scholarly journals Line antiderivations over local fields and their applications

2005 ◽  
Vol 2005 (2) ◽  
pp. 263-309 ◽  
Author(s):  
S. V. Ludkovsky
Keyword(s):  
Laurent Series ◽  
Differential Forms ◽  
Holomorphic Functions ◽  
Integral Representations ◽  
Local Fields ◽  
Cauchy Type ◽  
Series Representations

A non-Archimedean antiderivational line analog of the Cauchy-type line integration is defined and investigated over local fields. Classes of non-Archimedean holomorphic functions are defined and studied. Residues of functions are studied; Laurent series representations are described. Moreover, non-Archimedean antiderivational analogs of integral representations of functions and differential forms such as the Cauchy-Green, Martinelli-Bochner, Leray, Koppelman, and Koppelman-Leray formulas are investigated. Applications to manifold and operator theories are studied.

scholarly journals Generalization of I.Vekua's integral representations of holomorphic functions and their application to the Riemann–Hilbert–Poincaré problem

2011 ◽  
Vol 9 (3) ◽  
pp. 217-244 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Vakhtang Paatashvili
Keyword(s):  
Holomorphic Functions ◽  
Integral Representations ◽  
Closed Domain ◽  
Cauchy Type ◽  
Cauchy Type Integral ◽  
Smooth Curves ◽  
Type Integral ◽  
Cusp Points ◽  
Poincaré Problem ◽  
The Given

I. Vekua’s integral representations of holomorphic functions, whosem-th derivative (m≥0) is Hӧlder-continuous in a closed domain bounded by the Lyapunov curve, are generalized for analytic functions whosem-th derivative is representable by a Cauchy type integral whose density is from variable exponent Lebesgue spaceLp(⋅)(Γ;ω)with power weight. An integration curve is taken from a wide class of piecewise-smooth curves admitting cusp points for certainpandω. This makes it possible to obtain analogues ofI. Vekua’s results to the Riemann–Hilbert–Poincaré problem under new general assumptions about the desired and the given elements of the problem. It is established that the solvability essentially depends on the geometry of a boundary, a weight functionω(t)and a functionp(t).

scholarly journals SIMULATION OF OSCILLATORY PROCESSES USING DIFFERENTIAL EQUATIONS OF LIOUVILLE TYPE

2018 ◽  
pp. 117-123
Author(s):  
Albina Kuandykovna Ilyasova ◽  
Yuliia Vladimirovna Bulycheva
Keyword(s):  
Mathematical Modeling ◽  
Differential Equation ◽  
Differential Equations ◽  
Explicit Form ◽  
Physical Phenomenon ◽  
Integral Representations ◽  
Explicit Solutions ◽  
Cauchy Type ◽  
Partial Derivatives ◽  
Oscillatory Processes

The problems of mathematical modeling lead to the necessity to create computational algorithms directly related to finding solutions of differential equations with partial derivatives in explicit form. In this study, explicit solutions are original tests for approximate methods that reflect the essence of the general solution. Each explicit solution of the differential equation has great importance as an accurate representation of the physical phenomenon under study within the framework of this model, as an analysis of the verification of numerical methods, as a theoretical basis for further modeling of the researched process. There have been considered aspects of the application of mathematical modeling to the study of oscillatory processes. Methods of reducing the solution of differential equations to an explicit form are proposed. Solution is given through functions of real arguments. The possible field of application is the study of wave processes. There is being considered the problem of building a variety of explicit solutions of the nonlinear third-order differential equation with partial derivatives with two boundary singular planes in space and second-order equation of general form with hyper-singular lines in the plane. On the basis of the developed method there has been proved the uniqueness of the obtained integral representations, and the boundary value problem of Cauchy type is posed and solved. The results are formulated in the form of theorems.

Tensor Products of Holomorphic Discrete Series Representations

1979 ◽  
Vol 31 (4) ◽  
pp. 836-844 ◽  
Author(s):  
Joe Repka
Keyword(s):  
Discrete Series ◽  
Tensor Products ◽  
Holomorphic Functions ◽  
Verma Modules ◽  
Holomorphic Discrete Series ◽  
Spaces Of Holomorphic Functions ◽  
Highest Weight ◽  
Generalized Verma Modules ◽  
Series Representations ◽  
Discrete Series Representations

We discuss the decomposition of tensor products of holomorphic discrete series representations, generalizing a technique used in [9] for representations of SL2(R), based on a suggestion of Roger Howe. In the case of two representations with highest weights, the discussion is entirely algebraic, and is best formulated in the context of generalized Verma modules (see § 3). In the case when one representation has a highest weight and the other a lowest weight, the approach is more analytic, relying on the realization of these representations on certain spaces of holomorphic functions.For a simple group, these two cases exhaust the possibilities; for a nonsimple group, one has to piece together representations on the various factors.The author wishes to thank Roger Howe and Jim Lepowsky for very helpful conversations, and Nolan Wallach for pointing out the work of Eugene Gutkin (Thesis, Brandeis University, 1978), from which some of the results of this paper can be read off as easy corollaries.

SCHRÖDINGER OPERATORS ON LOCAL FIELDS: SELF-ADJOINTNESS AND PATH INTEGRAL REPRESENTATIONS FOR PROPAGATORS

2008 ◽  
Vol 11 (04) ◽  
pp. 495-512 ◽  
Author(s):  
TROND DIGERNES ◽  
V. S. VARADARAJAN ◽  
D. E. WEISBART
Keyword(s):  
Path Integral ◽  
Adjoint Operator ◽  
Hamiltonian Operator ◽  
Internal Symmetry ◽  
Integral Representations ◽  
Local Fields ◽  
Path Integral Representation ◽  
Finite Dimensional ◽  
Space Of Paths ◽  
Complex Coefficients

We consider quantum systems that have as their configuration spaces finite-dimensional vector spaces over local fields. The quantum Hilbert space is taken to be a space with complex coefficients and we include in our model particles with internal symmetry. The Hamiltonian operator is a pseudo-differential operator that is initially only formally defined. For a wide class of potentials we prove that this Hamiltonian is well-defined as an unbounded self-adjoint operator. The free part of the operator gives rise to a measure on the Skorokhod space of paths, D [0, ∞), and with respect to this measure there is a path integral representation for the semigroup associated to the Hamiltonian. We prove this Feynman–Kac formula in the local field setting as a consequence of the Hille–Yosida theory of semigroups.

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