This paper is concerned with spectral problems of second-order vector difference equations with two-point boundary value conditions, where the matrix-valued coefficient of the leading term may be singular. A concept of self-adjointness of the boundary value conditions is introduced. The self-adjointness of the corresponding difference operator is discussed on a suitable admissible function space, and fundamental spectral results are obtained. The dual orthogonality of eigenfunctions is shown in a special case. Rayleigh's principles and the minimax theorems in two linear spaces are given. As an application, a comparison theorem for eigenvalues of two Sturm᎐Liouville problems is presented. ᮊ 1999 Academic Press

A novel crosslinked conductive polyaniline (PANI) was prepared by chemically copolymerizing aniline (An) and p-phenylenediamine (PPDA) with triphenylamine (TPA) as crosslinker, using ammonium peroxydisulfate (APS) as an oxidant. The effects of different preparation conditions on the electrical conductivity of polymers were systematically investigated by adjusting acid kinds, concentration, the ratio of APS/An, the mounts of TPA and PPDA. The crosslinked PANI displayed a conductivity increase of up to 25% compared with the linear one. Their structures were characterized by Fourier-transformed infrared spectroscopy and X-ray photoelectron spectroscopy, and the electrical conductivity was also tested by a typical four-point probe (RTS-8) technique.

This article studies the eigenvalue problem of a fractional differential equation which is a foundation model of a bar of finite length with long-range interactions arising from non-local continuum mechanics. We show that this problem has countable simple real eigenvalues and the corresponding eigenfunctions form a complete orthogonal system in the Hilbert space L2. Furthermore, the asymptotic behavior of eigenvalues and the numbers of zeros of eigenfunctions are studied by using the analytic perturbation theory.

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