Generalized Functions Research Articles

Mollified Monte Carlo

Using a common technique [ 1, 2] for approximating distributions [generalized functions], we are able to use standard Monte Carlo methods to compute QFT quantities in Minkowski spacetime, under phase transitions, or when dealing with coalescing stationary points.

On the inversion of the convolution and Laplace transform

We present a new inversion formula for the classical, finite, and asymptotic Laplace transform f of continuous or generalized functions f. The inversion is given as a limit of a sequence of finite linear combinations of exponential functions whose construction requires only the values of f evaluated on a Muntz set of real numbers. The inversion sequence converges in the strongest possible sense. The limit is uniform if f is continuous, it is in L 1 if f E L 1 , and converges in an appropriate norm or Frechet topology for generalized functions f. As a corollary we obtain a new constructive inversion procedure for the convolution transform K: f → k * f; i.e., for given g and k we construct a sequence of continuous functions f n such that k *f n → g.

SYMMETRY GROUP ANALYSIS OF WEAK SOLUTIONS

Methods of Lie group analysis of differential equations are extended to weak solutions of (linear and non-linear) partial differential equations, where the term `weak solution' comprises the following settings: distributional solutions; solutions in generalized function algebras; solutions in the sense of association (corresponding to a number of weak or integral solution concepts in classical analysis). Factorization properties and infinitesimal criteria that allow the treatment of all three settings simultaneously are developed, thereby unifying and extending previous work in this area.2000 Mathematical Subject Classification: 46F30, 22E70, 35Dxx, 35A30.

δ -function converging sequences

We discuss the usefulness and physical interpretation of a simple and general way of constructing sequences of functions that converge to the Dirac delta function. The main result, which seems to have been largely overlooked, includes most of the δ-function converging sequences found in textbooks, is easily extended, and can be used to introduce many useful generalized functions to physics students with little mathematical background. We show that some interesting delta-function identities are simple consequences of the one discussed here. An illustrative example in electrodynamics is also analyzed, with the surprising result that the formalism allows as a limit an uncharged massless particle which creates no electromagnetic field, but has a nonzero electromagnetic energy–momentum tensor.

APPLICATION OF GENERALIZED SAMPLED-DATA HOLD FUNCTIONS TO DECENTRALIZED CONTROL STRUCTURE MODIFICATION

This paper investigates the decentralized control of LTI continuous-time plants using Generalized Sampled-data Hold Functions (GSHF). GSHFs can be used to modify the structure of the digraph of the resultant discrete plant, by removing certain interconnections in the equivalent discrete-time model to form a hierarchical system model of the plant. This is a new application of discretization, and has as its motivation, the design of decentralized controllers using centralized methods.

The effects of presenting functionally abstracted information in fault diagnosis tasks

Abstract With respect to the design of visual information display (VID) for process control, this study experimentally evaluated the effectiveness of functionally abstracted information in the task of fault diagnosis. The benefits of functional properties of work domain have been emphasized by ecological interface design (EID), a relatively new design framework for human–machine interfaces. According to the concept of EID, multilevel information representation based on abstraction hierarchy of work domain is expected to be advantageous for supporting the operator's problem solving. To investigate the advantage of EID application, an experiment was conducted using a computer-based simulation of the secondary cooling system of nuclear power plants. Three interfaces were compared: the first representing only the physical properties of the process, the second representing purpose-related generalized functions (GFs) in addition to the physical properties, and the third representing abstract functions (AFs) governing the GFs in addition to the physical properties. The results showed that the diagnostic performance was improved by displaying functionally abstracted information at both levels, and that the usefulness of the abstract information was dependent on the complexity of the diagnosis problems.

The Obrechkoff Transform on Spaces of Generalized Functions

The Obrechkoff Transform on Spaces of Generalized Functions

Solvability of Kolmogorov‐Fokker‐Planck equations for vector jump processes and occupation time on hypersurfaces

We study occupation time on hypersurface for Markov n‐dimensional jump processes. Solvability and uniqueness of integro‐differential Kolmogorov‐Fokker‐Planck with generalized functions in coefficients are investigated. Then these results are used to show that the occupation time on hypersurfaces does exist for the jump processes as a limit in variance for a wide class of piecewise smooth hypersurfaces, including some fractal type and moving surfaces. An analog of the Meyer‐Tanaka formula is presented.

Generalized Solutions of a Periodic Goursat Problem

Under reasonable assumptions on the data u , v and the function f , we show that the nonlinear periodic Goursat problem which cannot be posed in the general theory of distributions, may be studied and solved in a differential algebra of periodic new generalized functions on . This algebra contains, in a canonical way, the space of periodic distributions on as a linear subspace. When the data are Dirac measures, a particular study is done, showing the nature of the singularities involved in this case.

Rotation Inveriant Generalized Functioins*

We characterize the rotation invariant generalized functions including distributioins, ultradstributions and hyperfunctions. First, for a rotatioin invariant ultradistribution or hyperfunction u with supporty at the origin we find an expression for entire function F(ζ) in C satisfying secondly, we characterize a rotation invariant generalized function by its spherical average

Integration and Microlocal Analysis in Colombeau Algebras of Generalized Functions

We study integration and Fourier transform in the Colombeau algebra Gτ of tempered generalized functions using a general damping factor. This unifies different settings described earlier by J. F. Colombeau, M. Nedeljikov, S. Pilipović, and M. Damsma (for a simplified version). Further we prove characterizations of regularity for generalized functions in two situations: compactly supported or in the image of S′ inside Gτ. Finally we investigate the notion of wave front set in the Colombeau algebra G(Ω), Ω an open subset of Rn, and show that it is in fact independent of the damping measure used for Fourier transform.

Multiplication of Schwartz Distributions and Colombeau Generalized Functions

The differential ℂ-algebra of generalized functions of J.- F. Colombeau contains the space of Schwartz distributions as a ℂ-vector subspace and the notion of ‘association’ in that algebra generalizes the equality in . This is particularly useful for evaluation of distribution products, as they are embedded in , in terms of distributions again. The paper is devoted to results on particular products of distributions with coinciding singularities, as well as to a general property of Colombeau product of distributions together with some applications. All formulas obtained, when restricted to dimension one, are easily transformed into the setting of regularized model products in classical distribution theory.

Colombeau generalized functions: A theory based on harmonic regularizations

Colombeau generalized functions: A theory based on harmonic regularizations

An embedding of Schwartz distributions in the algebra of asymptotic functions

We present a solution of the problem of multiplication of Schwartz distributions by embedding the space of distributions into a differential algebra of generalized functions, called in the paper “asymptotic function,” similar to but different from J. F Colombeau′s algebras of new generalized functions.

On the Generalization of Vladimirov and Wightman Generalized Functions

In this article we generalize to Rn the Vladimirov generalized functions and the Wightman‐type generalized functions. Our results are the formulae (I,1;1), (I,1;3), (I,1;4), (I,1;5), (I,2;1), (II,1;2), (II,1;3), (II,1;4), and (II,1;6). Here, we also introduce an n‐ dimensional multiplicative products of the Wightman‐type generalized functions (cf. formula (II,2;4)).

Semiclassical-limit molecular dynamics on multiple electronic surfaces

In this paper, we present a new approach to treating many-body molecular dynamics on coupled electronic surfaces. The method is based on a semiclassical limit of the quantum Liouville equation. The formal result is a set of coupled classical-like partial differential equations for generalized distribution functions which describe both the nuclear probability densities on the coupled surfaces and the coherences between the electronic states. The Hamiltonian dynamics underlying the evolution of these distributions is augmented by nonclassical source and sink terms, which allow the flow of probability between the coupled surfaces and the corresponding formation and decay of electronic coherences. The formal results are shown analytically to reproduce the well-known Rabi and Landau–Zener results in appropriate limits. In addition, a direct numerical solution of the phase space partial differential equations is performed, and the results compared with exact quantum solutions for a model curve-crossing problem, yielding excellent agreement. Future trajectory-based implementation of the method in molecular dynamics simulations is also discussed.

An integral transform of generalized functions. II

An integral transform of generalized functions. II

Division problem and partial differential equations with constant coefficients in Colombeau's space of new generalized functions

We study the problem of division in Colombeau's space of new generalized functions in the associated sense: more precisely we solve the equation of the formF·G≈pH with adequate assumptions onF. By using the Fourier transformation we construct, by simple methods, the solution in the p-associated sense of a partial differential equation with constant coefficients.

Trivial solutions for a non-lineartwo-space dimensional wave equation perturbed by space-time white noise

Linear and nonlinear stochastic wave equations given by a space-time Gaussian white noise are considered in a space of dimension d≥2. In the linear case the solution is a random Schwartz distribution. In the nonlinear case existence and uniqueness of solutions is proven in the framework of Colombeau random generalized functions. In the case where the nonlinear drift is given by the Fourier transform ƒof a complex measure for instance when ƒ is a trigonometric function the solution is proven to be Lp-associated with the solution of the free equation, for any p ≥ 1

Cubature Formulas and Modern Analysis: An Introduction.

BASIC CONCEPTS AND FORMULATIONS - Concepts in Linear Algebra - Point Latices - Some Functional Spaces - Generalized Derivatives and Pdm@ve Functions - Densi@ of Fin4e Fundons - P@ncipal Embedding Theorems for We@hted Spaces - Fundons of a Discrete kgument - Generalized Functions - Fou@er Transformabon of Generalized Functions - Periodic Functions and Generalized Functions - Generalized Spherical Harmonics - POLYHARMONIC EQUATION - Green's Formula - Fundamental Solution of a Polyharmonic Equation - Differential Prope@ies of Solutons of a Polyharmonic Equation - Behavior of Polyharmonic Functions in the Neighborhood of Infinity - Almansi's Theorem and Kelvin Transformation - P@ncipal Solutions of a Polyharmonic Equation - Expansion of Polyharmonic Functions in Principal Solutions - Polyharmonic Functons from W,@ in the Neighborhood of Intini@ - Boundary Value Problems for a Polyharmonic Equation in a Bounded Domain - Kelvin of bodies adrature s revised of these s topics Problem in An Infinite Domain - Exterior Variational Problem for a Polyharmonic Equation - Extension of a Funct ion from the Region i2 on R' with the Least Norm - Values of Functions from Wpm) at the Lattice Points Explicit Method of Regularization of Divergent Integrals - TWO SIMPLE PROBLEMS OF THE THEORY OF COMPUTATIONS - Interpolation Construction oi Cubature Formulas - Functional Formulation of the Problems, Extremal Function of a Cubature Formula - Error Functional in W'2 n, (RI Square of the Norm of the Error Functional - Deviation of Error of a Cubature Formula from the Optimal - ORDER OF CONVERGENCE OF CUBATURE FORMULAS - Lower Estimate of the Norm of the Error Functional Approximate Upper Estimate of the Norm of the Error Functional CUBATURE FORMULAS CONSIDERING A REGULAR BOUNDARY LAYER Formulas for Periodic Functions - Norm of the Error Functional for Periodic Functions - Composition of Formulas with Small Suppoas - Error for Finite Functions - Construction of Formulas with a Regular Boundary Layer - Norm of the Error Functional of Cubature Formulas with a Regular Boundary Layer in the Space L(')(R) - Norm of the Error of Formulas with Regular Boundary 2 Layer in L(al(Q) - OPTIMAL FORMULAS - Formulation of the Problem on 2 Optimal Coefficients - Fourier Transformation of a Discrete Potential - Prope@ies of the Operator D(m@[pl' - Discrete Analog of a Polyharmonic h Operator - Optimal Coefficients of One-Dimensional Formulas - CONVERGENCE OF CUBATURE FORMULAS IN VARIOUS CLASSES AND DIFFERENT FUNCTIONS - Functional Class (D(PIA) - Functional Class T(plo) - Cubature Formulas for Infinitely Differentiable Functions - Convergence oi Cubature Formulas for an Arbitrary Function (P(x) c @ll - CUBATURE FORMULAS FOR RATIONAL POLYHEDRA - Convex Polyhedra Euler's Formula - Rational Polyhedra - Structure of Formulas for Rational Polyhedra - Cubature Formulas for a Polyhedron and Its Solid Angles - Formulas with a Formal Boundary Layer