Related papers: Generalized functions with infinitesimals
The normal or Gaussian distribution plays a prominent role in almost all fields of science. However, it is well known that the Gauss (or Euler--Poisson) integral over a finite boundary, as it is necessary for instance for the error function…
We introduce a new class of multiplications of distributions in one dimension merging together two different regularizations of distributions. Some of the features of these multiplications are discussed in a certain detail. We use our…
The applications of the recent results obtained in the theory of generalized Lambert functions, to the mean field theory of ferromagnetism are presented. As a consequence, all the predictions of the Weiss theory of ferromagnetism can be…
Under certain general conditions, an explicit formula to compute the greatest delta-epsilon function of a continuous function is given. From this formula, a new way to analyze the uniform continuity of a continuous function is given.…
Our main aim is to apply the theory of regularly varying functions to the asymptotical analysis at infinity of solutions of Friedmann cosmological equations. A new constant $\Gamma$ is introduced related to the Friedmann cosmological…
In a recent work [1, 2] Sjoberg remarked that generalization of the double reduction theory to partial differential equations of higher dimensions is still an open problem. In this note we have attempted to provide this generalization to…
We introduce and study new modules and spaces of generalized functions that are related to the classical Besov spaces. Various Schwartz distribution spaces are naturally embedded into our new generalized function spaces. We obtain precise…
The paper is devoted to a comprehensive second-order study of a remarkable class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to…
Possibilities for defining the radial derivative of the delta distribution $\delta(\underline{x})$ in the setting of spherical coordinates are explored. This leads to the introduction of a new class of continuous linear functionals similar…
A sharp, distribution free, non-asymptotic result is proved for the concentration of a random function around the mean function, when the randomization is generated by a finite sequence of independent data and the random functions satisfy…
The logarithmic representation of infinitesimal generators is generalized to the cases when the evolution operator is unbounded. The generalized result is applicable to the representation of infinitesimal generators of unbounded evolution…
Gaussian distributions can be generalized from Euclidean space to a wide class of Riemannian manifolds. Gaussian distributions on manifolds are harder to make use of in applications since the normalisation factors, which we will refer to as…
In functional analysis, there are different notions of limit for a bounded sequence of $L^1$ functions. Besides the pointwise limit, that does not always exist, the behaviour of a bounded sequence of $L^1$ functions can be described in…
In two previous papers the author introduced a multiplication of distributions in one dimension and he proved that two one-dimensional Dirac delta functions and their derivatives can be multiplied, at least under certain conditions. Here,…
We derive out naturally some important distributions such as high order normal distributions and high order exponent distributions and the Gamma distribution from a geometrical way. Further, we obtain the exact mean-values of integral form…
The primary goal of this paper is to introduce and investigate generalized incomplete exponential functions with matrix parameters. Integral representation, differential formula, addition formula, multiplication formula, and recurrence…
We generalize the usual exponential Boltzmann factor to any reasonable and potentially observable distribution function, $B(E)$. By defining generalized logarithms $\Lambda$ as inverses of these distribution functions, we are led to a…
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 contains…
This paper lays the foundations for a nonlinear theory of differential geometry that is developed in a subsequent paper which is based on Colombeau algebras of tensor distributions on manifolds. We adopt a new approach and construct a…
It is shown that the nonstandard representatives of Schwartz-distributions, as introduced by K.D. Stroyan and W.A.J. Luxemburg in their book 'Introduction to the theory of infinitesimals', are locally equal to a finite-order derivative of a…