Papers by Todor D. Todorov
arXiv (Cornell University), Aug 22, 2011
We present a characterization of the completeness of the field of real numbers in the form of a c... more 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.
arXiv (Cornell University), Jun 7, 2021
We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bas... more We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bases. As an application we prove the surjectivity of a large class of linear partial differential operators with smooth (C ∞ -coefficients) coefficients, called in the article regular, acting on the algebraic dual D * (Ω) of the space of test-functions D(Ω). The surjectivity of the partial differential operators guarantees solvability of the corresponding partial differential equations within D * (Ω). We discuss our result in contrast to and comparison with similar results about the restrictions of the regular operators on the space of Schwartz distribution D ′ (Ω), where these operators are often non-surjective.
Transactions of the American Mathematical Society, 1996
We prove the existence of solutions for essentially all linear partial differential equations wit... more 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.
Logic and Analysis, Jun 18, 2008
We construct an algebra of generalized functions endowed with a canonical embedding of the space ... more 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 J.F. Colombeau. This article provides a bridge between Colombeau theory of generalized functions and non-standard analysis.
arXiv (Cornell University), Aug 23, 2011
The usual ǫ, δ-definition of the limit of a function (whether presented at a rigorous or an intui... more The usual ǫ, 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 ǫ, 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.
arXiv (Cornell University), Sep 13, 2015
Dedicated to the memory of Christo Ya. Christov on the 100-th anniversary of his birth.***
The São Paulo Journal of Mathematical Sciences, 2013
We present a differential algebra of generalized functions over a field of generalized scalars by... more 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 C ∞ -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 Colombeau community; that is to say, those who do not necessarily work in the non-linear theory of generalized functions.
arXiv (Cornell University), Oct 18, 2010
In these lecture notes we present an introduction to non-standard analysis especially written for... more 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 * Work supported by START-project Y237 of the Austrian Science Fund MSC: Functional Analysis (46F30); Generalized Solutions of PDE (35D05). is the non-standard extension of Ω. 8.2 Proposition. * E(Ω) is a differential algebra over the field * C. 8.3 Definition (Sup and Support). Let f i ∈ * E(Ω) and let K ⊂⊂ Ω. Then
arXiv (Cornell University), Jan 30, 2006
Let * R be a nonstandard extension of R and ρ be a positive infinitesimal in * R. We show how to ... more Let * R be a nonstandard extension of R and ρ be a positive infinitesimal in * R. We show how to create a variety of isomorphisms between A. Robinson's field of asymptotic numbers ρ R and the Hahn field ρ R(t R ), where ρ R is the residue class field of ρ R. Then, assuming that * R is fully saturated we show that ρ R is isomorphic to * R and so ρ R contains a copy of * R. As a consequence (that is important for applications in non-linear theory of generalized functions) we show that every two fields of asymptotic numbers corresponding to different scales are isomorphic.
arXiv (Cornell University), Jan 29, 2011
We offer an axiomatic definition of a differential algebra of generalized functions over an algeb... more We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.
In these lecture notes we present an introduction to non-standard analysis especially written for... more 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. MSC: Functional Analysis (46F30); Generalized Solutions of PDE (35D05).
Novi Sad Journal of Mathematics, Jul 12, 2015
Dedicated to the memory of Christo Ya. Christov* on the 100-th anniversary of his birth and to Ja... more Dedicated to the memory of Christo Ya. Christov* on the 100-th anniversary of his birth and to James Vickers on the occasion his 60-th birthday.
arXiv (Cornell University), Aug 22, 2011
We present a characterization of the completeness of the field of real numbers in the form of a c... more 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.
Nonlinear Theory of Generalized Functions, 2022
The algebra of asymptotic functions ρ E(Ω) on an open set Ω ⊂ R d was introduced by M. Oberguggen... more The algebra of asymptotic functions ρ E(Ω) on an open set Ω ⊂ R d was introduced by M. Oberguggenberger and the author of this paper in the framework of A. Robinson's nonstandard analysis. It can be described as a differential associative and commutative ring (of generalized functions) which is an algebra over the field of A. Robinson's asymptotic numbers ρ C (A. Robinson [13] and A. H. Lightstone and A. Robinson [6]). Moreover, ρ E((Ω) is supplied with the chain of imbeddings: ) where E(Ω) denotes the differential ring of the C ∞ -functions (complex valued) on Ω, D(Ω) denotes the differential ring of the functions in E(Ω) with compact support in Ω and D ′ (Ω) denotes the differential linear space of Schwartz distributions on Ω. Here E(Ω) ⊂ D ′ (Ω) is the usual imbedding in the sense of distribution theory (V. Vladimirov [16]). The imbedding D ′ (Ω) ⊂ ρ E(Ω), constructed in (M. Oberguggenberger and T. Todorov [10]), preserves all linear operations, including partial differentiation of any order, and the pairing between D ′ (Ω) and D(Ω). Finally, the imbedding E(Ω) ⊂ ρ E(Ω) preserves all differential ring operations. In addition, if T d denotes the usual topology on R d , then the family { ρ E(Ω)} Ω∈T d is a sheaf of differential rings and the imbeddings in (0.1) are sheaf preserving. That means, in particular, that the restriction F |Ω ′ is a well defined element in ρ E(Ω ′ ) for any and any open Ω ′ ⊆ Ω. On these grounds we consider the asymptotic functions in ρ E(Ω) as "generalized functions on Ω": they are "generalized" functions (rather than "classical") because they are not mappings from Ω into C. On the other hand, they are still "functions" (although, "generalized" ones) : a) because of the imbeddings (0.1) and b) because of the sheaf properties mentioned above. As a result, ρ E(Ω), supplied with the imbeddings (0.1), offers a solution to
We present a characterization of the completeness of the field of real numbers in the form of a c... more 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 usual ǫ, δ-definition of the limit of a function (whether presented at a rigorous or an intui... more The usual ǫ, 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 ǫ, 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.
arXiv: Logic, 2015
We present a characterization of the completeness of the field of real numbers in the form of a \... more We present a characterization of the completeness of the field of real numbers in the form of a \emph{collection of several 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 the characterization of the reals to argue that the Leibniz-Euler infinitesimal calculus in the $17^\textrm{th}$-$18^\textrm{th}$ centuries 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. This article is directed to all mathematicians and scientists who are interested in the foundations of mathematics and the history of infinitesimal calculus.
Transactions of the American Mathematical Society, 1996
We prove the existence of solutions for essentially all linear partial differential equations wit... more We prove the existence of solutions for essentially all linear partial differential equations with C ∞ C^\infty -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 ∞ C^\infty -function.
Journal of Logic and Analysis, 2021
We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) b... more We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bases. As an application we prove the surjectivity of a large class of linear partial differential operators with smooth ($\mathcal C^\infty$-coefficients) coefficients, called in the article \emph{regular}, acting on the algebraic dual $\mathcal D^*(\Omega)$ of the space of test-functions $\mathcal D(\Omega)$. The surjectivity of the partial differential operators guarantees solvability of the corresponding partial differential equations within $\mathcal D^*(\Omega)$. We discuss our result in contrast to and comparison with similar results about the restrictions of the regular operators on the space of Schwartz distribution $\mathcal D^\prime(\Omega)$, where these operators are often non-surjective.
Novi Sad Journal of Mathematics, 2015
Dedicated to the memory of Christo Ya. Christov* on the 100-th anniversary of his birth and to Ja... more Dedicated to the memory of Christo Ya. Christov* on the 100-th anniversary of his birth and to James Vickers on the occasion his 60-th birthday.
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Papers by Todor D. Todorov