54 research outputs found
Algebraic Approach to Colombeau Theory
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 a copy of the space of Schwartz distributions, and the set of regular
distributions with -kernels forms a differential subalgebra.
We discuss the uniqueness of the field of scalars as well as the consistency
and independence of our axioms. This article is written mostly to satisfy the
interest of mathematicians and scientists who do not necessarily belong to the
\emph{Colombeau community}; that is to say, those who do not necessarily work
in the \emph{non-linear theory of generalized functions}.Comment: 16 page
A Lost Theorem: Definite Integrals in Asymptotic Setting
We present a simple yet rigorous theory of integration that is based on two
axioms rather than on a construction involving Riemann sums. With several
examples we demonstrate how to set up integrals in applications of calculus
without using Riemann sums. In our axiomatic approach even the proof of the
existence of the definite integral (which does use Riemann sums) becomes
slightly more elegant than the conventional one. We also discuss an interesting
connection between our approach and the history of calculus. The article is
written for readers who teach calculus and its applications. It might be
accessible to students under a teacher's supervision and suitable for senior
projects on calculus, real analysis, or history of mathematics
An Existence Result for Linear Partial Differential Equations with \u3cem\u3eC\u3csup\u3e∞\u3c/sup\u3e\u3c/em\u3e Coefficients in an Algebra of Generalized Functions
We prove the existence of solutions for essentially all linear partial differential equations with C∞-coefficients in an algebra of generalized functions, defined in the paper. In particular, we show that H. Lewy’s equation has solutions whenever its right-hand side is a classical C∞-function
Quasi-Extended Asymptotic Functions
The class F of quasi-extended asymptotic functions introduced in the present paper contains all extended asymptotic functions [8, (3.1)] (in particular, all examples constructed in [9, Sec. 1 ]). But F contains also some new asymptotic functions very similar to tht Schwartz distributions. On the other hand, every two quasi-extended asymptotic functions can be multiplied as opposed to the Schwartz distributions; in particular, the square &# 948;2 of an asymptotic function &# 948; similar to Dirac\u27s delta-function is constructed as an example. The connection with the asymptotic functions introduced in [2] and [4] is established
Lecture Notes: Non-Standard Approach to J.F. Colombeau’s Theory of Generalized Functions
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 applications. The main purpose of our non-standard approach to Colombeau’ theory is the improvement of the properties of the scalars of the varieties of spaces of generalized functions: in our non-standard approach the sets of scalars of the functional spaces always form algebraically closed non-archimedean Cantor complete fields. In contrast, the scalars of the functional spaces in Colombeau’s theory are rings with zero divisors. The improvement of the scalars leads to other improvements and simplifications of Colombeau’s theory such as reducing the number of quantifiers and possibilities for an axiomatization of the theory. Some of the algebras we construct in these notes have already counterparts in Colombeau’s theory, other seems to be without counterpart. We present applications of the theory to PDE and mathematical physics. Although our approach is directed mostly to Colombeau’s community, the readers who are already familiar with non-standard methods might also find a short and comfortable way to learn about Colombeau’s theory: a new branch of functional analysis which naturally generalizes the Schwartz theory of distributions with numerous applications to partial differential equations, differential geometry, relativity theory and other areas of mathematics and physics
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