We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions. We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from... more
Dedicated to the memory of Christo Ya. Christov on the 100-th anniversary of his birth.***
We present a differential algebra of generalized functions over a field of generalized scalars by means of several axioms in terms of general algebra and topology. Our differential algebra is of Colombeau type in the sense that it... more
In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau' theory of new generalized functions and its... more
We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz... more
In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau' theory of new generalized functions and its... more
Dedicated to the memory of Christo Ya. Christov* on the 100-th anniversary of his birth and to James Vickers on the occasion his 60-th birthday.
The algebra of asymptotic functions ρ E(Ω) on an open set Ω ⊂ R d was introduced by M. Oberguggenberger and the author of this paper in the framework of A. Robinson's nonstandard analysis. It can be described as a differential associative... more
Dedicated to the memory of Christo Ya. Christov* on the 100-th anniversary of his birth and to James Vickers on the occasion his 60-th birthday.
In these lecture notes we present an introduction to non-standard analysis especially written for the community of mathematicians, physicists and engineers who do research on J. F. Colombeau' theory of new generalized functions and its... more
We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions. We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from... more
We present a differential algebra of generalized functions over a field of generalized scalars by means of several axioms in terms of general algebra and topology. Our differential algebra is of Colombeau type in the sense that it... more
We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz... more
Recently, it is shown that the extended phase space formulation of quantum mechanics is a suitable technique for studying the quantum dissipative system. Here, as an application of this formalism, we consider a dissipative system of... more
Real analytic generalized functions are investigated as well as the analytic singular support and analytic wave front of a generalized function in G(Ω) are introduced and described. 2000 Mathematics Subject Classification. 46F30,... more
We investigate homogeneity in the special Colombeau algebra on R d as well as on the pierced space R d \ {0}. It is shown that strongly scaling invariant functions on R d are simply the constants. On the pierced space, strongly... more
We give necessary and sufficient conditions for a regularized net of a distribution in an open set Ω which imply that it is a smooth function or C k function in Ω. We also give necessary and sufficient conditions for an ultradistribution... more
A topological description of various generalized function algebras over corresponding basic locally convex algebras is given. The framework consists of algebras of sequences with appropriate ultra(pseudo)metrics defined by sequences of... more
Information measures arise in many disciplines, including forecasting (where scoring rules are used to provide incentives for probability estimation), signal processing (where information gain is measured in physical units of relative... more
This paper is devoted to introduce a new generalized 4 K -function in terms of some special functions. The differ-integration of this function is also investigated. A method for deriving the solution of the generalized fractional kinetic... more
The aim of this paper is to prove that the framework of generalized functions of Sobolev type is more suitable to pose and solve some PDEs problems with very irregular data, than the one introduced by J.-F. Colombeau, when C ∞ estimates... more
In the last two decades, many algebras of generalized functions have been constructed, particularly the so-called generalized Sobolev algebras. Our goal is to study the latter and some of their main properties. In this framework, we pose... more
The aim of this paper is to prove that the framework of generalized functions of Sobolev type is more suitable to pose and solve some PDEs problems with very irregular data, than the one introduced by J.-F. Colombeau, when C ∞ estimates... more
The exponential and Rayleigh are the two most commonly used distributions for analyzing lifetime data. These distributions have several desirable properties and nice physical interpretations. Unfortunately the exponential distribution... more
We define a type of generalized asymptotic series called v-asymptotic. We show that every function with moderate growth at infinity has a v-asymptotic expansion. We also describe the set of v-asymptotic series, where a given function with... more
In analogy to the classical isomorphism between L(D(R n), D (R m)) and D (R m+n) (resp. L(S(R n), S (R m)) and S (R m+n)), we show that a large class of moderate linear mappings acting between the space GC (R n) of compactly supported... more
We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of the Colombeau simplified model. This generalizes the notion of G ∞-regularity... more
We present some remarks about the embedding of spaces of Schwartz distributions into spaces of Colombeau generalized functions. Following ideas of M. Nedeljkov et al., we recall how a good choice of compactly supported mollifiers allows... more
In connection with the classical Schwartz kernel theorem, we show that in the framework of Colombeau generalized functions a large class of linear mappings admit integral kernels. To do this, we need to introduce news spaces of... more
We present new types of regularity for Colombeau nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of the simplified model. This generalizes the notion of G ∞-regularity... more
In literature, it is common to find problems which require a way to encode a finite set of information into a single data; usually means are used for that. An important generalization of means are the so called Aggregation Functions, with... more
Stochastic processes are regarded in the framework of Colombeau-type algebras of generalized functions. The notion of point values of Colombeau stochastic processes in compactly supported generalized points is established, which uniquely... more
1 2 The definitions of positivity and positive definiteness are extended to generalized function algebras in coherence with the corresponding notions for distributions. Versions of Bochner's theorem for a positive definite Colombeau... more
1 2 The definitions of positivity and positive definiteness are extended to generalized function algebras in coherence with the corresponding notions for distributions. Versions of Bochner's theorem for a positive definite Colombeau... more
The classical method of solving the equation P (D) g = f is adapted to a method of solving the family of equations with respect to ε with a prescribed growth rate. More precisely, the equation in the sense that the coefficients converge... more
By making use of some rather elementary techniques based upon certain inverse pairs of symbolic operators, the authors investigate several decomposition formulas associated with Lauricella's hypergeometric function F (r) A in r variables... more
We construct Euclidean random fields X over [Formula: see text] by convoluting generalized white noise F with some integral kernels G, as X=G*F. We study properties of Schwinger (or moment) functions of X. In particular, we give a general... more
In this article, we show that ratio estimates, and their asymptotic variances can be correctly obtained from a Poisson regression with appropriately chosen link function and Generalized Estimating Equation techniques. The sandwich... more
Based on a generalized Mexican-hat function proposed, the paper presents an algorithm for the design of a new class of continuous wavelets matched to arbitrary transient signals. While the generalized Mexican-hat wavelets are constructed... more
Euler's transformation formula for the Gauss hypergeometric function 2 F 1 is extended to hypergeometric functions of higher order. Unusually, the generalized transformation constrains the hypergeometric function parameters algebraically... more
Stochastic processes with paths in a generalized function algebra are defined and it is shown that there exists an embedding of generalized functional stochastic processes into such ones. Gaussian stochastic processes with paths in an... more
This paper extends the Meijer transformation, M, given by I /2K 2p f(t) (pt) (2 p)dt, (Mf) (p) (i+) 0 where f belongs to an appropriate function space, e (-1,) and K is the modified Bessel function of third kind of order , to certain... more
Recently two-parameter generalized exponential distribution has been introduced by the authors. In this paper we consider the Bayes estimators of the unknown parameters under the assumptions of gamma priors on both the shape and scale... more
In this article, we introduce a new estimator for the generalized Pareto distribution, which is based on the maximum likelihood estimation and the goodness of fit. The asymptotic normality of the new estimator is shown and a small... more
Properties of generalized convex functions, defined in terms of the generalized means introduced by Hardy, Littlewood, and Polya, are easily obtained by showing that generalized means and generalized convex functions are in fact ordinary... more
We introduce and analyze spaces and algebras of generalized functions which correspond to Hölder, Zygmund, and Sobolev spaces of functions. The main scope of the paper is the characterization of the regularity of distributions that are... more
A generalized quantum distribution function is introduced. The corresponding ordering rule for non-commuting operators is given in terms of a single parameter. The origin of this parameter is in the extended canonical transformations that... more
Stochastic processes are regarded in the framework of Colombeau-type algebras of generalized functions. The notion of point values of Colombeau stochastic processes in compactly supported generalized points is established, which uniquely... more
Our aim is to design a pattern classifier using fuzzy relational calculus (FRC) which was initially proposed by W. Pedrycz. In the course of doing this we introduce a new interpretation of multidimensional fuzzy implication (MFI) to... more