July 2003 - mathfun@mailman.xmission.com

RE: [math-fun] misc + (Riemann surfaces & string theory)
by Andy Latto 03 Jul '03

03 Jul '03

> -----Original Message----- > From: asimovd(a)aol.com [mailto:asimovd@aol.com] > Sent: Wednesday, July 02, 2003 8:05 PM > To: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] misc + (Riemann surfaces & string theory) > > > 1. Rich writes: > > << > One of my favorite word puzzles is finding "consecutive" words, > where the last letter is incremented to make a new word. > > CONVEX CONVEY > ... > >> > > Just wondering in what sense you mean "consecutive" (since e.g. > CONVEXITY would appear between these words alphabetically). Imagine an odometer-like device, with 26 letters instead of 10 digits on each reel. Just after "CONVEX" appears, the last letter increments from X to Y, producing "CONVEY". To look at it another way, it is consecutive among six-character strings. > ------------------------------------------------------------- > > 2. The NY Times had a nice short article by Dennis Overbye > in Sunday's Week in Review section (June 29) that is a paean > to the joys of math. (It's nice to have at least one math fan > writing for the public.) This is freely available through > next Saturday on the web; sorry I can't dig up the URL right > now (but just go to www.nytimes.com and it should be easy to > find, esp. with advanced search). http://www.nytimes.com/2003/06/29/weekinreview/29OVER.html Nice article; thanks for pointing it out. Andy

1 0

[math-fun] the negative-third digit of pi, etc.
by James Propp 03 Jul '03

03 Jul '03

Most of us know (and if you don't already, you will by the end of this sentence!) that there are models of computation that are universal (they can simulate a Turing machine running an arbitrary program with arbitrary input) and also reversible (each state of the computation determines its predecessor). My question is, what happens if one uses reversibility to run a mathematically interesting computation into negative time? E.g., suppose one has a reversible machine that, when run forward, computes the sequence of primes, or the digits of pi, or something like that. Is there anything interesting that happens when you run the machine backwards? To make sense of this question, one needs to be precise about what it means to say that a machine computes something. Often there is some non-trivial representation scheme; e.g., in (non-reversible) Life-simulations, a sequence might be represented by the spacings of some gliders. It could be that if you run a computation backwards from zero, properties of the representation that hold at positive times cease to hold, so that there's no canonical way to extend the _representation_ of the computation as a sequence of digits or whatever. My guess is that for most interesting mathematical questions with answers indexed by the natural numbers, and for most reversible computation schemes for computing those answers, you don't get anything interesting or canonical by running the computation backwards. But it would still be interesting to see what happens if one tries. I should mention that one slight exception to my overall pessimism: namely, there are reversible (albeit non-universal) arithmetic models of computation which when run forward can compute good rational approximations to a quadratic irrational and when run in reverse compute good approximations to its algebraic conjugate. But this is a very special situation. And indeed, if one uses a similar model of computation to try to compute rational approximations to e, one finds that if one runs the program in reverse, it goes into an infinite loop when it tries to divide 1 by 0. Jim Propp

1 0

Re: [math-fun] misc / astronomical gravitational simulations
by Dan Hoey 03 Jul '03

03 Jul '03

Richard Schroeppel <rcs(a)cs.arizona.edu> writes: > The finite speed of gravitational attraction should be an issue > for galactic simulations, even if other relativistic effects do > not arise. E.g., suppose that some configuration several light > years away suddenly flies apart. The observer (viewer _or_ body > effected by the gravitational change) wouldn't notice it for > quite a while. That always made me wonder. After all, if the configuration is far enough away to notice the delay, it ought to gravitate the observer much like a point mass. If the configuration flies apart, conservation of momentum keeps the center of mass fixed (or on the same inertial path), and conservation of mass keeps the magnitude the same. Now some of the mass needs to become energy to make the body fly apart, but I think the energy has the same gravitational effect, no? This is all very confusing to me, but I suppose we could turn all the mass into photons, half toward the observer and half away. The attraction will approach infinity when the photons arrive. But with lightspeed propagation of gravity, the gravity does not change until the photons arrive, so it essentially jumps to infinity rather than increasing gradually. Is that a sensible scenario for testing the difference, or is there something I've missed? Dan Hoey(a)AIC.NRL.Navy.Mil

1 0

Re: [math-fun] misc / astronomical gravitational simulations
by Henry Baker 02 Jul '03

02 Jul '03

The finite speed of gravitational attraction should be an issue for galactic simulations, even if other relativistic effects do not arise. E.g., suppose that some configuration several light years away suddenly flies apart. The observer (viewer _or_ body effected by the gravitational change) wouldn't notice it for quite a while. These issues should be less of a problem for solar system simulations, but they're still there a bit. Does anyone know what the current state of the art for galactic gravitational simulations is? At 04:39 PM 7/2/03 -0700, Richard Schroeppel wrote: >http://www.sciencedaily.com/releases/2003/06/030619075759.htm > >Speed of Gravity > >A few months ago, a group announced they'd measured the speed of >gravity by looking at starlight bent by Jupiter's gravity. >The above note is another group saying 'Tain't So.

1 0

Re: [math-fun] Re: Z*(SqrtN)
by Jpr2718＠aol.com 02 Jul '03

02 Jul '03

Marc LeBrun <mlb(a)fxpt.com> wrote: > Am I confused? No, of course not. I thought x and y could be any integers. From "{x + y SqrtD; integer x, y, D>=0}" I interpreted the ">=0" as applying only to D, not to x and y. Perhaps Z^{+}[\sqrd{D}] would be a better notation for what you have in mind. "*" often denotes something to do with multiplication, usually to just exclude zero. And ( ) usually allows quotients, while [ ] denotes a Z-module. Anyway, my answer applies only if x and y can be any integers, positive, negative, or zero. John Robertson

1 0

[math-fun] Dr. Matrix reference
by Jud McCranie 02 Jul '03

02 Jul '03

In Martin Gardner's book "The Incredible Dr. Matrix", in chapter 13, page 120-121 in my edition, "Dr. Matrix" says "You once quoted me as saying that the decimal expansion of pi, properly interpreted, contains the entire history of the human race." Since he says that he was previously quoted, I wanted to find the original passage. I skimmed all of the previous chapters twice, but I couldn't find it. Can anyone tell me where the original "quotation" is? Thanks.

1 0

Re: [math-fun] Re: Z*(SqrtN)
by Dan Hoey 02 Jul '03

02 Jul '03

Marc LeBrun wrote: > >> Regarding Z*(SqrtD) = {x + y SqrtD; integer x, y, D>=0}... > >> 2. What's best to call the whole algebraic structure? I'd say Z*(SqrtD) = {x + y SqrtD; integer x, y>=0}, for integer D>=0, just to keep the scope straight. I take it "Z*" is the nonnegative integers. Henry Baker <hbaker1(a)pipeline.com> wrote: > "algebraic natural numbers" or "natural algebraic numbers" ?? Cute name, but I'd rather use that for the positive elements of Z(SqrtD) (or the nonnegative ones, in France and set theory). Then the elements of Z(SqrtD) with both components positive could be called "supernatural", and those with at least one component positive could be called "subnatural". Dan Hoey(a)AIC.NRL.Navy.Mil

2 1

Re: [math-fun] Re: Z*(SqrtN)
by Henry Baker 02 Jul '03

02 Jul '03

"algebraic natural numbers" or "natural algebraic numbers" ?? At 08:12 PM 7/1/03 -0700, Marc LeBrun wrote: > >=Jpr2718(a)aol.com > >>=Marc LeBrun <mlb(a)fxpt.com> > >> Regarding Z*(SqrtD) = {x + y SqrtD; integer x, y, D>=0}... > >> 2. What's best to call the whole algebraic structure? > > > For D squarefree and congruent to 2 or 3 modulo 4, it's the ring of >algebraic integers of the field Q(\sqrt{D}) = {x + y \sqrt{D}; x and y >rational}. [...] > >Thanks for the informative answer. (Today I'm just looking at D=2, but >this will help me later...) > >Henry Baker also suggested "algebraic integer". However I objected on the >grounds that x,y are restricted here to be non-negative. I thought >"algebraic integer" connoted negative and even half-integer x&y. It also >didn't seem right to say "ring" when there's no additive inverses. > >Am I confused or just being obsessively pedantic about this? >Thanks!

1 0

[math-fun] any news on Gilbreath's Conjecture?
by Marc LeBrun 02 Jul '03

02 Jul '03

What is the current state of understanding, if any, of Gilbreath's Conjecture? (This is that every row after the first begins with a 1 in the following 2 3 5 7 11 13 17 19 23 29 31 ... 1 2 2 4 2 4 2 4 6 2 6 ... 1 0 2 2 2 2 2 2 4 4 2 ... 1 2 0 0 0 0 0 2 0 2 0 ... 1 2 0 0 0 0 2 2 2 2 0 ... 1 2 0 0 0 2 0 0 0 2 0 ... 1 2 0 0 2 2 0 0 2 2 2 ... 1 2 0 2 0 2 0 2 0 0 0 ... 1 2 2 2 2 2 2 2 0 0 0 ... 1 0 0 0 0 0 0 2 0 0 2 ... 1 0 0 0 0 0 2 2 0 2 0 ... ... where the top row is the primes and each lower row the absolute differences of that above it. It has been extensively computationally verified.) This seems MUCH more remarkable than, say, Goldbach's Conjecture!

2 1

[math-fun] Re: Z*(SqrtN)
by Jpr2718＠aol.com 01 Jul '03

01 Jul '03

Marc LeBrun <mlb(a)fxpt.com> wrote: > Regarding Z*(SqrtD) = {x + y SqrtD; integer x, y, D>=0}... > > 2. What's best to call the whole algebraic structure? For D squarefree and congruent to 2 or 3 modulo 4, it's the ring of algebraic integers of the field Q(\sqrt{D}) = {x + y \sqrt{D}; x and y rational}. For D squarefree and congruent to 1 modulo 4, it's the order of conductor 2 in the ring of algebraic integers of the field. In this case, the ring of algebraic integers is {x + y(1 + \sqrt{D})/2; x, y integers}. If D is not squarefree, then D=(f^2)d for some squarefree d, and, depending on what d is modulo 4, the algebraic structure is the order of conductor f or 2f in the ring of integers of the field Q(\sqrt{d}). John Robertson jpr2718(a)aol.com

2 1