail.com >> http://math.ucr.edu/~mike >> http://reperiendi.wordpress.com >> > > This ain't it, but, > due to its non-announcement at the time (ca 2001), some of you may have > missed that bilinear (Somos) sequences have addition formulae: > http://arxiv.org/PS_cache/math/pdf/0703/0703470v1.pdf > --rwg >

1 0

[math-fun] MIT prof, "quark soup", nuclear physics
by Henry Baker 09 Jul '09

09 Jul '09

Last night I attended a very entertaining talk by MIT Prof. Krishna Rajagopal entitled "quark soup" about the discovery that when nuclear particles are heated up to 2 trillion degrees K, they form a _liquid_ of quarks. Apparently, physicists previously thought that the quarks formed a gas, but this was shown to be incorrect. In the course of this talk, Krishna described a "string-like" theory in which only one additional space dimension was added, and in this space dimension there was a bizarre, but almost classical "gravity" generated by a "black hole" (really a "black sheet", since the field is uniform, not radial) which pulled on all objects, but since force of this "gravity" was orthogonal to the usual 3 dimensions, it had only indirect effects. In particular, a quark & an anti-quark could be thought of as tied together with a string which had enough mass that this "gravity" pulled on the string in an analogy to the catenary curve formed by a massy string in 3D with gravity. The force between the two quarks was that resulting from the projection into 3D of these force vectors -- presumably proportional cos(alpha), where alpha is the angle that the "string" made with usual 3D space. This model explains the usual quark-like behavior where the strength of the interaction gets _stronger_ with (usua l 3D) distance, instead of weaker. If you read the popular science literature, then the "man in the street" version of physics spends an inordinate amount of time worrying about "these extra dimensions" & attaches a mystical significance to them. However, how would such an extra "string-like" dimension be all that much different from the extra dimension used in 3D graphics "homogeneous coordinates", where the projection back into 3D is by dividing out the last coordinate ? After all, most "men in the street" ("street people?" :-) wouldn't attach much mystical significance to homogeneous coordinates, would they? --- I asked Krisha afterward if this "gravitational black sheet" obeyed the usual relativistic laws, including having an event horizon in which time slowed to essentially a stop, and he said yes. Apparently, this "black sheet" also supports Hawking radiation! This is all totally cool, and still 18 orders of magnitude larger than the Planck length, so traditional infinitesimal calculus still "works". I guess this really is a new century in physics.

2 1

Re: [math-fun] math, existence, and God
by Dan Asimov 08 Jul '09

08 Jul '09

This from a guy who can't even find the center of a triangle? --Dan, who is just teasing. Clark wrote: << Your PS raises a question. Are physicists now certain that each atom has an "individual identity" - that is, can we speak of a specific atom on the bottom of your shoe right now, and if so, will the "SAME atom" exist 100 years from now (assuming is hasn't split)? >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

5 4

Re: [math-fun] math, existence, and God
by James Propp 08 Jul '09

08 Jul '09

Andy Latto writes: >There is a sense in which phlogiston exists, because the >theory that things are flammable because they contain >phlogiston, which is released by burning, exists. Fred Lunnon writes: >The aether exists because Newtonian mechanics >exists, ditto. We hear a lot about the aether, but how come we never hear about what surely should be called the "naether"? This is the medium that darkness travels through. When a dark-sucker (sometimes called a light-source) sucks dark from a room, the dark has to travel through the naether to get there. See e.g. https://www.msu.edu/user/dynicrai/physics/dark.htm or http://members.dslextreme.com/users/rogermw/darksucker.html (If any of you know of more definitive web-sources for dark- sucker theory, or the original attribution, please send me this information --- or should I say, please extract this ignorance from me!) Jim

5 5

Re: [math-fun] math, existence, and God
by Dan Asimov 08 Jul '09

08 Jul '09

Hi, Mike. Just curious: Where are you? Mountain View? Riverside? Down Under? Up Over? Elsewhere? --Dan If you are in the Bay Area, maybe we could meet at some point. I'm currently in Berkeley. (Though my location is confidential to the general math-fun public.) (510) 420-8948. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

1 0

Re: [math-fun] math, existence, and God
by Dan Asimov 08 Jul '09

08 Jul '09

It's been a point of interest for me that although i and -i -- the complex square roots of -1 -- are distinct, there is no mathematical way to distinguish between them, if one starts from the field C of complex numbers. Which might mean that it should be disallowed to refer to one of them without the other. (Or perhaps this fact points to the pressing need for an Axiom of Choice for sets of size 2.) --Dan << I think one of the points of a Bose-Einstein condensate is that the atoms are in fact indistinguishable, so the number of quantum states is astronomically less than if the atoms have individual identity. >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

1 0

Re: [math-fun] math, existence, and God
by Steve Witham 08 Jul '09

08 Jul '09

>Back on March 25, Andy Latto wrote: > >>My view is that philosphers often worry too much about ontology (which > >things "exist" and which don't), [...] > >From: Dan Asimov <dasimov(a)earthlink.net> > >Personally, I find existence (of certain things discussed in >philosophy, anyway) >to be of the utmost interest to me, much as is the case in mathematics. One thread is about existence aka the verb to be, that menagerie of metaphorically-related predicates, and how sometimes people don't know, or don't care, or are being deliberately vague, heuristic or instinctive about what predicate they're talking about. I think philosophy has assumed confusions indicate Problems and problems must have solutions. Compare the questions in science or math of whether a problem is interesting, or a question is well-formed. (On the other hand I think positivism can get to a point like washing one's hands until they bleed.) Jim's thread is more interesting to me because it's less familiar. Does modern math really manage to be *less* concerned about existence than older math? In what way exactly? I mean, given that we had already stopped asking whether complex numbers "really exist," what more lessness does modern math do? I admit category theory gives me a strange sense of disconnection. I've heard a story from some East/West religion conference where the Easterers finally got exasperated with the West: "We're tired of you obsessing over what's *true*, the question is what *works*!" --Steve

1 0

Re: [math-fun] math, existence, and God
by Eric Angelini 08 Jul '09

08 Jul '09

> So, do holes exist? ... No, there are no h les. Best, É.

1 0