Lectures On Differential Galois Theory

Author : Andy R. Magid
Publisher : American Mathematical Soc.
Page : 105 pages
File Size : 45,9 Mb
Release : 1994
Category : Mathematics
ISBN : 9780821870044

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Lectures on Differential Galois Theory by Andy R. Magid Pdf

Differential Galois theory studies solutions of differential equations over a differential base field. In much the same way that ordinary Galois theory is the theory of field extensions generated by solutions of (one variable) polynomial equations, differential Galois theory looks at the nature of the differential field extension generated by the solutions of differential equations. An additional feature is that the corresponding differential Galois groups (of automorphisms of the extension fixing the base and commuting with the derivation) are algebraic groups. This book deals with the differential Galois theory of linear homogeneous differential equations, whose differential Galois groups are algebraic matrix groups.In addition to providing a convenient path to Galois theory, this approach also leads to the constructive solution of the inverse problem of differential Galois theory for various classes of algebraic groups. Providing a self-contained development and many explicit examples, this book provides a unique approach to differential Galois theory and is suitable as a textbook at the advanced graduate level.

Author : Anonim
Publisher : Unknown
Page : 105 pages
File Size : 44,8 Mb
Release : 1994
Category : Differential equations
ISBN : 1470421569

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Lectures on Differential Galois Theory by Anonim Pdf

Author : Emil Artin
Publisher : Courier Corporation
Page : 86 pages
File Size : 46,5 Mb
Release : 2012-05-24
Category : Mathematics
ISBN : 9780486158259

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Galois Theory by Emil Artin Pdf

Clearly presented discussions of fields, vector spaces, homogeneous linear equations, extension fields, polynomials, algebraic elements, as well as sections on solvable groups, permutation groups, solution of equations by radicals, and other concepts. 1966 edition.

Author : Jacques Sauloy
Publisher : American Mathematical Soc.
Page : 275 pages
File Size : 41,8 Mb
Release : 2016-12-07
Category : Galois theory
ISBN : 9781470430955

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Differential Galois Theory through Riemann-Hilbert Correspondence by Jacques Sauloy Pdf

Differential Galois theory is an important, fast developing area which appears more and more in graduate courses since it mixes fundamental objects from many different areas of mathematics in a stimulating context. For a long time, the dominant approach, usually called Picard-Vessiot Theory, was purely algebraic. This approach has been extensively developed and is well covered in the literature. An alternative approach consists in tagging algebraic objects with transcendental information which enriches the understanding and brings not only new points of view but also new solutions. It is very powerful and can be applied in situations where the Picard-Vessiot approach is not easily extended. This book offers a hands-on transcendental approach to differential Galois theory, based on the Riemann-Hilbert correspondence. Along the way, it provides a smooth, down-to-earth introduction to algebraic geometry, category theory and tannakian duality. Since the book studies only complex analytic linear differential equations, the main prerequisites are complex function theory, linear algebra, and an elementary knowledge of groups and of polynomials in many variables. A large variety of examples, exercises, and theoretical constructions, often via explicit computations, offers first-year graduate students an accessible entry into this exciting area.

Author : Teresa Crespo,Zbigniew Hajto
Publisher : American Mathematical Soc.
Page : 242 pages
File Size : 45,5 Mb
Release : 2011
Category : Differential algebraic groups
ISBN : 9780821853184

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Algebraic Groups and Differential Galois Theory by Teresa Crespo,Zbigniew Hajto Pdf

Differential Galois theory has seen intense research activity during the last decades in several directions: elaboration of more general theories, computational aspects, model theoretic approaches, applications to classical and quantum mechanics as well as to other mathematical areas such as number theory. This book intends to introduce the reader to this subject by presenting Picard-Vessiot theory, i.e. Galois theory of linear differential equations, in a self-contained way. The needed prerequisites from algebraic geometry and algebraic groups are contained in the first two parts of the book. The third part includes Picard-Vessiot extensions, the fundamental theorem of Picard-Vessiot theory, solvability by quadratures, Fuchsian equations, monodromy group and Kovacic's algorithm. Over one hundred exercises will help to assimilate the concepts and to introduce the reader to some topics beyond the scope of this book. This book is suitable for a graduate course in differential Galois theory. The last chapter contains several suggestions for further reading encouraging the reader to enter more deeply into different topics of differential Galois theory or related fields.

Author : Michio Kuga
Publisher : Springer Science & Business Media
Page : 147 pages
File Size : 54,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461203292

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Galois’ Dream: Group Theory and Differential Equations by Michio Kuga Pdf

First year, undergraduate, mathematics students in Japan have for many years had the opportunity of a unique experience---an introduction, at an elementary level, to some very advanced ideas in mathematics from one of the leading mathematicians of the world. English reading students now have the opportunity to enjoy this lively presentation, from elementary ideas to cartoons to funny examples, and to follow the mind of an imaginative and creative mathematician into a world of enduring mathematical creations.

Author : Marius van der Put,Michael F. Singer
Publisher : Springer Science & Business Media
Page : 438 pages
File Size : 44,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642557507

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Galois Theory of Linear Differential Equations by Marius van der Put,Michael F. Singer Pdf

From the reviews: "This is a great book, which will hopefully become a classic in the subject of differential Galois theory. [...] the specialist, as well as the novice, have long been missing an introductory book covering also specific and advanced research topics. This gap is filled by the volume under review, and more than satisfactorily." Mathematical Reviews

Author : Juan J. Morales Ruiz
Publisher : Birkhäuser
Page : 177 pages
File Size : 46,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034887182

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Differential Galois Theory and Non-Integrability of Hamiltonian Systems by Juan J. Morales Ruiz Pdf

This book is devoted to the relation between two different concepts of integrability: the complete integrability of complex analytical Hamiltonian systems and the integrability of complex analytical linear differential equations. For linear differential equations, integrability is made precise within the framework of differential Galois theory. The connection of these two integrability notions is given by the variational equation (i.e. linearized equation) along a particular integral curve of the Hamiltonian system. The underlying heuristic idea, which motivated the main results presented in this monograph, is that a necessary condition for the integrability of a Hamiltonian system is the integrability of the variational equation along any of its particular integral curves. This idea led to the algebraic non-integrability criteria for Hamiltonian systems. These criteria can be considered as generalizations of classical non-integrability results by Poincaré and Lyapunov, as well as more recent results by Ziglin and Yoshida. Thus, by means of the differential Galois theory it is not only possible to understand all these approaches in a unified way but also to improve them. Several important applications are also included: homogeneous potentials, Bianchi IX cosmological model, three-body problem, Hénon-Heiles system, etc. The book is based on the original joint research of the author with J.M. Peris, J.P. Ramis and C. Simó, but an effort was made to present these achievements in their logical order rather than their historical one. The necessary background on differential Galois theory and Hamiltonian systems is included, and several new problems and conjectures which open new lines of research are proposed. - - - The book is an excellent introduction to non-integrability methods in Hamiltonian mechanics and brings the reader to the forefront of research in the area. The inclusion of a large number of worked-out examples, many of wide applied interest, is commendable. There are many historical references, and an extensive bibliography. (Mathematical Reviews) For readers already prepared in the two prerequisite subjects [differential Galois theory and Hamiltonian dynamical systems], the author has provided a logically accessible account of a remarkable interaction between differential algebra and dynamics. (Zentralblatt MATH)

Author : Emil Artin
Publisher : American Mathematical Soc.
Page : 137 pages
File Size : 51,7 Mb
Release : 2007
Category : Algebra
ISBN : 9780821841297

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Algebra with Galois Theory by Emil Artin Pdf

'Algebra with Galois Theory' is based on lectures by Emil Artin. The book is an ideal textbook for instructors and a supplementary or primary textbook for students.

Author : George Janelidze
Publisher : American Mathematical Soc.
Page : 582 pages
File Size : 52,6 Mb
Release : 2004
Category : Mathematics
ISBN : 9780821832905

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Galois Theory, Hopf Algebras, and Semiabelian Categories by George Janelidze Pdf

This volume is based on talks given at the Workshop on Categorical Structures for Descent and Galois Theory, Hopf Algebras, and Semiabelian Categories held at The Fields Institute for Research in Mathematical Sciences (Toronto, ON, Canada). The meeting brought together researchers working in these interrelated areas. This collection of survey and research papers gives an up-to-date account of the many current connections among Galois theories, Hopf algebras, and semiabeliancategories. The book features articles by leading researchers on a wide range of themes, specifically, abstract Galois theory, Hopf algebras, and categorical structures, in particular quantum categories and higher-dimensional structures. Articles are suitable for graduate students and researchers,specifically those interested in Galois theory and Hopf algebras and their categorical unification.

Author : M.M. Postnikov
Publisher : Elsevier
Page : 123 pages
File Size : 49,9 Mb
Release : 2014-07-10
Category : Mathematics
ISBN : 9781483156477

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Foundations of Galois Theory by M.M. Postnikov Pdf

Foundations of Galois Theory is an introduction to group theory, field theory, and the basic concepts of abstract algebra. The text is divided into two parts. Part I presents the elements of Galois Theory, in which chapters are devoted to the presentation of the elements of field theory, facts from the theory of groups, and the applications of Galois Theory. Part II focuses on the development of general Galois Theory and its use in the solution of equations by radicals. Equations that are solvable by radicals; the construction of equations solvable by radicals; and the unsolvability by radicals of the general equation of degree n ? 5 are discussed as well. Mathematicians, physicists, researchers, and students of mathematics will find this book highly useful.

Author : Frédéric Fauvet,Jean-Pierre Ramis
Publisher : European Mathematical Society
Page : 284 pages
File Size : 44,6 Mb
Release : 2009
Category : Mathematics
ISBN : 3037190736

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Renormalization and Galois Theories by Frédéric Fauvet,Jean-Pierre Ramis Pdf

This volume is the outcome of a CIRM Workshop on Renormalization and Galois Theories held in Luminy, France, in March 2006. The subject of this workshop was the interaction and relationship between four currently very active areas: renormalization in quantum field theory (QFT), differential Galois theory, noncommutative geometry, motives and Galois theory. The last decade has seen a burst of new techniques to cope with the various mathematical questions involved in QFT, with notably the development of a Hopf-algebraic approach and insights into the classes of numbers and special functions that systematically appear in the calculations of perturbative QFT (pQFT). The analysis of the ambiguities of resummation of the divergent series of pQFT, an old problem, has been renewed, using recent results on Gevrey asymptotics, generalized Borel summation, Stokes phenomenon and resurgent functions. The purpose of the present book is to highlight, in the context of renormalization, the convergence of these various themes, orchestrated by diverse Galois theories. It contains three lecture courses together with five research articles and will be useful to both researchers and graduate students in mathematics and physics.

Author : Marius van der Put,Michael F. Singer
Publisher : Springer
Page : 182 pages
File Size : 40,9 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540692416

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Galois Theory of Difference Equations by Marius van der Put,Michael F. Singer Pdf

This book lays the algebraic foundations of a Galois theory of linear difference equations and shows its relationship to the analytic problem of finding meromorphic functions asymptotic to formal solutions of difference equations. Classically, this latter question was attacked by Birkhoff and Tritzinsky and the present work corrects and greatly generalizes their contributions. In addition results are presented concerning the inverse problem in Galois theory, effective computation of Galois groups, algebraic properties of sequences, phenomena in positive characteristics, and q-difference equations. The book is aimed at advanced graduate researchers and researchers.

Author : Ian Stewart
Publisher : CRC Press
Page : 386 pages
File Size : 47,9 Mb
Release : 2022-09-07
Category : Mathematics
ISBN : 9781000644067

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Galois Theory by Ian Stewart Pdf

Since 1973, Galois theory has been educating undergraduate students on Galois groups and classical Galois theory. In Galois Theory, Fifth Edition, mathematician and popular science author Ian Stewart updates this well-established textbook for today’s algebra students. New to the Fifth Edition Reorganised and revised Chapters 7 and 13 New exercises and examples Expanded, updated references Further historical material on figures besides Galois: Omar Khayyam, Vandermonde, Ruffini, and Abel A new final chapter discussing other directions in which Galois theory has developed: the inverse Galois problem, differential Galois theory, and a (very) brief introduction to p-adic Galois representations This bestseller continues to deliver a rigorous, yet engaging, treatment of the subject while keeping pace with current educational requirements. More than 200 exercises and a wealth of historical notes augment the proofs, formulas, and theorems.

Author : Charlotte Hardouin,Jacques Sauloy,Michael F. Singer
Publisher : American Mathematical Soc.
Page : 171 pages
File Size : 54,6 Mb
Release : 2016-04-27
Category : Difference and functional equations -- Difference equations -- Linear equations
ISBN : 9781470426552

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Galois Theories of Linear Difference Equations: An Introduction by Charlotte Hardouin,Jacques Sauloy,Michael F. Singer Pdf

This book is a collection of three introductory tutorials coming out of three courses given at the CIMPA Research School “Galois Theory of Difference Equations” in Santa Marta, Columbia, July 23–August 1, 2012. The aim of these tutorials is to introduce the reader to three Galois theories of linear difference equations and their interrelations. Each of the three articles addresses a different galoisian aspect of linear difference equations. The authors motivate and give elementary examples of the basic ideas and techniques, providing the reader with an entry to current research. In addition each article contains an extensive bibliography that includes recent papers; the authors have provided pointers to these articles allowing the interested reader to explore further.