For the eigenvalue function on symmetric matrices, we have gathered a number of it’s properties.We show that this map has the properties of continuity, strict continuity, directional differentiability, Frechet differentiability, continuous  differentiability. Eigenvalue function will be extended to a larger set of matrices and then the listed properties will prove again.

In this paper, we study synchronization of complex random networks of nonlinear oscillators, with specifiable expected degree distribution. We review a sufficient condition for synchronization and a sufficient condition for desynchronization, expressed in terms of the eigenvalue distribution of the Laplacian of the graph and the coupling strength. We then provide a general way to approximate th...

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semi-circle law. If the Gaussian entries are all shifted by a constant amount c/(2N)1/2, where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis of the secul...

We consider the eigenvalue problem for the case where the input matrix is symmetric and its entries perturb in some given intervals. We present a characterization of some of the exact boundary points, which allows us to introduce an inner approximation algorithm, that in many case estimates exact bounds. To our knowledge, this is the first algorithm that is able to guarantee exactness. We illus...

We explore singularly perturbed convection-diffusion equations in a circular domain. Considering boundary layer analysis of the singularly perturbed equations and we show convergence results. In view of numerical analysis, We discuss approximation schemes, error estimates and numerical computations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem...

The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $lambda$ of $A$ is a main eigenvalue if the eigenspace $epsilon(lambda)$ has an eigenvector $X$ such that $X^{t}jjneq 0$, where $jj$ is the all-one...

Harmonic extraction methods for the multiparameter eigenvalue problem willbe presented. These techniques are generalizations of their counterparts forthe standard and generalized eigenvalue problem. The methods aim to ap-proximate interior eigenpairs, generally more accurately than the standardextraction does. The process can be combined with any subspace expansionapproa...

The power of density functional theory is often limited by the high computational demand in solving an eigenvalue problem at each self-consistent-field (SCF) iteration. The method presented in this paper replaces the explicit eigenvalue calculations by an approximation of the wanted invariant subspace, obtained with the help of well-selected Chebyshev polynomial filters. In this approach, only ...

A result of Zyczkowski and Sommers [J. Phys. A 33, 2045–2057 (2000)] gives the eigenvalue probability density function for the top N ×N sub-block of a Haar distributed matrix from U(N + n). In the case n ≥ N , we rederive this result, starting from knowledge of the distribution of the sub-blocks, introducing the Schur decomposition, and integrating over all variables except the eigenvalues. The...

In this article we consider the a posteriori error estimation and adaptive mesh refinement of discontinuous Galerkin finite element approximations of the hydrodynamic stability problem associated with the incompressible Navier–Stokes equations. Particular attention is given to the reliable error estimation of the eigenvalue problem in channel and pipe geometries. Here, computable a posteriori e...