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 $\mathcal C^\infty$-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

Spread spectrum watermarking technique for information system securing

In this paper we consider a computer information system and a way to realize the security of the data in it with digital watermarking. A technique for spread spectrum watermarking is presented and its realization with MathLAB 6.5 is shown