Convex analysis” is developed for functions defined on integer lattice points. We investigate the class of functions which enjoy a variant of Steinitz’s exchange property. It includes linear functions on matroids, valuations on matroids (in the sense of Dress and Wenzel), and separable concave functions on the integral base polytope of submodular systems. It is shown that a function ω has the S...

The reciprocity theorem is a fundamental theorem of the Theory of Elasticity (Betti 1872, Rayleigh 1873). The theorem connects the body forces, the surface tractions and the displacements of two elastodynamic states in a domain of an elastic body, by a volume integral and a surface integral. In this talk it is shown that the reciprocity theorem of elastodynamics can be used to solve actual prob...

As a generalization of the famous four square theorem of Lagrange, Ramanujan and Willerding found all positive definite integral quaternary quadratic forms that represent all positive integers. In this paper, we find all positive definite integral quinary quadratic forms that represent all positive definite integral binary quadratic forms. We also discuss recent results on positive definite int...

We give an integral-geometric proof of the Gauss-Bonnet theorem for hypersurfaces in constant curvature spaces. As a tool, we obtain variation formulas in integral geometry with interest in its own.

We introduce the notion of free polygons as combinatorial building blocks for convex integral polygons; that is, polygons with vertices having integer coordinates. In this context, an Euler-type formula is derived for the number of integer points in the interior of an integral polygon. This leads in turn to a formula for the area of an integral polygon P via the enumeration of free integral tri...

We present a theorem concerning the representation of solutions of a first-order partial differential equation in terms of a complete integral of the equation. We discuss the geometrical significance of that theorem.

We present an existence theorem for integral equations of UrysohnVolterra type involving fuzzy set valued mappings. A fixed point theorem due to Schauder is the main tool in our analysis.

Krein-de Branges spectral theory establishes a correspondence between the class of differential operators called canonical Hamiltonian systems and measures on real line with finite Poisson integral. We further develop this area by giving description whose have logarithmic integral converging over line. This result can be viewed as version classical Szego theorem in polynomials orthogonal unit c...

In the present paper, we establish a fixed point theorem for a mapping and a common fixed point theorem for a pair of mappings. The mapping involved here generalizes various type of contractive mappings in integral setting.

We present a very accurate formula counting norms of normal integral bases in tame abelian extensions of the rational eld. The methods used include applications of Schmidt's Subspace Theorem, Baker's Theorem and the Hardy-Littlewood Method, all from diophantine approximation.