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Algebraic Modeling Systems: Modeling and Solving Real World Optimization Problems (Applied Optimization #104)
by Josef KallrathThis book Algebraic Modeling Systems - Modeling and Solving Real World Optimization Problems - deals with the aspects of modeling and solving real-world optimization problems in a unique combination. It treats systematically the major algebraic modeling languages (AMLs) and modeling systems (AMLs) used to solve mathematical optimization problems. AMLs helped significantly to increase the usage of mathematical optimization in industry. Therefore it is logical consequence that the GOR (Gesellschaft für Operations Research) Working Group Mathematical Optimization in Real Life had a second meeting devoted to AMLs, which, after 7 years, followed the original 71st Meeting of the GOR (Gesellschaft für Operations Research) Working Group Mathematical Optimization in Real Life which was held under the title Modeling Languages in Mathematical Optimization during April 23-25, 2003 in the German Physics Society Conference Building in Bad Honnef, Germany. While the first meeting resulted in the book Modeling Languages in Mathematical Optimization, this book is an offspring of the 86th Meeting of the GOR working group which was again held in Bad Honnef under the title Modeling Languages in Mathematical Optimization.
Algebraic Modeling of Topological and Computational Structures and Applications: THALES, Athens, Greece, July 1-3, 2015 (Springer Proceedings in Mathematics & Statistics #219)
by Louis H. Kauffman Sofia Lambropoulou Doros Theodorou Petros StefaneasThis interdisciplinary book covers a wide range of subjects, from pure mathematics (knots, braids, homotopy theory, number theory) to more applied mathematics (cryptography, algebraic specification of algorithms, dynamical systems) and concrete applications (modeling of polymers and ionic liquids, video, music and medical imaging). The main mathematical focus throughout the book is on algebraic modeling with particular emphasis on braid groups. The research methods include algebraic modeling using topological structures, such as knots, 3-manifolds, classical homotopy groups, and braid groups. The applications address the simulation of polymer chains and ionic liquids, as well as the modeling of natural phenomena via topological surgery. The treatment of computational structures, including finite fields and cryptography, focuses on the development of novel techniques. These techniques can be applied to the design of algebraic specifications for systems modeling and verification. This book is the outcome of a workshop in connection with the research project Thales on Algebraic Modeling of Topological and Computational Structures and Applications, held at the National Technical University of Athens, Greece in July 2015. The reader will benefit from the innovative approaches to tackling difficult questions in topology, applications and interrelated research areas, which largely employ algebraic tools.
Algebraic Monoids, Group Embeddings, and Algebraic Combinatorics (Fields Institute Communications #71)
by Benjamin Steinberg Mahir Can Zhenheng Li Qiang WangThis book contains a collection of fifteen articles and is dedicated to the sixtieth birthdays of Lex Renner and Mohan Putcha, the pioneers of the field of algebraic monoids. Topics presented include: structure and representation theory of reductive algebraic monoids monoid schemes and applications of monoids monoids related to Lie theory equivariant embeddings of algebraic groups constructions and properties of monoids from algebraic combinatorics endomorphism monoids induced from vector bundles Hodge-Newton decompositions of reductive monoids A portion of these articles are designed to serve as a self-contained introduction to these topics, while the remaining contributions are research articles containing previously unpublished results, which are sure to become very influential for future work. Among these, for example, the important recent work of Michel Brion and Lex Renner showing that the algebraic semi groups are strongly π-regular. Graduate students as well as researchers working in the fields of algebraic (semi)group theory, algebraic combinatorics and the theory of algebraic group embeddings will benefit from this unique and broad compilation of some fundamental results in (semi)group theory, algebraic group embeddings and algebraic combinatorics merged under the umbrella of algebraic monoids.
Algebraic Number Theory
by Edwin WeissCareful organization and clear, detailed proofs characterize this methodical, self-contained exposition of basic results of classical algebraic number theory from a relatively modem point of view. This volume presents most of the number-theoretic prerequisites for a study of either class field theory (as formulated by Artin and Tate) or the contemporary treatment of analytical questions (as found, for example, in Tate's thesis).Although concerned exclusively with algebraic number fields, this treatment features axiomatic formulations with a considerable range of applications. Modem abstract techniques constitute the primary focus. Topics include introductory materials on elementary valuation theory, extension of valuations, local and ordinary arithmetic fields, and global, quadratic, and cyclotomic fields.Subjects correspond to those usually covered in a one-semester, graduate level course in algebraic number theory, making this book ideal either for classroom use or as a stimulating series of exercises for mathematically minded individuals.
Algebraic Number Theory (Discrete Mathematics and Its Applications)
by Richard A. MollinBringing the material up to date to reflect modern applications, this second edition has been completely rewritten and reorganized to incorporate a new style, methodology, and presentation. It offers a more complete and involved treatment of Galois theory, a more comprehensive section on Pollard's cubic factoring algorithm, and more detailed explanations of proofs to provide a sound understanding of challenging material. This edition also studies binary quadratic forms and compares the ideal and form class groups. The text includes convenient cross-referencing, a comprehensive index, and numerous exercises and applications.
Algebraic Number Theory (Springer Undergraduate Mathematics Series)
by Frazer JarvisThe technical difficulties of algebraic number theory often make this subject appear difficult to beginners. This undergraduate textbook provides a welcome solution to these problems as it provides an approachable and thorough introduction to the topic. Algebraic Number Theory takes the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to perform. The middle section considers more general theory and results for number fields, and the book concludes with some topics which are more likely to be suitable for advanced students, namely, the analytic class number formula and the number field sieve. This is the first time that the number field sieve has been considered in a textbook at this level.
Algebraic Number Theory and Fermat's Last Theorem
by Ian Stewart David TallUpdated to reflect current research, Algebraic Number Theory and Fermat’s Last Theorem, Fourth Edition introduces fundamental ideas of algebraic numbers and explores one of the most intriguing stories in the history of mathematics—the quest for a proof of Fermat’s Last Theorem. The authors use this celebrated theorem to motivate a general study of the theory of algebraic numbers from a relatively concrete point of view. Students will see how Wiles’s proof of Fermat’s Last Theorem opened many new areas for future work.New to the Fourth Edition Provides up-to-date information on unique prime factorization for real quadratic number fields, especially Harper’s proof that Z(√14) is Euclidean Presents an important new result: Mihăilescu’s proof of the Catalan conjecture of 1844 Revises and expands one chapter into two, covering classical ideas about modular functions and highlighting the new ideas of Frey, Wiles, and others that led to the long-sought proof of Fermat’s Last Theorem Improves and updates the index, figures, bibliography, further reading list, and historical remarks Written by preeminent mathematicians Ian Stewart and David Tall, this text continues to teach students how to extend properties of natural numbers to more general number structures, including algebraic number fields and their rings of algebraic integers. It also explains how basic notions from the theory of algebraic numbers can be used to solve problems in number theory.
Algebraic Number Theory: A Brief Introduction (Textbooks in Mathematics)
by J.S. ChahalThis book offers the basics of algebraic number theory for students and others who need an introduction and do not have the time to wade through the voluminous textbooks available. It is suitable for an independent study or as a textbook for a first course on the topic. The author presents the topic here by first offering a brief introduction to number theory and a review of the prerequisite material, then presents the basic theory of algebraic numbers. The treatment of the subject is classical but the newer approach discussed at the end provides a broader theory to include the arithmetic of algebraic curves over finite fields, and even suggests a theory for studying higher dimensional varieties over finite fields. It leads naturally to the Weil conjecture and some delicate questions in algebraic geometry. About the Author Dr. J. S. Chahal is a professor of mathematics at Brigham Young University. He received his Ph.D. from Johns Hopkins University and after spending a couple of years at the University of Wisconsin as a post doc, he joined Brigham Young University as an assistant professor and has been there ever since. He specializes and has published several papers in number theory. For hobbies, he likes to travel and hike. His book, Fundamentals of Linear Algebra, is also published by CRC Press.
Algebraic Numbers and Algebraic Functions (Chapman Hall/crc Mathematics Ser.)
by P.M. CohnThis book is an introduction to the theory of algebraic numbers and algebraic functions of one variable. The basic development is the same for both using E Artin's legant approach, via valuations. Number Theory is pursued as far as the unit theorem and the finiteness of the class number. In function theory the aim is the Abel-Jacobi theorem describing the devisor class group, with occasional geometrical asides to help understanding. Assuming only an undergraduate course in algebra, plus a little acquaintance with topology and complex function theory, the book serves as an introduction to more technical works in algebraic number theory, function theory or algebraic geometry by an exposition of the central themes in the subject.
Algebraic Operads: An Algorithmic Companion
by Murray R. Bremner Vladimir DotsenkoThis book presents a systematic treatment of Grobner bases in several contexts. The book builds up to the theory of Grobner bases for operads due to the second author and Khoroshkin as well as various applications of the corresponding diamond lemmas in algebra. Throughout the book, both the mathematical theory and computational methods are emphasized and numerous algorithms, examples, and exercises are provided to clarify and illustrate the concrete meaning of abstract theory.
Algebraic Perspectives on Substructural Logics (Trends in Logic #55)
by Francesco Paoli Davide Fazio Antonio LeddaThis volume presents the state of the art in the algebraic investigation into substructural logics. It features papers from the workshop AsubL (Algebra & Substructural Logics - Take 6). Held at the University of Cagliari, Italy, this event is part of the framework of the Horizon 2020 Project SYSMICS: SYntax meets Semantics: Methods, Interactions, and Connections in Substructural logics. Substructural logics are usually formulated as Gentzen systems that lack one or more structural rules. They have been intensively studied over the past two decades by logicians of various persuasions. These researchers include mathematicians, philosophers, linguists, and computer scientists. Substructural logics are applicable to the mathematical investigation of such processes as resource-conscious reasoning, approximate reasoning, type-theoretical grammar, and other focal notions in computer science. They also apply to epistemology, economics, and linguistics. The recourse to algebraic methods -- or, better, the fecund interplay of algebra and proof theory -- has proved useful in providing a unifying framework for these investigations. The AsubL series of conferences, in particular, has played an important role in these developments. This collection will appeal to students and researchers with an interest in substructural logics, abstract algebraic logic, residuated lattices, proof theory, universal algebra, and logical semantics.
Algebraic Properties of Generalized Inverses (Developments in Mathematics #52)
by Dragana S. Cvetković Ilić Yimin WeiThis book addresses selected topics in the theory of generalized inverses. Following a discussion of the "reverse order law" problem and certain problems involving completions of operator matrices, it subsequently presents a specific approach to solving the problem of the reverse order law for {{1}} -generalized inverses. Particular emphasis is placed on the existence of Drazin invertible completions of an upper triangular operator matrix; on the invertibility and different types of generalized invertibility of a linear combination of operators on Hilbert spaces and Banach algebra elements; on the problem of finding representations of the Drazin inverse of a 2x2 block matrix; and on selected additive results and algebraic properties for the Drazin inverse. In addition to the clarity of its content, the book discusses the relevant open problems for each topic discussed. Comments on the latest references on generalized inverses are also included. Accordingly, the book will be useful for graduate students, PhD students and researchers, but also for a broader readership interested in these topics.
Algebraic Reasoning
by Paul Gray Jacqueline Weilmuenster Jennifer HylemonConsumable student worktext containing the Engage, Explore, and Explain sections of the Algebraic Reasoning textbook.
Algebraic Reasoning
by Paul Gray Jacqueline Weilmuenster Jennifer HylemonAlgebraic Reasoning by Paul Gray, Jacqueline Weilmuenster, Jennifer Hylemon
Algebraic Statistics: Computational Commutative Algebra in Statistics (Chapman & Hall/CRC Monographs on Statistics and Applied Probability)
by Henry P. Wynn Giovanni Pistone Eva RiccomagnoWritten by pioneers in this exciting new field, Algebraic Statistics introduces the application of polynomial algebra to experimental design, discrete probability, and statistics. It begins with an introduction to Grobner bases and a thorough description of their applications to experimental design. A special chapter covers the binary case
Algebraic Structure of String Field Theory (Lecture Notes in Physics #973)
by Martin Doubek Branislav Jurčo Martin Markl Ivo SachsThis book gives a modern presentation of modular operands and their role in string field theory. The authors aim to outline the arguments from the perspective of homotopy algebras and their operadic origin. Part I reviews string field theory from the point of view of homotopy algebras, including A-infinity algebras, loop homotopy (quantum L-infinity) and IBL-infinity algebras governing its structure. Within this framework, the covariant construction of a string field theory naturally emerges as composition of two morphisms of particular odd modular operads. This part is intended primarily for researchers and graduate students who are interested in applications of higher algebraic structures to strings and quantum field theory. Part II contains a comprehensive treatment of the mathematical background on operads and homotopy algebras in a broader context, which should appeal also to mathematicians who are not familiar with string theory.
Algebraic Structures and Applications: SPAS 2017, Västerås and Stockholm, Sweden, October 4-6 (Springer Proceedings in Mathematics & Statistics #317)
by Anatoliy Malyarenko Sergei Silvestrov Milica RančićThis book explores the latest advances in algebraic structures and applications, and focuses on mathematical concepts, methods, structures, problems, algorithms and computational methods important in the natural sciences, engineering and modern technologies. In particular, it features mathematical methods and models of non-commutative and non-associative algebras, hom-algebra structures, generalizations of differential calculus, quantum deformations of algebras, Lie algebras and their generalizations, semi-groups and groups, constructive algebra, matrix analysis and its interplay with topology, knot theory, dynamical systems, functional analysis, stochastic processes, perturbation analysis of Markov chains, and applications in network analysis, financial mathematics and engineering mathematics. The book addresses both theory and applications, which are illustrated with a wealth of ideas, proofs and examples to help readers understand the material and develop new mathematical methods and concepts of their own. The high-quality chapters share a wealth of new methods and results, review cutting-edge research and discuss open problems and directions for future research. Taken together, they offer a source of inspiration for a broad range of researchers and research students whose work involves algebraic structures and their applications, probability theory and mathematical statistics, applied mathematics, engineering mathematics and related areas.
Algebraic Structures in Natural Language
by Jean-Philippe BernardyAlgebraic Structures in Natural Language addresses a central problem in cognitive science concerning the learning procedures through which humans acquire and represent natural language. Until recently algebraic systems have dominated the study of natural language in formal and computational linguistics, AI, and the psychology of language, with linguistic knowledge seen as encoded in formal grammars, model theories, proof theories and other rule-driven devices. Recent work on deep learning has produced an increasingly powerful set of general learning mechanisms which do not apply rule-based algebraic models of representation. The success of deep learning in NLP has led some researchers to question the role of algebraic models in the study of human language acquisition and linguistic representation. Psychologists and cognitive scientists have also been exploring explanations of language evolution and language acquisition that rely on probabilistic methods, social interaction and information theory, rather than on formal models of grammar induction. This book addresses the learning procedures through which humans acquire natural language, and the way in which they represent its properties. It brings together leading researchers from computational linguistics, psychology, behavioral science and mathematical linguistics to consider the significance of non-algebraic methods for the study of natural language. The text represents a wide spectrum of views, from the claim that algebraic systems are largely irrelevant to the contrary position that non-algebraic learning methods are engineering devices for efficiently identifying the patterns that underlying grammars and semantic models generate for natural language input. There are interesting and important perspectives that fall at intermediate points between these opposing approaches, and they may combine elements of both. It will appeal to researchers and advanced students in each of these fields, as well as to anyone who wants to learn more about the relationship between computational models and natural language.
Algebraic Techniques and Their Use in Describing and Processing Uncertainty: To the Memory of Professor Elbert A. Walker (Studies in Computational Intelligence #878)
by Vladik Kreinovich Hung T. NguyenThis book discusses heuristic methods – methods lacking a solid theoretical justification – which are ubiquitous in numerous application areas, and explains techniques that can make heuristic methods more reliable. Focusing on algebraic techniques, i.e., those that use only a few specific features of a situation, it describes various state-of-the-art applications, ranging from fuzzy methods for dealing with imprecision to general optimization methods and quantum-based methods for analyzing economic phenomena. The book also includes recent results from leading researchers, which could (and hopefully will) provide the basis for future applications. As such, it is a valuable resource for mathematicians interested in potential applications of their algebraic results and ideas, as well as for application specialists wanting to discover how algebraic techniques can help in their domains.
Algebraic Theories (Dover Books on Mathematics)
by Leonard DicksonThis in-depth introduction to classical topics in higher algebra provides rigorous, detailed proofs for its explorations of some of mathematics' most significant concepts, including matrices, invariants, and groups. Algebraic Theories studies all of the important theories; its extensive offerings range from the foundations of higher algebra and the Galois theory of algebraic equations to finite linear groups (including Klein's "icosahedron" and the theory of equations of the fifth degree) and algebraic invariants. The full treatment includes matrices, linear transformations, elementary divisors and invariant factors, and quadratic, bilinear, and Hermitian forms, both singly and in pairs. The results are classical, with due attention to issues of rationality. Elementary divisors and invariant factors receive simple, natural introductions in connection with the classical form and a rational, canonical form of linear transformations. All topics are developed with a remarkable lucidity and discussed in close connection with their most frequent mathematical applications.
Algebraic Theory of Generalized Inverses
by Jianlong Chen Xiaoxiang ZhangMost of the existing monographs on generalized inverses are based on linear algebra tools and geometric methods of Banach (Hilbert) spaces to introduce generalized inverses of complex matrices and operators and their related applications, or focus on generalized inverses of matrices over special rings like division rings and integral domains, and does not include the results in general algebraic structures such as arbitrary rings, semigroups and categories, which are precisely the most general cases.In this book, five important generalized inverses are introduced in these algebraic structures. Moreover, noting that the (pseudo) core inverse was introduced in the last decade and has attracted much attention, this book also covers the very rich research results on it, so as to be a necessary supplement to the existing monographs. This book starts with decompositions of matrices, introduces the basic properties of generalized inverses of matrices, and then discusses generalizedinverses of elements in rings and semigroups, as well as morphisms in categories. The algebraic nature of generalized inverses is presented, and the behavior of generalized inverses are related to the properties of the algebraic system.Scholars and graduate students working on the theory of rings, semigroups and generalized inverses of matrices and operators will find this book helpful.
Algebraic Theory of Locally Nilpotent Derivations (Encyclopaedia of Mathematical Sciences #136.3)
by Gene FreudenburgThis book explores the theory and application of locally nilpotent derivations, a subject motivated by questions in affine algebraic geometry and having fundamental connections to areas such as commutative algebra, representation theory, Lie algebras and differential equations. The author provides a unified treatment of the subject, beginning with 16 First Principles on which the theory is based. These are used to establish classical results, such as Rentschler's Theorem for the plane and the Cancellation Theorem for Curves. More recent results, such as Makar-Limanov's theorem for locally nilpotent derivations of polynomial rings, are also discussed. Topics of special interest include progress in classifying additive actions on three-dimensional affine space, finiteness questions (Hilbert's 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem. A lot of new material is included in this expanded second edition, such as canonical factorization of quotient morphisms, and a more extended treatment of linear actions. The reader will also find a wealth of examples and open problems and an updated resource for future investigations.
Algebraic Theory of Numbers
by Pierre Samuel Allan J. SilbergerAlgebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics — algebraic geometry, in particular.This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Galois theory, Noetherian rings and modules, and rings of fractions. It covers the basics, starting with the divisibility theory in principal ideal domains and ending with the unit theorem, finiteness of the class number, and the more elementary theorems of Hilbert ramification theory. Numerous examples, applications, and exercises appear throughout the text.
Algebraic Theory of Quadratic Numbers (Universitext)
by Mak TrifkovićBy focusing on quadratic numbers, this advanced undergraduate or master's level textbook on algebraic number theory is accessible even to students who have yet to learn Galois theory. The techniques of elementary arithmetic, ring theory and linear algebra are shown working together to prove important theorems, such as the unique factorization of ideals and the finiteness of the ideal class group The book concludes with two topics particular to quadratic fields: continued fractions and quadratic forms The treatment of quadratic forms is somewhat more advanced than usual, with an emphasis on their connection with ideal classes and a discussion of Bhargava cubes. The numerous exercises in the text offer the reader hands-on computational experience with elements and ideals in quadratic number fields The reader is also asked to fill in the details of proofs and develop extra topics, like the theory of orders Prerequisites include elementary number theory and a basic familiarity with ring theory.
Algebraic Thinking Foundations (3rd Edition)
by Brian E. Enright Linda L. Mannhardt Leslie G. BakerAlgebraic thinking is much more than a formal course called Algebra and here it refers to mathematics of the type used every day both in the home and the workplace--from rates of auto fuel consumption to planning budgets.