Related papers: A note on sharp spectral estimates for periodic Ja…

We propose a novel estimation framework for quadratic functionals of precision matrices in high-dimensional settings, particularly in regimes where the feature dimension $p$ exceeds the sample size $n$. Traditional moment-based estimators…

We investigate Ambarzumian-type mixed inverse spectral problems for Jacobi matrices. Specifically, we examine whether the Jacobi matrix can be uniquely determined by knowing all but the first $m$ diagonal entries and a set of $m$ ordered…

In a high temperature regime where $\beta N \to 2c$, the empirical distribution of the eigenvalues of Gaussian beta ensembles, beta Laguerre ensembles and beta Jacobi ensembles converges to a limiting measure which is related to associated…

It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary $m\in\mathbb{N}$ one can construct a periodic non-compact Riemannian manifold $M$ with at least $m$ gaps in the spectrum of the corresponding Laplace-Beltrami operator…

A matrix polynomial is a polynomial in a complex variable $\lambda$ with coefficients in $n \times n$ complex matrices. The spectral curve of a matrix polynomial $P(\lambda)$ is the curve $\{ (\lambda, \mu) \in \mathbb{C}^2 \mid…

We address the computational spectral theory of Jacobi operators that are compact perturbations of the free Jacobi operator via the asymptotic properties of a connection coefficient matrix. In particular, for finite-rank perturbation we…

In this paper, we mainly consider arithmetic properties of the cyclotomic matrix $B_p(k)=\left[J_p(\chi^{ki},\chi^{kj})^{-1}\right]_{1\le i,j\le (p-1-k)/k}$, where $p$ is an odd prime, $1\le k<p-1$ is a divisor of $p-1$, $\chi$ is a…

It is well known from the Perron-Frobenius theory that the spectral gap of a positive square matrix is positive. In this paper, we give a more quantitative characterization of the spectral gap. More specifically, using a complex extension…

In this paper, we present several estimators of the diagonal elements of the inverse of the covariance matrix, called precision matrix, of a sample of iid random vectors. The focus is on high dimensional vectors having a sparse precision…

We describe an ensemble of (sparse) random matrices whose eigenvalues follow the Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature beta. Our approach combines elements from the theory of…

The $k$-th Laplacian spectral moment of a digraph $G$ is defined as $\sum_{i=1}^n \lambda_i^k$, where $\lambda_i$ are the eigenvalues of the Laplacian matrix of $G$ and $k$ is a nonnegative integer. For $k=2$, this invariant is better known…

In the following we are interested in the spectral gaps of discrete quasiperiodic Schr\"odinger operators when the frequency is Diophantine, the potential is analytic, and in the subcritical regime. The gap-labelling theorem asserts in this…

We consider positive Jacobi matrices $J$ with compact inverses and consequently with purely discrete spectra. A number of properties of the corresponding sequence of orthogonal polynomials is studied including the convergence of their…

Let $M_n$ be a class of symmetric sparse random matrices, with independent entries $M_{ij} = \delta_{ij} \xi_{ij}$ for $i \leq j$. $\delta_{ij}$ are i.i.d. Bernoulli random variables taking the value $1$ with probability $p \geq…

Classical Jacobi polynomials $P_{n}^{(\alpha,\beta)}$, with $\alpha, \beta>-1$, have a number of well-known properties, in particular the location of their zeros in the open interval $(-1,1)$. This property is no longer valid for other…

To any compact Riemann surface of genus g one may assign a principally polarized abelian variety of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus, and a Gram matrix of the lattice of a Jacobian is called…

For a graph $G=(V,E)$ and $v_{i}\in V$, denote by $d_{v_{i}}$ (or $d_{i}$ for short) the degree of vertex $v_{i}$. The $p$-Sombor matrix $\textbf{S}_{\textbf{p}}(G)$ ($p\neq0$) of a graph $G$ is a square matrix, where the $(i,j)$-entry is…

Let $A=(a_{ij})\in M_n(\R)$ be an $n$ by $n$ symmetric stochastic matrix. For $p\in [1,\infty)$ and a metric space $(X,d_X)$, let $\gamma(A,d_X^p)$ be the infimum over those $\gamma\in (0,\infty]$ for which every $x_1,...,x_n\in X$ satisfy…

We establish a sharp inequality between the blocks of positive partitioned matrices and conjecture a triangle type inequality for contractions: Given three contactions A,B,C, we conjecture that the constant c=3/4 is sharp in the triangle…

We develop direct and inverse spectral analysis for finite and semi-infinite non-self-adjoint Jacobi matrices with a rank one imaginary part. It is shown that given a set of $n$ not necessarily distinct non-real numbers in the open upper…