Basic Modern Theory Of Linear Complex Analytic Q Difference Equations
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Basic Modern Theory Of Linear Complex Analytic Q Difference Equations
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Author : Jacques Sauloy
language : en
Publisher: American Mathematical Society
Release Date : 2024-11-06
Basic Modern Theory Of Linear Complex Analytic Q Difference Equations written by Jacques Sauloy and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-11-06 with Mathematics categories.
The roots of the modern theories of differential and $q$-difference equations go back in part to an article by George D. Birkhoff, published in 1913, dealing with the three ?sister theories? of differential, difference and $q$-difference equations. This book is about $q$-difference equations and focuses on techniques inspired by differential equations, in line with Birkhoff's work, as revived over the last three decades. It follows the approach of the Ramis school, mixing algebraic and analytic methods. While it uses some $q$-calculus and is illustrated by $q$-special functions, these are not its main subjects. After a gentle historical introduction with emphasis on mathematics and a thorough study of basic problems such as elementary $q$-functions, elementary $q$-calculus, and low order equations, a detailed algebraic and analytic study of scalar equations is followed by the usual process of transforming them into systems and back again. The structural algebraic and analytic properties of systems are then described using $q$-difference modules (Newton polygon, filtration by the slopes). The final chapters deal with Fuchsian and irregular equations and systems, including their resolution, classification, Riemann-Hilbert correspondence, and Galois theory. Nine appendices complete the book and aim to help the reader by providing some fundamental yet not universally taught facts. There are 535 exercises of various styles and levels of difficulty. The main prerequisites are general algebra and analysis as taught in the first three years of university. The book will be of interest to expert and non-expert researchers as well as graduate students in mathematics and physics.
Galois Theories Of Linear Difference Equations An Introduction
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Author : Charlotte Hardouin
language : en
Publisher: American Mathematical Soc.
Release Date : 2016-04-27
Galois Theories Of Linear Difference Equations An Introduction written by Charlotte Hardouin and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-04-27 with Mathematics categories.
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.
Handbook Of Geometry And Topology Of Singularities Vi Foliations
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Author : Felipe Cano
language : en
Publisher: Springer Nature
Release Date : 2024-06-20
Handbook Of Geometry And Topology Of Singularities Vi Foliations written by Felipe Cano and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-06-20 with Mathematics categories.
This is the sixth volume of the Handbook of Geometry and Topology of Singularities, a series which aims to provide an accessible account of the state-of-the-art of the subject, its frontiers, and its interactions with other areas of research. Singularities are ubiquitous in mathematics and science in general, and singularity theory is a crucible where different types of mathematical problems converge, surprising connections are born and simple questions lead to ideas which resonate in other parts of the subject, and in other subjects. This Volume VI goes together with Volume V and focuses on singular holomorphic foliations, which is a multidisciplinary field and a whole area of mathematics in itself. Singular foliations arise, for instance, by considering: The fibers of a smooth map between differentiable manifolds, with singularities at the critical points. The integral lines of a vector field, or the action of a Lie group on a manifold. The singularities are the orbits with special isotropy. The kernel of appropriate 1-forms. The singularities are the zeroes of the form. Open books, which naturally appear in singularity theory, are foliations with singular set the binding. These important examples highlight the deep connections between foliations and singularity theory. This volume consists of nine chapters, authored by world experts, which provide in-depth and reader-friendly introductions to some of the foundational aspects of the theory. These introductions also give insights into important lines of further research. Volume VI ends with an Epilogue by one of the current world leaders in the theory of complex foliations, with plenty of open questions and ideas for further research. The book is addressed to graduate students and newcomers to the theory, as well as to specialists who can use it as a guidebook.
Degree Theory And Symmetric Equations Assisted By Gap System
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Author : Zalman Balanov
language : en
Publisher: American Mathematical Society
Release Date : 2025-01-30
Degree Theory And Symmetric Equations Assisted By Gap System written by Zalman Balanov and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-01-30 with Mathematics categories.
Symmetries are a common feature of real-world phenomena in many fields, including physics, biology, materials science, and engineering. They can help understand the behavior of a system and optimize engineering designs. Nonlinear effects such as delays, nonsmoothness, and hysteresis can have a significant impact on the dynamics and contribute to the increased complexity of symmetric systems. The goal of this book is to provide a complete theoretical and practical manual for studying a large class of dynamical problems with symmetries using degree theory methods. To study the impact of symmetries on the occurrence of periodic solutions in dynamical systems, special variants of the Brouwer degree, the Brouwer equivariant degree, and the twisted equivariant degree are developed to predict patterns, regularities, and symmetries of solutions. Applications to specific dynamical systems and examples are supported by a software package integrated with the GAP system, which provides assistance in the group-theoretic computations involved in equivariant analysis. This book is intended for readers with a basic knowledge of analysis and algebra, including researchers in pure and applied mathematical analysis, graduate students, and scientists interested in areas involving mathematical modeling of symmetric phenomena. The text is self-contained, and the necessary background material is provided in the appendices.
Iwasawa Theory And Its Perspective Volume 3
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Author : Tadashi Ochiai
language : en
Publisher: American Mathematical Society
Release Date : 2025-06-13
Iwasawa Theory And Its Perspective Volume 3 written by Tadashi Ochiai and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-06-13 with Mathematics categories.
Iwasawa theory began in the late 1950s with a series of papers by Kenkichi Iwasawa on ideal class groups in the cyclotomic tower of number fields and their relation to $p$-adic $L$-functions. The theory was later generalized by putting it in the context of elliptic curves and modular forms. The main motivation for writing this book was the need for a total perspective of Iwasawa theory that includes the new trends of generalized Iwasawa theory. Another motivation is to update the classical theory for class groups, taking into account the changed point of view on Iwasawa theory. The goal of this third part of the three-part publication is to present additional aspects of the Iwasawa theory of $p$-adic Galois deformations.
Symmetries And Related Topics In Differential And Difference Equations
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Author : David Blázquez-Sanz
language : en
Publisher: American Mathematical Soc.
Release Date : 2011
Symmetries And Related Topics In Differential And Difference Equations written by David Blázquez-Sanz and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011 with Mathematics categories.
The papers collected here discuss topics such as Lie symmetries, equivalence transformations and differential invariants, group theoretical methods in linear equations, and the development of some geometrical methods in theoretical physics. The reader will find new results in symmetries of differential and difference equations, applications in classical and quantum mechanics, two fundamental problems of theoretical mechanics, and the mathematical nature of time in Lagrangian mechanics.
Differential Galois Theory Through Riemann Hilbert Correspondence
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Author : Jacques Sauloy
language : en
Publisher: American Mathematical Soc.
Release Date : 2016-12-07
Differential Galois Theory Through Riemann Hilbert Correspondence written by Jacques Sauloy and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-12-07 with Mathematics categories.
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.
Complex Differential And Difference Equations
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Author : Galina Filipuk
language : en
Publisher: Walter de Gruyter GmbH & Co KG
Release Date : 2019-11-18
Complex Differential And Difference Equations written by Galina Filipuk and has been published by Walter de Gruyter GmbH & Co KG this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-11-18 with Mathematics categories.
With a balanced combination of longer survey articles and shorter, peer-reviewed research-level presentations on the topic of differential and difference equations on the complex domain, this edited volume presents an up-to-date overview of areas such as WKB analysis, summability, resurgence, formal solutions, integrability, and several algebraic aspects of differential and difference equations.
Discrete Painlev Equations
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Author : Nalini Joshi
language : en
Publisher: American Mathematical Soc.
Release Date : 2019-05-30
Discrete Painlev Equations written by Nalini Joshi and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-05-30 with Mathematics categories.
Discrete Painlevé equations are nonlinear difference equations, which arise from translations on crystallographic lattices. The deceptive simplicity of this statement hides immensely rich mathematical properties, connecting dynamical systems, algebraic geometry, Coxeter groups, topology, special functions theory, and mathematical physics. This book necessarily starts with introductory material to give the reader an accessible entry point to this vast subject matter. It is based on lectures that the author presented as principal lecturer at a Conference Board of Mathematical Sciences and National Science Foundation conference in Texas in 2016. Instead of technical theorems or complete proofs, the book relies on providing essential points of many arguments through explicit examples, with the hope that they will be useful for applied mathematicians and physicists.
Lectures On Analytic Differential Equations
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Author : I︠U︡. S. Ilʹi︠a︡shenko
language : en
Publisher: American Mathematical Soc.
Release Date : 2008
Lectures On Analytic Differential Equations written by I︠U︡. S. Ilʹi︠a︡shenko and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Mathematics categories.
The book combines the features of a graduate-level textbook with those of a research monograph and survey of the recent results on analysis and geometry of differential equations in the real and complex domain. As a graduate textbook, it includes self-contained, sometimes considerably simplified demonstrations of several fundamental results, which previously appeared only in journal publications (desingularization of planar analytic vector fields, existence of analytic separatrices, positive and negative results on the Riemann-Hilbert problem, Ecalle-Voronin and Martinet-Ramis moduli, solution of the Poincare problem on the degree of an algebraic separatrix, etc.). As a research monograph, it explores in a systematic way the algebraic decidability of local classification problems, rigidity of holomorphic foliations, etc. Each section ends with a collection of problems, partly intended to help the reader to gain understanding and experience with the material, partly drafting demonstrations of the mor The exposition of the book is mostly geometric, though the algebraic side of the constructions is also prominently featured. on several occasions the reader is introduced to adjacent areas, such as intersection theory for divisors on the projective plane or geometric theory of holomorphic vector bundles with meromorphic connections. The book provides the reader with the principal tools of the modern theory of analytic differential equations and intends to serve as a standard source for references in this area.