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We study dichotomous behavior of solutions to a non-autonomous linear difference equation in a Hilbert space. The evolution operator of this equation is not continuously invertible and the corresponding unstable subspace is of infinite dimension in general. We formulate a condition ensuring the dichotomy in terms of a sequence of indefinite metrics in the Hilbert space. We also construct an example of a difference equation in which dichotomous behavior of solutions is not compatible with the signature...
The concept of characteristic interval for piecewise monotone functions is introduced and used in the study of their iterative roots on a closed interval.
Equivalence of the spectral gap, exponential integrability of hitting times and Lyapunov conditions is well known. We give here the correspondence (with quantitative results) for reversible diffusion processes. As a consequence, we generalize results of Bobkov in the one dimensional case on the value of the Poincaré constant for log-concave measures to superlinear potentials. Finally, we study various functional inequalities under different hitting times integrability conditions (polynomial,…)....
Starting with the computation of the appropriate Poisson kernels, we review, complement, and compare results on drifted Laplace operators in two different contexts: homogeneous trees and the hyperbolic half-plane.
K. Nikodem and the present author proved in [3] a theorem concerning separation by affine functions. Our purpose is to generalize that result for polynomials. As a consequence we obtain two theorems on separation of an n-convex function from an n-concave function by a polynomial of degree at most n and a stability result of Hyers-Ulam type for polynomials.
The aim of this paper is to give a necessary and sufficient condition for a set-valued function to be a polynomial s.v. function of order at most 2.
We propose a matrix population modelling
approach in order to describe the dynamics of a grayling
(Thymallus thymallus, L. 1758) population living in the Ain
river (France). We built a Leslie like model, which integrates the
climate changes in terms of temperature and discharge. First, we
show how temperature and discharge can be related to life history
traits like survival and reproduction. Second, we show how to use
the population model to precisely examine the life cycle of
grayling : estimated...
Biological systems are able to switch their neural systems into inhibitory states and it is therefore important to build mathematical models that can explain such phenomena. If we interpret such inhibitory modes as `positive' or `negative' steady states of neural networks, then we will need to find the corresponding fixed points. This paper shows positive fixed point theorems for a particular class of cellular neural networks whose neuron units are placed at the vertices of a regular polygon. The...
In this paper, sharp a priori estimate of the periodic solutions is obtained for the discrete analogue of the continuous time ratio-dependent predator-prey system, which is governed by nonautonomous difference equations, modelling the dynamics of the competing preys and one predator having nonoverlapping generations. Based on more precise a priori estimate and the continuation theorem of the coincidence degree, an easily verifiable sufficient criterion of the existence of positive periodic solutions...
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