A new approach to the representation of trigonometric and hyperbolic functions by infinite products

An innovative technique is developed for obtaining infinite product representations for some elementary functions. The technique is based on the comparison of alternative expressions of Green's functions constructed by two different methods. Some standard boundary value problems are considered posed for two-dimensional Laplace equation on regions of a regular configuration. Classical closed analytic form of Green's functions for such problems are compared against those obtained by the method of images in the form of infinite products. This yields a number of new infinite product representations for trigonometric and hyperbolic functions.     

热弹性理论准静态问题解的一般公式及其应用

本文利用分解定理［１］求得热弹性理论的准静态问题在无限空间中的Ｇｒｅｅｎ函数组．从而，借助于文［２］中的互易定理建立了该问题解的一般公式．     

Green's Functions for Quasi-Hyperbolic Metrics on Degenerating Riemann Surfaces with a Separating Node

In this article, we consider a family of compact Riemann surfaces of genus q 2 degenerating to a Riemann surface with a separating node and many non-separating nodes. We obtain the asymptotic behavior of Green's functions associated to a continuous family of quasi-hyperbolic metrics on such degenerating Riemann surfaces.     

Sign properties of Green's functions for disconjugate dynamic equations on time scales

This paper continues the development of disconjugacy of higher order dynamic equations on time scales. Two-point conjugate type boundary value problems for general disconjugate dynamic equations on time scales are studied and the sign properties of associated Green's functions are established. As expected, the results unify known results from the theories of ordinary differential equations and finite difference equations.     

Green's function and comparison principles for first order periodic differential equations with piecewise constant arguments

In this paper we obtain the expression of the Green's function related with a first order periodic differential equation with piecewise constant argument. We derive comparison results for the treated linear operator by studying the sign of the obtained Green's function.     

Algorithmic Issues for Electromagnetic Scattering in Layered Media: Green's Functions, Current Basis, and Fast Solver

In this paper, we will address three computational issues for electromagnetic scattering in layered media: fast algorithm of calculating dyadic Green's functions, high order RWG current basis and fast solver for layered media. Numerical results will be included.     

A Variational Problem for Steady Vortices in a Shear Flow Past an Obstacle

We prove existence of maximizers for a variational problem for a steady vortex anomaly of bounded extent in a shear flow, past an obstacle, in a planar exterior domain. Kinetic energy is maximized subject to the vorticity being a rearrangement of a prescribed function.     

MULTIPLE POSITIVE SOLUTIONS OF BOUNDARY VALUE PROBLEM FOR A SYSTEM OF NONLINEAR THIRD ORDER DIFFERENTIAL EQUATIONS

This paper is concerned with the boundary value problem for a system of nonlinear third order differential equations. Under some suitable conditions, the existence and multiplicity of the positive solutions are established by using abstract fixed-point theorems.     

Multiple positive solutions of boundary value problems for systems of nonlinear second-order differential equations

This paper is concerned with boundary value problems for systems of nonlinear second-order differential equations. Under the suitable conditions, the existence and multiplicity of positive solutions are established by using abstract fixed-point theorems.     

Equivalence of Many-Photon Green's Functions in the Duffin–Kemmer–Petiau and Klein–Gordon–Fock Statistical Quantum Field Theories

Using the functional integral formalism for the statistical generating functional in the statistical (finite temperature) quantum field theory, we prove the equivalence of many-photon Green's functions in the Duffin–Kemmer–Petiau and Klein–Gordon–Fock statistical quantum field theories. As an illustration, we calculate the one-loop polarization operators in both theories and demonstrate their coincidence.     

Descending Chains and Antichains of the Unary, Linear, and Monotone Subfunction Relations

The -subfunction relations on the set of operations on a finite base set defined by function classes are examined. For certain clones on , it is determined whether the partial orders induced by the respective -subfunction relations have infinite descending chains or infinite antichains. More specifically, we investigate the subfunction relations defined by the clone of all functions on , the clones of essentially at most unary operations, the clones of linear functions on a finite field, and the clones of monotone functions with respect to the various partial orders on .     

Explicit approximation of the symmetric rearrangement by polarizations

We give an explicit sequence of polarizations such that for every measurable function, the sequence of iterated polarizations converge to the symmetric rearrangement of the initial function.