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Apr
7
comment Can anyone give an example of Ricci flat Riemannian or Lorentzian Manifold that is not flat?
Many gravitational-wave vacuum solution to the Einstein field equations are geodesically complete. Many exact solutions of this type are known. The standard reference for this sort of thing is Stefani, Exact Solutions of Einstein's Field Equations.
Mar
15
comment Which mathematical ideas have done most to change history?
@Max: I was intrigued by the quote and made an effort to track it down. The source most people refer to is Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer, translated from French by David Vellos, E. F. Harding, Sophie Wood and Ian Monk, John Wiley and Sons, New York, 2000, p 577. I don't have the Ifrah book to hand, but the version of the quote that people give in English maybe have been translated from German to French by Ifrah, and then to English.
Feb
15
comment Is there a physical intuition for Darboux's theorem?
This seems like a perfectly fine answer, but the question asked for physical intuition based on the application to Hamiltonian systems in which the phase space represents the motion of particles. I don't see any physics in this answer.
Feb
15
revised Is there a physical intuition for Darboux's theorem?
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Feb
15
answered Is there a physical intuition for Darboux's theorem?
Feb
12
awarded  Yearling
Feb
8
comment Why does bosonic string theory require 26 spacetime dimensions?
all modern physical doctrines suggest that out world is NOT 4-dimensional, but higher Not all modern physical theories, just string theory -- which is probably wrong.
Feb
6
awarded  Revival
Jan
4
awarded  Nice Question
Nov
25
revised Periods and commas in mathematical writing
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Nov
25
revised Periods and commas in mathematical writing
added 104 characters in body
Nov
25
answered Periods and commas in mathematical writing
Nov
25
comment Periods and commas in mathematical writing
I use \qquad. Of course this is a matter of taste.
Aug
17
comment What proportion of chess positions that one can set up on the board, using a legal collection of pieces, can actually arise in a legal chess game?
let us continue this discussion in chat
Aug
17
comment What proportion of chess positions that one can set up on the board, using a legal collection of pieces, can actually arise in a legal chess game?
Re the comment beginning with "To simplify things," I've stated three times now, once in the revised answer and twice in comments, that I don't think it's particularly important whether my conjecture about $P_2$ holds. I'm now getting the message reading "Please avoid extended discussions in comments," which is probably good advice. I would be happy to move this to chat, but I would ask that before we do that, you take a look at my revised answer and acknowledge my repeated statements that the hypothesis about $P_2$ is not relevant to the more fundamental issues.
Aug
17
comment What proportion of chess positions that one can set up on the board, using a legal collection of pieces, can actually arise in a legal chess game?
What is the difference you are making? By random sampling I mean generating $n$ random positions on a computer, using the computer to determine whether each position is reachable, counting the number $m$ of unreachable positions, and estimating $P_i=m/n$.
Aug
17
comment What proportion of chess positions that one can set up on the board, using a legal collection of pieces, can actually arise in a legal chess game?
If you can move from positions A to B ignoring checks, and A and B do not have a check, then what stops you from "homotoping" the path to one with no checks? This would be an argument against the hypothesis that $P_2$ is close to 1. I don't understand the argument (e.g., I don't know what you mean by "homotoping"), but in any case, as I've said repeatedly now, I don't think it's particularly important whether this particular conjecture about $P_2$ holds.
Aug
17
comment What proportion of chess positions that one can set up on the board, using a legal collection of pieces, can actually arise in a legal chess game?
@DouglasZare: What exactly do you mean by, "It's pretty difficult to get that many powerful pieces on the board without causing a checkmate." What I meant by that is simply that I conjectured that $P_2$ was close to 1.
Aug
17
comment What proportion of chess positions that one can set up on the board, using a legal collection of pieces, can actually arise in a legal chess game?
@DouglasZare: Your skepticism about what I call $M_2$ may be entirely correct, and it may be true that my $P_2$ is small rather than close to 1 as I conjectured. I don't know. But the point that I've tried to make clear in my revised answer is that there are fundamental reasons that if one tries to attack this problem using this general approach of listing mechanisms and multiplying probabilities, there are fundamental reasons why nobody will be able to verify that the resulting estimate is correct.
Aug
15
revised What proportion of chess positions that one can set up on the board, using a legal collection of pieces, can actually arise in a legal chess game?
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