35 research outputs found
Existence, uniqueness and decay rates for evolution equations on trees
We study evolution equations governed by an averaging operator on a directed tree, showing existence and uniqueness of solutions. In addition we find conditions of the initial condition that allows us to find the asymptotic decay rate of the solutions as . It turns out that this decay rate is not uniform, it strongly depends on how the initial condition goes to zero as one goes down in the tree.Fil: del Pezzo, Leandro Martin. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas; ArgentinaFil: Mosquera, Carolina Alejandra. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas; ArgentinaFil: Rossi, Julio Daniel. Universidad de Alicante; España. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentin
Local state space geometry and thermal metastability in complex landscapes: the spin-glass case
A simple geometrical characterization of configuration space neighborhoods of
local energy minima in spin glass landscapes is found by exhaustive search.
Combined with previous Monte Carlo investigations of thermal domain growth, it
allows a discussion of the connection between real and configuration space
descriptions of low temperature relaxational dynamics. We argue that the part
of state-space corresponding to a single growing domain is adequately modeled
by a hierarchically organized set of states and that thermal (meta)stability in
spin glasses is related to the nearly exponential local density of states
present within each trap.Comment: 16 pages, 8 figures, RevTeX, to appear in Physica A The figures have
been improved and the text somewhat shortened. New references have been adde
On unique continuation for solutions of the Schr{\"o}dinger equation on trees
We prove that if a solution of the time-dependent Schr{\"o}dinger equation on
an homogeneous tree with bounded potential decays fast at two distinct times
then the solution is trivial. For the free Schr{\"o}dinger operator, we use the
spectral theory of the Laplacian and complex analysis and obtain a
characterization of the initial conditions that lead to a sharp decay at any
time. We then use the recent spectral decomposition of the Schr{\"o}dinger
operator with compactly supported potential due to Colin de Verdi{\`e}rre and
Turc to extend our results in the presence of such potentials. Finally, we use
real variable methods first introduced by Escauriaza, Kenig, Ponce and Vega to
establish a general sharp result in the case of bounded potentials
Dispersive Properties for Discrete Schrödinger Equations
In this paper we prove dispersive estimates for the system formed by two coupled discrete Schrödinger equations. We obtain estimates for the resolvent of the discrete operator and prove that it satisfies the limiting absorption principle. The decay of the solutions is proved by using classical and some new results on oscillatory integrals
Testing decipherability of directed figure codes with domino graphs
Various kinds of decipherability of codes, weaker than unique decipherability, have been studied since mid-1980s. We consider decipherability
of directed gure codes, where directed gures are de ned as labelled polyomi-
noes with designated start and end points, equipped with catenation operation
that may use a merging function to resolve possible con
icts. This setting ex-
tends decipherability questions from words to 2D structures. In the present
paper we develop a (variant of) domino graph that will allow us to decide some
of the decipherability kinds by searching the graph for speci c paths. Thus the
main result characterizes directed gure decipherability by graph properties
Propagation reversal on trees in the large diffusion regime
In this work we study travelling wave solutions to bistable reaction
diffusion equations on bi-infinite -ary trees in the continuum regime where
the diffusion parameter is large. Adapting the spectral convergence method
developed by Bates and his coworkers, we find an asymptotic prediction for the
speed of travelling front solutions. In addition, we prove that the associated
profiles converge to the solutions of a suitable limiting reaction-diffusion
PDE. Finally, for the standard cubic nonlinearity we provide explicit formula's
to bound the thin region in parameter space where the propagation direction
undergoes a reversal