Kernel Of Integral Representation Research Articles

Modelling of rectangular Kirchhoff plate oscillations under unsteady elastodiffusive perturbations

We study unsteady elastic-diffusion vibrations of a rectangular isotropic Kirchhoff plate. The coupled elastic-diffusion multicomponent continuum model is used to formulate the problem. We are using the d’Alembert variational principle to obtain from the model the equations of longitudinal and transverse vibrations of a rectangular isotropic elastodiffusive Kirchhoff plate. The problem solution is sought in integral form. The kernels of integral representations are the Green’s functions. For their determination, the Laplace transform and expansion in double Fourier series are used. As an example, we consider the bending of a simply supported plate. The effect of bending on the diffusion processes in the plate is investigated.

Algorithm for construction of volume forms on toric varieties starting from a convex integer polytope

This paper presents a method and a corresponding algorithm for constructing volume forms (and related forms that act as kernels of integral representations) on toric varieties from a convex integer polytope. The algorithm is implemented in the Maple computer algebra system. The constructed volume forms are similar to the volume forms of the Fubini---Study metric on a complex projective space and can be used for constructing integral representations of holomorphic functions in polycircular regions of a multidimensional complex space.

Factorization of integro-differential equations of optics of dispersive anisotropic media and tensor integral operators of wave packet velocities

The integro-differential one-dimensional tensor wave equations of the electrodynamics of dispersive anisotropic media are factorized. The resulting first-order partial differential equations contain tensor integral functionals that describe light velocities of polarized plane-wave pulses. The velocity operators are written in a compact coordinate-free form. They are defined by kernels of integral representations and provide a general description of the kinematics and dynamics of groups of electromagnetic waves for arbitrary propagation directions and arbitrary initial polarization states in tensor media exhibiting frequency and rotatory dispersion. Some branches of the solutions of wave equations and their spectral representations are analyzed and the relations of these equations to the general systems of Whitham’s wave equations are found.

Influence functions for 2-D compound regions of complex configuration

This paper presents a detailed description of the algorithm for construction of influence (Green's) functions for a two-dimensional equation of potential over regions compound in nature. Several particular formulations are considered in the Cartesian, polar, and geographical coordinate systems. This paper also deals with an application of the Green's function method to the development of influence functions for compound multi-connected regions of complex configuration. Influence functions for appropriate simply connected regions, which have been constructed in advance, are utilized as the kernels of integral representations in the development.

Generally Covariant Integral Formulation of Einstein's Field Equations

As a basis for a generally covariant theory of Mach's principle, we express Einstein's field equations in integral form. The nonlinearity of these equations is reflected in the kernel of the integral representation, which is a functional of the metric tensor. The functional dependence is so constructed that, subject to supplementary conditions, the kernel may be regarded as remaining unchanged to the first order when a small change in the source produces a corresponding change in the potential. To obtain this kernel, a linear differential operator is derived by varying a particular form of Einstein's field equations. The elementary solution corresponding to this linear operator provides the kernel of an approximate integral representation which becomes exact in the limit of vanishing variations. This representation is in a certain sense unique. Our discussion is confined to a normal neighborhood of the field point.