Related papers: Multiplication of Distributions in one dimension: …
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,…
In a previous paper we have introduced a class of multiplications of distributions in one dimension. Here we furnish different generalizations of the original definition and we discuss some applications of these procedures to the…
We present the construction of a theory of distributions (generalized functions) with a ``thick submanifold'', that is, a new theory of thick distributions on $\mathbb{R}^n$ whose domain contains a smooth submanifold on which the test…
As follows from the Schwartz Impossibility Theorem, multiplication of two distributions is in general impossible. Nevertheless, often one needs to multiply a distribution by a discontinuous function, not by an arbitrary distribution. In the…
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 generalization of a distribution increases the flexibility particularly in studying of a phenomenon and its properties. Many generalizations of continuous univariate distributions are available in literature. In this study, an…
A new definition of a fractional derivative has recently been developed, making use of a fractional Dirac delta function as its integral kernel. This derivative allows for the definition of a distributional fractional derivative, and as…
The theory of thick distributions (both in dimension 1 and in higher dimensions) was constructed in recent years [7, 27]. However this theory of distributions with one thick point in dimension one is very different from that in higher…
The article presents, in an elementary way, but with mathematical precision and without harm to the intuition, the path from the integral representation to the Dirac delta, starting with Schwartz's functional approach. Next, the considered…
Probabilistic submeasures generalizing the classical (numerical) submeasures are introduced and discussed in connection with some classes of aggregation functions. A special attention is paid to triangular norm-based probabilistic…
Mathematical justifications are given for several integral and series representations of the Dirac delta function which appear in the physics literature. These include integrals of products of Airy functions, and of Coulomb wave functions;…
Dirac delta function (delta-distribution) approach can be used as efficient method to derive identities for number series and their reciprocals. Applying this method, a simple proof for identity relating prime counting function…
A theoretical framework is developed to describe the transformation that distributes probability density functions uniformly over space. In one dimension, the cumulative distribution can be used, but does not generalize to higher…
In this article we consider regularizations of the Dirac delta distribution with applications to prototypical elliptic and hyperbolic partial differential equations (PDEs). We study the convergence of a sequence of distributions…
Methods for generating new distributions from old can be thought of as techniques for simplifying integrals used in reverse. Hence integrating a probability density function (pdf) by parts provides a new way of modifying distributions; the…
We develop a theory of multiplicative functions (with values inside or on the unit circle) in arithmetic progressions analogous to the well-known theory of primes in arithmetic progressions.
In this paper an iterated function system on the space of distribution functions is built. The inverse problem is introduced and studied by convex optimization problems. Some applications of this method to approximation of distribution…
In this article, we study the distribution of values of Dirichlet $L$-functions, the distribution of values of the random models for Dirichlet $L$-functions, and the discrepancy between these two kinds of distributions. For each question,…
In this pedagogical text aimed at those wanting to start thinking about or brush up on probabilistic inference, I review the rules by which probability distribution functions can (and cannot) be combined. I connect these rules to the…
Dirac delta function of matrix argument is employed frequently in the development of diverse fields such as Random Matrix Theory, Quantum Information Theory, etc. The purpose of the article is pedagogical, it begins by recalling detailed…