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We investigate singular perturbation problems caused by small delays in the view of pseudo-exponential dichotomy. For a general linear non-autonomous retarded differential equation with small delay, previous works established the existence…
This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian unitary ensemble. In particular, the kth…
Boundary value problem for complete second order elliptic equation is considered in Banach space. The equation and boundary conditions involve a small and spectral parameter. The uniform L_{p}-regularity properties with respect to space…
We consider the approximation of elliptic eigenvalue problem with an immersed interface. The main aim of this paper is to prove the stability and convergence of an immersed finite element method (IFEM) for eigenvalues using Crouzeix-Raviart…
We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional…
We study the semigroup of the symmetric $\alpha$-stable process in bounded domains in $\R^2$. We obtain a variational formula for the spectral gap, i.e. the difference between two first eigenvalues of the generator of this semigroup. This…
Spectral embedding of graphs uses the top k non-trivial eigenvectors of the random walk matrix to embed the graph into R^k. The primary use of this embedding has been for practical spectral clustering algorithms [SM00,NJW02]. Recently,…
We consider computing eigenspaces of an elliptic self-adjoint operator depending on a countable number of parameters in an affine fashion. The eigenspaces of interest are assumed to be isolated in the sense that the corresponding…
Spectral methods include a family of algorithms related to the eigenvectors of certain data-generated matrices. In this work, we are interested in studying the geometric landscape of the eigendecomposition problem in various spectral…
In the analysis of elastic-scattering experimental data, optical-model parameters (usually, depths of real and imaginary potentials) are fitted and conclusions are drawn analyzing their variations at bombardment energies close to the…
This paper is concerned with the study of constrained statistical learning problems, the unconstrained version of which are at the core of virtually all of modern information processing. Accounting for constraints, however, is paramount to…
In this paper, we investigate the Dirichlet problem of Laplacian on complete Riemannian manifolds. By constructing new trial functions, we obtain a sharp upper bound of the gap of the consecutive eigenvalues in the sense of the order, which…
We consider the homogenization of a singularly perturbed self-adjoint fourth order elliptic equation with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The…
Spectral independence is a recently-developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal $O(n \log n)$ sampling algorithms on bounded-degree graphs…
Parameter-elliptic boundary-value problems are investigated on the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces with respect to the Hilbert Sobolev scale. The latter are the H\"ormander…
In this paper we deal with an eigenvalue problem in an interface elliptic equation. We characterize the set of principal eigenvalues as a level set of a concave and regular function. As application, we study a problem arising in population…
We derive optimal estimates in stochastic homogenization of linear elliptic equations in divergence form in dimensions $d\ge 2$. In previous works we studied the model problem of a discrete elliptic equation on $\mathbb{Z}^d$. Under the…
This paper studies a Sturm--Liouville boundary value problem in which one of the boundary conditions depends bilinearly on the spectral parameter. The differential equation is considered on the interval $(0,1)$ with a classical boundary…
To a unitary matrix U we associate a doubly stochastic matrix M by taking the modulus squared of each element of U. To study the connection between onset of quantum chaos on graphs and ergodicity of the underlying Markov chain, specified by…
Homogenization of a spectral problem in a bounded domain with a high contrast in both stiffness and density is considered. For a special critical scaling, two-scale asymptotic expansions for eigenvalues and eigenfunctions are constructed.…