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Completeness Of The Leibniz Field And Rigorousness Of Infinitesimal Calculus, James F. Hall, Todor D. Todorov
Completeness Of The Leibniz Field And Rigorousness Of Infinitesimal Calculus, James F. Hall, Todor D. Todorov
Mathematics
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 18th 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 …
Full Algebra Of Generalized Functions And Non-Standard Asymptotic Analysis, Todor D. Todorov, Hans Vernaeve
Full Algebra Of Generalized Functions And Non-Standard Asymptotic Analysis, Todor D. Todorov, Hans Vernaeve
Mathematics
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 …