3,836 research outputs found
A bound for the rank-one transient of inhomogeneous matrix products in special case
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
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
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
Asynchronous iterative solution for dominant eigenvectors with applications in performance modelling and PageRank
Imperial Users onl
Bounds for long max-plus matrix products
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
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
We consider large random matrices with centered, independent entries but
possibly different variances. We compute the normalized trace of
for functions analytic on the spectrum of . 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 .Comment: 20 pages, Corrected a typo in Assumption (1) [after final
publication] and made other irrelevant revision
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