4,666 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
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