Functions of Brownian motion

Cauchy-subregular functions vis-à-vis different types of continuity

A Universal Approximation Theorem for Gaussian-Gated Mixture of Experts Models

The problem of determining source term in a semilinear wave equation is considered. The source term is represented as the product ƒ(u(x, t))p(x), where ƒ(s) is a given function, u(x, t) is a solution to Cauchy problem for wave equation, p(x) is an unknown function. To determine p(x) the additional information on the solution of the Cauchy problem u(α(t), t) = g(t), u(β(t), t) = h(t) is used. Theorems of existence and uniqueness of solution to an inverse problem in the class of continuous functions p(x) and in the class of functions p(x) = po + xq(x), where q(x) is continuous, are proved.

The problem of determining source term in a semilinear wave equation is considered. The source term is represented as the product ƒ(u(x, t))p(x), where ƒ(s) is a given function, u(x, t) is a solution to Cauchy problem for wave equation, p(x) is an unknown function. To determine p(x) the additional information on the solution of the Cauchy problem u(α(t), t) = g(t), u(β(t), t) = h(t) is used. Theorems of existence and uniqueness of solution to an inverse problem in the class of continuous functions p(x) and in the class of functions p(x) = po + xq(x), where q(x) is continuous, are proved.

A new class of closed sets, known as semi-regular weekly closed sets, is introduced in ideal topological space. In addition, we define a new class of continuous functions known as semi-regular weekly -continuous functions is introduced and studied. Some of their characteristics have also been examined and discussed interesting properties of semi-regular weekly -irresolute functions, strongly semi-regular weekly -continuous functions, and semi- regular weekly -continuous functions.

In this paper, we introduce two new classes of continuous functions in topological spaces called almost Ip-continuous and weakly Ip-continuous functions. We obtain several characterizations of them and some of their properties. Also, we investigate their relationships with some other types of functions.

This paper provides a proof that the class of those real functions f for which there exists a change of variable g so that f ∘ g f \circ g is differentiable coincides with the class of continuous functions which are of generalized bounded variation in the restricted sense.

We study the relationship between two characteristics of functional classes, pseudodimension and bracketing entropy. (Pseudodimension is a generalization of VC-dimension to classes of functions. Bracket entropy characterizes the L1-error of one-sided approximation of a class by finite sets.) It is shown that classes of continuous functions with finite pseudodimension possess a finite bracketing ɛ-entropy for any ɛ > 0. We establish a general result concerning the relationship between the VC-dimension of classes of sets and their bracketing entropy.

We study the asymptotic properties, consistency, asymptotic normality, of the least squares estimator in a non linear regression problem. The model uses a parametric class Л of functions, but we do not assume that the unknown function belongs to that class. Л is here a class of continuous functions with a discontinuity in the first derivative. The problem of making a choice between two classes of that type is also studied.

The main aim of this work is to show the solvability of the homogeneous and inhomogeneous boundary conjugation problems for piecewise analytic functions in the Besov space, which is embedded into the class of continuous functions.

We consider the Darboux problem for a differential equation of fractional order that contains a regularized mixed derivative. Sufficient conditions for the existence and uniqueness of a solution of this problem are obtained in the class of continuous functions. We also propose a method for finding an approximate solution of this problem and prove the convergence of this method.

An existence theorem for the “gap equation” in the superconductivity theory is given, as a consequence of the Schauder-Tychonoff theorem. Sufficient conditions on the kernel are given, which insure the existence of a solution amongst a particular class of continuous functions. The case of a positive kernel is studied in detail.

We consider a backward stochastic differential equation, whose data (the final condition and the coefficient) are given functions of a jump-diffusion process. We prove that under mild conditions the solution of the BSDE provides a viscosity solution of a system of parabolic integral-partial differential equations. Under an additional assumption, that system of equations is proved to have a unique solution, in a given class of continuous functions

In 1979, Pisier proved remarkably that a sequence of independent and identically distributed standard Gaussian random variables determines, via random Fourier series, a homogeneous Banach algebra P strictly contained in C(T), the class of continuous functions on the unit circle T and strictly containing the classical Wiener algebra A(T), that is, A(T)⫋P⫋C(T). This improved some previous results obtained by Zafran in solving a long-standing problem raised by Katznelson. In this paper, we extend Pisier’s result by showing that any probability measure on the unit circle defines a homogeneous Banach algebra contained in C(T). Thus Pisier algebra is not an isolated object but rather an element in a large class of Pisier-type algebras. We consider the case of spectral measures of stationary sequences of Gaussian random variables and obtain a sufficient condition for the boundedness of the random Fourier series ∑ n∈Zfˆ(n)ξnexp(2πint) in the general setting of dependent random variables (ξn).

In this paper two theorems are proved on the existence of a null Chaplygin approximation for a solution of the generalized Cauchy problem, from which the solvability of this problem in the class of continuous functions follows.