Cochran's statistical theorem for outer inverses of matrices and matrix quadratic forms

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.     

Some geometrical aspects of semidefinite linear complementarity problems

This article studies some geometrical aspects of the semidefinite linear complementarity problem (SDLCP), which can be viewed as a generalization of the well-known linear complementarity problem (LCP). SDLCP is a special case of a complementarity problem over a closed convex cone, where the cone considered is the closed convex cone of positive semidefinite matrices. It arises naturally in the unified formulation of a pair of primal-dual semidefinite programming problems. In this article, we introduce the notion of complementary cones in the semidefinite setting using the faces of the cone of positive semidefinite matrices and show that unlike complementary cones induced by an LCP, semidefinite complementary cones need not be closed. However, under R₀-property of the linear transformation, closedness of all the semidefinite complementary cones induced by L is ensured. We also introduce the notion of a principal subtransformation with respect to a face of the cone of positive semidefinite matrices and show that for a self-adjoint linear transformation, strict copositivity is equivalent to strict semimonotonicity of each principal subtransformation. Besides the above, various other solution properties of SDLCP will be interpreted and studied geometrically.     

局域模、简正模的李代数表示

本文介绍李代数的非简谐振子的表示。运用这样的表示可以方便地讨论局域模与简正模之关系及其半经典图象。     

Analysis of glucopyranoside trinitrate IR spectrum

The IR spectrum of 4-O-methyl-2,3,6-tri-O-nitromethyl-β-D-glucopyranoside has been interpreted in detail for the first time based on complete calculation of the normal vibration frequencies and absolute intensities of its IR absorption bands and a comparison of them with the corresponding experimental data. The results indicate that the location of the nitrates in cellulose nitrates can be determined experimentally from separate components of complicated absorption bands in the ranges 1700–1600 and 900–800 cm⁻¹ of the IR spectrum. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 75, No. 1, pp. 43–51, January–February, 2008.     

TWO POSITIVE SOLUTIONS FOR A CLASS OF SINGULAR BOUNDARY VALUE PROBLEMS

An existence theorem of two positive solutions of the singular BVP1/(p(t))(p(t)y′(t))′+λα(t)f(y(t))=0,t∈(0,1),αy(0)-βp(t)y′(t)=0=γy(1)+δp(t)y′(t)was established by using topological degree theory.     

Calculus of sequential normal compactness in variational analysis

In this paper we study some properties of sets, set-valued mappings, and extended-real-valued functions unified under the name of “sequential normal compactness.” These properties automatically hold in finite-dimensional spaces, while they play a major role in infinite-dimensional variational analysis. In particular, they are essential for calculus rules involving generalized differential constructions, for stability and metric regularity results and their broad applications, for necessary optimality conditions in constrained optimization and optimal control, etc. This paper contains principal results ensuring the preservation of sequential normal compactness properties under various operations over sets, set-valued mappings, and functions.     

Divisorial Linear Algebra of Normal Semigroup Rings

We investigate the minimal number of generators and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound C there exist, up to isomorphism, only finitely many divisorial ideals I such that (I)C. It follows that there exist only finitely many Cohen–Macaulay divisor classes. Moreover, we determine the minimal depth of all divisorial ideals and the behaviour of and depth in arithmetic progressions in the divisor class group.The results are generalized to more general systems of linear inequalities whose homogeneous versions define the semigroup in a not necessarily irredundant way. The ideals arising this way can also be considered as defined by the nonnegative solutions of an inhomogeneous system of linear diophantine equations.We also give a more ring-theoretic approach to the theorem on minimal number of generators of divisorial ideals: it turns out to be a special instance of a theorem on the growth of multigraded Hilbert functions.     

Lattice dynamics of some binary alloys

Summary The phonon dispersion frequencies for Cu₃Au, Cu₃Zn and Ni₃Fe alloys have been calculated using the model potential approach. The contribution of a short-range three-body interaction to the dynamical matrix has also been included in the present study. The theoretically computed phonon dispersion curves of these alloys are found to be in good agreement with the experimental data.