‎This paper deals with the singular Sturm-Liouville expressions‎ ‎$ ‎ell y =‎ -‎y''+q(x)y=lambda y‎ ‎$‎ ‎on a finite interval‎, ‎where the potential function $q$ is real and‎ ‎has a singularity inside the interval‎. ‎Using the asymptotic estimates of a‎ ‎spectral fundamental system of solutions of Sturm-Liouville‎ ‎equation‎, ‎the asymptotic form of the solution of the‎ ‎equation (0.1) and the ...

Whitham theory of modulations is developed for periodic waves described by nonlinear wave equations integrable by the inverse scattering transform method associated with 2 × 2 matrix or second order scalar spectral problems. The theory is illustrated by derivation of the Whitham equations for perturbed Korteweg-de Vries equation and nonlinear Schrödinger equation with linear damping.

An M ∨-matrix has the form A = sI − B, where s ≥ ρ(B) ≥ 0 and B is eventually nonnegative; i.e., B k is entrywise nonnegative for all sufficiently large integers k. A theory of M ∨-matrices is developed here that parallels the theory of M-matrices, in particular as it regards exponential nonnegativity, spectral properties, semipositivity, monotonicity, inverse nonnegativity and diagonal dominance.

An M∨ matrix has the form A = sI − B, where s ≥ ρ(B) ≥ 0 and B is eventually nonnegative; i.e., Bk is entrywise nonnegative for all sufficiently large integers k. A theory of M∨ matrices is developed here that parallels the theory of M-matrices, in particular as it regards exponential nonnegativity, spectral properties, semipositivity, monotonicity, inverse nonnegativity and diagonal dominance.

Abstract. We introduce a spectral transform for the finite relativistice Toda lattice (RTL) in generalized form. In the nonrelativistic case, Moser constructed a spectral transform from the spectral theory of symmetric Jacobi matrices. Here we use a nonsymmetric generalized eigenvalue problem for a pair of bidiagonal matrices (L, M) to define the spectral transform for the RTL. The inverse spec...

Parallel to the signless Laplacian spectral theory, we introduce and develop the nonlinear spectral theory of signless 1-Laplacian on graphs. Again, the first eigenvalue μ1 of the signless 1-Laplacian precisely characterizes the bipartiteness of a graph and naturally connects to the maxcut problem. However, the dual Cheeger constant h+, which has only some upper and lower bounds in the Laplacia...