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Economic and ecological models can be extremely complex, with a large number of agents/species each featuring multiple interacting dynamical quantities. In an attempt to understand the generic stability properties of such systems, we define…
We study an indefinite spectral problem for a second-order self-adjoint elliptic operator in an asymptotically thin cylinder. The operator coefficients and the spectral density function are assumed to be locally periodic in the axial…
Numerical multiscale methods usually rely on some coupling between a macroscopic and a microscopic model. The macroscopic model is incomplete as effective quantities, such as the homogenized material coefficients or fluxes, are missing in…
We investigate the relationship between the spectral gap delta E_0 and the localization length xi in general one-particle systems. A relationship for many-body systems between the spectral gap and the exponential clustering has been derived…
We study the spectral linear elasticity problem in an unbounded periodic waveguide, which consists of a sequence of identical bounded cells connected by thin ligaments of diameter of order $ h >0$. The essential spectrum of the problem is…
This course is intended as an introduction to the analysis of elliptic partial differential equations. The objective is to provide a large overview of the different aspects of elliptic partial differential equations and their modern…
A finite dimensional abstract approximation and convergence theory is developed for estimation of the distribution of random parameters in infinite dimensional discrete time linear systems with dynamics described by regularly dissipative…
We consider the homogeneous equation ${\mathcal A} u=0$, where ${\mathcal A}$ is a symmetric and coercive elliptic operator in $H^1(\Omega)$ with $\Omega$ bounded domain in ${{\mathbb R}}^d$. The boundary conditions involve fractional power…
We study the spectral gap of the Erd\H{o}s--R\'enyi random graph through the connectivity threshold. In particular, we show that for any fixed $\delta > 0$ if $$p \ge \frac{(1/2 + \delta) \log n}{n},$$ then the normalized graph Laplacian of…
We consider a second order self-adjoint operator in a domain which can be bounded or unbounded. The boundary is partitioned into two parts with Dirichlet boundary condition on one of them, and Neumann condition on the other. We assume that…
For a large class of symmetric random matrices with correlated entries, selected from stationary random fields of centered and square integrable variables, we show that the limiting distribution of eigenvalue counting measure always exists…
Inverse problems of recovering the coefficients of Sturm-Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2)…
Maximization and minimization problems of the principle eigenvalue for divergence form second order elliptic operators with the Dirichlet boundary condition are considered. The principal eigen map of such elliptic operators is introduced…
For elliptic systems with block structure in the upper half-space and t-independent coefficients, we settle the study of boundary value problems by proving compatible well-posedness of Dirichlet, regularity and Neumann problems in optimal…
The goal of this paper is to study the Dirichlet eigenvalues of bounded domains $\Omega\subset \Omega'$. With a local spectral stability requirement on $\Omega$, we show that the difference of the Dirichlet eigenvalues of $\Omega'$ and…
The aim of this paper is to study, in the infinite dimensional framework, the existence and uniqueness for the solution of the following multivalued generalized backward stochastic differential equation, considered on a random, possibly…
In this paper, we study a generalized finite element method for solving second-order elliptic partial differential equations with rough coefficients. The method uses local approximation spaces computed by solving eigenvalue problems on…
Inspired by the quantum computing algorithms for Linear Algebra problems [HHL,TaShma] we study how the simulation on a classical computer of this type of "Phase Estimation algorithms" performs when we apply it to solve the Eigen-Problem of…
This work concerns the minimization of the pseudospectral abscissa of a matrix-valued function dependent on parameters analytically. The problem is motivated by robust stability and transient behavior considerations for a linear control…
In this paper, we propose the unfitted spectral element method for solving elliptic interface and corresponding eigenvalue problems. The novelty of the proposed method lies in its combination of the spectral accuracy of the spectral element…