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Abstract Algebra with Applications: Volume 1: Vector Spaces and Groups (Pure and Applied Mathematics)
by Karlheinz SpindlerA comprehensive presentation of abstract algebra and an in-depth treatment of the applications of algebraic techniques and the relationship of algebra to other disciplines, such as number theory, combinatorics, geometry, topology, differential equations, and Markov chains.
Abstract Algebra with Applications: Volume 2: Rings and Fields (Pure and Applied Mathematics)
by Karlheinz SpindlerA comprehensive presentation of abstract algebra and an in-depth treatment of the applications of algebraic techniques and the relationship of algebra to other disciplines, such as number theory, combinatorics, geometry, topology, differential equations, and Markov chains.
Abstract Algebra: A Comprehensive Treatment (Pure and Applied Mathematics #267)
by Freddy Van Oystaeyen Claudia MeniniIn one exceptional volume, Abstract Algebra covers subject matter typically taught over the course of two or three years and offers a self-contained presentation, detailed definitions, and excellent chapter-matched exercises to smooth the trajectory of learning algebra from zero to one. Field-tested through advance use in the ERASMUS educational project in Europe, this ambitious, comprehensive book includes an original treatment of representation of finite groups that avoids the use of semisimple ring theory and explains sets, maps, posets, lattices, and other essentials of the algebraic language; Peano's axioms and cardinality; groupoids, semigroups, monoids, groups; and normal subgroups.
Abstract Algebra: A First Course
by Dan SaracinoThis book is intended for use in a junior-senior level course in abstract algebra. The Second Edition of this text maintains the clear exposition, logical organization, and accessible breadth of coverage that have been its hallmarks. It plunges directly into algebraic structures and incorporates a large number of examples to clarify abstract concepts as they arise. Proofs of theorems do more than just prove the stated results; Saracino examines them so readers gain a better impression of where the proofs come from and why they proceed as they do. Most of the exercises range from easy to moderately difficult and ask for understanding of ideas rather than flashes of insight. The new edition introduces five new sections on field extensions and Galois theory, increasing its versatility by making it appropriate for a two-semester as well as a one-semester course.
Abstract Algebra: A First Course (Textbooks in Mathematics)
by Stephen LovettWhen a student of mathematics studies abstract algebra, he or she inevitably faces questions in the vein of, "What is abstract algebra" or "What makes it abstract?" Algebra, in its broadest sense, describes a way of thinking about classes of sets equipped with binary operations. In high school algebra, a student explores properties of operations (+, −, ×, and ÷) on real numbers. Abstract algebra studies properties of operations without specifying what types of number or object we work with. Any theorem established in the abstract context holds not only for real numbers but for every possible algebraic structure that has operations with the stated properties. This textbook intends to serve as a first course in abstract algebra. The selection of topics serves both of the common trends in such a course: a balanced introduction to groups, rings, and fields; or a course that primarily emphasizes group theory. The writing style is student-centered, conscientiously motivating definitions and offering many illustrative examples. Various sections or sometimes just examples or exercises introduce applications to geometry, number theory, cryptography and many other areas. This book offers a unique feature in the lists of projects at the end of each section. the author does not view projects as just something extra or cute, but rather an opportunity for a student to work on and demonstrate their potential for open-ended investigation. The projects ideas come in two flavors: investigative or expository. The investigative projects briefly present a topic and posed open-ended questions that invite the student to explore the topic, asking and to trying to answer their own questions. Expository projects invite the student to explore a topic with algebraic content or pertain to a particular mathematician’s work through responsible research. The exercises challenge the student to prove new results using the theorems presented in the text. The student then becomes an active participant in the development of the field.
Abstract Algebra: A Gentle Introduction (Textbooks in Mathematics)
by Gary L. Mullen James A. SellersAbstract Algebra: A Gentle Introduction advantages a trend in mathematics textbook publishing towards smaller, less expensive and brief introductions to primary courses. The authors move away from the ‘everything for everyone’ approach so common in textbooks. Instead, they provide the reader with coverage of numerous algebraic topics to cover the most important areas of abstract algebra. Through a careful selection of topics, supported by interesting applications, the authors Intend the book to be used for a one-semester course in abstract algebra. It is suitable for an introductory course in for mathematics majors. The text is also very suitable for education majors who need to have an introduction to the topic. As textbooks go through various editions and authors employ the suggestions of numerous well-intentioned reviewers, these book become larger and larger and subsequently more expensive. This book is meant to counter that process. Here students are given a "gentle introduction," meant to provide enough for a course, yet also enough to encourage them toward future study of the topic. Features Groups before rings approach Interesting modern applications Appendix includes mathematical induction, the well-ordering principle, sets, functions, permutations, matrices, and complex nubers. Numerous exercises at the end of each section Chapter "Hint and Partial Solutions" offers built in solutions manual
Abstract Algebra: An Inquiry-Based Approach (Textbooks in Mathematics)
by Jonathan K. Hodge Steven Schlicker Ted SundstromAbstract Algebra: An Inquiry-Based Approach, Second Edition not only teaches abstract algebra, but also provides a deeper understanding of what mathematics is, how it is done, and how mathematicians think. The second edition of this unique, flexible approach builds on the success of the first edition. The authors offer an emphasis on active learning, helping students learn algebra by gradually building both their intuition and their ability to write coherent proofs in context. The goals for this text include: Allowing the flexibility to begin the course with either groups or rings. Introducing the ideas behind definitions and theorems to help students develop intuition. Helping students understand how mathematics is done. Students will experiment through examples, make conjectures, and then refine or prove their conjectures. Assisting students in developing their abilities to effectively communicate mathematical ideas. Actively involving students in realizing each of these goals through in-class and out-of-class activities, common in-class intellectual experiences, and challenging problem sets. Changes in the Second Edition Streamlining of introductory material with a quicker transition to the material on rings and groups. New investigations on extensions of fields and Galois theory. New exercises added and some sections reworked for clarity. More online Special Topics investigations and additional Appendices, including new appendices on other methods of proof and complex roots of unity. Encouraging students to do mathematics and be more than passive learners, this text shows students the way mathematics is developed is often different than how it is presented; definitions, theorems, and proofs do not simply appear fully formed; mathematical ideas are highly interconnected; and in abstract algebra, there is a considerable amount of intuition to be found.
Abstract Algebra: An Interactive Approach (Second Edition) (Textbooks in Mathematics)
by William Paulsen<p>The new edition of Abstract Algebra: An Interactive Approach presents a hands-on and traditional approach to learning groups, rings, and fields. It then goes further to offer optional technology use to create opportunities for interactive learning and computer use. This new edition offers a more traditional approach offering additional topics to the primary syllabus placed after primary topics are covered. This creates a more natural flow to the order of the subjects presented. This edition is transformed by historical notes and better explanations of why topics are covered. <p>This innovative textbook shows how students can better grasp difficult algebraic concepts through the use of computer programs. It encourages students to experiment with various applications of abstract algebra, thereby obtaining a real-world perspective of this area. Each chapter includes, corresponding Sage notebooks, traditional exercises, and several interactive computer problems that utilize Sage and Mathematica® to explore groups, rings, fields and additional topics. This text does not sacrifice mathematical rigor. It covers classical proofs, such as Abel’s theorem, as well as many topics not found in most standard introductory texts. The author explores semi-direct products, polycyclic groups, Rubik’s Cube®-like puzzles, and Wedderburn’s theorem. The author also incorporates problem sequences that allow students to delve into interesting topics, including Fermat’s two square theorem.</p>
Abstract Algebra: An Interactive Approach (Textbooks in Mathematics)
by William PaulsenAbstract Algebra: An Interactive Approach, Third Edition is a new concept in learning modern algebra. Although all the expected topics are covered thoroughly and in the most popular order, the text offers much flexibility. Perhaps more significantly, the book gives professors and students the option of including technology in their courses. Each chapter in the textbook has a corresponding interactive Mathematica notebook and an interactive SageMath workbook that can be used in either the classroom or outside the classroom. Students will be able to visualize the important abstract concepts, such as groups and rings (by displaying multiplication tables), homomorphisms (by showing a line graph between two groups), and permutations. This, in turn, allows the students to learn these difficult concepts much more quickly and obtain a firmer grasp than with a traditional textbook. Thus, the colorful diagrams produced by Mathematica give added value to the students. Teachers can run the Mathematica or SageMath notebooks in the classroom in order to have their students visualize the dynamics of groups and rings. Students have the option of running the notebooks at home, and experiment with different groups or rings. Some of the exercises require technology, but most are of the standard type with various difficulty levels.The third edition is meant to be used in an undergraduate, single-semester course, reducing the breadth of coverage, size, and cost of the previous editions. Additional changes include: Binary operators are now in an independent section. The extended Euclidean algorithm is included. Many more homework problems are added to some sections. Mathematical induction is moved to Section 1.2. Despite the emphasis on additional software, the text is not short on rigor. All of the classical proofs are included, although some of the harder proofs can be shortened by using technology.
Abstract Algebra: An Introductory Course (Springer Undergraduate Mathematics Series #First Edition)
by Gregory T. Lee<P>This carefully written textbook offers a thorough introduction to abstract algebra, covering the fundamentals of groups, rings and fields. <P> The first two chapters present preliminary topics such as properties of the integers and equivalence relations. The author then explores the first major algebraic structure, the group, progressing as far as the Sylow theorems and the classification of finite abelian groups. An introduction to ring theory follows, leading to a discussion of fields and polynomials that includes sections on splitting fields and the construction of finite fields. The final part contains applications to public key cryptography as well as classical straightedge and compass constructions. <P> Explaining key topics at a gentle pace, this book is aimed at undergraduate students. It assumes no prior knowledge of the subject and contains over 500 exercises, half of which have detailed solutions provided.
Abstract Algebra: Structure and Application (Springer Undergraduate Texts in Mathematics and Technology)
by David R. Finston Patrick J. MorandiThis text seeks to generate interest in abstract algebra by introducing each new structure and topic via a real-world application. The down-to-earth presentation is accessible to a readership with no prior knowledge of abstract algebra. Students are led to algebraic concepts and questions in a natural way through their everyday experiences. Applications include: Identification numbers and modular arithmetic (linear) error-correcting codes, including cyclic codes ruler and compass constructions cryptography symmetry of patterns in the real plane Abstract Algebra: Structure and Application is suitable as a text for a first course on abstract algebra whose main purpose is to generate interest in the subject or as a supplementary text for more advanced courses. The material paves the way to subsequent courses that further develop the theory of abstract algebra and will appeal to students of mathematics, mathematics education, computer science, and engineering interested in applications of algebraic concepts.
Abstract Algebra: Suitable for Self-Study or Online Lectures (Mathematics Study Resources #7)
by Marco HienThis book contains the basics of abstract algebra. In addition to elementary algebraic structures such as groups, rings and solids, Galois theory in particular is developed together with its applications to the cyclotomic fields, finite fields or the question of the resolution of polynomial equations.Special attention is paid to the natural development of the contents. Numerous intermediate explanations support this basic idea, show connections and help to better penetrate the underlying concepts. The book is therefore particularly suitable for learning algebra in self-study or accompanying online lectures.
Abstract Analytic Number Theory (Dover Books on Mathematics #Volume 12)
by John Knopfmacher"This book is well-written and the bibliography excellent," declared Mathematical Reviews of John Knopfmacher's innovative study. The three-part treatment applies classical analytic number theory to a wide variety of mathematical subjects not usually treated in an arithmetical way. The first part deals with arithmetical semigroups and algebraic enumeration problems; Part Two addresses arithmetical semigroups with analytical properties of classical type; and the final part explores analytical properties of other arithmetical systems.Because of its careful treatment of fundamental concepts and theorems, this text is accessible to readers with a moderate mathematical background, i.e., three years of university-level mathematics. An extensive bibliography is provided, and each chapter includes a selection of references to relevant research papers or books. The book concludes with an appendix that offers several unsolved questions, with interesting proposals for further development.
Abstract Calculus: A Categorical Approach (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)
by Francisco Javier Garcia-PachecoAbstract Calculus: A Categorical Approach provides an abstract approach to calculus. It is intended for graduate students pursuing PhDs in pure mathematics but junior and senior researchers in basically any field of mathematics and theoretical physics will also be interested. Any calculus text for undergraduate students majoring in engineering, mathematics or physics deals with the classical concepts of limits, continuity, differentiability, optimization, integrability, summability, and approximation. This book covers the exact same topics, but from a categorical perspective, making the classification of topological modules as the main category involved. Features Suitable for PhD candidates and researchers Requires prerequisites in set theory, general topology, and abstract algebra, but is otherwise self-contained Dr. Francisco Javier García-Pacheco is a full professor and Director of the Departmental Section of Mathematics at the College of Engineering of the University of Cádiz, Spain.
Abstract Cauchy Problems: Three Approaches (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)
by Irina V. Melnikova Alexei FilinkovAlthough the theory of well-posed Cauchy problems is reasonably understood, ill-posed problems-involved in a numerous mathematical models in physics, engineering, and finance- can be approached in a variety of ways. Historically, there have been three major strategies for dealing with such problems: semigroup, abstract distribution, and regularizat
Abstract Lie Algebras
by David J WinterSolid but concise, this account of Lie algebra emphasizes the theory's simplicity and offers new approaches to major theorems. Author David J. Winter, a Professor of Mathematics at the University of Michigan, also presents a general, extensive treatment of Cartan and related Lie subalgebras over arbitrary fields.Preliminary material covers modules and nonassociate algebras, followed by a compact, self-contained development of the theory of Lie algebras of characteristic 0. Topics include solvable and nilpotent Lie algebras, Cartan subalgebras, and Levi's radical splitting theorem and the complete reducibility of representations of semisimple Lie algebras. Additional subjects include the isomorphism theorem for semisimple Lie algebras and their irreducible modules, automorphism of Lie algebras, and the conjugacy of Cartan subalgebras and Borel subalgebras. An extensive theory of Cartan and related subalgebras of Lie algebras over arbitrary fields is developed in the final chapter, and an appendix offers background on the Zariski topology.
Abstract Methods in Partial Differential Equations
by Robert W. CarrollDetailed and self-contained, this treatment is directed to graduate students with some previous exposure to classical partial differential equations. The author examines a variety of modern abstract methods in partial differential equations, especially in the area of abstract evolution equations. Additional topics include the theory of nonlinear monotone operators applied to elliptic and variational problems. 1969 edition.
Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality I: Abstract Theory (SpringerBriefs in Mathematics)
by Atsushi YagiThe classical Łojasiewicz gradient inequality (1963) was extended by Simon (1983) to the infinite-dimensional setting, now called the Łojasiewicz–Simon gradient inequality. This book presents a unified method to show asymptotic convergence of solutions to a stationary solution for abstract parabolic evolution equations of the gradient form by utilizing this Łojasiewicz–Simon gradient inequality.In order to apply the abstract results to a wider class of concrete nonlinear parabolic equations, the usual Łojasiewicz–Simon inequality is extended, which is published here for the first time. In the second version, these abstract results are applied to reaction–diffusion equations with discontinuous coefficients, reaction–diffusion systems, and epitaxial growth equations. The results are also applied to the famous chemotaxis model, i.e., the Keller–Segel equations even for higher-dimensional ones.
Abstract Parabolic Evolution Equations and Łojasiewicz–Simon Inequality II: Applications (SpringerBriefs in Mathematics)
by Atsushi YagiThis second volume continues the study on asymptotic convergence of global solutions of parabolic equations to stationary solutions by utilizing the theory of abstract parabolic evolution equations and the Łojasiewicz–Simon gradient inequality. In the first volume of the same title, after setting the abstract frameworks of arguments, a general convergence theorem was proved under the four structural assumptions of critical condition, Lyapunov function, angle condition, and gradient inequality. In this volume, with those abstract results reviewed briefly, their applications to concrete parabolic equations are described.Chapter 3 presents a discussion of semilinear parabolic equations of second order in general n-dimensional spaces, and Chapter 4 is devoted to treating epitaxial growth equations of fourth order, which incorporate general roughening functions. In Chapter 5 consideration is given to the Keller–Segel equations in one-, two-, and three-dimensional spaces. Some of these results had already been obtained and published by the author in collaboration with his colleagues. However, by means of the abstract theory described in the first volume, those results can be extended much more.Readers of this monograph should have a standard-level knowledge of functional analysis and of function spaces. Familiarity with functional analytic methods for partial differential equations is also assumed.
Abstract Sets and Finite Ordinals: An Introduction to the Study of Set Theory
by G. B. KeeneThis text unites the logical and philosophical aspects of set theory in a manner intelligible both to mathematicians without training in formal logic and to logicians without a mathematical background. It combines an elementary level of treatment with the highest possible degree of logical rigor and precision.Starting with an explanation of all the basic logical terms and related operations, the text progresses through a stage-by-stage elaboration that proves the fundamental theorems of finite sets. It focuses on the Bernays theory of finite classes and finite sets, exploring the system's basis and development, including Stage I and Stage II theorems, the theory of finite ordinals, and the theory of finite classes and finite sets. This volume represents an excellent text for undergraduates studying intermediate or advanced logic as well as a fine reference for professional mathematicians.
Abstract State Machines, Alloy, B, TLA, VDM, and Z: 5th International Conference, ABZ 2016, Linz, Austria, May 23-27, 2016, Proceedings (Lecture Notes in Computer Science #9675)
by Michael Butler Klaus-Dieter Schewe Atif Mashkoor Miklos BiroThis bookconstitutes the refereed proceedings of the 5th International Conference on AbstractState Machines, Alloy, B, TLA, VDM, and Z, ABZ 2016, held in Linz, Austria, inMay 2016. The 17 full and 15 short papers presented in this volume were carefullyreviewed and selected from 61 submissions. They record the latest researchdevelopments in state-based formal methods Abstract State Machines, Alloy, B,Circus, Event-B, TLS+, VDM and Z.
Abstract State Machines, Alloy, B, TLA, VDM, and Z: 5th International Conference, Abz 2016, Linz, Austria, May 23-27, 2016, Proceedings (Lecture Notes in Computer Science #9675)
by Michael Butler Atif Mashkoor Miklos Biro Laus-Dieter ScheweThis book constitutes the refereed proceedings of the 5th International Conference on Abstract State Machines, Alloy, B, TLA, VDM, and Z, ABZ 2016, held in Linz, Austria, in May 2016. <P><P> The 17 full and 15 short papers presented in this volume were carefully reviewed and selected from 61 submissions. They record the latest research developments in state-based formal methods Abstract State Machines, Alloy, B, Circus, Event-B, TLS+, VDM and Z.
Abzähltheorie nach Pólya (essentials)
by Karl-Heinz ZimmermannIm Zentrum dieses essentials steht der gefeierte Abzählsatz von Pólya. Damit lassen sich kombinatorische Objekte mit Symmetrien abzählen, wie etwa Halsketten mit bunten Perlen und Würfel mit gefärbten Seiten, aber auch Graphen und Bäume. Die Gruppentheorie wird dafür benutzt, die Symmetrien der abzuzählenden Figuren zu beschreiben. Darauf aufbauend kann anhand der Operation der jeweiligen Symmetriegruppe auf den gefärbten Figuren die Anzahl der verschiedenen Muster ermittelt werden. Grundlegend hierfür ist das Lemma von Burnside. Aus seiner gewichteten Fassung wird unter Einbeziehung der Zyklenindexpolynome von Symmetriegruppen der berühmte Pólyasche Satz hergeleitet. Einige Beispiele runden die Darstellung ab.
Accelerate: The Science of Lean Software and DevOps: Building and Scaling High Performing Technology Organizations
by Jez Humble Gene Kim Nicole Forsgren, PhDWinner of the Shingo Publication AwardAccelerate your organization to win in the marketplace.How can we apply technology to drive business value? For years, we've been told that the performance of software delivery teams doesn't matter―that it can't provide a competitive advantage to our companies. Through four years of groundbreaking research to include data collected from the State of DevOps reports conducted with Puppet, Dr. Nicole Forsgren, Jez Humble, and Gene Kim set out to find a way to measure software delivery performance―and what drives it―using rigorous statistical methods. This book presents both the findings and the science behind that research, making the information accessible for readers to apply in their own organizations.Readers will discover how to measure the performance of their teams, and what capabilities they should invest in to drive higher performance. This book is ideal for management at every level.
Accelerated Lattice Boltzmann Model for Colloidal Suspensions: Rheology and Interface Morphology
by Hassan Farhat Joon Sang Lee Sasidhar KondarajuColloids are ubiquitous in the food, medical, cosmetics, polymers, water purification, and pharmaceutical industries. The thermal, mechanical, and storage properties of colloids are highly dependent on their interface morphology and their rheological behavior. Numerical methods provide a convenient and reliable tool for the study of colloids. Accelerated Lattice Boltzmann Model for Colloidal Suspensions introduce the main building-blocks for an improved lattice Boltzmann-based numerical tool designed for the study of colloidal rheology and interface morphology. This book also covers the migrating multi-block used to simulate single component, multi-component, multiphase, and single component multiphase flows and their validation by experimental, numerical, and analytical solutions. Among other topics discussed are the hybrid lattice Boltzmann method (LBM) for surfactant-covered droplets; biological suspensions such as blood; used in conjunction with the suppression of coalescence for investigating the rheology of colloids and microvasculature blood flow. The presented LBM model provides a flexible numerical platform consisting of various modules that could be used separately or in combination for the study of a variety of colloids and biological flow deformation problems.