Galois Connections And Applications
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Author | : K. Denecke |
Publisher | : Springer Science & Business Media |
Total Pages | : 511 |
Release | : 2013-11-11 |
Genre | : Mathematics |
ISBN | : 1402018983 |
Galois connections provide the order- or structure-preserving passage between two worlds of our imagination - and thus are inherent in hu man thinking wherever logical or mathematical reasoning about cer tain hierarchical structures is involved. Order-theoretically, a Galois connection is given simply by two opposite order-inverting (or order preserving) maps whose composition yields two closure operations (or one closure and one kernel operation in the order-preserving case). Thus, the "hierarchies" in the two opposite worlds are reversed or transported when passing to the other world, and going forth and back becomes a stationary process when iterated. The advantage of such an "adjoint situation" is that information about objects and relationships in one of the two worlds may be used to gain new information about the other world, and vice versa. In classical Galois theory, for instance, properties of permutation groups are used to study field extensions. Or, in algebraic geometry, a good knowledge of polynomial rings gives insight into the structure of curves, surfaces and other algebraic vari eties, and conversely. Moreover, restriction to the "Galois-closed" or "Galois-open" objects (the fixed points of the composite maps) leads to a precise "duality between two maximal subworlds".
Author | : Jörg Bewersdorff |
Publisher | : American Mathematical Soc. |
Total Pages | : 202 |
Release | : 2006 |
Genre | : Mathematics |
ISBN | : 0821838172 |
Galois theory is the culmination of a centuries-long search for a solution to the classical problem of solving algebraic equations by radicals. This book follows the historical development of the theory, emphasizing concrete examples along the way. It is suitable for undergraduates and beginning graduate students.
Author | : Kesav V. Nori |
Publisher | : |
Total Pages | : 519 |
Release | : 1986 |
Genre | : Artificial intelligence |
ISBN | : 9780387171623 |
Author | : David A. Cox |
Publisher | : John Wiley & Sons |
Total Pages | : 602 |
Release | : 2012-03-27 |
Genre | : Mathematics |
ISBN | : 1118218426 |
Praise for the First Edition ". . .will certainly fascinate anyone interested in abstractalgebra: a remarkable book!" —Monatshefte fur Mathematik Galois theory is one of the most established topics inmathematics, with historical roots that led to the development ofmany central concepts in modern algebra, including groups andfields. Covering classic applications of the theory, such assolvability by radicals, geometric constructions, and finitefields, Galois Theory, Second Edition delves into noveltopics like Abel’s theory of Abelian equations, casusirreducibili, and the Galois theory of origami. In addition, this book features detailed treatments of severaltopics not covered in standard texts on Galois theory,including: The contributions of Lagrange, Galois, and Kronecker How to compute Galois groups Galois's results about irreducible polynomials of primeor prime-squared degree Abel's theorem about geometric constructions on thelemniscates Galois groups of quartic polynomials in allcharacteristics Throughout the book, intriguing Mathematical Notes andHistorical Notes sections clarify the discussed ideas andthe historical context; numerous exercises and examples use Mapleand Mathematica to showcase the computations related to Galoistheory; and extensive references have been added to provide readerswith additional resources for further study. Galois Theory, Second Edition is an excellent book forcourses on abstract algebra at the upper-undergraduate and graduatelevels. The book also serves as an interesting reference for anyonewith a general interest in Galois theory and its contributions tothe field of mathematics.
Author | : Jacques Sauloy |
Publisher | : American Mathematical Soc. |
Total Pages | : 303 |
Release | : 2016-12-07 |
Genre | : Mathematics |
ISBN | : 1470430959 |
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 | : V. I. Arnold |
Publisher | : Cambridge University Press |
Total Pages | : 91 |
Release | : 2010-12-02 |
Genre | : Mathematics |
ISBN | : 1139493442 |
V. I. Arnold reveals some unexpected connections between such apparently unrelated theories as Galois fields, dynamical systems, ergodic theory, statistics, chaos and the geometry of projective structures on finite sets. The author blends experimental results with examples and geometrical explorations to make these findings accessible to a broad range of mathematicians, from undergraduate students to experienced researchers.
Author | : Francis Borceux |
Publisher | : Cambridge University Press |
Total Pages | : 360 |
Release | : 2001-02-22 |
Genre | : Mathematics |
ISBN | : 9780521803090 |
Develops Galois theory in a more general context, emphasizing category theory.
Author | : Michael Cochez |
Publisher | : Springer Nature |
Total Pages | : 158 |
Release | : 2021-04-16 |
Genre | : Computers |
ISBN | : 3030723089 |
This open access book constitutes the thoroughly refereed post-conference proceedings of the 6th International Workshop on Graph Structures for Knowledge Representation and Reasoning, GKR 2020, held virtually in September 2020, associated with ECAI 2020, the 24th European Conference on Artificial Intelligence. The 7 revised full papers presented together with 2 invited contributions were reviewed and selected from 9 submissions. The contributions address various issues for knowledge representation and reasoning and the common graph-theoretic background, which allows to bridge the gap between the different communities.
Author | : Brendan Fong |
Publisher | : Cambridge University Press |
Total Pages | : 351 |
Release | : 2019-07-18 |
Genre | : Computers |
ISBN | : 1108482295 |
Category theory reveals commonalities between structures of all sorts. This book shows its potential in science, engineering, and beyond.
Author | : George M. Bergman |
Publisher | : Springer |
Total Pages | : 574 |
Release | : 2015-02-05 |
Genre | : Mathematics |
ISBN | : 3319114786 |
Rich in examples and intuitive discussions, this book presents General Algebra using the unifying viewpoint of categories and functors. Starting with a survey, in non-category-theoretic terms, of many familiar and not-so-familiar constructions in algebra (plus two from topology for perspective), the reader is guided to an understanding and appreciation of the general concepts and tools unifying these constructions. Topics include: set theory, lattices, category theory, the formulation of universal constructions in category-theoretic terms, varieties of algebras, and adjunctions. A large number of exercises, from the routine to the challenging, interspersed through the text, develop the reader's grasp of the material, exhibit applications of the general theory to diverse areas of algebra, and in some cases point to outstanding open questions. Graduate students and researchers wishing to gain fluency in important mathematical constructions will welcome this carefully motivated book.