we consider a mixed boundary value problem for a quarter-plane with a robin condition on one edge. we have developed two procedures, one based on the advanced theory of dual integral equations and the other, in our opinion simpler technique, relying on conformal mapping. both of the procedures are of interest, because the former may be easier to adapt to other boundary value problems.

There is no doubt that the molecular imaging is an extremely important technique in diagnosing diseases. Dual imaging is emerging as a step forward in molecular imaging technique because it can provide us with more information useful for diagnosing diseases than single imaging. Therefore, diverse dual imaging modalities should be developed. Molecular imaging generally relies on imaging agents. ...

— We present a constructive proof of Alexandrov’s theorem on the existence of a convex polytope with a given metric on the boundary. The polytope is obtained by deforming certain generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a connection with weighted Delaunay triangulations of polyhedral surfaces. The existence of the defo...

1. Volumes of (2n − 1)-dimensional hyperbolic manifolds and the Borel regulator on K2n−1(Q). Let M be an n-dimensional hyperbolic manifold with finite volume vol(M). If n = 2m is an even number, then by the Gauss-Bonnet theorem ([Ch]) vol(M) = −c2m · χ(M) where c2m = 1/2×(volume of sphere S of radius 1) and χ(M) is the Euler characteristic of M. This is straightforward for compact manifolds and...

We deal with the iterative solution of linear systems arising from so-called dual-dual mixed finite element formulations. The linear systems are of a two-fold saddle point structure; they are indefinite and ill-conditioned. We define a special inner product that makes matrices of the two-fold saddle point structure, after a specific transformation, symmetric and positive definite. Therefore, th...

Two mixed symmetric dual models for a class of non-differentiable multiobjective nonlinear programming problems with multiple arguments are introduced in this paper. These two mixed symmetric dual models unify the four existing multiobjective symmetric dual models in the literature. Weak and strong duality theorems are established for these models under some mild assumptions of generalized conv...

Mixed volumes, which are the polarization of volume with respect to the Minkowski addition, are fundamental objects in convexity. In this note we announce the construction of mixed integrals, which are functional analogs of mixed volumes. We build a natural addition operation ⊕ on the class of quasi-concave functions, such that every class of α-concave functions is closed under ⊕. We then defin...

This thesis consists of two parts. In the first part, we start by investigating the classical permutohedra as Minkowski sums of the hypersimplices. Their volumes can be expressed as polynomials whose coefficients the mixed Eulerian numbers are given by the mixed volumes of the hypersimplices. We build upon results of Postnikov and derive various recursive and combinatorial formulas for the mixe...

A pair of Mond-Weir type second order mixed symmetric duals is presented for a class of non-differentiable multi-objective nonlinear programming problems with multiple arguments. We establish duality theorems for the new pair of dual models under second order generalized convexity assumptions. This mixed second order dual formulation unifies the two existing second order symmetric dual formulat...