Todor D. Todorov

Asymptotic Functions as Kernels of the Schwartz Distributions

Todor D. Todorov — Thu, 01 Dec 2011 11:41:19 PST

Using a version of the sequential method we introduce a class of generalized functions called here "asymptotic functions''. This class contains kernels of all Schwartz distributions and is equipped with a correctly defined multiplication operation. So, in a sense, one solves the problem of "multiplication of Schwartz distributions" although the solution refers to the class of the asymptotic functions and not to the Schwartz distributions themselves. The paper is a continuation of a series of works [1-10] but here only part of the results of [5], [6] and [8] will be needed.

Operations with Distribution Vectors

Brian Fisher et al. — Mon, 14 Nov 2011 13:05:30 PST

Asymptotic Numbers: Algebraic Operations with Them

Christo Ya. Christov et al. — Mon, 14 Nov 2011 13:05:27 PST

The main subject of the present paper is to define the four algebraic operations - additions, subtraction, multiplication and division in the set of the asymptotic numbers A [7] and to deduce the corresponding formulas for the components of the asymptotic number, representing the result as functions of the components of the arguments. The definitions of the operations, in fact, are introduced as a special case of the more general notion of a quasiclassical function - one special class of functions defined on A. The discussion of the algebraic and some other properties of the asymptotic numbers is put off for a next paper.

The set of asymptotic numbers, introduced by the same authors in [7], is a generalization of the system of real (complex) numbers, comprising infinitely small and infinitely large numbers [1], [2]. The reasons for introducing these numbers are connected with concrete problems of the quantum mechanics [5], [6], [8], but it seems to us that they are also interesting for themselves.

The definition of the asymptotic numbers and some of their properties are reminded in the introductory chapter, by which we achieve logical independence of [7].

Monads and Realcompactness

Sergio Salbany et al. — Mon, 14 Nov 2011 13:05:25 PST

We give a quantifier free characterization of realcompactness and ordered realcompactness in terms of monads. We also present simple proofs of some topological facts concerning realcompact spaces.

Quasi-Extended Asymptotic Functions

Todor D. Todorov — Mon, 14 Nov 2011 13:05:22 PST

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's 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

Todor D. Todorov — Mon, 14 Nov 2011 13:05:20 PST

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.

A Lost Theorem: Definite Integrals in An Asymptotic Setting

Ray Cavalcante et al. — Mon, 14 Nov 2011 13:05:18 PST

Another Proof of the Existence a Dedekind Complete Totally Ordered Field

James F. Hall et al. — Mon, 14 Nov 2011 13:05:16 PST

We describe the Dedekind cuts explicitly in terms of non-standard rational numbers. This leads to another construction of a Dedekind complete totally ordered field or, equivalently, to another proof of the consistency of the axioms of the real numbers. We believe that our construction is simpler and shorter than the classical Dedekind construction and Cantor construction of such fields assuming some basic familiarity with non-standard analysis.

Asymptotic Numbers: I. Algebraic Properties

Todor D. Todorov — Mon, 14 Nov 2011 13:05:14 PST

The set of asymptotic numbers A introduced in Refs. [1] and [3] is a system of generalized numbers including the system of real numbers R, as well as infinitely small (infinitesimals) and infinitely large numbers. The purpose of this paper is to study in detail the algebraic properties of A which are a little unusual, in a cenain sense, as compared with the known algebraic structures (rings. fields, etc.) This is necessary for the investigation of the class of asymptotic functions [2.4], which are on their part, generalized functions similar to the distributions of Schwartz but allowing the operation of multiplication.

The motivations of this work are in fact, connected with some physical problems [1, 2, 3, 4, 5, 8]; we are going to use the asymptotic numbers and asymptotic functions in the quantum theory in some cases when the multiplication of the distributions is desirable but not possible. Our methods are analogous, in a certain sense, to the methods of the non-standard analysis* [9, 10, 11, 12]. For the sake or convenience we have exposed briefly the most important results of Refs. [1] and [2] and the paper could be read independently of them.

Completeness of the Leibniz Field and Rigorousness of Infinitesimal Calculus

James F. Hall et al. — Mon, 14 Nov 2011 13:05:12 PST

We present a characterization of the completeness of the field of real numbers in the form of a collection of ten equivalent statements borrowed from algebra, real analysis, general topology and non-standard analysis. We also discuss the completeness of non-Archimedean fields and present several examples of such fields. As an application we exploit one of our results to argue that the Leibniz infinitesimal calculus in the 18^th century was already a rigorous branch of mathematics – at least much more rigorous than most contemporary mathematicians prefer to believe. By advocating our particular historical point of view, we hope to provoke a discussion on the importance of mathematical rigor in mathematics and science in general. We believe that our article will be of interest for those readers who teach courses on abstract algebra, real analysis, general topology, logic and the history of mathematics.

The Products δn²(x), δ(x). X^-n, ϴ(x). X^-n, etc. in the Class of the Asymptotic Functions

Todor D. Todorov — Mon, 14 Nov 2011 13:05:09 PST

Several products like δⁿ(x), δ(x)ϴ(x), δ^(m)(x). X^-n, ϴ(x). X^-n, etc., where δ(x), ϴ(x), X^-n, etc., are kernels of the corresponding Schwartz distributions, are studied in the framework of the class of the asymptotic functions F₀ introduced in a previous paper [11]. In some particular cases many formulae are derived and several examples are presented. The work is of mathematical type but its motivations lie in some problems in quantum theory. It is closely connected with a series of previous works [1 - 11] and first of all with [11].

Asymptotic Functions and the Problem of Multiplication of Distributions

Todor D. Todorov — Mon, 14 Nov 2011 13:05:07 PST

The asymptotic functions are a new type of generalized functions. But they are not functionals on some space of test-functions as the Schwartz distributions. They are mappings of the set of the asymptotic numbers (1, 3, 5, 6) into itself. On its part, the set of the asymptotic numbers is a totally-ordered set of generalized numbers including the systems of real and complex numbers, as well as infinitesimals and infinitely large numbers. Every two asymptotic functions can be multiplied. On the other hand, the Schwartz distributions have realizations, in a certain sense, as asymptotic functions. The motivations of this work are connected with some physical problems of quantum theory [18, 25].

Asymptotic Numbers: II. Order Relation, Infinitesimals and Interval Topology

Todor D. Todorov — Mon, 14 Nov 2011 13:05:05 PST

It has been shown in [8] that the set of asympototic numbers A is a system of generalized numbers including isomorphically the set of real numbers R, as well as the field of formal power (asymptotic) series. In the present paper, which is a continuation of [8], an order relation in A is introduced due to A turning out to be a totally-ordered set. The consistency between the order relation and the algebraic operations in A is investigated and in particular, it is shown that the inequalities in A can be added and multiplied as in the set of the real numbers. The notions of infinitesimals (infinitely small numbers), finite and infinitely large numbers are introduced; A turns out to be a nonarchimedean set. The usage of infinitesimals as infinitely large numbers along with the real numbers is the reason why the terms and the notions introduced in this paper are very much like those of the non-standard analysis (Robinson's theory of infinitesimals) [7]*. In connection with the order relation, an interval topology of A is introduced and some of its properties are established. The theory of asymptotic functions, as well as the applications to the quantum theory, are put off for a future paper.

The notions or the asymptotic numbers [2, 4, 8] and those of the asymptotic functions [3, 5] are introduced as a subsidiary device for investigation of some problems in quantum theory. For further details about the motivation of this work we advise the reader to refer to [2, 3, 4, 5, 8]. But the knowledge of [8] is quite sufficient for the understanding of the present paper.

Nonstandard and Standard Compactifications of Ordered Topological Spaces

Sergio Salbany et al. — Mon, 14 Nov 2011 13:05:02 PST

We construct the Nachbin ordered compactification and the ordered realcompactification, a notion defined in the paper, of a given ordered topological space as nonstandard ordered hulls. The maximal ideals in the algebras of the differences of monotone continuous functions are completely described. We give also a characterization of the class of completely regular ordered spaces which are closed subspaces of products of copies of the ordered real line, answering a question of T.H. Choe and Y.H. Hong. The methods used are topological (standard) and nonstandard.

An Embedding of Schwartz Distributions in the Algebra of Asymptotic Functions

Michael Oberguggenberger et al. — Fri, 04 Nov 2011 10:24:48 PDT

We present a solution of the problem of multiplication of Schwartz distributions by embedding the space of distributions into a differential algebra of generalized functions, called in the paper "asymptotic function," similar to but different from J. F Colombeau's algebras of new generalized functions.

An Existence Result for Linear Partial Differential Equations with C^∞ Coefficients in an Algebra of Generalized Functions

Todor D. Todorov — Fri, 04 Nov 2011 10:24:46 PDT

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.

Full algebra of generalized functions and non-standard asymptotic analysis

Todor D. Todorov et al. — Fri, 04 Nov 2011 10:24:45 PDT

We construct an algebra of generalized functions endowed with a canonical embedding of the space of Schwartz distributions. We offer a solution to the problem of multiplication of Schwartz distributions similar to but different from Colombeau’s solution. We show that the set of scalars of our algebra is an algebraically closed field unlike its counterpart in Colombeau theory, which is a ring with zero divisors. We prove a Hahn–Banach extension principle which does not hold in Colombeau theory. We establish a connection between our theory with non-standard analysis and thus answer, although indirectly, a question raised by Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis.

Radius of convergence of a power series

Todor D. Todorov — Fri, 04 Nov 2011 10:24:42 PDT

We derive two simple and memorizable formulas for the radius of convergence of a power series which seem to be appropriate for teaching in an introductory calculus course.

Back to Classics: Teaching Limits Through Infinitesimals

Todor D. Todorov — Fri, 04 Nov 2011 10:24:41 PDT

The usual ϵ, δ-definition of the limit of a function (whether presented at a rigorous or an intuitive level) requires a “candidate L” for the limit value. Thus, we have to start our first calculus course with “guessing” instead of “calculating”. In this paper we criticize the method of using calculators for the purpose of selecting candidates for L. We suggest an alternative: a working formula for calculating the limit value L of a real function in terms of infinitesimals. Our formula, if considered as a definition of limit, is equivalent to the usual ϵ, δ-definition but does not involve a candidate L for the limit value. As a result, the Calculus becomes to “calculate” again as it was originally designed to do.

Nonstandard Analysis in Topology: Nonstandard and Standard Compactifications

S. Salbany et al. — Fri, 04 Nov 2011 10:24:39 PDT

Let (X, T) be a topological space, and ^∗X a non–standard extension of X. There is a natural “standard” topology ^ST on ^∗X generated by ^∗G,where G ∈ T . The topological space (^∗X,^ST) will be used to study, in a systematic way, compactifications of (X, T).