The statistical properties of trajectories of eigenvalues of Gaussian complex matrices whose Hermitian condition is progressively broken are investigated. It is shown how the ordering on the real axis of the real eigenvalues is reflected in the structure of the trajectories and also in the final distribution of the eigenvalues in the complex plane.

Symplectic eigenvalues are conventionally defined for symmetric positive-definite matrices via Williamson's diagonal form. Many properties of standard eigenvalues, including the trace minimization theorem, have been extended to case symplectic eigenvalues. In this note, we will generalize form positive-semidefinite matrices, which allows us define and prove theorem in new setting.

‎‎‎This paper presents a remarkable formula for spectral distance of a given block normal matrix $G_{D_0} = begin{pmatrix}‎ ‎A & B \‎ ‎C & D_0‎ ‎end{pmatrix} $ to set of block normal matrix $G_{D}$ (as same as $G_{D_0}$ except block $D$ which is replaced by block $D_0$)‎, ‎in which $A in mathbb{C}^{ntimes n}$ is invertible‎, ‎$ B in mathbb{C}^{ntimes m}‎, ‎C in mathbb{C}^{mti...

We study a formulation of euclidean general relativity in which the dynamical variables are given by a sequence of real numbers λn, representing the eigenvalues of the Dirac operator on the curved spacetime. These quantities are diffeomorphisminvariant functions of the metric and they form an infinite set of “physical observables” for general relativity. Recent work of Connes and Chamseddine su...

There is a fundamental limit on the capacity of fibre optical communication system (Shannon Limit). This limit can be potentially overcome via using Nonlinear Frequency Division Multiplexing. Dealing with noises in these systems is one of the most critical parts in implementing a practical system. In this paper, we discover and characterize the correlations among the NFT channels. It is demonst...

Here k = ω/co is the wave number, ω denotes the frequency of excitation, S2 = {x ∈ R3, |x| = 1} is the unit sphere in R3, and co is such that its real and imaginary parts are respectively <(co)> 0 and =(co)60. The support ofD is assumed to be such that R3 \D is connected, and that ∂D is of Lipschitz type. As a canonical example of the scattering by a penetrable obstacle, consider next the case ...

Linear expanders have numerous applications to theoretical computer science. Here we show that a regular bipartite graph is an expander ifandonly if the second largest eigenvalue of its adjacency matrix is well separated from the first. This result, which has an analytic analogue for Riemannian manifolds enables one to generate expanders randomly and check efficiently their expanding properties...

The eigenvalues of a symmetric matrix depend on the matrix nons-moothly. This paper describes the nonsmooth analysis of these eigen-values. In particular, I present a simple formula for the approximate (limiting Fr echet) subdiierential of an arbitrary function of the eigen-values, subsuming earlier results on convex and Clarke subgradients. As an example I compute the subdiierential of the k't...

Let G be a connected graph of order n. The diameter of a graph is the maximum distance between any two vertices of G. In this paper, we will give some bounds on the diameter of G in terms of eigenvalues of adjacency matrix and Laplacian matrix, respectively.