Related papers: Block subspace expansions for eigenvalues and eige…

We consider the problem of determining the optimal block (or subsample) size for a spatial subsampling method for spatial processes observed on regular grids. We derive expansions for the mean square error of the subsampling variance…

We consider the standard overlap $\mathcal{O}_{ij}: =\langle \mathbf{r}_j, \mathbf{r}_i\rangle\langle \mathbf{l}_j, \mathbf{l}_i\rangle$ of any bi-orthogonal family of left and right eigenvectors of a large random matrix $X$ with centred…

We demonstrate the equivalence of two classes of $D$-invariant polynomial subspaces introduced in [8] and [9], i.e., these two classes of subspaces are different representations of the breadth-one $D$-invariant subspace. Moreover, we solve…

An eigenvalue problem arising in optimal insulation related to the minimization of the heat decay rate of an insulated body is adapted to enforce a positive lower bound imposed on the distribution of insulating material. We prove the…

We formulate the issue of minimality of self-adjoint operators on a Hilbert space as a semi-definite problem, linking the work by Overton in [1] to the characterization of minimal hermitian matrices. This motivates us to investigate the…

We give an explicit (in particular, deterministic polynomial time) construction of subspaces X of R^N of dimension (1-o(1))N such that for every element x in X, |x|_1 and N^{1/2} |x|_2 are equivalent up to a factor of (log N)^{log log log…

Quasi-normal-eigenvalue optimization is studied under constraints $b_1(x) \le B(x) \le b_2 (x)$ on structure functions $B$ of 2-side open optical or mechanical resonators. We prove existence of various optimizers and provide an example when…

We make progress on some questions related to polynomial approximations of ${\rm AC}^0$. It is known, by works of Tarui (Theoret. Comput. Sci. 1993) and Beigel, Reingold, and Spielman (Proc. $6$th CCC, 1991), that any ${\rm AC}^0$ circuit…

We present a novel block reduction method for the study of a general class of random band matrices (RBM) defined on the $d$-dimensional lattice $\mathbb{Z}_{L}^d:=\{1,2,\ldots,L\}^{d}$ for $d\in \{1,2\}$, with band width $W$ and an almost…

We apply the Wiener Hermite (WH) expansion to the non-linear evolution of Large-Scale Structure, and obtain an approximate expression for the matter power spectrum in full order of the expansion. This method allows us to expand any random…

We establish Hermite expansion characterizations for several subspaces of the Fr\'{e}chet space of functions on the real line satisfying \begin{equation*} |f(x)| \lesssim e^{-(\frac{1}{2} - \lambda ) x^{2}} , \qquad | \widehat{f}(\xi )|…

We study a shape optimization problem associated with the first eigenvalue of a nonlinear spectral problem involving a mixed operator ($p-$Laplacian and Laplacian) with a constraint on the volume. First, we prove the existence of a…

We consider the well-known shape optimization problem with spectral cost: minimizing the first eigenvalue of the Dirichlet Laplacian among all subdomains $\Omega$ having prescribed volume and contained in a fixed box $D$; equivalently, we…

Let $\D_{v,b,k}$ denote the family of all connected block designs with $v$ treatments and $b$ blocks of size $k$. Let $d\in\D_{v,b,k}$. The replication of a treatment is the number of times it appears in the blocks of $d$. The matrix…

We consider growth-optimal e-variables with maximal e-power, both in an absolute and relative sense, for simple null hypotheses for a $d$-dimensional random vector, and multivariate composite alternatives represented as a set of…

Let A be an n x n symmetric random matrix whose upper-triangular entries are independent and follow possibly non-identical subgaussian distributions. This paper investigates the spectral properties of A, including its eigenvalues and…

We prove sharp embedding inequalities for certain reduced Sobolev spaces that arise naturally in the context of Dirichlet problems with $L^1$ data. We also find the optimal target spaces for such embeddings, which in dimension 2 could be…

Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalues, and is of practical interest because of wide range of applications in fields such as structural…

We prove a structure theorem for the feasible solutions of the Arora-Rao-Vazirani SDP relaxation on low threshold rank graphs and on small-set expanders. We show that if G is a graph of bounded threshold rank or a small-set expander, then…

We consider a linear symmetric operator in a Hilbert space that is neither bounded from above nor from below, admits a block decomposition corresponding to an orthogonal splitting of the Hilbert space and has a variational gap property…