3,836 research outputs found

    A bound for the rank-one transient of inhomogeneous matrix products in special case

    Get PDF
    We consider inhomogeneous matrix products over max-plus algebra, where the matrices in the product satisfy certain assumptions under which the matrix products of sufficient length be rank-one, as it was shown in [6][L. Shue, B.D.O. Anderson, S. Dey: On steady state properties of certain max-plus products. Proceedings of the American Control Conference, Philadelphia, Pensylvania, (June 1998), 1909 1913.]. We establish a bound on the transient after which this starts to happen for any product of matrices whose length exceeds that bound

    Martingales arising from minimal submanifolds and other geometric contexts

    Full text link
    We consider a class of martingales on Cartan-Hadamard manifolds that includes Brownian motion on a minimal submanifold. We give sufficient conditions for such martingales to be transient, extending previous results on the transience of minimal submanifolds. We also give conditions for the almost sure convergence of the angular component (in polar coordinates) of a martingale in this class, including both the negatively pinched case (using earlier results on martingales of bounded dilation), and the radially symmetric case with quadratic decay of the upper curvature bound. Applied to minimal submanifolds, this gives curvature conditions on the ambient Cartan-Hadamard manifold under which any minimal submanifold admits a non-constant, bounded, harmonic function. Though our discussion is primarily motivated by minimal submanifolds, this class of martingales includes diffusions naturally associated to ancient solutions of mean curvature flow and to certain sub-Riemannian structures, and we briefly discuss these contexts as well. Our techniques are elementary, consisting mainly of comparison geometry and Ito's rule.Comment: Accepted version (some mistakes corrected from the previous), to appear in Illinois Journal of Mathematic

    Reheating in the Presence of Noise

    Get PDF
    Explosive particle production due to parametric resonance is a crucial feature of reheating in inflationary cosmology. Coherent oscillations of the inflaton field act as a periodically varying mass in the evolution equation for matter fields which couple to the inflaton. This in turn results in the parametric resonance instability. Thermal and quantum noise will lead to a nonperiodic perturbation in the mass. We study the resulting equation for the evolution of matter fields and demonstrate that noise (at least if it is temporally uncorrelated) will increase the rate of particle production. We also estimate the limits on the magnitude of the noise for which the resonant behavior is qualitatively unchanged.Comment: 26 pages, 2 figures, uses LATE

    Bounds for long max-plus matrix products

    Get PDF
    We consider long matrix products over max-plus algebra and develop bounds on the transient of their length after which they admit a certain decomposition as the product length exceeds these bounds. First we build on the weak CSR approach for max-plus powers of a matrix by Merlet, Nowak, and Sergeev [68] and consider the case when the products are tropical matrix powers of just one matrix. For this case we obtain new bounds on the above mentioned transient that make use of the cyclicity of the associated digraph and the tropical factor rank. Next, we develop a CSR decomposition for tropical inhomogeneous matrix products and establish bounds in which certain matrix products become CSR. We also critically examine the limitations of the developed theory by presenting a number of counterexamples in the cases where no bound exists for a matrix product to be CSR

    Observations on Unstable Quantons, Hyperplane Dependence and Quantum Fields

    Get PDF
    There is persistent heterodoxy in the physics literature concerning the proper treatment of those quantons that are unstable against spontaneous decay. Following a brief litany of this heterodoxy, I develop some of the consequences of assuming that such quantons can exist, undecayed and isolated, at definite times and that their treatment can be carried out within a standard quantum theoretic state space. This assumption requires hyperplane dependence for the unstable quanton states and leads to clarification of some recent results concerning deviations from relativistic time dilation of decay lifetimes. In the course of the discussion I make some observations on the relationship of unstable quantons to quantum fields.Comment: 29 pages, 4 figures, revised with added references, section 4 revise

    Power law decay for systems of randomly coupled differential equations

    Full text link
    We consider large random matrices XX with centered, independent entries but possibly different variances. We compute the normalized trace of f(X)g(X∗)f(X) g(X^*) for f,gf,g functions analytic on the spectrum of XX. We use these results to compute the long time asymptotics for systems of coupled differential equations with random coefficients. We show that when the coupling is critical the norm squared of the solution decays like t−1/2t^{-1/2}.Comment: 20 pages, Corrected a typo in Assumption (1) [after final publication] and made other irrelevant revision
    • 

    corecore