Related papers: Frame bound, spectral gap and Plus space
This paper introduces the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We then show that the condition of local spectral expansion for a complex yields various…
Given a frame in a finite dimensional Hilbert space we construct additive perturbations which decrease the condition number of the frame. By iterating this perturbation, we introduce an algorithm that produces a tight frame in a finite…
Starting from the formulation of pseudo-Riemannian generalisation of real spectral triples we develop the data of geometries over finite-dimensional algebras with indefinite metric and their Riemannian parts. We then discuss the Standard…
This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames $\mathcal{F} = \{f_i\}_{i \in I}$ and $\mathcal{E} =…
We consider averages $\kappa$ of spectral measures of rank one perturbations with respect to a $\sigma$-finite measure $\nu$. It is examined how various degrees of continuity of $\nu$ with respect to $\alpha$-dimensional Hausdorff measures…
An analysis is given of the local phase space of gravity coupled to matter to second order in perturbation theory. Working in local regions with boundaries at finite distance, we identify matter, Coulomb, and additional boundary modes. The…
We characterize the potential-energy functions $V(x)$ that minimize the gap $\Gamma$ between the two lowest Sturm-Liouville eigenvalues for \[ H(p,V) u := -\frac{d}{dx} \left(p(x)\frac{du}{dx}\right)+V(x) u = \lambda u, \quad\quad x\in…
We prove bounds for a class of unital homomorphisms arising in the study of spectral sets, by involving extremal functions and vectors. These are used to recover three celebrated results on spectral constants by Crouzeix--Palencia,…
The paper is devoted to frame expansions in Fr\'echet spaces. First we review some results which concern series expansions in general Fr\'echet spaces via Fr\'echet and General Fr\'echet frames. Then we present some new results on series…
We obtain new bounds for the Laplacian spectral radius of a signed graph. Most of these new bounds have a dependence on edge sign, unlike previously known bounds, which only depend on the underlying structure of the graph. We then use some…
The matching basis in field extentions is introduced by S. Eliahou and C. Lecouvey in [2]. In this paper we define the minimal and maximal linear matching property for field extensions and prove that if K is not algebraically closed, then K…
We investigate the averaging problem in cosmology as the problem of introducing a distance between spaces. We first introduce the spectral distance, which is a measure of closeness between spaces defined in terms of the spectra of the…
We study the spectral gap behavior of an operator obtained by summing a random permutation $M$ and a deterministic bistochastic matrix $Q$. We are interested in the asymptotic in terms of dimension. In the case where $(M,Q)$ are…
The aim of this paper is to give an upper bound for the intrinsic diameter of a surface with boundary immersed in a conformally flat three dimensional Riemannian manifold in terms of the integral of the mean curvature and of the length of…
A central challenge in machine learning is to understand how noise or measurement errors affect low-rank approximations, particularly in the spectral norm. This question is especially important in differentially private low-rank…
We study spectral theory for bounded Borel subsets of $\br$ and in particular finite unions of intervals. For Hilbert space, we take $L^2$ of the union of the intervals. This yields a boundary value problem arising from the minimal operator…
We give an improvement of sharp Berezin type bounds on the Riesz means $\sum_k(\Lambda-\lambda_k)_+^\sigma$ of the eigenvalues $\lambda_k$ of the Dirichlet Laplacian in a domain if $\sigma\geq 3/2$. It contains a correction term of the…
In this paper, we study the entries of the principal eigenvector of the signless Laplacian matrix of a hypergraph. More precisely, we obtain bounds for this entries. These bounds are computed trough other important parameters, such as…
We discuss the necessity and the emergence of quantum reference frames when attempting to define a rest frame for mixed particles. We analyze the corresponding concept of frame dependent entanglement and how it could affect measurements on…
Using spectral embedding based on the signless Laplacian, we obtain bounds on the spectrum of transition matrices on graphs. As a consequence, we bound return probabilities and the uniform mixing time of simple random walk on graphs. In…