Let [WTHX]A[WTBZ]=([WTBX]aij)∈Cn×n, if there exists α∈(0,1) which can make |aii|≥Rαi(A)S1-αi(A)  be right for i∈N={1,2,…,n}, then [WTHX]A[WTBZ] is called an Ostrowski diagonally dominant matrix. We extended the concept to generalized Ostrowski diagonally dominant matrix，and obtained a new criteria conditions for a matrix to be a nonsingular [WTHX]H[WTBZ]-matrix. The theory of Ostrowski diagonally dominant matrix and nonsingular [WTHX]H[WTBZ]-matrix was improved and completed. These conclusions provide strong basis for the research of relative fields, such as computational mathematics, matrix theory, control theory, mathematical economics, etc.

In order to analyze the energy loss of a beam at crystal surfaces, a method was proposed to calculate the corresponding reflectivity and transmissivity. According to the electromagnetic boundary conditions and the phase matching conditions, reflection and refraction of a beam incident from a crystal upon an isotropic medium were studied, and the Fresnel formulas at the interface between the crystal and the isotropic medium were deduced. As the orientation of optical axis is arbitrary, the polarization states and the amplitudes of the o ray and e ray and the refracted ray were given, and the theoretical expressions of the reflectivity and the transmissivity representing energy relations between the rays were obtained, which provide a theoretical basis for developing and applying crystal devices. Numerical simulations show that the results are accordant with conservation of energy; electrical field or magnetic field of the ray refracted to the isotropic medium does not parallel to the electric field or the magnetic field of original incident ray; the direction of optical axis and the magnitude of incident angle have a great effect on the energy and the polarization direction of the refracted ray.

Let G be a simple graph with n vertices, the new upper bound on the sum of the Q spectral radius of a graph and its complement were given by its m edges, minimal degree of a vertex δ, maximal degree of a vertex Δ and chromatic number k． When graph G has no isolated vertex ,we have 2(n-1)≤ρ(Q(G))+ρ(Q([AKG-]))≤2(Δ-δ+n-1) and ρ(Q(G))+ρQ([AKG-]))≤2n-3+[KF(][JB((]2-[SX(]1[]2[SX)][JB))](n-1)n[KF)]，where t=min{k,[AKk-]}. When graph [AKG-] has l isolated vertices ,we have ρ(Q(G))+ρ(Q([AKG-]))≤2n-3+[KF(][JB((]2-[SX(]1[]k[SX)][JB))](n-1)2+l[KF)]. At the same time the upper bound on the sum of the Laplace spectral radius of a graph and its complement was given.