The nondegenerate Nevanlinna-Pick-Carathéodory-Fejer interpolation problem with finitely many interpolation conditions always has infinitely many solutions in a generalized Schur class Sκ for every κ ≥ κmin where the integer κmin equals the number of negative eigenvalues of the Pick matrix associated to the problem and completely determined by interpolation data. A linear fractional description...

The purpose of this note is to recall the theory of the (homogenized) spectral problem for a Schrödinger equation with a polynomial potential developed in the 60’s by M. Evgrafov with M. Fedoryuk, and, by Y. Sibuya and its relation with quadratic differentials. We derive from these results that the accumulation rays of the eigenvalues of this problem are in 1−1-correspondence with the short geo...

The bi-Cayley graph of a finite group G with respect to a subset S ⊆ G, which is denoted by BCay(G,S), is the graph with vertex set G× {1, 2} and edge set {{(x, 1), (sx, 2)} | x ∈ G, s ∈ S}. A finite group G is called a bi-Cayley integral group if for any subset S of G, BCay(G,S) is a graph with integer eigenvalues. In this paper we prove that a finite group G is a bi-Cayley integral group if a...

The analog of the principal SO(3) subalgebra of a finite dimensional simple Lie algebra can be defined for any hyperbolic Kac Moody algebra g(A) associated with a symmetrizable Cartan matrix A, and coincides with the non-compact algebra SO(1, 2). We exhibit the decomposition of g(A) into representations of SO(1, 2); with the exception of the adjoint SO(1, 2) algebra itself, all of these represe...

An integral tree is a tree whose adjacency matrix has only integer eigenvalues. While most previous work by other authors has been focused either on the very restricted case of balanced trees or on finding trees with diameter as large as possible, we study integral trees of diameter 4. In particular, we characterize all diameter 4 integral trees of the form T (m1, t1) • T (m2, t2). In addition ...

0. Introduction Let ∆ be the Laplacian on R, l > 0 an integer and V ≥ 0 a measurable function (“weight-function”). Consider the eigenvalue problem (0.1) λ(−∆)u = V u. The following result was proved by Rosenblum [Roz]: Theorem 0.1. Let 2l < d and V ∈ L d 2l (R). Then the non-zero spectrum of the problem (0.1) consists of positive eigenvalues λk (counted according to their multiplicities), and f...

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. Zhang et al (Discrete Appl. Math., 92(1999), 71-84) characterized the trees with a perfect matching having the minimal and the second minimal energies, which solved a conjecture proposed by Gutman (J. Math. Chem., 1(1987), 123-143). In this letter, for a given positive integer d we characterize t...

The purpose of this short paper is to recall the theory of the (homogenized) spectral problem for a Schrödinger equation with a polynomial potential developed in the 60’s by M. Evgrafov with M. Fedoryuk, and, by Y. Sibuya and its relation with quadratic differentials. We derive from these results that the accumulation rays of the eigenvalues of this problem are in 1 − 1-correspondence with the ...

Given a positive definite matrix M and an integer Nm ≥ 1, Oja’s subspace algorithm will provide convergent estimates of the first Nm eigenvalues of M along with the corresponding eigenvectors. It is a common approach to principal component analysis. This paper introduces a normalized stochastic-approximation implementation of Oja’s subspace algorithm, as well as new applications to the spectral...

In this paper, we consider the application of the homotopy perturbation method (HPM) to compute the eigenvalues of the Sturm-Liouville problem (SLP) which is called non-definite SLP. Two important Examples show that HPM is reliable method for computing the eigenvalues of SLP.