Galois Connections And Applications
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Galois Connections And Applications
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Author : K. Denecke
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-11-11
Galois Connections And Applications written by K. Denecke and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-11-11 with Mathematics categories.
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".
Galois Connections And Applications
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Author : Klaus Denecke
language : en
Publisher: Springer Science & Business Media
Release Date : 2004-03-31
Galois Connections And Applications written by Klaus Denecke and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-03-31 with Computers categories.
This book presents the main ideas of General Galois Theory as a generalization of Classical Galois Theory. It sketches the development of Galois connections through the last three centuries. Examples of Galois connections as powerful tools in Category Theory and Universal Algebra are given. Applications of Galois connections in Linguistic and Data Analysis are presented.
Galois Connections And Applications
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Author : Klaus Denecke
language : en
Publisher:
Release Date : 2014-01-15
Galois Connections And Applications written by Klaus Denecke and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-15 with categories.
Universal Algebra And Applications In Theoretical Computer Science
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Author : Klaus Denecke
language : en
Publisher: CRC Press
Release Date : 2018-10-03
Universal Algebra And Applications In Theoretical Computer Science written by Klaus Denecke and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-10-03 with Mathematics categories.
Over the past 20 years, the emergence of clone theory, hyperequational theory, commutator theory and tame congruence theory has led to a growth of universal algebra both in richness and in applications, especially in computer science. Yet most of the classic books on the subject are long out of print and, to date, no other book has integrated these theories with the long-established work that supports them. Universal Algebra and Applications in Theoretical Computer Science introduces the basic concepts of universal algebra and surveys some of the newer developments in the field. The first half of the book provides a solid grounding in the core material. A leisurely pace, careful exposition, numerous examples, and exercises combine to form an introduction to the subject ideal for beginning graduate students or researchers from other areas. The second half of the book focuses on applications in theoretical computer science and advanced topics, including Mal'cev conditions, tame congruence theory, clones, and commutators. The impact of the advances in universal algebra on computer science is just beginning to be realized, and the field will undoubtedly continue to grow and mature. Universal Algebra and Applications in Theoretical Computer Science forms an outstanding text and offers a unique opportunity to build the foundation needed for further developments in its theory and in its computer science applications.
Galois Theory
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Author : David A. Cox
language : en
Publisher: John Wiley & Sons
Release Date : 2012-03-27
Galois Theory written by David A. Cox and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-03-27 with Mathematics categories.
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.
Galois Theory For Beginners
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Author : Jörg Bewersdorff
language : en
Publisher: American Mathematical Soc.
Release Date : 2006
Galois Theory For Beginners written by Jörg Bewersdorff 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 2006 with Mathematics categories.
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.
Mordell Weil Lattices
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Author : Matthias Schütt
language : en
Publisher: Springer Nature
Release Date : 2019-10-17
Mordell Weil Lattices written by Matthias Schütt and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-10-17 with Mathematics categories.
This book lays out the theory of Mordell–Weil lattices, a very powerful and influential tool at the crossroads of algebraic geometry and number theory, which offers many fruitful connections to other areas of mathematics. The book presents all the ingredients entering into the theory of Mordell–Weil lattices in detail, notably, relevant portions of lattice theory, elliptic curves, and algebraic surfaces. After defining Mordell–Weil lattices, the authors provide several applications in depth. They start with the classification of rational elliptic surfaces. Then a useful connection with Galois representations is discussed. By developing the notion of excellent families, the authors are able to design many Galois representations with given Galois groups such as the Weyl groups of E6, E7 and E8. They also explain a connection to the classical topic of the 27 lines on a cubic surface. Two chapters deal with elliptic K3 surfaces, a pulsating area of recent research activity which highlights many central properties of Mordell–Weil lattices. Finally, the book turns to the rank problem—one of the key motivations for the introduction of Mordell–Weil lattices. The authors present the state of the art of the rank problem for elliptic curves both over Q and over C(t) and work out applications to the sphere packing problem. Throughout, the book includes many instructive examples illustrating the theory.
Graphs On Surfaces And Their Applications
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Author : Sergei K. Lando
language : en
Publisher: Springer Science & Business Media
Release Date : 2003-12-01
Graphs On Surfaces And Their Applications written by Sergei K. Lando and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-12-01 with Mathematics categories.
Graphs drawn on two-dimensional surfaces have always attracted researchers by their beauty and by the variety of difficult questions to which they give rise. The theory of such embedded graphs, which long seemed rather isolated, has witnessed the appearance of entirely unexpected new applications in recent decades, ranging from Galois theory to quantum gravity models, and has become a kind of a focus of a vast field of research. The book provides an accessible introduction to this new domain, including such topics as coverings of Riemann surfaces, the Galois group action on embedded graphs (Grothendieck's theory of "dessins d'enfants"), the matrix integral method, moduli spaces of curves, the topology of meromorphic functions, and combinatorial aspects of Vassiliev's knot invariants and, in an appendix by Don Zagier, the use of finite group representation theory. The presentation is concrete throughout, with numerous figures, examples (including computer calculations) and exercises, and should appeal to both graduate students and researchers.
Information Processing And Management Of Uncertainty In Knowledge Based Systems
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Author : Davide Ciucci
language : en
Publisher: Springer Nature
Release Date : 2022-07-04
Information Processing And Management Of Uncertainty In Knowledge Based Systems written by Davide Ciucci and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-07-04 with Computers categories.
This two-volume set (CCIS 1601-1602) constitutes the proceedings of the 19th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2021, held in Milan, Italy, in July 2022. The 124 papers were carefully reviewed and selected from 188 submissions. The papers are organized in topical sections as follows: aggregation theory beyond the unit interval; formal concept analysis and uncertainty; fuzzy implication functions; fuzzy mathematical analysis and its applications; generalized sets and operators; information fusion techniques based on aggregation functions, pre-aggregation functions, and their generalizations; interval uncertainty; knowledge acquisition, representation and reasoning; logical structures of opposition and logical syllogisms; mathematical fuzzy logics; theoretical and applied aspects of imprecise probabilities; data science and machine learning; decision making modeling and applications; e-health; fuzzy methods in data mining and knowledge discovery; soft computing and artificia intelligence techniques in image processing; soft methods in statistics and data analysis; uncertainty, heterogeneity, reliability and explainability in AI; weak and cautious supervised learning.
Galois Theories
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Author : Francis Borceux
language : en
Publisher: Cambridge University Press
Release Date : 2001-02-22
Galois Theories written by Francis Borceux and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-02-22 with Mathematics categories.
Starting from the classical finite-dimensional Galois theory of fields, this book develops Galois theory in a much more general context, presenting work by Grothendieck in terms of separable algebras and then proceeding to the infinite-dimensional case, which requires considering topological Galois groups. In the core of the book, the authors first formalize the categorical context in which a general Galois theorem holds, and then give applications to Galois theory for commutative rings, central extensions of groups, the topological theory of covering maps and a Galois theorem for toposes. The book is designed to be accessible to a wide audience: the prerequisites are first courses in algebra and general topology, together with some familiarity with the categorical notions of limit and adjoint functors. The first chapters are accessible to advanced undergraduates, with later ones at a graduate level. For all algebraists and category theorists this book will be a rewarding read.