Lyapunov-type inequalities for nonlinear impulsive systems with applications

We obtain new Lyapunov-type inequalities for systems of nonlinear impulsive differential equations, special cases of which include the impulsive Emden-Fowler equations and half-linear equations. By applying these inequalities, sufficient conditions are derived for the disconjugacy of solutions and the boundedness of weakly oscillatory solutions

Weak order for the discretization of the stochastic heat equation

In this paper we study the approximation of the distribution of $X_t$ Hilbert--valued stochastic process solution of a linear parabolic stochastic partial differential equation written in an abstract form as $$dX_t+AX_t dt = Q^{1/2} d W_t, \quad X_0=x \in H, \quad t\in[0,T],$$ driven by a Gaussian space time noise whose covariance operator $Q$ is given. We assume that $A^{-\alpha}$ is a finite trace operator for some $\alpha>0$ and that $Q$ is bounded from $H$ into $D(A^\beta)$ for some $\beta\geq 0$. It is not required to be nuclear or to commute with $A$. The discretization is achieved thanks to finite element methods in space (parameter $h>0$) and implicit Euler schemes in time (parameter $\Delta t=T/N$). We define a discrete solution $X^n_h$ and for suitable functions $\phi$ defined on $H$, we show that |\E \phi(X^N_h) - \E \phi(X_T) | = O(h^{2\gamma} + \Delta t^\gamma) \noindent where $\gamma<1- \alpha + \beta$. Let us note that as in the finite dimensional case the rate of convergence is twice the one for pathwise approximations

Positivity, rational Schur functions, Blaschke factors, and other related results in the Grassmann algebra

We begin a study of Schur analysis in the setting of the Grassmann algebra, when the latter is completed with respect to the $1$-norm. We focus on the rational case. We start with a theorem on invertibility in the completed algebra, and define a notion of positivity in this setting. We present a series of applications pertaining to Schur analysis, including a counterpart of the Schur algorithm, extension of matrices and rational functions. Other topics considered include Wiener algebra, reproducing kernels Banach modules, and Blaschke factors.Comment: 35 page

Fast algorithms for spectral differentiation matrices

Recently Olver and Townsend presented a fast spectral method that relies on bases of ultraspherical polynomials to give differentiation matrices that are almost banded. The almost-banded structure allowed them to develop efficient algorithms for solving ceratin discretized systems in linear time. We show that one can also design fast algorithms for standard spectral methods because the underlying matrices, though dense, have the same rank structure as those of Olver and Townsend

Sum rules and spectral measures of Schrödinger operators with L^2 potentials

Necessary and sufficient conditions are presented for a positive measure to be the spectral measure of a half-line Schrödinger operator with square integrable potential

Corner Singularities and Analytic Regularity for Linear Elliptic Systems. Part I: Smooth domains.

211 pagesThis is a preliminary version of the first part of a book project that will consist of four parts. We are making it available in electronic form now, because there is a demand for some of the technical tools it provides, in particular a detailed presentation of analytic elliptic regularity estimates in the neighborhood of smooth boundary points. In our proof of analytic a priori estimates, besides the classical Morrey-Nirenberg techniques of nested open sets and difference quotients, a new ingredient is a Cauchy-type estimate for coordinate transformations based on the Faà di Bruno formula for derivatives of composite functions. This first part can also serve as a general introduction into the subject of regularity for linear elliptic systems with smooth coefficients on smooth domains. We treat regularity in Sobolev spaces for a general class of second order elliptic systems and corresponding boundary operators that cover, in particular, many elliptic problems in variational form. Starting from the regularity of the variational solution, we follow the improvement of the regularity of the solution as the regularity of the data is raised, first for low regularity, and then going to ever higher regularity and finally to analytic regularity. Supported by the discussion of many examples, some of them new, such as the variational formulation of the electromagnetic impedance problem, we hope to provide new insight into this classical subject. We hope to be able to finish the whole project soon and to publish all four parts, but in the meantime this first part can be used as a starting point for proofs of elliptic regularity estimates in more complicated situations