Description:
Discussion of current mathematical research. (Moderated)
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Fermat's not-so-little theorem
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I haven't got around to TeXing out a proof of this thing, but I'm eager to show it off anyhow. Theorem: Let f(A,B,...) be a polynomial in any number of variables over Z, and let n be a positive integer. Define a polynomial F by F(A,B,...) = sum_{d|n} M(d) f(A^d, B^d, ...)^{n/d} where M denotes the Mobius function. Then all the coefficients of F are... more »
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Necessity of least action principle
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I've developed the Feynman Path Integral from first principles, apart from physical requirements. And I'm trying to make contact with physics. It would help if there were a requirement that the variation of the action be zero. Then Euler-Lagrange equations of motion would procede from that. So does the evaluation of the path integral require... more »
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Graphs for an adjacency matrix in Z_2
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Hi, Let us consider the matrix $S$ of a Sudoku, $S$ being $n\times n$, with $n$ being a square (i.e. 9, 16, 25, etc.). Let's say that $S_{i,j}:=S_{i,j}\pmod{2}$, i.e. we now consider $S$ with all its elements being taken in $1/\mathbb Z_2$ (i.e. the 2-adic numbers), for $1\leq i\leq n$, $1\leq j\leq n$.... more »
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recurerent map
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Hi A continuous map $f$ is recurrent if for all $a>0$ there exists an integer $n$ such that $d(f^n,id)<a$. It is say to be equicontinuous if ${f,f^2,.....f^n,...} $ is equicontinuous. Is there an implication between the two notions? If not is there example showing that none of the two implication is true.... more »
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Mayer-Vietoris homotopy groups sequence of a pull-back of a fibration
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This must be an elementary question: could somebody tell me a reference for the Mayer-Vietoris homotopy groups sequence of a pull- back of a fibration? I'm working in the category of pointed simplicial sets. So I've a pull- back of a (Kan) fibration of pointed simplicial sets, and I've read that in this situation you have an associated Mayer-Vietoris sequence... more »
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"C^1" isometric deformations of S^2 in R^3
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It's known that the standard sphere S^2 in R^3 is rigid with respect to C^2 deformations through C^2 surfaces. (See, e.g., Edgar Kann, "A new method for infinitesimal rigidity of surfaces with K > 0," J. Diff. Geom., 1970 pp. 5-12.) But have there been results about isometric deformation of the standard S^2 in R^3, through surfaces that need be only C^1... more »
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This Week's Finds in Mathematical Physics (Week 282)
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Also available at [link] October 29, 2009 This Week's Finds in Mathematical Physics (Week 282) John Baez This week I'll get back to explaining some serious math: the relation between associative, commutative, Lie and Poisson algebras, and how this relates to quantization. There's some beautiful algebra and... more »
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Higher order "trigonometric" functions
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Dear Colleagues, I have been studying some very interesting functions that arise as higher order analogues of the trig functions. The idea is quite simple. Consider the Fermat curve x^n + y^n = 1. For any parametrization we must have x^{n-1}x' + y^{n-1}y' = 0. Surely the simplest and most natural possibility is x' = -y^{n-1} and y' = x^... more »
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