Re: [math-fun] Exactly Distributed Digits in the Initial Digits of Pi
Yes, that is a famous result of George Polya from 1921 that is explained in Feller vol. 1. I found it amazing at first that 1 and 2 dimensions should differ from all higher dimensions in that respect. This also has implications for heat flow in those dimensions: If n-space is a (homogenous and isotropic) heat-conducting medium with the unit ball kept at unit temperature and the rest of n-space beginning at temperature 0 at t = 0, then for n = 1 and n = 2, n-space asymptotically warms up to unit temperature but not for n >= 3 (cf. "There's no space like home", The Sciences, September-October 1995). —Dan
From: Hans Havermann <gladhobo@bell.net> Sent: Nov 26, 2016 9:46 AM I have been informed by Warren D. Smith that it is known that 1- and 2-dimensional random walks are recurrent, while 3-dimensional (and higher) such walks are transient. From that perspective, it strikes me that the sequence 15, 18, ... (third term > 8*10^9) [the number of initial digits of ternary pi where the digits 0, 1, and 2 appear exactly the same number of times] should be recurrent because it can be modeled by a 2-dimensional (presumably random) walk (just as I had done in my blog article). The implication would be that this sequence is infinite!
Wow, I was not aware of the heating result and how it follows from the random walk results! I found this PDF on the MAA site if anyone wants to read Dan's fascinating article: https://www.maa.org/sites/default/files/pdf/horizonsarchive/february_1996.pd... -tom On Sat, Nov 26, 2016 at 11:00 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Yes, that is a famous result of George Polya from 1921 that is explained in Feller vol. 1. I found it amazing at first that 1 and 2 dimensions should differ from all higher dimensions in that respect. This also has implications for heat flow in those dimensions: If n-space is a (homogenous and isotropic) heat-conducting medium with the unit ball kept at unit temperature and the rest of n-space beginning at temperature 0 at t = 0, then for n = 1 and n = 2, n-space asymptotically warms up to unit temperature but not for n >= 3 (cf. "There's no space like home", The Sciences, September-October 1995).
—Dan
From: Hans Havermann <gladhobo@bell.net> Sent: Nov 26, 2016 9:46 AM I have been informed by Warren D. Smith that it is known that 1- and 2-dimensional random walks are recurrent, while 3-dimensional (and higher) such walks are transient. From that perspective, it strikes me that the sequence 15, 18, ... (third term > 8*10^9) [the number of initial digits of ternary pi where the digits 0, 1, and 2 appear exactly the same number of times] should be recurrent because it can be modeled by a 2-dimensional (presumably random) walk (just as I had done in my blog article). The implication would be that this sequence is infinite!
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My favorite version of this is that if you have an infinite grid of resistors, with the origin connected to one volt and grounded at infinity, the resistance is infinite in 1 or 2 dimensions, but finite in 3 or more. There’s a beautiful monograph by Doyle and Snell about this: https://math.dartmouth.edu/~doyle/docs/walks/walks.pdf <https://math.dartmouth.edu/~doyle/docs/walks/walks.pdf> - Cris
On Nov 26, 2016, at 12:12 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
Wow, I was not aware of the heating result and how it follows from the random walk results! I found this PDF on the MAA site if anyone wants to read Dan's fascinating article:
https://www.maa.org/sites/default/files/pdf/horizonsarchive/february_1996.pd...
-tom
On Sat, Nov 26, 2016 at 11:00 AM, Dan Asimov <dasimov@earthlink.net> wrote:
Yes, that is a famous result of George Polya from 1921 that is explained in Feller vol. 1. I found it amazing at first that 1 and 2 dimensions should differ from all higher dimensions in that respect. This also has implications for heat flow in those dimensions: If n-space is a (homogenous and isotropic) heat-conducting medium with the unit ball kept at unit temperature and the rest of n-space beginning at temperature 0 at t = 0, then for n = 1 and n = 2, n-space asymptotically warms up to unit temperature but not for n >= 3 (cf. "There's no space like home", The Sciences, September-October 1995).
—Dan
From: Hans Havermann <gladhobo@bell.net> Sent: Nov 26, 2016 9:46 AM I have been informed by Warren D. Smith that it is known that 1- and 2-dimensional random walks are recurrent, while 3-dimensional (and higher) such walks are transient. From that perspective, it strikes me that the sequence 15, 18, ... (third term > 8*10^9) [the number of initial digits of ternary pi where the digits 0, 1, and 2 appear exactly the same number of times] should be recurrent because it can be modeled by a 2-dimensional (presumably random) walk (just as I had done in my blog article). The implication would be that this sequence is infinite!
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participants (3)
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Cris Moore -
Dan Asimov -
Tomas Rokicki