April 2010 - mathfun@mailman.xmission.com

[math-fun] Dumb astronomy question
by Henry Baker 11 Apr '10

11 Apr '10

The earth's axis is inclined ("axial tilt") at approx 23.5 degrees to its orbit around the Sun. How did it get that way? If most of the stuff in the solar system originally had the same axis of angular momentum, then wouldn't the earth most likely end up with a zero degree axial tilt? What is the current best explanation for the tilt? Did the earth get hit by another body from an odd angle -- e.g., when the Moon was formed? Wouldn't tidal forces eventually get rid of the tilt? Thanks in advance for any pointers.

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[math-fun] Penrose Tilings/Algebraic Number Theory
by Henry Baker 11 Apr '10

11 Apr '10

This young fellow is looking to spend more time in the U.S. due to a romantic interest. Does anyone here have any suggestions about places he might start looking ? "I have a B.S in mathematics and computer science. My bachelor thesis was on algebraic aspects of Penrose Tilings. My upcoming thesis is in the field of Algebraic Number Theory, more specifically in the theory of valuations and Galois representations, even more specifically the higher ramification groups and The Artin Conductor."

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[math-fun] Fwd: Sticky Towers of Hanoi
by Thane Plambeck 09 Apr '10

09 Apr '10

---------- Forwarded message ---------- From: <rjn(a)mathstat.dal.ca> Date: Friday, April 9, 2010 Subject: [math-fun] Sticky Towers of Hanoi To: Thane Plambeck <tplambeck(a)gmail.com> Cc: finka(a)math.berkeley.edu Thane, I remember Alex Fink did a lot of work with Conway on this problem. Richard > I remember writing a program for it, similar to one that Erik Demaine > (who was also there) wrote, but I don't remember the rules. Erik? > > Sent from my iPad > > On Apr 8, 2010, at 9:49 PM, Fred lunnon <fred.lunnon(a)gmail.com> wrote: > >> Sticky Towers of Hanoi was apparently invented by John Conway, >> while attending a Combinatorial Theory Workshop held at "BIRS" >> (Berkeley?) in June 2005, and attended by (amongst others) >> Richard Guy and (Thane?) Plambeck [we know what you did >> last summer]. See >> https://www.birs.ca/workshops/2005/05w5048/report05w5048.pdf >> >> Is anyone in a position to disclose to us the rules pertaining to >> this mysterious diversion? >> >> Fred Lunnon >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > -- Richard Nowakowski Dept. Mathematics & Statistics Dalhousie University, Halifax NS, B3H 3J5 Canada Phone (902)-494-6635 FAX (902)-494-5130 -- Richard Nowakowski Dept. Mathematics & Statistics Dalhousie University, Halifax NS, B3H 3J5 Canada Phone (902)-494-6635 FAX (902)-494-5130 -- Thane Plambeck tplambeck(a)gmail.com http://thaneplambeck.typepad.com/

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Re: [math-fun] Sticky Towers of Hanoi
by Dan Asimov 09 Apr '10

09 Apr '10

<< . . . while attending a Combinatorial Theory Workshop held at "BIRS" (Berkeley?) . . . Banff International Research Station. The acronym is appropriate: in March '09 I had the privilege of attending a workshop there, when the temperature hit -22 F. (-27 C.), and there were lots of brrs. <<< http://www.birs.ca/ >>> --Dan ________________________________________________________________________________________ "Outside of a dog, a book is man's best friend. Inside of a dog, it's too dark to read." --Groucho Marx

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[math-fun] Sticky Towers of Hanoi
by Fred lunnon 09 Apr '10

09 Apr '10

Sticky Towers of Hanoi was apparently invented by John Conway, while attending a Combinatorial Theory Workshop held at "BIRS" (Berkeley?) in June 2005, and attended by (amongst others) Richard Guy and (Thane?) Plambeck [we know what you did last summer]. See https://www.birs.ca/workshops/2005/05w5048/report05w5048.pdf Is anyone in a position to disclose to us the rules pertaining to this mysterious diversion? Fred Lunnon

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Re: [math-fun] faster and faster
by James Propp 08 Apr '10

08 Apr '10

>Look at the literature on Zeno machines / hypercomputation The phrase "Zeno machines" let me to the literature I wanted; thanks, Jeff! Incidentally, apropos of Andy Latto's suggestion about ordinal time: in the article "Rotor Walks and Markov Chains" (arXiv:0904.4507), which is related to my work on infinite decimals, Ander Holroyd and I explicit use countable ordinals to index the steps of a computation (though curiously only the "linear ordinals" of the form a omega + b turn out to be relevant). In a sense, Zeno machines are just one way of embedding ordinals into R. Jim Propp

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Re: [math-fun] faster and faster
by Jeffrey Shallit 08 Apr '10

08 Apr '10

Look at the literature on Zeno machines / hypercomputation http://en.wikipedia.org/wiki/Zeno_machine > From: James Propp <jpropp(a)cs.uml.edu> > To: math-fun(a)mailman.xmission.com > > I wasn't able to attend Gathering for Gardner 9, but > my student Linda Zayas-Palmer did, and she presented > there a joint paper called "Carrying On with Infinite > Decimals" (http://jamespropp.org/carrying.pdf) I'd be > interested in people's comments (since at some point > I plan to publish this in one of the Gathering for > Gardner volumes). > > One comment I've already received can be paraphrased as > follows: "You describe these computations with infinite > decimals as taking place in ordinary time, so that even > though each individual digit eventually stabilizes, you > have to wait till time omega to be able to read off the > answer. Why not imagine a computer in which each carry > in the 10^{-n} position takes 10^{-n} seconds, so that > the computation finishes in finite time?" > > I was wondering if anyone had seen any proposals for > (unrealistic) models of computation along these lines. > > One needs to take some care to make sure that the > state of the system remains well-defined even at > limit-times. For instance, if one switches a light > switch on and off at times 1/2, 3/4, 7/8, ..., the > state of the light at time 1 is undefined. > > Still, it seems that even if one imposes constraints > to avoid the aforementioned "light switch problem", > one can get idealized computers of this kind to do > all kinds of things ordinary computers can't do, > such as solve the halting problem. > > There must be existing literature on computers that > can be made to run faster and faster. > > Jim Propp > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >

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[math-fun] faster and faster
by James Propp 08 Apr '10

08 Apr '10

I wasn't able to attend Gathering for Gardner 9, but my student Linda Zayas-Palmer did, and she presented there a joint paper called "Carrying On with Infinite Decimals" (http://jamespropp.org/carrying.pdf) I'd be interested in people's comments (since at some point I plan to publish this in one of the Gathering for Gardner volumes). One comment I've already received can be paraphrased as follows: "You describe these computations with infinite decimals as taking place in ordinary time, so that even though each individual digit eventually stabilizes, you have to wait till time omega to be able to read off the answer. Why not imagine a computer in which each carry in the 10^{-n} position takes 10^{-n} seconds, so that the computation finishes in finite time?" I was wondering if anyone had seen any proposals for (unrealistic) models of computation along these lines. One needs to take some care to make sure that the state of the system remains well-defined even at limit-times. For instance, if one switches a light switch on and off at times 1/2, 3/4, 7/8, ..., the state of the light at time 1 is undefined. Still, it seems that even if one imposes constraints to avoid the aforementioned "light switch problem", one can get idealized computers of this kind to do all kinds of things ordinary computers can't do, such as solve the halting problem. There must be existing literature on computers that can be made to run faster and faster. Jim Propp

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Re: [math-fun] fun function
by Steve Witham 07 Apr '10

07 Apr '10

>From: rcs(a)xmission.com > >What's wrong with f(x) = x / (1+x^2)? >And f(z) = z / (1+|z^2|) ? Doesn't that look like this? ^ ----\ / \----- v Do you mean f(x) = x / ( 1 - x^2 )? That's more like it but I want it to be superexponential as it approaches -1 and +1. x sb(x) x / ( 1 - x^2 ) 1/2 1 2/3 3/4 2 12/7 7/8 4 56/15 15/16 16 240/31 31/32 65536 992/63 63/64 2^65536 4032/127 Something involving x/(1-x^2) may do what I want. sinh( x/(1-x^2) ) or sinh( z/|1-x^2| ) is more like it but not it. Also, sb(x) is really flat near zero, e.g. sb(1/64) = 2^-65536, which is also something I want, whereas x/(1-x^2) is linearish there. My sb() function is only like a tower, and there are all sorts of largeness beyond that, but it's what I was going for. --Steve

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[math-fun] A different fun function
by Allan Wechsler 06 Apr '10

06 Apr '10

Steve Witham's function reminded me that I have never learned the name of the following function, similar in some ways to Steve's example, but closely related to Farey sequences and continued fractions. Let f(0) = 0/1, and f(1) = 1/0. Forgive me for playing fast and loose with infinities; you'll see what I'm about in a moment. I'm going to give a recipe for calculating f at dyadic rationals; then I'm pretty sure f can be "completed" continuously to all reals between 0 and 1. f is going to be strictly increasing, and its range is all positive reals. Suppose we have a dyadic rational p/2^n in lowest terms (so p is odd). p/2^n is bracketed by simpler dyadic rationals, L = (p-1)/2^n and R = (p+1)/2^n, which when reduced to lowest terms will have denominators less than 2^n. Suppose we know f(L) = a/b, f(R) = c/d; then define f(p/2^n) = f((L+R)/2) = (a+c)/(b+d). This is the standard Farey mediant between f(L) and f(R). Note that 1/0 works perfectly well as a "bookend" for these mediants, which are guaranteed to be in lowest terms. Example: Find f(3/8). First we must find f(1/4) and f(1/2). f(1/2) is the mediant of 0/1 and 1/0, or 1/1. Now f(1/4) = 1/2, and f(3/8) = (1+1)/(2+1) = 2/3. Exercise: show that f(2/3) = (sqrt(5) + 1)/2. The function f stretches the dyadic rationals to cover all the rationals; it stretches the rationals to cover all the quadratic rationals. I have never seen a graph of it, but I'm sure it's fractally delicious.

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