Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form

In this paper we consider minimizers of the functional$\min \{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }\subset D\text{ open}\}$ where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets ${\mathrm{\Omega }}^{⁎}$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{⁎}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-regular function, and a singular part, which is empty if $d<d^{⁎}$, discrete if $d=d^{⁎}$ and of Hausdorff dimension at most $d-d^{⁎}$ if $d>d^{⁎}$, for some $d^{⁎}\in \{5,6,7\}$.     

Approximations in Sobolev spaces by prolate spheroidal wave functions

Recently, there is a growing interest in the spectral approximation by the Prolate Spheroidal Wave Functions (PSWFs) ${\psi }_{n,c},c>0$. This is due to the promising new contributions of these functions in various classical as well as emerging applications from Signal Processing, Geophysics, Numerical Analysis, etc. The PSWFs form a basis with remarkable properties not only for the space of band-limited functions with bandwidth c, but also for the Sobolev space $H^s([?1,1])$. The quality of the spectral approximation and the choice of the parameter c when approximating a function in $H^s([?1,1])$ by its truncated PSWFs series expansion, are the main issues. By considering a function $f\in H^s([?1,1])$ as the restriction to $[?1,1]$ of an almost time-limited and band-limited function, we try to give satisfactory answers to these two issues. Also, we illustrate the different results of this work by some numerical examples.     

Exploring artificial neural networks to model interatomic and intermolecular potential energy surfaces

We demonstrate how a back-propagation artificial neural network can be trained to represent a potential energy surface (PES) in a formless manner with limited data points and exploited to predict interaction energies for configurations not included in the training set. A similar exercise is undertaken for predicting the eigenvalues and eigenvectors of a model Hamiltonian matrix that delicately depends on parameters and exhibits crossing of eigen values.     

On graphs with exactly one anti-adjacency eigenvalue and beyond

The anti-adjacency matrix of a graph is constructed from the distance matrix of a graph by keeping each row and each column only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping in each row and each column only the distances equal to 1. The (anti-)adjacency eigenvalues of a graph are those of its (anti-)adjacency matrix. Employing a novel technique introduced by Haemers (2019) [9], we characterize all connected graphs with exactly one positive anti-adjacency eigenvalue, which is an analog of Smith's classical result that a connected graph has exactly one positive adjacency eigenvalue iff it is a complete multipartite graph. On this basis, we identify the connected graphs with all but at most two anti-adjacency eigenvalues equal to ?2 and 0. Moreover, for the anti-adjacency matrix we determine the HL-index of graphs with exactly one positive anti-adjacency eigenvalue, where the HL-index measures how large in absolute value may be the median eigenvalues of a graph. We finally propose some problems for further study.     

On signed graphs with just two distinct adjacency eigenvalues

In this paper, all simple connected signed graphs with maximum degree at most 4 and with just two distinct adjacency eigenvalues are completely characterized, there exists an infinite family of 4-regular signed graphs with just two distinct adjacency eigenvalues.     

Principal Submatrices of a Hermitian Matrix

Suppose k 1 ,<, k m and n are positive integers such that k 1 + … + k m h n . We characterize those k i × k i Hermitian matrices A i , i = 1, < , m that can appear as diagonal blocks of an n × n Hermitian matrix C with prescribed eigenvalues. The characterization will be given in terms of the eigenvalues of C and A i , i = 1, <, m . Our results extend those of Thompson and Freede, Horn, Fan and Pall.     

Anisotropic eigenvalues upper bounds for hypersurfaces in weighted Euclidean spaces

We prove anisotropic Reilly-type upper bounds for divergence-type operators on hypersurfaces of the Euclidean space in presence of a weighted measure.     

The eigenpairs of a Sylvester–Kac type matrix associated with a simple model for one-dimensional deposition and evaporation

A straightforward model for deposition and evaporation on discrete cells of a finite array of any dimension leads to a matrix equation involving a Sylvester–Kac type matrix. The eigenvalues and eigenvectors of the general matrix are determined for an arbitrary number of cells. A variety of models to which this solution may be applied are discussed.