Diagonal Entries Research Articles

Some essential spectra of unbounded operator matrices pencils with non-diagonal domain and application

In this paper, we investigate the stability of some essential spectra of a 2x2 block operator matrices pencil with unbounded entries and with non-diagonal domain (i.e a domain consisting of vectors which satisfy certain relations between their components) by using the resolvent of this kind of matrix operator in terms of the union of the essential spectra of the restriction of its diagonal operators entries. Furthermore, an example of two-group transport operators pencils is presented to illustrate the validity of the main results.

Approximations for the von Neumann and Rényi entropies of graphs with circulant type Laplacians

<abstract><p>In this note, we approximate the von Neumann and Rényi entropies of high-dimensional graphs using the Euler-Maclaurin summation formula. The obtained estimations have a considerable degree of accuracy. The performed experiments suggest some entropy problems concerning graphs whose Laplacians are $ g $-circulant matrices, i.e., circulant matrices with $ g $-periodic diagonals, or quasi-Toeplitz matrices. Quasi means that in a Toeplitz matrix the first two elements in the main diagonal, and the last two, differ from the remaining diagonal entries by a perturbation.</p></abstract>

From symplectic eigenvalues of positive definite matrices to their pseudo-orthogonal eigenvalues

Williamson's theorem states that every real symmetric positive definite matrix $A$ of even order can be brought to diagonal form via a symplectic $T$-congruence transformation. The diagonal entries of the resulting diagonal form are called the symplectic eigenvalues of $A$. We point at an analog of this classical result related to Hermitian positive definite matrices, *-congruences, and another class of transformation matrices, namely, pseudo-unitary matrices. This leads to the concept of pseudo-unitary (or pseudo-orthogonal, in the real case) eigenvalues of positive definite matrices.

The p-norm of circulant matrices via Fourier analysis

Abstract A recent work derived expressions for the induced p-norm of a special class of circulant matrices A(n, a, b) ∈ ℝ n × n , with the diagonal entries equal to a ∈ ℝ and the off-diagonal entries equal to b ≥ 0. We provide shorter proofs for all the results therein using Fourier analysis. The key observation is that a circulant matrix is diagonalized by a DFT matrix. The results comprise an exact expression for ǁAǁ p , 1 ≤ p ≤ ∞, where A = A(n, a, b), a ≥ 0 and for ǁAǁ2 where A = A(n, −a, b), a ≥ 0; for the other p-norms of A(n, −a, b), 2 < p < ∞, upper and lower bounds are derived.

Improved Diagonal Hessian Approximations for Large-Scale Unconstrained Optimization

We consider some diagonal quasi-Newton methods for solving large-scale unconstrained optimization problems. A simple and effective approach for diagonal quasi-Newton algorithms is presented by proposing new updates of diagonal entries of the Hessian. Moreover, we suggest employing an extra BFGS update of the diagonal updating matrix and use its diagonal again. Numerical experiments on a collection of standard test problems show, in particular, that the proposed diagonal quasi-Newton methods perform substantially better than certain available diagonal methods.

On the Determinants of the Square-Type Stirling Matrix and Bell Matrix

We study determinants of the square-type Stirling matrix S ∗ and the square-type Bell matrix B ∗ . For this purpose, we prove that S ∗ and B ∗ have LU factorizations S ∗ = L S U S and B ∗ = L B U B where the diagonal entries of U S are k k − 1 , while those of U B are k ! ( k ≥ 1 ).

An inverse formula for the distance matrix of a fan graph

Let be the fan graph with vertices. The distance between any two distinct vertices i and j of is the length of the shortest path connecting i and j. Let be the symmetric matrix with diagonal entries equal to zero and off-diagonal entries equal to . In this paper, we find two positive semidefinite matrices and such that , all row sums of and are equal to zero, and find two rank one matrices and such that The interlacing property between the eigenvalues of and (resp., ) is also established.

EV Charging in Case of Limited Power Resource

In the case of the widespread adoption of electric vehicles (EV), it is well known that their use and charging could affect the network distribution system, with possible repercussions including line overload and transformer saturation. In consequence, during periods of peak energy demand, the number of EVs that can be simultaneously charged, or their individual power consumption, should be controlled, particularly if the production of energy relies solely on renewable sources. This requires the adoption of adaptive and/or intelligent charging strategies. This paper focuses on public charging stations and proposes methods of attribution of charging priority based on the level of charge required and premiums. The proposed solution is based on model predictive control (MPC), which maintains total current/power within limits (which can change with time) and imparts real-time priority charge scheduling of multiple charging bays. The priority is defined in the diagonal entry of the quadratic form matrix of the cost function. In all simulations, the order of EV charging operation matched the attributed priorities for the cases of ten cars within the available power. If two or more EVs possess similar or equal diagonal entry values, then the car with the smallest battery capacitance starts to charge its battery first. The method is also shown to readily allow participation in Demand Side Response (DSR) schemes by reducing the current temporarily during the charging operation.

Further results on A-numerical radius inequalities

Let A be a bounded linear positive operator on a complex Hilbert space {mathcal {H}}. Furthermore, let {mathcal {B}}_Amathcal {(H)} denote the set of all bounded linear operators on {mathcal {H}} whose A-adjoint exists, and {mathbb {A}} signify a diagonal operator matrix with diagonal entries are A. Very recently, several {mathbb {A}}-numerical radius inequalities of 2times 2 operator matrices were established. In this paper, we prove a few new {mathbb {A}}-numerical radius inequalities for 2times 2 and ntimes n operator matrices. We also provide a new proof of an existing result by relaxing a sufficient condition “A is strictly positive”. Our proofs show the importance of the theory of the Moore–Penrose inverse of a bounded linear operator in this field of study.

Path integral based convolution and pooling for graph neural networks* *This article is an updated version of: Ma Z, Xuan J, Wang Y G, Li M and Liò P 2020 Path integral based convolution and pooling for graph neural networks Advances in Neural Information Processing Systems vol 33 ed H Larochelle, M Ranzato, R Hadsell, M F Balcan and H

Graph neural networks (GNNs) extend the functionality of traditional neural networks to graph-structured data. Similar to CNNs, an optimized design of graph convolution and pooling is key to success. Borrowing ideas from physics, we propose path integral-based GNNs (PAN) for classification and regression tasks on graphs. Specifically, we consider a convolution operation that involves every path linking the message sender and receiver with learnable weights depending on the path length, which corresponds to the maximal entropy random walk. It generalizes the graph Laplacian to a new transition matrix that we call the maximal entropy transition (MET) matrix derived from a path integral formalism. Importantly, the diagonal entries of the MET matrix are directly related to the subgraph centrality, thus leading to a natural and adaptive pooling mechanism. PAN provides a versatile framework that can be tailored for different graph data with varying sizes and structures. We can view most existing GNN architectures as special cases of PAN. Experimental results show that PAN achieves state-of-the-art performance on various graph classification/regression tasks, including a new benchmark dataset from statistical mechanics that we propose to boost applications of GNN in physical sciences.

Hadamard-type inequalities for k-positive matrices

We establish Hadamard-type inequalities for a class of symmetric matrices called k-positive matrices for which the m-th elementary symmetric functions of their eigenvalues are positive for all m≤k. These matrices arise naturally in the study of k-Hessian equations in Partial Differential Equations. For each k-positive matrix, we show that the sum of its principal minors of size k is not larger than the k-th elementary symmetric function of their diagonal entries. The case k=n corresponds to the classical Hadamard inequality for positive definite matrices. Some consequences are also obtained.

Fully nontrivial solutions to elliptic systems with mixed couplings

We study the existence of fully nontrivial solutions to the system −Δui+λiui=∑j=1ℓβij|uj|p|ui|p−2uiinΩ,i=1,…,ℓ,in a bounded or unbounded domain Ω in RN,N≥3. The λi’s are real numbers, and the nonlinear term may have subcritical (1<p<NN−2), critical (p=NN−2), or supercritical growth (p>NN−2). The matrix (βij) is symmetric and admits a block decomposition such that the diagonal entries βii are positive, the interaction forces within each block are attractive (i.e., all entries βij in each block are non-negative) and the interaction forces between different blocks are repulsive (i.e., all other entries are non-positive). We obtain new existence and multiplicity results of fully nontrivial solutions, i.e., solutions where every component ui is nontrivial. We also find fully synchronized solutions (i.e., ui=ciu1 for all i=2,…,ℓ) in the purely cooperative case whenever p∈(1,2).

Corrected trapezoidal rules for boundary integral equations in three dimensions

The manuscript describes a quadrature rule that is designed for the high order discretization of boundary integral equations (BIEs) using the Nyström method. The technique is designed for surfaces that can naturally be parameterized using a uniform grid on a rectangle, such as deformed tori, or channels with periodic boundary conditions. When a BIE on such a geometry is discretized using the Nyström method based on the Trapezoidal quadrature rule, the resulting scheme tends to converge only slowly, due to the singularity in the kernel function. The key finding of the manuscript is that the convergence order can be greatly improved by modifying only a very small number of elements in the coefficient matrix. Specifically, it is demonstrated that by correcting only the diagonal entries in the coefficient matrix, \(O(h^{3})\) convergence can be attained for the single and double layer potentials associated with both the Laplace and the Helmholtz kernels. A nine-point correction stencil leads to an \(O(h^5)\) scheme. The method proposed can be viewed as a generalization of the quadrature rule of Duan and Rokhlin, which was designed for the 2D Lippmann–Schwinger equation in the plane. The techniques proposed are supported by a rigorous error analysis that relies on Wigner-type limits involving the Epstein zeta function and its parametric derivatives.

Design of Four-DoF Compliant Parallel Manipulators Considering Maximum Kinematic Decoupling for Fast Steering Mirrors

Laser beams can fluctuate in four directions, which requires active compensation by a fast steering mirror (FSM) motion system. This paper deals with the design of four-degrees-of-freedom (DoF) compliant parallel manipulators, for responding to the requirements of the FSM. In order to simplify high-precision control in parallel manipulators, maximum kinematic decoupling is always desired. A constraint map method is used to propose the four required DoF with the consideration of maximum kinematic decoupling. A specific compliant mechanism is presented based on the constraint map, and its kinematics is estimated analytically. Finite element analysis demonstrates the desired qualitative motion and provides some initial quantitative analysis. A normalization-based compliance matrix is finally derived to verify and demonstrate the mobility of the system clearly. In a case study, the results of normalization-based compliance matrix modelling show that the diagonal entries corresponding to the four DoF directions are about 10 times larger than those corresponding to the two-constraint directions, validating the desired mobility.

The selective regularization of a linear regression model

This paper discusses constructing a linear regression model with regularization of the system matrix of normal equations. In contrast to the conventional ridge regression, where positive parameters are added to all diagonal terms of a matrix, in the method proposed only those matrix diagonal entries that correspond to the data with a high correlation are increased. This leads to a decrease in the matrix conditioning and, therefore, to a decrease in the corresponding coefficients of the regression equation. The selection of the entries to be increased is based on the triangular decomposition of the correlation matrix of the original dataset. The effectiveness of the method is tested on a known dataset, and it is performed not only with a ridge regression, but also with the results of applying the widespread algorithms LARS and Lasso.

Ambarzumian-type Problems for Discrete Schr\xf6dinger Operators

We discuss the problem of unique determination of the finite free discrete Schrödinger operator from its spectrum, also known as the Ambarzumian problem, with various boundary conditions, namely any real constant boundary condition at zero and Floquet boundary conditions of any angle. Then we prove the following Ambarzumian-type mixed inverse spectral problem: diagonal entries except the first and second ones and a set of two consecutive eigenvalues uniquely determine the finite free discrete Schrödinger operator.

Calculating a function of a matrix with a real spectrum

Let T be a square matrix with a real spectrum, and let f be an analytic function. The problem of the approximate calculation of f(T) is discussed. Applying the Schur triangular decomposition and the reordering, one can assume that T is triangular and its diagonal entries tii are arranged in increasing order. To avoid calculations using the differences tii − tjj with close (including equal) tii and tjj, it is proposed to represent T in a block form and calculate the two main block diagonals using interpolating polynomials. The rest of the f(T) entries can be calculated using the Parlett recurrence algorithm. It is also proposed to perform some scalar operations (such as the building of interpolating polynomials) with an enlarged number of significant decimal digits.

On local spectral properties of operator matrices

In this paper, we focus on a 2 times 2 operator matrix T_{epsilon _{k}} as follows: \t\t\tTϵk=(ACϵkDB),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} T_{\\epsilon _{k}}= \\begin{pmatrix} A & C \\\\ \\epsilon _{k} D & B\\end{pmatrix}, \\end{aligned}$$ \\end{document} where epsilon _{k} is a positive sequence such that lim_{krightarrow infty }epsilon _{k}=0. We first explore how T_{epsilon _{k}} has several local spectral properties such as the single-valued extension property, the property (beta ), and decomposable. We next study the relationship between some spectra of T_{epsilon _{k}} and spectra of its diagonal entries, and find some hypotheses by which T_{epsilon _{k}} satisfies Weyl’s theorem and a-Weyl’s theorem. Finally, we give some conditions that such an operator matrix T_{epsilon _{k}} has a nontrivial hyperinvariant subspace.

DOA Estimations With Limited Snapshots Based on Improved Rank-One Correlation Model in Unknown Nonuniform Noise

It is well known that classic direction-of-arrival (DOA) estimation techniques yield unsatisfactory performance with limited snapshots in unknown nonuniform noise. In this article, a sparse reconstruction (SR) DOA estimation method combining vectorized and reduced signal covariance matrix (SCM) is proposed. The new approach is based on an improved rank-one correlation model for denoising. With an extended virtual aperture array, this technique is able to provide a high-resolution DOA estimation with limited snapshots and is robust to nonuniform noise. In addition, two effective algorithms are also derived to obtain noise-free SCMs. One is to obtain diagonal entries of the noise-free SCM from the off-diagonal entries based on the fact that diagonal entries of SCM across the array are identical. The other is based on matrix completion (MC) theory which uses the low rank property of the SCM. Numerical results demonstrate that the new approaches yield superior performance than existing techniques in DOA estimations for multiple and compactly placed sources.

Inverse spectral problems for arrowhead matrices

The problem of constructing a matrix by its spectral information is called inverse eigenvalue problem (IEP) which arises in a variety of applications. In this paper, we study an IEP for arrowhead matrices in different cases. The problem involves constructing of the matrix by some eigenvalues of each of the leading principal submatrices and one eigenpair. We will also investigate this problem and its variants in the cases of matrix entries being real, nonnegative, positive definite, complex and equal diagonal entries. To solve the problems, a new method to establish a relationship between the IEP and properties of symmetric and general form of matrices is developed. The necessary and sufficient conditions of the solvability of the problems are obtained. Finally, some numerical examples are presented.