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 R ₀-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.     

Invariant factors,Wiener–Hopf factorization indices,and invariant factors at infinity of rational matrices

The relationship between the finite structure, the infinite structure, and the Wiener–Hopf factorization indices of any rectangular rational matrix is studied.     

Identifying a spherically symmetric conductivity in a nonlinear parabolic equation

In this article we determine, i.e. we prove existence and uniqueness of a radially symmetric operator conductivity in a nonlinear heat equation of the form (1.1) related to a spherical corona Ω ? R ³ under a suitable additional information.     

Perturbation of Volterra Difference Equations

A fixed point theorem is used to investigate nonlinear Volterra difference equations that are perturbed versions of linear equations. Sufficient conditions are established to ensure that the stability properties of linear Volterra difference equations are preserved under perturbation. The existence of asymptotically periodic solutions of perturbed Volterra difference equations is also proved.     

The Synthesis and Spectroscopic Characterization of Heterobimetallic Bu2Sn(IV) Complexes Derived from Some Chelating Ligands

The interaction of Bu₂Sn(OPrⁱ)₂ with a trifunctional tetradentate Schiff base (LH₃) (where H₃L = HOC₆H₄CH═NCH₃C(CH₂OH)₂) yields the precursor complex Bu₂Sn(LH) 1, which, on equimolar reactions with different metal alkoxides [Al(OPrⁱ)₃, Bu₃Sn(OPrⁱ), Ge(OEt)₄]; Al(Medea)(OPrⁱ) (where Medea = CH₃N- (CH₂CH₂O)₂); and Me₃SiCl in the presence of Et₃N], affords, respectively, the complexes Bu₂Sn(L)Al(OPrⁱ)₂ 2, Bu₂Sn(L)Al(Medea) 3, Bu₂Sn(L)Bu₃Sn 4, Bu₂Sn(L)Ge(OEt)₃ 5, and Bu₂Sn(L)SiMe₃ 6. The reactions of 2 with 2,5-dimethyl-2,5-hexanediol in a 1:1 ratio and with acetylacetone (acacH) in a 1:2 molar ratio afforded derivatives Bu₂Sn(L)Al(OC(CH₃)₂CH₂CH₂C(CH₃)₂ O) 7 and Bu₂Sn(L)Al(acac)₂ 8, respectively. All of the derivatives 1– 8 have been characterized by elemental analyses, molecular weight measurements, and spectroscopic [IR and NMR (¹H, ¹¹⁹Sn, ²⁹Si, and ²⁷Al)] studies.