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## ‣ The Infinite within Descartes’ Mathematical Physics; Lo indefinido en la física matemática de Descartes [Inglés]

Fonte: Universidad del Norte
Publicador: Universidad del Norte

Tipo: article; publishedVersion
Formato: application/pdf; application/pdf

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#Philosophy#History of Philosophy#Descartes, infinity , mathematical physics , space , matter#Infinity#Descartes

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...

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## ‣ Quaternionic factorization of the Schroedinger operator and its applications to some first order systems of mathematical physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/05/2003
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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.

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## ‣ Exterior and evolutionary skew-symmetric differential forms and their role in mathematical physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/10/2003
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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...

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## ‣ Mathematical Physics : Problems and Solutions of The Students Training Contest Olympiad in Mathematical and Theoretical Physics (May 21st - 24th, 2010)

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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...

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## ‣ Spencer Operator and Applications: From Continuum Mechanics to Mathematical physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Analysis of PDEs#Mathematical Physics#Mathematics - Commutative Algebra#Mathematics - Differential Geometry#Mathematics - Group Theory

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...

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## ‣ Idempotent and tropical mathematics and problems of mathematical physics (Volume II)

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/09/2007
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#Mathematics - Rings and Algebras#Mathematical Physics#Mathematics - Algebraic Geometry#Mathematics - Analysis of PDEs#Mathematics - Operator Algebras#Mathematics - Optimization and Control#Mathematics - Representation Theory#00B10, 81Q20, 06F07, 35Q99, 49L90, 46S99, 81S99, 52B20, 52A41, 14P99

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

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## ‣ Holomorphic Methods in Mathematical Physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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## ‣ Experimental Mathematics and Mathematical Physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/05/2010
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#Mathematical Physics#High Energy Physics - Phenomenology#Mathematics - Classical Analysis and ODEs#Mathematics - Number Theory

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

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## ‣ Skew-symmetric forms: On integrability of equations of mathematical physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/10/2009
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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...

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## ‣ Idempotent and tropical mathematics and problems of mathematical physics (Volume I)

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/10/2007
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#Mathematics - Rings and Algebras#Mathematical Physics#Mathematics - Algebraic Geometry#Mathematics - Analysis of PDEs#Mathematics - Operator Algebras#Mathematics - Optimization and Control#Mathematics - Representation Theory#00B10, 81Q20, 06F07, 35Q99, 49L90, 46S99, 81S99, 52B20,52A41, 14P99

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

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## ‣ Adeles in Mathematical Physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/07/2007
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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

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## ‣ Orthogonal polynomials, special functions and mathematical physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 02/02/2004
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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)

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## ‣ p-Adic Mathematical Physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/04/2009
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#Mathematical Physics#General Relativity and Quantum Cosmology#High Energy Physics - Theory#Quantum Physics

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

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## ‣ An Overview of the Relationship between Group Theory and Representation Theory to the Special Functions in Mathematical Physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/09/2013
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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...

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## ‣ Specific features of differential equations of mathematical physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/02/2007
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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

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## ‣ Role of exterior and evolutionary skew-symmetric differential forms in mathematical physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/10/2005
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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

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## ‣ Analysis of the equations of mathematical physics and foundations of field theories with the help of skew-symmetric differential forms

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/12/2005
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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

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## ‣ Introduction to Mathematical Physics. Calculus of Variations and Boundary-value Problems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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## ‣ Functional self-similarity and renormalization group symmetry in mathematical physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/01/2000
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#Mathematical Physics#High Energy Physics - Theory#Nonlinear Sciences - Exactly Solvable and Integrable Systems

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

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## ‣ Modern Mathematical Physics: what it should be?

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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

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