21,947 research outputs found

    On the Topological Characterization of Near Force-Free Magnetic Fields, and the work of late-onset visually-impaired Topologists

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    The Giroux correspondence and the notion of a near force-free magnetic field are used to topologically characterize near force-free magnetic fields which describe a variety of physical processes, including plasma equilibrium. As a byproduct, the topological characterization of force-free magnetic fields associated with current-carrying links, as conjectured by Crager and Kotiuga, is shown to be necessary and conditions for sufficiency are given. Along the way a paradox is exposed: The seemingly unintuitive mathematical tools, often associated to higher dimensional topology, have their origins in three dimensional contexts but in the hands of late-onset visually impaired topologists. This paradox was previously exposed in the context of algorithms for the visualization of three-dimensional magnetic fields. For this reason, the paper concludes by developing connections between mathematics and cognitive science in this specific context.Comment: 20 pages, no figures, a paper which was presented at a conference in honor of the 60th birthdays of Alberto Valli and Paolo Secci. The current preprint is from December 2014; it has been submitted to an AIMS journa

    Cycle expansions for intermittent maps

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    In a generic dynamical system chaos and regular motion coexist side by side, in different parts of the phase space. The border between these, where trajectories are neither unstable nor stable but of marginal stability, manifests itself through intermittency, dynamics where long periods of nearly regular motions are interrupted by irregular chaotic bursts. We discuss the Perron-Frobenius operator formalism for such systems, and show by means of a 1-dimensional intermittent map that intermittency induces branch cuts in dynamical zeta functions. Marginality leads to long-time dynamical correlations, in contrast to the exponentially fast decorrelations of purely chaotic dynamics. We apply the periodic orbit theory to quantitative characterization of the associated power-law decays.Comment: 22 pages, 5 figure

    Parametric Polyhedra with at least kk Lattice Points: Their Semigroup Structure and the k-Frobenius Problem

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    Given an integral d×nd \times n matrix AA, the well-studied affine semigroup \mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be stratified by the number of lattice points inside the parametric polyhedra PA(b)={x:Ax=b,x≥0}P_A(b)=\{x: Ax=b, x\geq0\}. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{ Sg}(A) such that PA(b)∩ZnP_A(b) \cap {\mathbb Z}^n has at least kk solutions. We demonstrate that this set is finitely generated, it is a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computations. Related results can be derived for those right-hand-side vectors bb for which PA(b)∩ZnP_A(b) \cap {\mathbb Z}^n has exactly kk solutions or fewer than kk solutions. (2) A computational complexity theory. We show that, when nn, kk are fixed natural numbers, one can compute in polynomial time an encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors of bounded norm that have at least kk solutions. (3) Applications and computation for the kk-Frobenius numbers. Using Generating functions we prove that for fixed n,kn,k the kk-Frobenius number can be computed in polynomial time. This generalizes a well-known result for k=1k=1 by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of kk-Frobenius numbers and their relatives

    Some Comments on Branes, G-flux, and K-theory

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    This is a summary of a talk at Strings2000 explaining three ways in which string theory and M-theory are related to the mathematics of K-theory.Comment: 10pp., late
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