85,293 research outputs found

    On the Head and the Tail of the Colored Jones Polynomial

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    The colored Jones polynomial is a series of one variable Laurent polynomials J(K,n) associated with a knot K in 3-space. We will show that for an alternating knot K the absolute values of the first and the last three leading coefficients of J(K,n) are independent of n when n is sufficiently large. Computation of sample knots indicates that this should be true for any fixed leading coefficient of the colored Jones polynomial for alternating knots. As a corollary we get a Volume-ish Theorem for the colored Jones Polynomial.Comment: 14 pages, 6 figure

    Spectral Theory of Discrete Processes

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    We offer a spectral analysis for a class of transfer operators. These transfer operators arise for a wide range of stochastic processes, ranging from random walks on infinite graphs to the processes that govern signals and recursive wavelet algorithms; even spectral theory for fractal measures. In each case, there is an associated class of harmonic functions which we study. And in addition, we study three questions in depth: In specific applications, and for a specific stochastic process, how do we realize the transfer operator TT as an operator in a suitable Hilbert space? And how to spectral analyze TT once the right Hilbert space H\mathcal{H} has been selected? Finally we characterize the stochastic processes that are governed by a single transfer operator. In our applications, the particular stochastic process will live on an infinite path-space which is realized in turn on a state space SS. In the case of random walk on graphs GG, SS will be the set of vertices of GG. The Hilbert space H\mathcal{H} on which the transfer operator TT acts will then be an L2L^{2} space on SS, or a Hilbert space defined from an energy-quadratic form. This circle of problems is both interesting and non-trivial as it turns out that TT may often be an unbounded linear operator in H\mathcal{H}; but even if it is bounded, it is a non-normal operator, so its spectral theory is not amenable to an analysis with the use of von Neumann's spectral theorem. While we offer a number of applications, we believe that our spectral analysis will have intrinsic interest for the theory of operators in Hilbert space.Comment: 34 pages with figures removed, for the full version with all the figures please go to http://www.siue.edu/~msong/Research/spectrum.pd
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