Descartes’ philosophy contains an intriguing notion of the infinite, a concept labeled by the philosopher as indefinite. Even though Descartes clearly defined this term on several occasions in the correspondence with his contemporaries, as well as in his Principles of Philosophy, numerous problems about its meaning have arisen over the years. Most commentators reject the view that the indefinite could mean a real thing and, instead, identify it with an Aristotelian potential infinite. In the first part of this article, I show why there is no numerical infinity in Cartesian mathematics, as such a concept would be inconsistent with the main fundamental attribute of numbers: to be comparable with each other. In the second part, I analyze the indefinite in the context of Descartes’ mathematical physics. It is my contention that, even with no trace of infinite in his mathematics, Descartes does refer to an actual indefinite because of its application to the material world within the system of his physics. This fact underlines a discrepancy between his mathematics and physics of the infinite, but does not lead a difficulty in his mathematical physics. Thus, in Descartes’ physics, the indefinite refers to an actual dimension of the world rather than to an Aristotelian mathematical potential infinity. In fact...

We consider the following first order systems of mathematical physics. 1.The Dirac equation with scalar potential. 2.The Dirac equation with electric potential. 3.The Dirac equation with pseudoscalar potential. 4.The system describing non-linear force free magnetic fields or Beltrami fields with nonconstant proportionality factor. 5.The Maxwell equations for slowly changing media. 6.The static Maxwell system. We show that all this variety of first order systems reduces to a single quaternionic equation the analysis of which in its turn reduces to the solution of a Schroedinger equation with biquaternionic potential. In some important situations the biquaternionic potential can be diagonalized and converted into scalar potentials.

At present the theory of skew-symmetric exterior differential forms has been developed. The closed exterior forms possess the invariant properties that are of great importance. The operators of the exterior form theory lie at the basis of the differential and integral operators of the field theory. However, the theory of exterior forms, being invariant one, does not answer the questions related to the evolutionary processes. In the work the readers are introduced to the skew-symmetric differential forms that possess evolutionary properties. They were called evolutionary ones. The radical distinction between the evolutionary forms and the exterior ones consists in the fact that the exterior forms are defined on manifolds with closed metric forms, whereas the evolutionary forms are defined on manifolds with unclosed metric forms. The mathematical apparatus of exterior and evolutionary forms allows description of discrete transitions, quantum steps, evolutionary processes, generation of various structures. These are radically new possibilities of the mathematical physics. A role of exterior and evolutionary forms in the mathematical physics is conditioned by the fact that they reflect properties of the conservation laws and allow elucidate a mechanism of evolutionary processes in material media...

The present issue of the series <> represents the Proceedings of the Students Training Contest Olympiad in Mathematical and Theoretical Physics and includes the statements and the solutions of the problems offered to the participants. The contest Olympiad was held on May 21st-24th, 2010 by Scientific Research Laboratory of Mathematical Physics of Samara State University, Steklov Mathematical Institute of Russia's Academy of Sciences, and Moscow Institute of Physics and Technology (State University) in cooperation. The present Proceedings is intended to be used by the students of physical and mechanical-mathematical departments of the universities, who are interested in acquiring a deeper knowledge of the methods of mathematical and theoretical physics, and could be also useful for the persons involved in teaching mathematical and theoretical physics.; Comment: 68 pages, Proceedings of the statements and solutions of the problems of the Students Training Contest Olympiad in Mathematical and Theoretical Physics. The subjects covered by the problems include classical mechanics, integrable nonlinear systems, probability, integral equations, PDE, quantum and particle physics, cosmology...

The Spencer operator, introduced by D.C. Spencer fifty years ago, is rarely used in mathematics today and, up to our knowledge, has never been used in engineering applications or mathematical physics. The main purpose of this paper, an extended version of a lecture at the second workshop on Differential Equations by Algebraic Methods (DEAM2, february 9-11, 2011, Linz, Austria) is to prove that the use of the Spencer operator constitutes the common secret of the three following famous books published about at the same time in the beginning of the last century, though they do not seem to have anything in common at first sight as they are successively dealing with elasticity theory, commutative algebra, electromagnetism and general relativity: (C) E. and F. COSSERAT: "Th\'eorie des Corps D\'eformables", Hermann, Paris, 1909. (M) F.S. MACAULAY: "The Algebraic Theory of Modular Systems", Cambridge University Press, 1916. (W) H. WEYL: "Space, Time, Matter", Springer, Berlin, 1918 (1922, 1958; Dover, 1952). Meanwhile, we shall point out the importance of (M) for studying control identifiability and of (C)+(W) for the group theoretical unification of finite elements in engineering sciences, recovering in a purely mathematical way well known field-matter coupling phenomena (piezzoelectricity...

This volume contains the proceedings of an International Workshop on Idempotent and Tropical Mathematics and Problems of Mathematical Physics, held at the Independent University of Moscow, Russia, on August 25-30, 2007.; Comment: This volume contains the proceedings of an International Workshop on Idempotent and Tropical Mathematics and Problems of Mathematical Physics, held at the Independent University of Moscow, Russia, on August 25-30, 2007

This set of lecture notes gives an introduction to holomorphic function spaces as used in mathematical physics. The emphasis is on the Segal-Bargmann space and the canonical commutation relations. Later sections describe more advanced topics such as the Segal-Bargmann transform for compact Lie groups and the infinite-dimensional theory.; Comment: Final version

One of the most effective techniques of experimental mathematics is to compute mathematical entities such as integrals, series or limits to high precision, then attempt to recognize the resulting numerical values. Recently these techniques have been applied with great success to problems in mathematical physics. Notable among these applications are the identification of some key multi-dimensional integrals that arise in Ising theory, quantum field theory and in magnetic spin theory.; Comment: 18 pages, 2 figures

The study of integrability of the mathematical physics equations showed that the differential equations describing real processes are not integrable without additional conditions. This follows from the functional relation that is derived from these equations. Such a relation connects the differential of state functional and the skew-symmetric form. This relation proves to be nonidentical, and this fact points to the nonintegrability of the equations. In this case a solution to the equations is a functional, which depends on the commutator of skew-symmetric form that appears to be unclosed. However, under realization of the conditions of degenerate transformations, from the nonidentical relation it follows the identical one on some structure. This points out to the local integrability and realization of a generalized solution. In doing so, in addition to the exterior forms, the skew-symmetric forms, which, in contrast to exterior forms, are defined on nonintegrable manifolds (such as tangent manifolds of differential equations, Lagrangian manifolds and so on), were used. In the present paper, the partial differential equations, which describe any processes, the systems of differential equations of mechanics and physics of continuous medium and field theory equations are analyzed.; Comment: 8 pages...

This volume contains the proceedings of an International Workshop on Idempotent and Tropical Mathematics and Problems of Mathematical Physics, held at the Independent University of Moscow, Russia, on August 25-30, 2007.; Comment: This volume contains the proceedings of an International Workshop on Idempotent and Tropical Mathematics and Problems of Mathematical Physics, held at the Independent University of Moscow, Russia, on August 25-30, 2007

Application of adeles in modern mathematical physics is briefly reviewed. In particular, some adelic products are presented.; Comment: 9 pages. Invited talk at the international conference Actual Problems of Mathematics and Computer Modeling, 18-22 June 2007, Grodno, Belarus. To appear in the Proceedings

In the 6th Int. Symposium on OPSFA there were several communications dealing with concrete applications of orthogonal polynomials to experimental and theoretical physics, chemistry, biology and statistics. Here I make suggestions concerning the use of powerful apparatus of orthogonal polynomials and special functions in several lines of research in mathematical physics; Comment: LaTeX, 4 pages, Comunication presented to the 6th International Symposium on Orthogonal Polynomials, Special Functions and their Applications, Rome, June 2001 (late submission to arxiv.org)

A brief review of some selected topics in p-adic mathematical physics is presented.; Comment: 36 pages

Advances in mathematical physics during the 20th century led to the discovery of a relationship between group theory and representation theory with the theory of special functions. Specifically, it was discovered that many of the special functions are (1) specific matrix elements of matrix representations of Lie groups, and (2) basis functions of operator representations of Lie algebras. By viewing the special functions in this way, it is possible to derive many of their properties that were originally discovered using classical analysis, such as generating functions, differential relations, and recursion relations. This relationship is of interest to physicists due to the fact that many of the common special functions, such as Hermite polynomials and Bessel functions, are related to remarkably simple Lie groups used in physics. Unfortunately, much of the literature on this subject remains inaccessible to undergraduate students. The purpose of this project is to research the existing literature and to organize the results, presenting the information in a way that can be understood at the undergraduate level. The primary objects of study will be the Heisenberg group and its relationship to the Hermite polynomials, as well as the Euclidean group in the plane and its relationship to the Bessel functions. The ultimate goal is to make the results relevant for undergraduate students who have studied quantum mechanics.; Comment: Drexel undergraduate senior thesis in physics advised by Robert Gilmore. 43 pages...

Three types of equations of mathematical physics, namely, the equations, which describe any physical processes, the equations of mechanics and physics of continuous media, and field-theory equations are studied in this paper. In the first and second case the investigation is reduced to the analysis of the nonidentical relations of the skew-symmetric differential forms that are obtained from differential equations. It is shown that the integrability of equations and the properties of their solutions depend on the realization of the conditions of degenerate transformations under which the identical relations are obtained from the nonidentical relation. The field-theory equations, in contrast to the equations of first two types, are the relations made up by skew-symmetric differential forms or their analogs (differential or integral ones). This is due to the fact that the field-theory equations have to describe physical structures (to which closed exterior forms correspond) rather than physical quantities. The equations that correspond to field theories are obtained from the equations that describe the conservation laws (of energy, linear momentum, angular momentum, and mass) of material systems (of continuous media). This disclose a connection between field theories and the equations for material systems (and points to that material media generate physical fields).; Comment: 12pages

A role of skew-symmetric differential forms in mathematical physics relates to the fact that they reflect the properties of conservation laws. The closed exterior forms correspond to the conservation laws for physical fields, whereas the evolutionary forms correspond to the conservation laws for material media. Skew-symmetric differential forms can describe a conjugacy of any objects (that correspond to the conservation laws). The closed exterior forms describe conjugated objects. And the evolutionary forms, whose basis are deforming manifolds, describe the process of conjugating objects and obtaining conjugated objects. From the evolutionary forms the closed exterior forms are obtained. This shows that material media generate physical fields. The relation between evolutionary and closed exterior forms discloses the relation between the equations of mathematical physics and field theories. This explains the field theory postulates. Conjugacy is possible if there is symmetry. Symmetries of closed exterior forms, which are conditions of fulfilment of the conservation laws for physical fields, are interior symmetries of field theories. And symmetries of dual forms (due to the degrees of freedom of material media) are external symmetries of the equations of field theories. This shows connection between internal and external symmetries of field theories.; Comment: 16 pages

In the paper it is shown that, even without a knowledge of the concrete form of the equations of mathematical physics and field theories, with the help of skew-symmetric differential forms one can see specific features of the equations of mathematical physics, the relation between mathematical physics and field theory, to understand the mechanism of evolutionary processes that develop in material media and lead to emergency of physical structures forming physical fields. This discloses a physical meaning of such concepts like "conservation laws", "postulates" and "causality" and gives answers to many principal questions of mathematical physics and general field theory. In present paper, beside the exterior forms, the skew-symmetric differential forms, whose basis (in contrast to the exterior forms) are deforming manifolds, are used. Mathematical apparatus of such differential forms(which were named evolutionary ones) includes nontraditional elements like nonidentical relations and degenerate transformations and this enables one to describe discrete transitions, quantum steps, evolutionary processes, and generation of various structures.; Comment: 36 pages

This book considers posing and the methods of solving simple linear boundary-value problems in classical mathematical physics. The questions encompassed include: the fundamentals of calculus of variations; one-dimensional boundary-value problems in the oscillation and heat conduction theories, with a detailed analysis of the Sturm-Liouville boundary-value problem and substantiation of the Fourier method; sample solutions of the corresponding problems in two and three dimensions, with essential elements of the special function theory. The text is designed for Physics, Engineering, and Mathematics majors.; Comment: A university textbook (in Ukrainian), 380 pages, 40 figures, ISBN 978-966-190-912-9

The result from developing and applying the notions of functional self-similarity and the Bogoliubov renormalization group to boundary-value problems in mathematical physics during the last decade are reviewed. The main achievement is the regular algorithm for finding renormalization group-type symmetries using the contemporary theory of Lie groups of transformations.; Comment: Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol.121, No.1, pp.66-88, October, 1999

Personal view of author on goals and content of Mathematical Physics.; Comment: 9 pages Latex file