The Sturm sequence computation is used by the bisection method to compute eigenvalues of real symmetric tridiagonal matrices. Let Tn be a symmetric tridiagonal matrix with the diagonal elements α1, α2, . . . , αn and the off-diagonal elements β1, β2, . . . , βn−1. Given a number λ, the sequence of characteristic polynomials pj(λ) for the leading j × j principal submatrices of Tn can be computed...

The equation of motion a microparticle in various force fields is the Schr¨odinger wave equation.Many questions quantum mechanics, particular thermal radiation electromagneticwaves, lead to problem separability singular differential operators. One such operator isthe above Schr?dinger operator. In this paper, named studied by methods offunctional analysis. Found sufficient conditions for existe...

We study finite difference approximations of solutions of direct and inverse Sturm–Liouville problems, in a finite or infinite interval on the real line. The discretization is done on optimal grids, with a three-point finite difference stencil. The optimal location of the grid points is calculated via a rational approximation of the Neumann-to-Dirichletmap and the latter converges exponentially...

In the paper we prove that any nonconvex quadratic problem over some set K ⊆Rn with additional linear and binary constraints can be rewritten as linear problem over the cone, dual to the cone of K-semidefinite matrices. We show that when K is defined by one quadratic constraint or by one concave quadratic constraint and one linear inequality, then the resulting K-semidefinite problem is actuall...

The aim of this article is to give Sturm-Picone type theorems for the pair of second order nonlinear elliptic differential equations div(p1(x)|∇u|∇u) + q1(x)f1(u) + r1(x)g1(u) = 0, div(p2(x)|∇v|∇v) + q2(x)f2(v) + r2(x)g2(v) = 0, where | · | denotes the Euclidean length and ∇ = ( ∂ ∂x1 , . . . , ∂ ∂xn )T (the superscript T denotes the transpose). Our results include some earlier results and gene...