Schwartz Distributions Research Articles

A Monte-Carlo study of equilibrium polymers in a shear flow

We use an off-lattice microscopic model for solutions of equilibrium polymers (EP) in a lamellar shear flow generated by means of a self-consistent external field between parallel hard walls. The individual conformations of the chains are found to elongate in flow direction and shrink perpendicular to it while the average polymer length decreases with increasing shear rate. The Molecular Weight Distribution of the chain lengths retains largely its exponential form in dense solutions whereas in dilute solutions it changes from a power-exponential Schwartz distribution to a purely exponential one upon an increase of the shear rate. With growing shear rate the system becomes increasingly inhomogeneous so that a characteristic variation of the total monomer density, the diffusion coefficient, and the center-of-mass distribution of polymer chains of different contour length with the velocity of flow is observed. At higher temperature, as the average chain length decreases significantly, the system is shown to undergo an order-disorder transition into a state of nematic liquid crystalline order with an easy direction parallel to the hard walls. The influence of shear flow on this state is briefly examined.

Extending the Martingale Measure Stochastic Integral With Applications to Spatially Homogeneous S.P.D.E.'s

We extend the definition of Walsh's martingale measure stochastic integral so as to be able to solve stochastic partial differential equations whose Green's function is not a function but a Schwartz distribution. This is the case for the wave equation in dimensions greater than two. Even when the integrand is a distribution, the value of our stochastic integral process is a real-valued martingale. We use this extended integral to recover necessary and sufficient conditions under which the linear wave equation driven by spatially homogeneous Gaussian noise has a process solution, and this in any spatial dimension. Under this condition, the non-linear three dimensional wave equation has a global solution. The same methods apply to the damped wave equation, to the heat equation and to various parabolic equations.

Abstract Differential Geometry, Differential Algebras of Generalized Functions, and de Rham Cohomology

Abstract differential geometry is a recent extension of classical differential geometry on smooth manifolds which, however, does no longer use any notion of Calculus. Instead of smooth functions, one starts with a sheaf of algebras, i.e., the structure sheaf, considered on an arbitrary topological space, which is the base space of all the sheaves subsequently involved. Further, one deals with a sequence of sheaves of modules, interrelated with appropriate ‘differentials’, i.e., suitable ‘Leibniz’ sheaf morphisms, which will constitute the ‘differential complex’. This abstract approach captures much of the essence of classical differential geometry, since it places a powerful apparatus at our disposal which can reproduce and, therefore, extend fundamental classical results. The aim of this paper is to give an indication of the extent to which this apparatus can go beyond the classical framework by including the largest class of singularities dealt with so far. Thus, it is shown that, instead of the classical structure sheaf of algebras of smooth functions, one can start with a significantly larger, and nonsmooth, sheaf of so-called nowhere dense differential algebras of generalized functions. These latter algebras, which contain the Schwartz distributions, also provide global solutions for arbitrary analytic nonlinear PDEs. Moreover, unlike the distributions, and as a matter of physical interest, these algebras can deal with the vastly larger class of singularities which are concentrated on arbitrary closed, nowhere dense subsets and, hence, can have an arbitrary large positive Lebesgue measure. Within the abstract differential geometric context, it is shown that, starting with these nowhere dense differential algebras as a structure sheaf, one can recapture the exactness of the corresponding de Rham complex, and also obtain the short exponential sequence. These results are the two fundamental ingredients in developing differential geometry along classical, as well as abstract lines. Although the commutative framework is used here, one can easily deal with a class of singularities which is far larger than any other one dealt with so far, including in noncommutative theories.

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.

Positive definiteness of temperature functions

Positive definite temperature functions u( x,t) in are characterized by where μ is a positive measure satisfying that for every ∈ > 0 is finite. A transform is introduced to give an isomorphism between the class of all positive definite temperature functions in [ILM0002] Then correspondence given by [ILM0003] generalizes the Bochner-Schwartz theorem for the Schwartz distributions and extends Widder's correspondence characterizing some subclass of the positive temperature functions by Fourier-Stieltjes transform.

Harmonic analysis in the schwartz distribution spaces and some applications to nonclassical problems of mathematical physics

We construct a special apparatus of harmonic analysis in the Schwartz distribution space which generalizes the main facts of harmonic analysis in\(L_2 (\mathbb{R}^n )\) to the indicated spaces. Using this appaparatus, we describe the structure of some distribution in\(Z'(\mathbb{R}^n )\) with nonalgebraic singularities at isolated singular points and study well-posedness of some nonclassical boundary value problems of mathematical physics in this class.

Inverse problem in planar magnetron sputtering

Mathematical consideration is given to the optimal source distribution which produces a film thickness profile having the least deviation from the flat in axisymmetric planar magnetron sputtering. When the definition domain of the solution is expanded to the Schwartz distribution space, the optimal solution is found to be represented by a finite number of Dirac’s δ-functions, that is, discretized erosion rings of zero width. Numerical analysis yields the bifurcation diagram of the optimal solution. With increasing target radius for a fixed target-substrate distance, the number of δ-functions increases infinitely. A newly bifurcated δ-function first appears at the center of the target remains there for a time, and then begins to move outwards.

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.

A Few Remarks about the Hilbert Transform

A generalized integral similar to integrability B is used to study the Hilbert transform onX=∑1⩽p<∞Lp(R), with a view to obtaining (i) a mollifier which commutes with the Hilbert transform onXand coincides with the Friedrichs mollifier onLloc1(R); (ii) estimates for nonlinear equations; (iii) an integral representation for the Hilbert transform of a regular Schwartz distribution; (iv) a generalized multiplier representation of the Hilbert transform onL1(R); (v) an elementary proof of the injectivity of H onX.

Positive definite temperature functions and a correspondence to positive temperature functions

SynopsisPositive definite temperature functionsu(x, t) in ℝn+1= {(x, t)|x∈ ℝn,t&gt; 0} are characterised bywhere μ is a positive measure satisfying that for every ℰ &gt; 0,is finite. A transformis introduced to give an isomorphism between the class ofall positive definite temperature functions and the class of all possible temperature functions inThen correspondence given bygeneralises the Bochner–Schwartz Theorem for the Schwartz distributions and extends Widder's correspondence characterising some subclass of the positive temperature functions by the Fourier-Stieltjes transform.

Stochastic integration on generalized function spaces and its applications

We define stochastic integration for processes taking values in Colombeau generalized function spaces with respect to a real brownian motion. We extend Ito's formula to any Schwartz's distribution and give, as an application, a probabilistic solution to the partial differential equation: , where the symbol denotes the weak equality used in generalized functions theory. We also solve the following nonlinear partial differential equation: which is satisfied by the variance of a process with a mean function solving the heat equation. At the end of the paper, we give a parallel between our results and the representation of distributions in the sense of Watanabe on Wiener space

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

Financial Systems in Continuous Time: Modelling Cash Stream

Despite the emphasis on securities rather than on cash streams in general financial theories, the concept of cash stream remains useful in modelling realistic financial decision problems. The paper presents a unified view on cash stream modelling, valid in both continuous and discrete time, in the standard analytical framework of Schwartz distributions. Abstract cash stream spaces are defined by stipulating certain general properties, and are then identified as some well-known Lebesgue lattices of measures on the real line. All basic financial notions such as discounting or preference ordering carry over to the general case, and the linear decision problems become instances of infinite-dimensional linear programming.

Stability properties of stochastic partial differential equations

Stochastic partial differential equations (SPDEs) often have solutions that are known to be pure Schwartz distributions i.e. not functions. To make sense of such equations one needs to introduce some kind of smoothing parameters. This paper is concerned with stability properties of the solutions as one lets the smoothing parameters approach some kind of delta function. The first part of the paper concentrates on linear functionals in connection with SPDEs. In the second part we adress similar problems related to functionals of Hida distributions

Tempered Boehmians and ultradistributions

An extension of the Fourier transform which is a one-to-one continuous mapping from the space of tempered Boehmians onto the space of Schwartz distributions is introduced. This shows that the space of tempered Boehmians can be identified with the space Z ′ \mathcal {Z}’ of ultradistributions.

Mixed Limit Theorems for Pattern Analysis

Limit theorems are derived for probability measures of random configurations over graphs which are used as prior distributions in pattern theory. For one-dimensional graphs, these limits can be viewed as distributions of certain stochastic processes, while in higher dimensions the limits will in some cases have to be interpreted as belonging to Schwartz distributions. Such limit distributions are easy to use in pattern analysis, and greatly reduce the computing effort required in comparison with stochastic relaxation methods.

Frequency domain Volterra filters in terms of distributions

The Fourier transform of the output of a Volterra filter is given in terms of Schwartz distributions. This theory is developed in order to handle input signals that do not possess a Fourier transform in the usual sense. To illustrate this, the authors apply the result to the case of a sinusoidal input and to the case of a sampled and periodized signal.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the solution of the functional equation

The functional equation was originally proposed by Daroczy[1]under the assumption that f,gand h are continuous. Later on this equation has been discussed under other assumptions. These discussions are dealt with by Maksa [4]. In this paper we will study a generalization of the above equation, namely We shall show that equation (0.1)can be reformulated into the following functional equation in distributions: Where Q∗ R∗ E∗ are E∗ are operators defined on the space of Schwartz distributions .First we shall show that equation (0.2)reduces to equation(0.1) when the solutions are regular distribuions,i.e. locally Lebesgue integrable functions. With the help of distributions techniques we shall then solve (0.2) in the domain of distributions.

Group invariance of generalized solutions obtained through the algebraic method

Three results are presented as a first in the literature: (1) large classes of nonlinear Lie groups of transformations are defined on algebras of generalized functions containing the L. Schwartz distributions; the definitions do not involve infinite dimensional manifolds, (2) the projectable Lie groups of symmetry for classical solutions of nonlinear PDEs can be extended to symmetry groups for global generalized solutions, (3) applications are given of the nonlinear group invariance for delta wave solutions of semilinear hyperbolic equations and for Riemann type solvers of the nonlinear shock wave equation.

Generalized solutions of a stochastic partial differential equation

We discuss the Cauchy problem of a certain stochastic parabolic partial differential equation arising in the nonlinear filtering theory, where the initial data and the nonhomogeneous noise term of the equation are given by Schwartz distributions. The generalized (distributional) solution is represented by a partial (conditional) generalized expectation ofT(t)°ϕ −1 , whereT(t) is a stochastic process with values in distributions and ϕ s,t is a stochastic flow generated by a certain stochastic differential equation. The representation is used for getting estimates of the solution with respect to Sobolev norms. Further, by applying the partial Malliavin calculus of Kusuoka-Stroock, we show that any generalized solution is aC ∞-function under a condition similar to Hormander's hypoellipticity condition.