Algebra Books

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Book SynopsisThe Integers.- Groups.- Rings.- Polynomials.- Vector Spaces and Modules.- Some Linear Groups.- Field Theory.- Finite Fields.- The Real and Complex Numbers.- Sets.Trade ReviewFrom the reviews of the third edition: "As is very typical for Professor Lang’s self demand and style of publishing, he has tried to both improve and up-date his already well-established text. … Numerous examples and exercises accompany this now already classic primer of modern algebra, which as usual, reflects the author’s great individuality just as much as his unrivalled didactic mastery and his care for profound mathematical education at any level. … The present textbook … will remain one of the great standard introductions to the subject for beginners." (Werner Kleinert, Zentralblatt MATH, Vol. 1063, 2005)Table of Contents* Foreword * The Integers * Groups * Rings * Polynomials * Vector Spaces and Modules * Some Linear Groups * Field Theory * Finite Fields * The Real and Complex Numbers * Sets * Appendix * Index

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Book Synopsisthis section gained proofs of the Schroeder–Bernstein theorem and the Trichotomy Law for Sets, and lost most of the material about finite and countable sets, which has now been moved to a new section devoted to those two types of sets.Trade Review“This is a well-written book, based on very sound pedagogical ideas. It would be an excellent choice as a textbook for a ‘transition’ course.” (Margret Höft, zbMATH 1012.00013, 2021)“The contents of the book is organized in three parts … . this is a nice book, which also this reviewer has used with profit in his teaching of beginner students. It is written in a highly pedagogical style and based upon valuable didactical ideas.” (R. Steinbauer, Monatshefte für Mathematik, Vol. 174, 2014)“Books in this category are meant to teach mathematical topics and techniques that will become valuable in more advanced courses. This book meets these criteria. … This book is well suited as a textbook for a transitional course between calculus and more theoretical courses. I also recommend it for academic libraries.” (Edgar R. Chavez, ACM Computing Reviews, February, 2012)“This is an improved edition of a good book that can serve in the undergraduate curriculum as a bridge between computationally oriented courses like calculus and more abstract courses like algebra.” (Teun Koetsier, Zentralblatt MATH, Vol. 1230, 2012)Table of ContentsPreface to the Second Edition Preface to the First Edition To the Student To the Instructor Part I. Proofs 1. Informal Logic 2. Strategies for Proofs Part II. Fundamentals 3. Sets 4. Functions 5. Relations 6. Finite and Infinite Sets Part III. Extras 7. Selected Topics 8. Explorations Appendix: Properties of Numbers Bibliography Index

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Book SynopsisThis book is intended to present group representation theory at a level accessible to mature undergraduate students and beginning graduate students. This is achieved by mainly keeping the required background to the level of undergraduate linear algebra, group theory and very basic ring theory. Module theory and Wedderburn theory, as well as tensor products, are deliberately avoided. Instead, we take an approach based on discrete Fourier Analysis.Applications to the spectral theory of graphs are given to help the student appreciate the usefulness of the subject. A number of exercises are included. This book is intended for a 3rd/4th undergraduate course or an introductory graduate course on group representation theory. However, it can also be used as a reference for workers in all areas of mathematics and statistics.Trade ReviewFrom the reviews:“The aim of the author is to give an introductory text on (ordinary) representation theory of finite groups which is accessible for advanced undergraduates already. And Steinberg manages this outstandingly well. … a book which is ideally suited for beginners in math as well as physicists, engineers and so on, who need a concise, well conceived and easy comprehensible introduction into representation theory.” (G. Kowol, Monatshefte für Mathematik, Vol. 167 (3-4), September, 2012)“Steinberg … provides a one-semester course on representation theory with just linear algebra and a beginning course in abstract algebra (primarily group theory) as prerequisites. … the author covers most of the standard introductory topics in representation theory. The exercises provide more examples and further common results. It is the applications that Steinberg uses to motivate the subject that make this text both interesting and valuable. … Overall, a very user-friendly text with many examples and copious details. Summing Up: Recommended. Upper-division undergraduates through researchers/faculty.” (J. T. Zerger, Choice, Vol. 49 (11), August, 2012)“The book consists of 157 pages spread over 11 chapters. … This book is an introductory course and it could be used by mathematicians and students who would like to learn quickly about the representation theory and character theory of finite groups, and for non-algebraists, statisticians and physicists who use representation theory.” (Jamshid Moori, Mathematical Reviews, Issue 2012 j)“The required background as to this introductory course on group representations, is in the level of linear algebra, group theory and some ring theory. … the book under review is a welcome one for students at an advanced undergraduate or introductory graduate level course, also for those people like physicists, statisticians and non-algebraically oriented mathematicians who need representation theory in their work.” (R. W. van der Waall, Zentralblatt MATH, Vol. 1243, 2012)“The author has, by combining clear writing with an accessible and minimal-prerequisite approach to group representations, created a book that may well help bring group representation theory into the undergraduate curriculum. This is an impressive and useful text, and should be looked at by anybody with an interest in the subject.” (Mark Hunacek, The Mathematical Association of America, February, 2012)Table of Contents-Preface.-Introduction.-Review of Linear Algebra.-Group Representations.-Character Theory.-Fourier Analysis on Finite Groups.-Burnside's Theorem.-Permutation Representations.-Induced Representations.-Another Theorem of Burnside.-The Symmetric Group.-Bibliography.-Index.

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Book SynopsisA Classroom-Tested, Alternative Approach to Teaching Math for Liberal Arts Puzzles, Paradoxes, and Problem Solving: An Introduction to Mathematical Thinking uses puzzles and paradoxes to introduce basic principles of mathematical thought. The text is designed for students in liberal arts mathematics courses. Decision-making situations that progress from recreational problems to important contemporary applications develop the critical-thinking skills of non-science and non-technical majors. The logical underpinnings of this textbook were developed and refined throughout many years of classroom feedback and in response to commentary from presentations at national conferences. The text's five units focus on graphs, logic, probability, voting, and cryptography. The authors also cover related areas, such as operations research, game theory, number theory, combinatorics, statistics, and circuit design. The text uses a core set oTrade Review"… an interesting approach to the world of mathematics. … The layout is good, as is the coverage. … The reinforcing exercises are excellent. Using logic and other techniques, the text lays out methods to help students learn to think in a mathematical manner. Summing up: Recommended. General readers, lower- and upper-division undergraduates."—M. D. Sanford, Felician College, Lodi, New Jersey, USA for CHOICE, November 2015Table of ContentsGraphs: Puzzles and Optimization. Logic: Rational Inference and Computer Circuits. Probability: Predictions and Expectations. Counting: Voting Methods and Apportionment. Numbers: Cryptosystems and Security. Appendices. Index.

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Book SynopsisThe Essentials of a First Linear Algebra Course and MoreLinear Algebra, Geometry and Transformation provides students with a solid geometric grasp of linear transformations. It stresses the linear case of the inverse function and rank theorems and gives a careful geometric treatment of the spectral theorem.An Engaging Treatment of the Interplay among Algebra, Geometry, and MappingsThe text starts with basic questions about images and pre-images of mappings, injectivity, surjectivity, and distortion. In the process of answering these questions in the linear setting, the book covers all the standard topics for a first course on linear algebra, including linear systems, vector geometry, matrix algebra, subspaces, independence, dimension, orthogonality, eigenvectors, and diagonalization. A Smooth Transition to the Conceptual Realm of Higher MathematicsThis book guides students on a journey from Trade Review"All the standard topics of a first course are covered, but the treatment omits abstract vector spaces. … What is unusual is the author's aim to interpret every concept and result geometrically, thus motivating the student to learn to visualize what is going on, rather than just relying on calculations. This is a strong and useful feature. … The book has very many practice sections with over 500 exercises, most of them numerical. … As the author mentions in the preface, it was his aim to provide a sound mathematical introduction, and in the reviewer's opinion he has succeeded in doing this."—Zentralblatt MATH 1314Table of ContentsVectors, Mappings, and Linearity. Solving Linear Systems. Linear Geometry. The Algebra of Matrices. Subspaces. Orthogonality. Linear Transformation. Appendices. Index.

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Book SynopsisThe Discrete.- Integers.- Natural Numbers and Induction.- Some Points of Logic.- Recursion.- Underlying Notions in Set Theory.- Equivalence Relations and Modular Arithmetic.- Arithmetic in Base Ten.- The Continuous.- Real Numbers.- Embedding Z in R.- Limits and Other Consequences of Completeness.- Rational and Irrational Numbers.- Decimal Expansions.- Cardinality.- Final Remarks.- Further Topics.- Continuity and Uniform Continuity.- Public-Key Cryptography.- Complex Numbers.- Groups and Graphs.- Generating Functions.- Cardinal Number and Ordinal Number.- Remarks on Euclidean Geometry.Trade ReviewFrom the reviews:"The Art of Proof is a surprising union of rigor with taste and wit. The authors take a hard-core axiomatic approach, but the writing is never dry. Instead, topics are carefully chosen and meticulously developed with grace and humor, careful attention to detail, and just the right number of skill-building exercises and thought-provoking problems."The text is spare—well under two hundred pages—but contains a thorough axiomatic development of the integers and the reals, along with non-standard optional topics such as Cayley graphs and generating functions. Instead of the standard scattershot "symbolic logic-set theory-functions-proof by contradiction-zzzz..." books, this text keeps its focus on just a few fundamental ideas, of which induction is the most important. This helps my students to feel that they are participants in a grand undertaking—the construction of a number system—rather than passive victims of one proof technique after another." —Paul Zeitz (Mathematics Professor at the University of San Francisco)“This qualitative transition presents a most acute pedagogical challenge. … This book does feature definite mathematical content, contrasting with works that aim at decoupling purely logical apparatus from strictly mathematical concerns. … The authors write with the authority of research mathematicians and clearly mean to open that avenue to students. Summing Up: Recommended. Upper-division undergraduates through professionals.” (D. V. Feldman, Choice, Vol. 48 (8), April, 2011)“This book offers an approach well-balanced between rigor and clarifying simplification. Dilbert and Foxtrot cartoons with philosophical quotes presage the introduction of axioms and preliminary propositions. This graceful and witty blend succeeds well in a textbook for a post-calculus course transitioning a student to higher mathematics. The Art of Proof can also well serve independent readers looking for a solitary path to a vista on higher mathematics.” (Tom Schulte, The Mathematical Association of America, November, 2010)“This is an undergraduate text to extend, in a deeper and formal way, the usual initial knowledge of mathematics. The book deals with classical topics like integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, uncountable sets … . The publication may be useful for people using the book to teach a course on the above mentioned topics. … The aim behind this textbook is teaching how to read and write mathematics as well as understanding key methods and concepts.” (Claudi Alsina, Zentralblatt MATH, Vol. 1198, 2010)Table of ContentsPreface.- Notes for the Student.- Notes for Instructors.- Part I: The Discrete.- 1 Integers.- 2 Natural Numbers and Induction.- 3 Some Points of Logic.- 4 Recursion.- 5 Underlying Notions in Set Theory.- 6 Equivalence Relations and Modular Arithmetic.- 7 Arithmetic in Base Ten.- Part II: The Continuous.- 8 Real Numbers.- 9 Embedding Z in R.- 10. Limits and Other Consequences of Completeness.- 11 Rational and Irrational Numbers.- 12 Decimal Expansions.- 13 Cardinality.- 14 Final Remarks.- Further Topics.- A Continuity and Uniform Continuity.- B Public-Key Cryptography.- C Complex Numbers.- D Groups and Graphs.- E Generating Functions.- F Cardinal Number and Ordinal Number.- G Remarks on Euclidean Geometry.- List of Symbols.- Index.

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Book SynopsisThe book reviews inequalities for weighted entry sums of matrix powers. Applications range from mathematics and CS to pure sciences. It unifies and generalizes several results for products and powers of sesquilinear forms derived from powers of Hermitian, positive-semidefinite, as well as nonnegative matrices. It shows that some inequalities are valid only in specific cases. How to translate the Hermitian matrix results into results for alternating powers of general rectangular matrices? Inequalities that compare the powers of the row and column sums to the row and column sums of the matrix powers are refined for nonnegative matrices. Lastly, eigenvalue bounds and derive results for iterated kernels are improved.Table of ContentsIntroduction. Notation and Basic Facts. Motivation. Diagonalization and Spectral Decomposition. Undirected Graphs / Hermitian Matrices. General Results. Restricted Graph Classes. Directed Graphs / Nonsymmetric. Walks and Alternating Walks in Directed Graphs. Powers of Row and Column Sums. Applications. Bounds for the Largest Eigenvalue. Iterated Kernels. Conclusion. Bibliography. Index.

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Book SynopsisInvitation to Linear Algebra is an informative, clearly written, flexible textbook for instructors and students. Based on over 30 years of experience as a mathematics professor, the author invites students to develop a more informed understanding of complex algebraic concepts using innovative, easy-to-follow methods. The book is organized into lessons rather than chapters. This limits the size of the mathematical morsels that students must digest, making it easier for instructors to budget class time. Each definition is carefully explained with detailed proofs of key theorems, including motivation for each step. This makes the book more flexible, allowing instructors to choose material that reflects their and their students' interests. A larger than normal amount of exercises illustrate how linear and nonlinear algebra apply in the students' areas of study. Features The book's unique lesson format enabTable of ContentsMatrices and Linear Systems. Introduction to Matrices. Matrix Multiplication. Additional Topics in Matrix Algebra. Introduction to Linear Systems. The Inverse of a Matrix. Determinants. Introduction to Determinants. Properties of Determinants. Applications of Determinants. A First Look at Vector Spaces. Introduction to Vector Spaces. Subspaces of Vector Spaces. Linear Dependence and Independence. Basis and Dimension. The Rank of a Matrix. Linear Systems Revisited. More About Vector Spaces. Sums and Direct Sums of Subspaces. Quotient Spaces. Change of Basis. Euclidean Spaces. Orthonormal Bases. Linear Transformations. Introduction to Linear Transformations. Isomorphisms of Vector Spaces. The Kernel and Range of a Linear Transformation. Matrices of Linear Transformations. Similar Matrices. Matrix Diagonalization. Eigenvalues and Eigenvectors. Diagonalization of Square Matrices. Diagonalization of Symmetric Matrices. Complex Vector Spaces. Complex Vector Spaces. Unitary and Hermitian Matrices. Advanced Topics. Powers of Matrices. Functions of a Square Matrix. Matrix Power Series. Minimal Polynomials. Direct Sum Decompositions. Jordan Canonical Form. Applications. Systems of First Order Differential Equations. Stability Analysis of First Order Systems. Coupled Oscillations. Appendix. Solutions and Hints to Selected Exercises.

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Book SynopsisWhat comes to mind when you think about algebra? For many of us, it's memories of dull or frustrating classes in high school. Award-winning mathematics professor G. Arnell Williams is here to change that. Algebra the Beautiful is a journey into the heart of fundamental math that proves just how amazing this subject really is. Drawing on lessons from twenty-five years of teaching mathematics, Williams blends metaphor, history, and storytelling to uncover algebra's hidden grandeur. Whether you're a teacher looking to make math come alive for your students, a parent hoping to get your children engaged, a student trying to come to terms with a sometimes bewildering subject, or just a lover of mathematics, this book has something for you. With a passion that's contagious, G. Arnell Williams shows how each of us can grasp the beauty and harmony of algebra.

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Book SynopsisComprising a selection of expository and research papers, Harmonic Analysis and Integral Geometry grew from presentations offered at the July 1998 Summer University of Safi, Morocco-an annual, advanced research school and congress. This lively and very successful event drew the attendance of many top researchers, who offered both individual lectures and coordinated courses on specific research topics within this fast growing subject.Harmonic Analysis and Integral Geometry presents important recent advances in the fields of Radon transforms, integral geometry, and harmonic analysis on Lie groups and symmetric spaces. Several articles are devoted to the new theory of Radon transforms on trees.With its related presentations addressing recent developments in various aspects of these intriguing areas of study, Harmonic Analysis and Integral Geometry becomes an important addition not only to the Research Notes in Mathematics series, but to the general mathematics literature.Table of ContentsJohn's Equation and the Plane to Line Transform on R3. Radon Transforms on Compact Grassmann Manifolds and Invariant Differential Operators of Determinantal Type. Invariant Berezin Transforms. Integral Geometry on Hyperbolic Spaces. On Laguerre Polynomials of Two Variables. A Topological Obstruction for the Real Radon Transform. Integral Geometry in the Sphere Sd. The Distribution-Valued Horocyclic Radon Transform on Trees. The Geodesic Radon Transform on Trees. Integral Geometry on Affine Buildings. Poisson Transform on H3. Realization of a Holomorphic Discete-Series of the Lie Group SU(1,2) as Star-Representation. q-Analogue of Watanabe Unitary Transform Associated to the q-Continuous Gegenbauer Polynomials.

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Book SynopsisAbout the book…In honor of Edgar Enochs and his venerable contributions to a broad range of topics in Algebra, top researchers from around the world gathered at Auburn University to report on their latest work and exchange ideas on some of today's foremost research topics. This carefully edited volume presents the refereed papers of the participants of these talks along with contributions from other veteran researchers who were unable to attend.These papers reflect many of the current topics in Abelian Groups, Commutative Algebra, Commutative Rings, Group Theory, Homological Algebra, Lie Algebras, and Module Theory. Accessible even to beginning mathematicians, many of these articles suggest problems and programs for future study. This volume is an outstanding addition to the literature and a valuable handbook for beginning as well as seasoned researchers in Algebra. about the editors…H. PAT GOETERS completed his undergraduate studies in mathematics and computer science at Southern Connecticut State University and received his Ph.D. in 1984 from the University of Connecticut under the supervision of William J. Wickless. After spending one year in a post-doctoral position in Wesleyan University under the tutelage of James D. Reid, Goeters was invited for a tenure track position in Auburn University by Ulrich F. Albrecht. Soon afterwards, William Ullery and Overtoun Jenda were hired, and so began a lively Algebra group.OVERTOUN M. G. JENDA received his bachelor's degree in Mathematics from Chancellor College, the University of Malawi. He moved to the U.S. 1977 to pursue graduate studies at University of Kentucky, earning his Ph.D. in 1981 under the supervision of Professor Edgar Enochs. He then returned to Chancellor College, where he was a lecturer (assistant professor) for three years. He moved to the University of Botswana for another three-year stint as a lecturer before moving back to the University of Kentucky as a visiting assistant professor in 1987. In 1988, he joined the Algebra research group at Auburn University.Table of ContentsSome Aspects of Noncommutative Geometry. Omega Groups. Cancellation for Quotient Divisible Mixed Abelian Groups. Abelian Groups and Topological Structures. Gorenstein Homological Algebra. Applications of Module Approximations. Picard Groups of Certain Module Categories. Some Trends in the Theory of Covers and Envelopes. On Degenerate B2 Groups. The Loewy Length of Modules over Almost Perfect Domains. Half-factorial Domains. Projective Presentations of Finitely Generated Modules over Integral Domains. E-locally Cyclic Abelian Groups and Maximal Near-rings of Mappings. Approximately Simultaneously Diagnoalizable Matrices. FI-extending Hulls for Abelian Groups. Torsion-free Modules over a Discrete Valuation Ring. Forcing a Finite Group to be Abelian. Torsion-free Modules over Non-commutative Rings. Axiom 3 and Coverings. Divisibility Properties in Ultrapowers of Commutative Rings. Localizations of Torsion-free Abelian Groups. Modular Representations of Algebraic Groups, Finite Groups, and Frobenius Kernels. Homological Properties of Universal Enveloping Algebras.

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Book SynopsisAlthough there are many types of ring extensions, simple extensions have yet to be thoroughly explored in one book. Covering an understudied aspect of commutative algebra, Simple Extensions with the Minimum Degree Relations of Integral Domains presents a comprehensive treatment of various simple extensions and their properties. In particular, it examines several properties of simple ring extensions of Noetherian integral domains. As experts who have been studying this field for over a decade, the authors present many arguments that they have developed themselves, mainly exploring anti-integral, super-primitive, and ultra-primitive extensions. Within this framework, they study certain properties, such as flatness, integrality, and unramifiedness. Some of the topics discussed include Sharma polynomials, vanishing points, Noetherian domains, denominator ideals, unit groups, and polynomial rings. Presenting a complete treatment of each topic, Simple Extensions with the Minimum Degree Relations of Integral Domains serves as an ideal resource for graduate students and researchers involved in the area of commutative algebra.Trade Review"All topics … are developed in a clear way and illustrated by many examples." – EMS Newsletter, September 2008Table of ContentsBirational Simple Extensions. Simple Extensions of High Degree. Subrings of Anti-Integral Extensions. Denominator Ideals and Excellent Elements. Unramified Extensions. The Unit Groups of Extensions. Exclusive Extensions of Noetherian Domains. Ultra-Primitive Extensions and Their Generators. Flatness and Contractions of Ideals. Anti-Integral Ideals and Super Primitive Polynomials. Semi Anti-Integral and Pseudo-Simple Extensions. References. Index.

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Book SynopsisRecent years have seen a significant rise of interest in max-linear theory and techniques. Specialised international conferences and seminars or special sessions devoted to max-algebra have been organised. This book aims to provide a first detailed and self-contained account of linear-algebraic aspects of max-algebra for general (that is both irreducible and reducible) matrices. Among the main features of the book is the presentation of the fundamental max-algebraic theory (Chapters 1-4), often scattered in research articles, reports and theses, in one place in a comprehensive and unified form. This presentation is made with all proofs and in full generality (that is for both irreducible and reducible matrices). Another feature is the presence of advanced material (Chapters 5-10), most of which has not appeared in a book before and in many cases has not been published at all. Intended for a wide-ranging readership, this book will be useful for anyone with basic mathematical knowledge (including undergraduate students) who wish to learn fundamental max-algebraic ideas and techniques. It will also be useful for researchers working in tropical geometry or idempotent analysis.Table of ContentsMax-algebra: Two Special Features.- One-sided Max-linear Systems and Max-algebraic Subspaces.- Eigenvalues and Eigenvectors.- Maxpolynomials. The Characteristic Maxpolynomial.- Linear Independence and Rank. The Simple Image Set.- Two-sided Max-linear Systems.- Reachability of Eigenspaces.- Generalized Eigenproblem.- Max-linear Programs.- Conclusions and Open Problems.

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Book SynopsisThis book offers an introduction to the algorithmic-numerical thinking using basic problems of linear algebra. By focusing on linear algebra, it ensures a stronger thematic coherence than is otherwise found in introductory lectures on numerics. The book highlights the usefulness of matrix partitioning compared to a component view, leading not only to a clearer notation and shorter algorithms, but also to significant runtime gains in modern computer architectures. The algorithms and accompanying numerical examples are given in the programming environment MATLAB, and additionally – in an appendix – in the future-oriented, freely accessible programming language Julia. This book is suitable for a two-hour lecture on numerical linear algebra from the second semester of a bachelor's degree in mathematics.Table of ContentsPreface.- I Computing with Matrices.- II Matrix Factorization.- III Error Analysis.- IV Least Squares.- V Eigenvalue Problems.- Appendix.- Notation.- Index.

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Book SynopsisAlgebra und Diskrete Mathematik gehören zu den wichtigsten mathematischen Grundlagen der Informatik. In diese mathematischen Teilgebiete führt Band 1 des zweibändigen Lehrbuchs umfassend ein. Dabei ermöglichen klar herausgearbeitete Lösungsalgorithmen, viele Beispiele und ausführliche Beweise einen raschen Zugang zum Thema. Die umfangreiche Sammlung von Übungsaufgaben hilft bei der Erarbeitung des Stoffs und zeigt darüber hinaus, welche unterschiedlichen Anwendungsmöglichkeiten es gibt. Die 3. Auflage wurde korrigiert und erweitert.Table of ContentsTeil I Grundbegriffe der Mathematik und Algebraische Strukturen.- Teil II Lineare Algebra und analytische Geometrie.- Teil III Numerische Algebra und Kombinatorik.- Teil IV Übungsaufgaben.

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Book SynopsisDas Lehrbuch bietet umfangreiche Anleitungen und Übungsaufgaben zum Band „Mathematik für Physiker“ desselben Autors. Die Studienanleitungen mit Fragen und Kontrollaufgaben erleichtern Lesern das eigenständige Erarbeiten des Stoffs. In zusätzlichen Erläuterungen vertieft der Autor einzelne Themenfelder und geht auf individuelle Lernschwierigkeiten ein. Band 2 des Übungswerks enthält über 700 Aufgaben mit ausführlichen Lösungen und ist der ideale Begleiter für Bachelor-Studierende der Physik während des zweiten Semesters.Table of ContentsFunktionen mehrerer Variablen.- Partielle Ableitung, totales Differential.- Mehrfachintegrale, Parameterdarstellung.- Oberflächenintegrale.- Divergenz und Rotation.- Koordinatentransformation, Matrizen.- Lineare Gleichungssysteme.- Eigenwerte, Eigenvektoren.- Fourierreihen.- Fourier-Integrale, Fourier-Transformation.- Laplace-Transformation.- Wellengleichungen.

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Book SynopsisVI of Oregon lectures in 1962, Bass gave simplified proofs of a number of "Morita Theorems", incorporating ideas of Chase and Schanuel. One of the Morita theorems characterizes when there is an equivalence of categories mod-A R::! mod-B for two rings A and B. Morita's solution organizes ideas so efficiently that the classical Wedderburn-Artin theorem is a simple consequence, and moreover, a similarity class [AJ in the Brauer group Br(k) of Azumaya algebras over a commutative ring k consists of all algebras B such that the corresponding categories mod-A and mod-B consisting of k-linear morphisms are equivalent by a k-linear functor. (For fields, Br(k) consists of similarity classes of simple central algebras, and for arbitrary commutative k, this is subsumed under the Azumaya [51]1 and Auslander-Goldman [60J Brauer group. ) Numerous other instances of a wedding of ring theory and category (albeit a shot­ gun wedding!) are contained in the text. Furthermore, in. my attempt to further simplify proofs, notably to eliminate the need for tensor products in Bass's exposition, I uncovered a vein of ideas and new theorems lying wholely within ring theory. This constitutes much of Chapter 4 -the Morita theorem is Theorem 4. 29-and the basis for it is a corre­ spondence theorem for projective modules (Theorem 4. 7) suggested by the Morita context. As a by-product, this provides foundation for a rather complete theory of simple Noetherian rings-but more about this in the introduction.Table of Contentsto Volume.- Foreword on Set Theory.- I Introduction to the Operations: Monoid, Semigroup, Group, Category, Ring, and Module.- 1. Operations: Monoid, Semigroup, Group, and Category.- 2. Product and Coproduct.- 3. Ring and Module.- 4. Correspondence Theorems for Projective Modules and the Structure of Simple Noetherian Rings.- 5. Limits, Adjoints, and Algebras.- 6. Abelian Categories.- II Structure of Noetherian Semiprime Rings.- 7. General Wedderburn Theorems.- 8. Semisimple Modules and Homological Dimension.- 9. Noetherian Semiprime Rings.- 10. Orders in Semilocal Matrix Rings.- III Tensor Algebra.- 11. Tensor Products and Flat Modules.- 12. Morita Theorems and the Picard Group.- 13. Algebras over Fields.- IV Structure of Abelian Categories.- 14. Grothendieck Categories.- 15. Quotient Categories and Localizing Functors.- 16. Torsion Theories, Radicals, and Idempotent, Topologizing, and Multiplicative Sets.

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Book SynopsisDieses „Lern-und Lesebuch“ gibt eine erste Einführung in die grundlegenden Methoden und Ergebnisse der Algebra. Wie in einführenden Vorlesungen üblich, besteht es aus den drei Teilen Gruppen-Ringe-Körper, das sind die tragenden Säulen der Algebra. Höhepunkt im dritten Kapitel ist die klassische Galoistheorie in zeitgemäßer Darstellung, bei der viele der zuvor erzielten Ergebnisse zusammengefügt werden. Neben den üblichen Inhalten enthält das Buch aber auch Exkurse zu weiterführenden Themen, wie Symmetrien Platonischer Körper, quadratische Zahlringe oder Wurzelausdrücke für Einheitswurzel nach der Methode von Gauss. Ein ausführlicher Anhang schildert die Entwicklung der axiomatischen Methode von Euklid bis Bourbaki. Um Studierende der Algebra behutsam mit den subtilen Methoden und dem engmaschigen Netz von Begriffen vertraut zu machen, werden viele motivierende Vorbemerkungen, zahlreiche charakteristische Beispiele und auch – was in der Algebra nicht sehr üblich ist – mit Bildern zur Illustration von manchen Rechnungen eingefügt. Damit soll erreicht werden, dass die Studierenden neben einer Vorlesung einen Begleittext zur Hand haben, der ihnen nicht nur hilft die Schwierigkeiten zu meistern, sondern auch ein Gefühl für die Klarheit und Schönheit der Algebra vermitteln kann. Auch ohne den Besuch einer Vorlesung ist das Buch wegen seiner ausführlichen Darstellung für ein Selbststudium gut geeignet. Viele der Beispiele sind als Übungsaufgaben mit Anleitung gestaltet. Table of ContentsGruppen: Grundlegende Begriffe, Symmetriegruppen (insbesondere von Platonischen Körpern), Struktursätze, einfache und auflösbare Gruppen.- Ringe: Normalteiler, Ideale, Restklassenringe, Teilbarkeit, elementare Zahlentheorie, quadratische Zahlringe.- Körpererweiterungen: Zerfällungskörper, Vielfachheit von Nullstellen, Resultanten und Diskriminanten, Galois-Erweiterungen, Lösung von Polynomgleichungen, Konstruktionen mit Zirkel und Lineal.

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Book SynopsisDieses „Lern-und Lesebuch“ gibt eine erste Einführung in die grundlegenden Methoden und Ergebnisse der Algebra. Wie in einführenden Vorlesungen üblich, besteht es aus den drei Teilen Gruppen-Ringe-Körper, das sind die tragenden Säulen der Algebra. Höhepunkt im dritten Kapitel ist die klassische Galoistheorie in zeitgemäßer Darstellung, bei der viele der zuvor erzielten Ergebnisse zusammengefügt werden. Neben den üblichen Inhalten enthält das Buch aber auch Exkurse zu weiterführenden Themen, wie Symmetrien Platonischer Körper, quadratische Zahlringe oder Wurzelausdrücke für Einheitswurzel nach der Methode von Gauss. Ein ausführlicher Anhang schildert die Entwicklung der axiomatischen Methode von Euklid bis Bourbaki. Um Studierende der Algebra behutsam mit den subtilen Methoden und dem engmaschigen Netz von Begriffen vertraut zu machen, werden viele motivierende Vorbemerkungen, zahlreiche charakteristische Beispiele und auch – was in der Algebra nicht sehr üblich ist – mit Bildern zur Illustration von manchen Rechnungen eingefügt. Damit soll erreicht werden, dass die Studierenden neben einer Vorlesung einen Begleittext zur Hand haben, der ihnen nicht nur hilft die Schwierigkeiten zu meistern, sondern auch ein Gefühl für die Klarheit und Schönheit der Algebra vermitteln kann. Auch ohne den Besuch einer Vorlesung ist das Buch wegen seiner ausführlichen Darstellung für ein Selbststudium gut geeignet. Viele der Beispiele sind als Übungsaufgaben mit Anleitung gestaltet. Table of ContentsGruppen: Grundlegende Begriffe, Symmetriegruppen (insbesondere von Platonischen Körpern), Struktursätze, einfache und auflösbare Gruppen.- Ringe: Normalteiler, Ideale, Restklassenringe, Teilbarkeit, elementare Zahlentheorie, quadratische Zahlringe.- Körpererweiterungen: Zerfällungskörper, Vielfachheit von Nullstellen, Resultanten und Diskriminanten, Galois-Erweiterungen, Lösung von Polynomgleichungen, Konstruktionen mit Zirkel und Lineal.

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Book SynopsisIntroduction.- Principle of deep cognitive engagement for learning with instructionalvideos.- Principle of conceptual focus for the case of understanding variables andalgebraic expressions.- Research questions and research approaches.- Design of a learning environment with interactive video in the knowledge organization phase.- Qualitative study on how interactive features can scaffold cognitive engagement and conceptual focus.- Randomized controlled trial on how interactive videos enhance understanding of generalizing.- Summary and discussion of the results.

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Book SynopsisDer zweite Band der linearen Algebra führt den mit Lineare Algebra 1 und der Einführung in die Algebra begonnenen Kurs dieses Gegenstandes weiter und schliesst ihn weitgehend ab. Hierzu gehört die Theorie der sesquilinearen und quadratischen Formen sowie der unitären und euklidischen Vektorräume in Kapitel III. Kapitel IV enthält einen Abriss von Methoden und Ergebnissen der mulitlinearen Algebra, so wie sie für Anwendungen gebraucht werden; in Kapitel V wird gezeigt, wie die lineare und multilineare Algebra zur Begründung und Diskussion der linear-analytischen Geometrie verwendet werden kann. Auch hier sind den einzelnen Paragraphen zur inhaltlichen Vertiefung und Einübung der Gegenstände jeweils umfangreiche Ergänzungen und Aufgabensammlungen beigefügt.Table of ContentsI K-Vektorräume und ihre Homomorphismen.- §1. Algebraische Grundbegriffe.- Ergänzungen zu §1.- Aufgaben zu §1.- §2. Die Modulstruktur von K-Vektorräumén.- Ergänzungen zu §2.- Aufgaben zu §2.- §3. Lineare Unabhängigkeit, Vektorraumbasen.- Ergänzungen zu §3.- Aufgaben zu §3.- §4. Linearformen, Bilinearformen, Dualität.- Ergänzungen zu §4.- Aufgaben zu §4.- II K-Endomorphismen, Elementarteiler und Normalformenprobleme.- §5. Algebraische Eigenschaften von K-Homomorphismen, K-Endomorphismen und zugeordneten Matrizen.- Ergänzungen zu §5.- Aufgaben zu §5.- §6. Moduln über Hauptidealringen, Elementarteilersatz.- Ergänzungen zu §6.- Aufgaben zu §6.- §7. Normalformen von Matrizen und Anwendungen.- Ergänzungen zu §7.- Aufgaben zu §7.- Ergänzende Literatur.- Verzeichnis der Symbole.

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Book Synopsis(Autor) Jürgen Appell / Kristina Appell (Titel) Mengen - Zahlen - Zahlbereiche (Untertitel) Eine elementare Einführung in die Mathematik (HL) Mathematik für Erstsemester im Lehramt (USP) > beispielorientierter Aufbau > Vielzahl von Lösungsaufgaben inkl. Lösungshinweisen (copy) Das Buch dient nicht nur der Einführung der wichtigsten Zahlbereiche von den natürlichen bis zu den komplexen Zahlen und darüber hinaus, sondern behandelt auch ausführlich den in der Mahtmatik fundamentalen Mengen- und Funktionsbegriff. Zudem können SIe sich schon an Begriffe und Techniken gewöhnen, di ein den beiden mathematischen Grundvorlesungen Analysis und Lineare Algebra zentral sind. Ein besonderes Merkmal ist die Vielzahl der Beispiele und Gegenbeispiele an Hand derer neue BEgriffe eingeführt werden. Darüberhinaus enthält das Buch ein Kapitel mit etwa 300 Übungsaufgaben. (Biblio) 2005.256 S., kart. € 20,- / sFr 32,- ISBN 3-8274-1660-4 (Störer) neu!Trade ReviewDas Ziel des Bandes ist, zukünftigen studierenden der Mathematik den Übergang von der Schule zur Hochschule zu erleichtern. Der sehr verständlich formulierte Text ist hervorragend dazu geeignet, Schüler auf das Studium vorzubereiten. ekz-InformationsdienstTable of ContentsEinleitung 1 Aussagen und Mengen 1.1 Etwas Aussagenlogik 1.2 Einige Beweistechniken 1.3 Das Rechnen mit reellen Zahlen 1.4 Operationen mit Mengen 2 Funktionen und Mächtigkeiten 2.1 Injektive und surjektive Funktionen 2.2 Monotone Funktionen 2.3 Zahlenfolgen 2.4 Gleichmächtige Mengen und Abzählbarkeit 3 Natürliche und ganze Zahlen 3.1 Halbgruppen und Gruppen 3.2 Das Prinzip der vollständigen Induktion 3.3 Fakultäten und Binomialkoeffizienten 3.4 Teilbarkeit und Primzahlen 3.5 Der Hauptsatz der Arithmetik 3.6 Pythagoreische Tripel 3.7 p-adische Zahlensysteme 4 Rationale, reelle und komplexe Zahlen 4.1 Ringe und Körper 4.2 Homomorphismen und Isomorphismen 4.3 Rationale und reelle Zahlen 4.4 Komplexe Zahlen 4.5 Wichtige transzendente Zahlen 4.6 Höhere Zahlbereiche 5 Äquivalenzrelationen und Ordnungsrelationen 5.1 Äquivalenzrelationen und -klassen 5.2 Restklassenringe 5.3 Ordnungsrelationen 5.4 Boole'sche Verbände 6 Aufbau des Zahlensystems 6.1 Die Peano-Axiome 6.2 Konstruktion der ganzen Zahlen 6.3 Konstruktion der rationalen Zahlen 6.4 Konstruktion der reellen Zahlen 6.5 Einzigkeit der Menge der reellen Zahlen 7 Aufgaben 7.1 Aufgaben zum 1. Kapitel 7.2 Aufgaben zum 2. Kapitel 7.3 Aufgaben zum 3. Kapitel 7.4 Aufgaben zum 4. Kapitel 7.5 Aufgaben zum 5. Kapitel 7.6 Aufgaben zum 6. Kapitel 7.7 Lösungshinweise zu ausgewählten Aufgaben Anhang Einige Bezeichnungen und Abkürzungen Englische mathematische Ausdrücke Literaturverzeichnis Symbol-Index Stichwort-Index

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Table of ContentsFrontmatter -- Introduction -- Contents -- 1. Elements of Mathematics (165) -- Preface (1) (164) -- Preface (2) (94a) -- 1. Introduction, on Mathematics in General -- 2. Sequences -- 3. The Fundamental Operations in Algebra -- 4. Factors -- 5. Negative Numbers -- 6. Fractional Quantities -- 7. Simple Equations -- 8. Ratios and Proportions -- 9. Surds -- 10. Topical Geometry -- 11. Graphics and Perspective -- 2. New Elements of Geometry Based on Benjamin Peirce’s Works and Teachings (94) -- Preface -- Book I. Fundamental Properties of Space -- Book II. Topology -- Book III. Graphics -- Book IV. Metrics -- 3. Topical Geometry (137) -- Topical Geometry (137) -- 4. Appendices -- A. On the Quadratic Equation (86) -- B. Rational Fractions (278a fragment) -- C. Numerical Equations (69) -- D. [Additional Definitions] (from 166 and 150) -- E. [A Gloss. Elliptic, Hyperbolic, and Parabolic Measurement] (150) -- F. A Geometrico-Logical Discussion (126) -- G. [Projective Space] (part of 114) -- H. [Logic of Number – Le Fevre] (229) -- I. [Additional Theorems] (150 and 266) -- J. Promptuarium of Analytical Geometry (part of 102) -- K. Pythagorean Triangles (109) -- L. Analysis of Time (part of 138) -- M. Plan of Geometry (132) -- N. [Non-Euclidean Representation] (105) -- O. An Attempt to State Systematically the Doctrine of the Census in Geometrical Topics or Topical Geometry, More Commonly known as ‘Topology’; Being, a Mathematico- Logical Recreation of C. S. Peirce. Following the Lead of J. B. Listing’s Paper in the Göttinger Abhandlungen (from 145) -- P. [“Census” from Century Dictionary with Model] -- Q. Chapter III The Nature of Logical Inquiry (608) -- R. Three Problems -- S. Comments on Cayley’s Memoir on Abstract Geometry from the Point of View of the Logic of Relatives (546) -- T. [Obituary. Prof. Arthur Cayley] -- U. [From an Address to National Academy of Sciences] (95) -- Key to Greek Terms -- Key to Contents of MS. 165 -- Key to Contents of MS. 94 -- Index of Names -- Subject Index

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Book SynopsisIn this book, I. Ya. Bakel'man introduces inversion transformations in the Euclidean plane and discusses the interrelationships among more general mathematical concepts. The author begins by defining and giving examples of the concept of a transformation in the Euclidean plane, and then explains the point of infinity and the stereographic projection of the sphere onto the plane. With this preparation, the student is capable of applying the theory of inversions to classical construction problems in the plane. The author also discusses the theory of pencils of circles, and he uses the acquired techniques in a proof of Ptolemy's theorem. In the final chapter, the idea of a group is introduced with applications of group theory to geometry. The author demonstrates the group-theoretic basis for the distinction between Euclidean and Lobachevskian geometry.

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Book SynopsisExplores the role of group actions and rigidity in several areas of mathematics, including ergodic theory, dynamics, geometry, topology, and the algebraic properties of representation varieties.

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a huge range and FREE tracked UK delivery on ALL orders.

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Book SynopsisEssential mathematical tools for the study of modern quantum theory. Linear Algebra for Quantum Theory offers an excellent survey of those aspects of set theory and the theory of linear spaces and their mappings that are indispensable to the study of quantum theory.Table of ContentsElements of Set Theory. Linear Spaces. Binary Product Spaces. Axioms of Quantum Theory Formulated as a Trace Algebra. References. Appendices. Index.

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Book SynopsisThrough its presentation of the essentials, this text is briefer than some and has been carefully edited and designed to meet the specific needs of a one-semester course at the appropriate level for a Senior.Following extensive student testing for readability and understandability, examples have been intermixed with the theory throughout the book to introduce, motivate, and extend the main text. Although readability is emphasized, proofs are provided to promote logical thinking. Finally, the author''s conversational style holds the reader''s interest while exploring several important topics that traditionally have been reserved for graduate courses. The result is that students can apply theory that is sometimes a sterile subject in other courses, and can hit the ground running in advanced courses in feedback control design, dynamics of power systems, communications, and signal processing.Trade Review"...an excellent collection of examples...a good addition to the field of linear systems..." (Int Jnl of Robust & Nonlinear Control, May 2002)Table of ContentsSolution of State-Space Equations. Transform Methods. Writing State-Space Equations. Matrices Over a Field. Vector Spaces. Similarity Transformations. Stability. Minimality via Similarity Transformations. Poles and Zeros. References. Appendix. Index.

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Book SynopsisMultivariable Mathematics combines linear algebra and multivariable calculus in a rigorous approach. The material is integrated to emphasize the role of linearity in all of calculus and the recurring theme of implicit versus explicit that persists in linear algebra and analysis. In the text, the author addresses all of the standard computational material found in the usual linear algebra and multivariable calculus courses, and more, interweaving the material as effectively as possible and also including complete proofs. By emphasizing the theoretical aspects and reviewing the linear algebra material quickly, the book can also be used as a text for an advanced calculus or multivariable analysis course culminating in a treatment of manifolds, differential forms, and the generalized Stokes's Theorem.Table of ContentsPreface. Chapter 1. Vectors and Matrices. 1.1 Vectors in Rn.. 1.2 Dot Product. 1.3 Subspaces of Rn. 1.4 Linear Transformations and Matrix Algebra. 1.5 Introduction to Determinates and the Cross Product. Chapter 2. Functions, Limits, and Continuity. 2.1. Scalar- and Vector-Valued Functions. 2.2. A Bit of Topology in Rn. 2.3. Limits and Continuity. Chapter 3. The Derivative. 3.1. Partial Derivatives and Directional Derivatives. 3.2. Differentiability. 3.3. Differentiation Rules. 3.4. The Gradient. 3.5. Curves. 3.6. Higher-Order Partial Derivatives. Chapter 4. Implicit and Explicit Solutions of Linear Systems. 4.1. Gaussian Elimination and the Theory of Linear Systems. 4.2. Elementary Matrices and Calculating Inverse Matrices. 4.3. Linear Independence, Basis, and Dimension. 4.4. The Four Fundamental Subspaces. 4.5. The Nonlinear Case: Introduction to Manifolds. Chapter 5. Extremum Problems. 5.1. Compactness and the Maximum Value Theorem. 5.2. Maximum/Minimum Problems. 5.3. Quadratic Forms and the Second Derivative Test. 5.4. Lagrange Multipliers. 5.5. Projections, Least Squares, and Inner Product Spaces. Chapter 6. Solving Nonlinear Problems. 6.1. The Contraction Mapping Principle. 6.2. The Inverse and Implicit Function Theorems. 6.3. Manifolds Revisited. Chapter 7. Integration. 7.1. Multiple Integrals. 7.2. Iterated Integrals and Fubini’s Theorem. 7.3. Polar, Cylindrical, and Spherical Coordinates. 7.4. Physical Applications. 7.5. Determinants and n-Dimensional Volume. 7.6. Change of Variables Theorem. Chapter 8. Differential Forms and Integration on Manifolds. 8.1. Motivation. 8.2. Differential Forms. 8.3. Line Integrals and Green’s Theorem. 8.4. Surface Integrals and Flux. 8.5. Stokes’s Theorem. 8.6. Applications to Physics. 8.7. Applications to Topology. 9. Eigenvalues, Eigenvectors, and Applications. 9.1. Linear Transformations and Change of Basis. 9.2. Eigenvalues, Eigenvectors, and Diagonalizability. 9.3. Difference Equations and Ordinary Differential Equations. 9.4. The Spectral Theorem. Glossary of Notations and Results from Single-Variable Calculus. For Further Reading. Answers to Selected Exercises. Index.

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Book SynopsisDetails the basic theory of polynomial and fractional representation methods for algebraic analysis and synthesis of linear multivariable control systems. It also serves as a self-contained treatise of the mathematical theory so that results and techniques of the ``state space approaches'''' for regular and singular systems appear as special cases of a general theory covering the wider class of PMDs of linear systems. Among the topics covered are: real rational vector spaces and rational matrices, pole and zero structure of rational matrices at infinity, proper and omega stable rational fuctions and matrices.Table of ContentsReal Rational Vector Spaces and Rational Matrices. Polynomial Matrix Models of Linear Multivariable Systems. Pole and Zero Structure of Rational Matrices at Infinity. Dynamics of Polynomial Matrix Models. Proper and Omega-Stable Rational Functions and Matrices. Feedback System Stability and Stabilization. Some Algebraic Design Problems. Notations. Appendices. Index.

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Book SynopsisQuarternionic calculus covers a branch of mathematics which uses computational techniques to help solve problems from a wide variety of physical systems which are mathematically modelled in 3, 4 or more dimensions. Examples of the application areas include thermodynamics, hydrodynamics, geophysics and structural mechanics. Focusing on the Clifford algebra approach the authors have drawn together the research into quarternionic calculus to provide the non-expert or research student with an accessible introduction to the subject. This book fills the gap between the theoretical representations and the requirements of the user.Table of ContentsQuanternions and Multivectors. Clifford Valued Functions and Forms. Clifford Operator Calculus. Boundary Value Problems. Numerical Clifford Analysis. Further Results and Research Problems. Appendices. Index.

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Book SynopsisExplores fundamental concepts in arithmetic. This book begins with the definitions and properties of algebraic fields. It then discusses the theory of divisibility from an axiomatic viewpoint, rather than by the use of ideals. It also gives an introduction to p-adic numbers and their uses, which are important in modern number theory.Table of ContentsCh. I Algebraic Fields 1 Ch. II Theory of Divisibility (Kronecker, Dedekind) 33 Ch. III Local Primadic Analysis (Kummer, Hensel) 71 Ch. IV Algebraic Number Fields 141 Amendments 223

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Book SynopsisThis book contains accounts of talks held at a symposium in honor of John C. Moore in October 1983 at Princeton University, The work includes papers in classical homotopy theory, homological algebra, rational homotopy theory, algebraic K-theory of spaces, and other subjects.Table of Contents*Frontmatter, pg. i*Contents, pg. vii*Preface, pg. ix*I. Exponents in Homotopy Theory, pg. 1*II. The Exponent of a Moore Space, pg. 35*III. The Space of Maps of Moore Spaces Into Spheres, pg. 72*IV. The Adams Spectral Sequence of OMEGA2S3 and Brown Gitlet Spectra, pg. 101*V. Homotopy Groups of Some Mapping Telescopes, pg. 126*VI. Mapping Telescopes and K*-Localization, pg. 152*VII. The Geometric Realization of The Chromatic Resolution, pg. 168*VIII. Equivalences Between Homotopy Theories of Diagrams, pg. 180*IX. The Role of The Steenrod Algebra in The Mod 2 Cohomology of a Finite H-Space, pg. 206*X. Maps Between Classifying Spaces, pg. 228*XI. Generic Algebras and CW Complexes, pg. 247*XII. Deformation Theory and The Little Constructions of Cartan and Moore, pg. 322*XIII. Free (Z/2)3 - Actions on Finite Complexes, pg. 332*XIV. Equivariant Constructions of Nonequivariant Spectra, pg. 345*XV. A Decomposition of The Space of Generalized Morse Functions, pg. 365*XVI. Algebriac K-Theory of Spaces, Concordance, and Stable Homotopy Theory, pg. 392*XVII. The Map BSG --> A(*) --> QS , pg. 418*XVIII. Vector Bundles, Projective Modules and The K-Theory of Spheres, pg. 432*XIX. Limits Of Infinitesimal Group Cohomology, pg. 523*XX. Algebraic K-Theory of Group Scheme Actions, pg. 539*Backmatter, pg. 564

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Book SynopsisAn elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Developing, with many examples, the elementary theory of elliptic curves, this book goes on to the subject of modular forms and the first connections with elliptic curves.

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Book SynopsisOffers a survey of representation theory of semisimple Lie groups. Suitable for both graduate students and researchers, this book states the theorems precisely, and gives many illustrative examples or classes of examples. It includes for the reader a useful 300-item bibliography and an extensive section of notes.Trade ReviewWinner of the 1997 Leroy P. Steele Prize, American Mathematics Society "Anthony Knapp has written a marvelous book... Written with accuracy, style, and a genuine desire to communicate the materials... This is one of the finest books I have ever had the pleasure to read, and I recommend it in the strongest possible terms to anyone wishing to appreciate the intricate beauty and technical difficulty of representation theory of semisimple Lie groups."--R. J. Plymen, Bulletin of the London Mathematical Society "Each [theme] is developed carefully and thoroughly, with beautifully worked examples and proofs that reflect long experience in teaching and research... This result is delightful: a readable text that loses almost none of its value as a reference work."--David A. Vogan Jr., Bulletin of the American Mathematical SocietyTable of ContentsPreface to the Princeton Landmarks in Mathematics Edition xiii Preface xv Acknowledgments xix CHAPTER I. SCOPE OF THE THEORY 1. The Classical Groups 3 2. Cartan Decomposition 7 3. Representations 10 4. Concrete Problems in Representation Theory 14 5. Abstract Theory for Compact Groups 14 6. Application of the Abstract Theory to Lie Groups 23 7. Problems 24 CHAPTER II. REPRESENTATIONS OF SU(2), SL(2,R), AND SL(2,C) l. The Unitary Trick 28 2. Irreducible Finite-Dimensional Complex-Linear Representations of 91(2,C) 30 3. Finite-Dimensional Representations of 91(2,C) 31 4. Irreducible Unitary Representations of SL(2,C) 33 5. Irreducible Unitary Representations of SL(2,08) 35 6. Use of SU(1,1) 39 7. Plancherel Formula 41 8. Problems 42 CHAPTER III. C VECTORS AND THE UNIVERSAL ENVELOPING ALGEBRA l. Universal Enveloping Algebra 46 2. Actions on Universal Enveloping Algebra 50 3. C Vectors 55 4. Garding Subspace. Problems 57 CHAPTER IV. REPRESENTATIONS OF COMPACT LIE GROUPS 1. Examples of Root Space Decompositions 60 2. Roots 65 3. Abstract Root Systems and Positivity 72 4. Weyl Group, Algebraically 78 5. Weights and Integral Forms 81 6. Centalizers of Tori 86 7. Theorem of the Highest Weight 89 8. Verma Modules 93 9. Weyl Group, Analytically 100 10. Weyl Character Formula 104 11. Problems 109 CHAPTER V. STRUCTURE THEORY FOR NONCOMPACT GROUPS l. Cartan Decomposition and the Unitary Trick 113 2. Iwasawa Decomposition 116 3. Regular Elements, Weyl Chambers, and the Weyl Group 121 4. Other Decompositions 126 5. Parabolic Subgroups 132 6. Integral Formulas 137 7. Borel-Weil Theorem 142 8. Problems 147 CHAPTER VI. HOLOMORPHIC DISCRETE SERIES 1. Holomorphic Discrete Series for SU(1,1) 150 2. Classical Bounded Symmetric Domains 152 3. Harish-Chandra Decomposition 153 4. Holomorphic Discrete Series 158 5. Finiteness of an Integral 161 6. Problems 164 CHAPTER VII. INDUCED REPRESENTATIONS 1. Three Pictures 167 2. Elementary Properties 169 3. Bruhat Theory 172 4. Formal Intertwining Operators 174 5. Gindikin-Karpelevic Formula 177 6. Estimates on Intertwining Operators, Part I 181 7. Analytic Continuation of Intertwining Operators, Part I 183 8. Spherical Functions 185 9. Finite-Dimensional Representations and the H function 191 10. Estimates on Intertwining Operators, Part II 196 11. Tempered Representations and Langlands Quotients 198 12. Problems 201 CHAPTER VIII. ADMISSIBLE REPRESENTATIONS l. Motivation 203 2. Admissible Representations 205 3. Invariant Subspaces 209 4. Framework for Studying Matrix Coefficients 215 5. Harish-Chandra Homomorphism 218 6. Infinitesimal Character 223 7. Differential Equations Satisfied by Matrix Coefficients 226 8. Asymptotic Expansions and Leading Exponents 234 9. First Application: Subrepresentation Theorem 238 10. Second Application: Analytic Continuation of Interwining Operators, Part II 239 11. Third Application: Control of K-Finite Z(gc)-Finite Functions 242 12. Asymptotic Expansions near the Walls 247 13. Fourth Application: Asymptotic Size of Matrix Coefficients 253 14. Fifth Application: Identification of Irreducible Tempered Representations 258 15. Sixth Application: Langlands Classification of Irreducible Admissible Representations 266 16. Problems 276 CHAPTER IX. CONSTRUCTION OF DISCRETE SERIES 1. Infinitesimally Unitary Representations 281 2. A Third Way of Treating Admissible Representations 282 3. Equivalent Definitions of Discrete Series 284 4. Motivation in General and the Construction in SU(1,1) 287 5. Finite-Dimensional Spherical Representations 300 6. Duality in the General Case 303 7. Construction of Discrete Series 309 8. Limitations on K Types 320 9. Lemma on Linear Independence 328 10. Problems 330 CHAPTER X. GLOBAL CHARACTERS l. Existence 333 2. Character Formulas for SL(2,R) 338 3. Induced Characters 347 4. Differential Equations 354 5. Analyticity on the Regular Set, Overview and Example 355 6. Analyticity on the Regular Set, General Case 360 7. Formula on the Regular Set 368 8. Behavior on the Singular Set 371 9. Families of Admissible Representations 374 10. Problems 383 CHAPTER XI. INTRODUCTION TO PLANCHEREL FORMULA l. Constructive Proof for SU(2) 385 2. Constructive Proof for SL(2,C) 387 3. Constructive Proof for SL(2,R) 394 4. Ingredients of Proof for General Case 401 5. Scheme of Proof for General Case 404 6. Properties of F f 407 7. Hirai's Patching Conditions 421 8. Problems 425 CHAPTER XII. EXHAUSTION OF DISCRETE SERIES 1. Boundedness of Numerators of Characters 426 2. Use of Patching Conditions 432 3. Formula for Discrete Series Characters 436 4. Schwartz Space 447. 5. Exhaustion of Discrete Series 452 6. Tempered Distributions 456 7. Limits of Discrete Series 460 8. Discrete Series of M 467 9. Schmid's Identity 473 10. Problems 476 CHAPTER XIII. PLANCHEREL FORMULA 1. Ideas and Ingredients 482 2. Real-Rank-One Groups, Part I 482 3. Real-Rank-One Groups, Part II 485 4. Averaged Discrete Series 494 5. Sp (2,R) 502 6. General Case 511 7. Problems 512 CHAPTER XIV. IRREDUCIBLE TEMPERED REPRESENTATIONS l. SL(2,R) from a More General Point of View 515 2. Eisenstein Integrals 520 3. Asymptotics of Eisenstein Integrals 526 4. The il Functions for Intertwining Operators 535 5. First Irreducibility Results 540 6. Normalization of Intertwining Operators and Reducibility 543 7. Connection with Plancherel Formula when dim A = 1 547 8. Harish-Chandra's Completeness Theorem 553 9. R Group 560 10. Action by Weyl Group on Representations of M 568 11. Multiplicity One Theorem 577 12. Zuckerman Tensoring of Induced Representations 584 13. Generalized Schmid Identities 587 14. Inversion of Generalized Schmid Identities 595 15. Complete Reduction of Induced Representations 599 16. Classification 606 17. Revised Langlands Classification 614 18. Problems 621 CHAPTER XV. MINIMAL K TYPES l. Definition and Formula 626 2. Inversion Problem 635 3. Connection with Intertwining Operators 641 4. Problems 647 CHAPTER XVI. UNITARY REPRESENTATIONS 1. SL(2,U8) and SL(2,C) 650 2. Continuity Arguments and Complementary Series 653 3. Criterion for Unitary Representations 655 4. Reduction to Real Infinitesimal Character 660 5. Problems 665 APPENDIX A: ELEMENTARY THEORY OF LIE GROUPS l. Lie Algebras 667 2. Structure Theory of Lie Algebras 668 3. Fundamental Group and Covering Spaces 670 4. Topological Groups 673 5. Vector Fields and Submanifolds 674 6. Lie Groups 679 APPENDIX B: REGULAR SINGULAR POINTS OF PARTIAL DIFFERENTIAL EQUATIONS 1. Summary of Classical One-Variable Theory 685 2. Uniqueness and Analytic Continuation of Solutions in Several Variables 690 3. Analog of Fundamental Matrix 693 4. Regular Singularities 697 5. Systems of Higher Order 700 6. Leading Exponents and the Analog of the Indicial Equation 703 7. Uniqueness of Representation 710 APPENDIX C: ROOTS AND RESTRICTED ROOTS FOR CLASSICAL GROUPS 1. Complex Groups 713 2. Noncompact Real Groups 713 3. Roots vs. Restricted Roots in Noncompact Real Groups 715 NOTES 719 REFERENCES 747 INDEX OF NOTATION 763 INDEX 767

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Book SynopsisPresents an introduction to the theory of algebraic curves over a finite field, a subject that has applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. This book emphasizes the algebraic geometry rather than the function field approach to algebraic curves.Trade Review"This book is well-written and I greatly enjoyed reading it. The wealth of information and examples in this book give the reader a firm foundation and develop an intuition for the subject. The authors have used it as a textbook for a two-year course, and it would be a fine introduction to any advanced undergraduate or graduate student wanting to learn this subject."--Thomas Hagedorn, MAA ReviewsTable of ContentsPreface xi PART 1. GENERAL THEORY OF CURVES 1 Chapter 1. Fundamental ideas 3 1.1 Basic definitions 3 1.2 Polynomials 6 1.3 Affine plane curves 6 1.4 Projective plane curves 9 1.5 The Hessian curve 13 1.6 Projective varieties in higher-dimensional spaces 18 1.7 Exercises 18 1.8 Notes 19 Chapter 2. Elimination theory 21 2.1 Elimination of one unknown 21 2.2 The discriminant 30 2.3 Elimination in a system in two unknowns 31 2.4 Exercises 35 2.5 Notes 36 Chapter 3. Singular points and intersections 37 3.1 The intersection number of two curves 37 3.2 B'ezout's Theorem 45 3.3 Rational and birational transformations 49 3.4 Quadratic transformations 51 3.5 Resolution of singularities 55 3.6 Exercises 61 3.7 Notes 62 Chapter 4. Branches and parametrisation 63 4.1 Formal power series 63 4.2 Branch representations 75 4.3 Branches of plane algebraic curves 81 4.4 Local quadratic transformations 84 4.5 Noether's Theorem 92 4.6 Analytic branches 99 4.7 Exercises 107 4.8 Notes 109 Chapter 5. The function field of a curve 110 5.1 Generic points 110 5.2 Rational transformations 112 5.3 Places 119 5.4 Zeros and poles 120 5.5 Separability and inseparability 122 5.6 Frobenius rational transformations 123 5.7 Derivations and differentials 125 5.8 The genus of a curve 130 5.9 Residues of differential forms 138 5.10 Higher derivatives in positive characteristic 144 5.11 The dual and bidual of a curve 155 5.12 Exercises 159 5.13 Notes 160 Chapter 6. Linear series and the Riemann-Roch Theorem 161 6.1 Divisors and linear series 161 6.2 Linear systems of curves 170 6.3 Special and non-special linear series 177 6.4 Reformulation of the Riemann-Roch Theorem 180 6.5 Some consequences of the Riemann-Roch Theorem 182 6.6 The Weierstrass Gap Theorem 184 6.7 The structure of the divisor class group 190 6.8 Exercises 196 6.9 Notes 198 Chapter 7. Algebraic curves in higher-dimensional spaces 199 7.1 Basic definitions and properties 199 7.2 Rational transformations 203 7.3 Hurwitz's Theorem 208 7.4 Linear series composed of an involution 211 7.5 The canonical curve 216 7.6 Osculating hyperplanes and ramification divisors 217 7.7 Non-classical curves and linear systems of lines 228 7.8 Non-classical curves and linear systems of conics 230 7.9 Dual curves of space curves 238 7.10 Complete linear series of small order 241 7.11 Examples of curves 254 7.12 The Linear General Position Principle 257 7.13 Castelnuovo's Bound 257 7.14 A generalisation of Clifford's Theorem 260 7.15 The Uniform Position Principle 261 7.16 Valuation rings 262 7.17 Curves as algebraic varieties of dimension one 268 7.18 Exercises 270 7.19 Notes 271 PART 2. CURVES OVER A FINITE FIELD 275 Chapter 8. Rational points and places over a finite field 277 8.1 Plane curves defined over a finite field 277 8.2 Fq-rational branches of a curve 278 8.3 Fq-rational places, divisors and linear series 281 8.4 Space curves over Fq 287 8.5 The Stohr-Voloch Theorem 292 8.6 Frobenius classicality with respect to lines 305 8.7 Frobenius classicality with respect to conics 314 8.8 The dual of a Frobenius non-classical curve 326 8.9 Exercises 327 8.10 Notes 329 Chapter 9. Zeta functions and curves with many rational points 332 9.1 The zeta function of a curve over a finite field 332 9.2 The Hasse-Weil Theorem 343 9.3 Refinements of the Hasse-Weil Theorem 348 9.4 Asymptotic bounds 353 9.5 Other estimates 356 9.6 Counting points on a plane curve 358 9.7 Further applications of the zeta function 369 9.8 The Fundamental Equation 373 9.9 Elliptic curves over Fq 378 9.10 Classification of non-singular cubics over Fq 381 9.11 Exercises 385 9.12 Notes 388 PART 3. FURTHER DEVELOPMENTS 393 Chapter 10. Maximal and optimal curves 395 10.1 Background on maximal curves 396 10.2 The Frobenius linear series of a maximal curve 399 10.3 Embedding in a Hermitian variety 407 10.4 Maximal curves lying on a quadric surface 421 10.5 Maximal curves with high genus 428 10.6 Castelnuovo's number 431 10.7 Plane maximal curves 439 10.8 Maximal curves of Hurwitz type 442 10.9 Non-isomorphic maximal curves 446 10.10 Optimal curves 447 10.11 Exercises 453 10.12 Notes 454 Chapter 11. Automorphisms of an algebraic curve 458 11.1 The action of K-automorphisms on places 459 11.2 Linear series and automorphisms 464 11.3 Automorphism groups of plane curves 468 11.4 A bound on the order of a K-automorphism 470 11.5 Automorphism groups and their fixed fields 473 11.6 The stabiliser of a place 476 11.7 Finiteness of the K-automorphism group 480 11.8 Tame automorphism groups 483 11.9 Non-tame automorphism groups 486 11.10 K-automorphism groups of particular curves 501 11.11 Fixed places of automorphisms 509 11.12 Large automorphism groups of function fields 513 11.13 K-automorphism groups fixing a place 532 11.14 Large p-subgroups fixing a place 539 11.15 Notes 542 Chapter 12. Some families of algebraic curves 546 12.1 Plane curves given by separated polynomials 546 12.2 Curves with Suzuki automorphism group 564 12.3 Curves with unitary automorphism group 572 12.4 Curves with Ree automorphism group 575 12.5 A curve attaining the Serre Bound 585 12.6 Notes 587 Chapter 13. Applications: codes and arcs 590 13.1 Algebraic-geometry codes 590 13.2 Maximum distance separable codes 594 13.3 Arcs and ovals 599 13.4 Segre's generalisation of Menelaus' Theorem 603 13.5 The connection between arcs and curves 607 13.6 Arcs in ovals in planes of even order 611 13.7 Arcs in ovals in planes of odd order 612 13.8 The second largest complete arc 615 13.9 The third largest complete arc 623 13.10 Exercises 625 13.11 Notes 625 Appendix A. Background on field theory and group theory 627 A.1 Field theory 627 A.2 Galois theory 633 A.3 Norms and traces 635 A.4 Finite fields 636 A.5 Group theory 638 A.6 Notes 649 Appendix B. Notation 650 Bibliography 655 Index 689

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Book SynopsisCovers stability and stabilization of equilibria for both linear and nonlinear time-invariant systems of ordinary differential equations. Suitable for advanced undergraduates and beginning graduate students in the sciences, engineering, and mathematics, this book takes a modern approach that bridges the gap between linear and nonlinear systems.Trade Review"This book takes a unique modern approach that bridges the gap between linear and nonlinear systems... Clear formulated definitions and theorems, correct proofs and many interesting examples and exercises make this textbook very attractive."--Ferenc Szenkovits, MathematicaTable of ContentsList of Figures xi Preface xiii Chapter 1: Introduction 1 1.1 Open Loop Control 1 1.2 The Feedback Stabilization Problem 2 1.3 Chapter and Appendix Descriptions 5 1.4 Notes and References 11 Chapter 2: Mathematical Background 12 2.1 Analysis Preliminaries 12 2.2 Linear Algebra and Matrix Algebra 12 2.3 Matrix Analysis 17 2.4 Ordinary Differential Equations 30 2.4.1 Phase Plane Examples: Linear and Nonlinear 35 2.5 Exercises 44 2.6 Notes and References 48 Chapter 3: Linear Systems and Stability 49 3.1 The Matrix Exponential 49 3.2 The Primary Decomposition and Solutions of LTI Systems 53 3.3 Jordan Form and Matrix Exponentials 57 3.3.1 Jordan Form of Two-Dimensional Systems 58 3.3.2 Jordan Form of n-Dimensional Systems 61 3.4 The Cayley-Hamilton Theorem 67 3.5 Linear Time Varying Systems 68 3.6 The Stability Definitions 71 3.6.1 Motivations and Stability Definitions 71 3.6.2 Lyapunov Theory for Linear Systems 73 3.7 Exercises 77 3.8 Notes and References 81 Chapter 4: Controllability of Linear Time Invariant Systems 82 4.1 Introduction 82 4.2 Linear Equivalence of Linear Systems 84 4.3 Controllability with Scalar Input 88 4.4 Eigenvalue Placement with Single Input 92 4.5 Controllability with Vector Input 94 4.6 Eigenvalue Placement with Vector Input 96 4.7 The PBH Controllability Test 99 4.8 Linear Time Varying Systems: An Example 103 4.9 Exercises 105 4.10 Notes and References 108 Chapter 5: Observability and Duality 109 5.1 Observability, Duality, and a Normal Form 109 5.2 Lyapunov Equations and Hurwitz Matrices 117 5.3 The PBH Observability Test 118 5.4 Exercises 121 5.5 Notes and References 123 Chapter 6: Stabilizability of LTI Systems 124 6.1 Stabilizing Feedbacks for Controllable Systems 124 6.2 Limitations on Eigenvalue Placement 128 6.3 The PBH Stabilizability Test 133 6.4 Exercises 134 6.5 Notes and References 136 Chapter 7: Detectability and Duality 138 7.1 An Example of an Observer System 138 7.2 Detectability, the PBH Test, and Duality 142 7.3 Observer-Based Dynamic Stabilization 145 7.4 Linear Dynamic Controllers and Stabilizers 147 7.5 LQR and the Algebraic Riccati Equation 152 7.6 Exercises 156 7.7 Notes and References 159 Chapter 8: Stability Theory 161 8.1 Lyapunov Theorems and Linearization 161 8.1.1 Lyapunov Theorems 162 8.1.2 Stabilization from the Jacobian Linearization 171 8.1.3 Brockett's Necessary Condition 172 8.1.4 Examples of Critical Problems 173 8.2 The Invariance Theorem 176 8.3 Basin of Attraction 181 8.4 Converse Lyapunov Theorems 183 8.5 Exercises 183 8.6 Notes and References 187 Chapter 9: Cascade Systems 189 9.1 The Theorem on Total Stability 189 9.1.1 Lyapunov Stability in Cascade Systems 192 9.2 Asymptotic Stability in Cascades 193 9.2.1 Examples of Planar Systems 193 9.2.2 Boundedness of Driven Trajectories 196 9.2.3 Local Asymptotic Stability 199 9.2.4 Boundedness and Global Asymptotic Stability 202 9.3 Cascades by Aggregation 204 9.4 Appendix: The Poincar'e-Bendixson Theorem 207 9.5 Exercises 207 9.6 Notes and References 211 Chapter 10: Center Manifold Theory 212 10.1 Introduction 212 10.1.1 An Example 212 10.1.2 Invariant Manifolds 213 10.1.3 Special Coordinates for Critical Problems 214 10.2 The Main Theorems 215 10.2.1 Definition and Existence of Center Manifolds 215 10.2.2 The Reduced Dynamics 218 10.2.3 Approximation of a Center Manifold 222 10.3 Two Applications 225 10.3.1 Adding an Integrator for Stabilization 226 10.3.2 LAS in Special Cascades: Center Manifold Argument 228 10.4 Exercises 229 10.5 Notes and References 231 Chapter 11: Zero Dynamics 233 11.1 The Relative Degree and Normal Form 233 11.2 The Zero Dynamics Subsystem 244 11.3 Zero Dynamics and Stabilization 248 11.4 Vector Relative Degree of MIMO Systems 251 11.5 Two Applications 254 11.5.1 Designing a Center Manifold 254 11.5.2 Zero Dynamics for Linear SISO Systems 257 11.6 Exercises 263 11.7 Notes and References 267 Chapter 12: Feedback Linearization of Single-Input Nonlinear Systems 268 12.1 Introduction 268 12.2 Input-State Linearization 270 12.2.1 Relative Degree n 271 12.2.2 Feedback Linearization and Relative Degree n 272 12.3 The Geometric Criterion 275 12.4 Linearizing Transformations 282 12.5 Exercises 285 12.6 Notes and References 287 Chapter 13: An Introduction to Damping Control 289 13.1 Stabilization by Damping Control 289 13.2 Contrasts with Linear Systems: Brackets, Controllability, Stabilizability 296 13.3 Exercises 299 13.4 Notes and References 300 Chapter 14: Passivity 302 14.1 Introduction to Passivity 302 14.1.1 Motivation and Examples 302 14.1.2 Definition of Passivity 304 14.2 The KYP Characterization of Passivity 306 14.3 Positive Definite Storage 309 14.4 Passivity and Feedback Stabilization 314 14.5 Feedback Passivity 318 14.5.1 Linear Systems 321 14.5.2 Nonlinear Systems 325 14.6 Exercises 327 14.7 Notes and References 330 Chapter 15: Partially Linear Cascade Systems 331 15.1 LAS from Partial-State Feedback 331 15.2 The Interconnection Term 333 15.3 Stabilization by Feedback Passivation 336 15.4 Integrator Backstepping 349 15.5 Exercises 355 15.6 Notes and References 357 Chapter 16: Input-to-State Stability 359 16.1 Preliminaries and Perspective 359 16.2 Stability Theorems via Comparison Functions 364 16.3 Input-to-State Stability 366 16.4 ISS in Cascade Systems 372 16.5 Exercises 374 16.6 Notes and References 376 Chapter 17: Some Further Reading 378 Appendix A: Notation: A Brief Key 381 Appendix B: Analysis in R and Rn 383 B.1 Completeness and Compactness 386 B.2 Differentiability and Lipschitz Continuity 393 Appendix C: Ordinary Differential Equations 393 C.1 Existence and Uniqueness of Solutions 393 C.2 Extension of Solutions 396 C.3 Continuous Dependence 399 Appendix D: Manifolds and the Preimage Theorem; Distributions and the Frobenius Theorem 403 D.1 Manifolds and the Preimage Theorem 403 D.2 Distributions and the Frobenius Theorem 410 Appendix E: Comparison Functions and a Comparison Lemma 420 E.1 Definitions and Basic Properties 420 E.2 Differential Inequality and Comparison Lemma 424 Appendix F: Hints and Solutions for Selected Exercises 430 Bibliography 443 Index 451

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Book SynopsisWhat is algebra? For some, it is an abstract language of x's and y's. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and fields. This title considers how these two seemingly different types of algebra evolved and how they relate.Trade Review"An excellent book; its accurate historical and pedagogical purpose offers an accessible read for historians and mathematicians."--Raffaele Pisano, Metascience "Well written and engaging with a wealth of useful material and a substantial bibliography for further reading, this book is a valuable resource for anyone with a serious interest in the history of algebra. With Taming the Unknown, Victor Katz and Karen Parshall have created a comprehensive synthesis of recent research on the subject, accessible to mathematicians, historians of mathematics and anyone involved in the teaching of algebra."--Adrian Rice, BSHM Bulletin "The authors have ... pitched their writing perfectly for their intended audience. The broad outline of the story is expressed in clear prose, combined with a judicious use of that other 'native tongue' of the college mathematics graduate, symbolic algebra... There is an extensive bibliography presenting the more detailed historical research that has been carried out... You could base a really nice third-year course on this book."--John Hannah, AestimatioTable of ContentsAcknowledgments xi 1 Prelude: What Is Algebra? 1 Why This Book? 3 Setting and Examining the Historical Parameters 4 The Task at Hand 10 2 Egypt and Mesopotamia 12 Proportions in Egypt 12 Geometrical Algebra in Mesopotamia 17 3 The Ancient Greek World 33 Geometrical Algebra in Euclid's Elements and Data 34 Geometrical Algebra in Apollonius's Conics 48 Archimedes and the Solution of a Cubic Equation 53 4 Later Alexandrian Developments 58 Diophantine Preliminaries 60 A Sampling from the Arithmetica: The First Three Greek Books 63 A Sampling from the Arithmetica: The Arabic Books 68 A Sampling from the Arithmetica: The Remaining Greek Books 73 The Reception and Transmission of the Arithmetica 77 5 Algebraic Thought in Ancient and Medieval China 81 Proportions and Linear Equations 82 Polynomial Equations 90 Indeterminate Analysis 98 The Chinese Remainder Problem 100 6 Algebraic Thought in Medieval India 105 Proportions and Linear Equations 107 Quadratic Equations 109 Indeterminate Equations 118 Linear Congruences and the Pulverizer 119 The Pell Equation 122 Sums of Series 126 7 Algebraic Thought in Medieval Islam 132 Quadratic Equations 137 Indeterminate Equations 153 The Algebra of Polynomials 158 The Solution of Cubic Equations 165 8 Transmission, Transplantation, and Diffusion in the Latin West 174 The Transplantation of Algebraic Thought in the Thirteenth Century 178 The Diffusion of Algebraic Thought on the Italian Peninsula and Its Environs from the Thirteenth Through the Fifteenth Centuries 190 The Diffusion of Algebraic Thought and the Development of Algebraic Notation outside of Italy 204 9 The Growth of Algebraic Thought in Sixteenth-Century Europe 214 Solutions of General Cubics and Quartics 215 Toward Algebra as a General Problem-Solving Technique 227 10 From Analytic Geometry to the Fundamental Theorem of Algebra 247 Thomas Harriot and the Structure of Equations 248 Pierre de Fermat and the Introduction to Plane and Solid Loci 253 Albert Girard and the Fundamental Theorem of Algebra 258 Rene Descartes and The Geometry 261 Johann Hudde and Jan de Witt, Two Commentators on The Geometry 271 Isaac Newton and the Arithmetica universalis 275 Colin Maclaurin's Treatise of Algebra 280 Leonhard Euler and the Fundamental Theorem of Algebra 283 11 Finding the Roots of Algebraic Equations 289 The Eighteenth-Century Quest to Solve Higher-Order Equations Algebraically 290 The Theory of Permutations 300 Determining Solvable Equations 303 The Work of Galois and Its Reception 310 The Many Roots of Group Theory 317 The Abstract Notion of a Group 328 12 Understanding Polynomial Equations in n Unknowns 335 Solving Systems of Linear Equations in n Unknowns 336 Linearly Transforming Homogeneous Polynomials in n Unknowns: Three Contexts 345 The Evolution of a Theory of Matrices and Linear Transformations 356 The Evolution of a Theory of Invariants 366 13 Understanding the Properties of "Numbers" 381 New Kinds of "Complex" Numbers 382 New Arithmetics for New "Complex" Numbers 388 What Is Algebra?: The British Debate 399 An "Algebra" of Vectors 408 A Theory of Algebras, Plural 415 14 The Emergence of Modern Algebra 427 Realizing New Algebraic Structures Axiomatically 430 The Structural Approach to Algebra 438 References 449 Index 477

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Book SynopsisProvides an exploration of standard numerical analysis topics, as well as non-traditional ones, including mathematical modeling, Monte Carlo methods, Markov chains, and fractals. This textbook considers modern application areas, such as information retrieval and animation, and classical topics from physics and engineering.Trade Review"Distinguishing features are the inclusion of many recent applications of numerical methods and the extensive discussion of methods based on Chebyshev interpolation. This book would be suitable for use in courses aimed at advanced undergraduate students in mathematics, the sciences, and engineering."--Choice "An instructor could assemble several different one-semester courses using this book--numerical linear algebra and interpolation, or numerical solutions of differential equations--or perhaps a two-semester sequence. This is a charming book, well worth consideration for the next numerical analysis course."--William J. Satzer, MAA FocusTable of ContentsPreface xiii Chapter 1: MATHEMATICAL MODELING 1 1.1 Modeling in Computer Animation 2 1.1.1 A Model Robe 2 1.2 Modeling in Physics: Radiation Transport 4 1.3 Modeling in Sports 6 1.4 Ecological Models 8 1.5 Modeling a Web Surfer and Google 11 1.5.1 The Vector Space Model 11 1.5.2 Google's PageRank 13 1.6 Chapter 1 Exercises 14 Chapter 2: BASIC OPERATIONS WITH MATLAB 19 2.1 Launching MATLAB 19 2.2 Vectors 20 2.3 Getting Help 22 2.4 Matrices 23 2.5 Creating and Running .m Files 24 2.6 Comments 25 2.7 Plotting 25 2.8 Creating Your Own Functions 27 2.9 Printing 28 2.10 More Loops and Conditionals 29 2.11 Clearing Variables 31 2.12 Logging Your Session 31 2.13 More Advanced Commands 31 2.14 Chapter 2 Exercises 32 Chapter 3: MONTE CARLO METHODS 41 3.1 A Mathematical Game of Cards 41 3.1.1 The Odds in Texas Holdem 42 3.2 Basic Statistics 46 3.2.1 Discrete Random Variables 48 3.2.2 Continuous Random Variables 51 3.2.3 The Central Limit Theorem 53 3.3 Monte Carlo Integration 56 3.3.1 Buffon's Needle 56 3.3.2 Estimating pi 58 3.3.3 Another Example of Monte Carlo Integration 60 3.4 Monte Carlo Simulation of Web Surfing 64 3.5 Chapter 3 Exercises 67 Chapter 4: SOLUTION OF A SINGLE NONLINEAR EQUATION IN ONE UNKNOWN 71 4.1 Bisection 75 4.2 Taylor's Theorem 80 4.3 Newton's Method 83 4.4 Quasi-Newton Methods 89 4.4.1 Avoiding Derivatives 89 4.4.2 Constant Slope Method 89 4.4.3 Secant Method 90 4.5 Analysis of Fixed Point Methods 93 4.6 Fractals, Julia Sets, and Mandelbrot Sets 98 4.7 Chapter 4 Exercises 102 Chapter 5: FLOATING-POINT ARITHMETIC 107 5.1 Costly Disasters Caused by Rounding Errors 108 5.2 Binary Representation and Base 2 Arithmetic 110 5.3 Floating-Point Representation 112 5.4 IEEE Floating-Point Arithmetic 114 5.5 Rounding 116 5.6 Correctly Rounded Floating-Point Operations 118 5.7 Exceptions 119 5.8 Chapter 5 Exercises 120 Chapter 6: CONDITIONING OF PROBLEMS; STABILITY OF ALGORITHMS 124 6.1 Conditioning of Problems 125 6.2 Stability of Algorithms 126 6.3 Chapter 6 Exercises 129 Chapter 7: DIRECT METHODS FOR SOLVING LINEAR SYSTEMS AND LEAST SQUARES PROBLEMS 131 7.1 Review of Matrix Multiplication 132 7.2 Gaussian Elimination 133 7.2.1 Operation Counts 137 7.2.2 LU Factorization 139 7.2.3 Pivoting 141 7.2.4 Banded Matrices and Matrices for Which Pivoting Is Not Required 144 7.2.5 Implementation Considerations for High Performance 148 7.3 Other Methods for Solving Ax = b 151 7.4 Conditioning of Linear Systems 154 7.4.1 Norms 154 7.4.2 Sensitivity of Solutions of Linear Systems 158 7.5 Stability of Gaussian Elimination with Partial Pivoting 164 7.6 Least Squares Problems 166 7.6.1 The Normal Equations 167 7.6.2 QR Decomposition 168 7.6.3 Fitting Polynomials to Data 171 7.7 Chapter 7 Exercises 175 Chapter 8: POLYNOMIAL AND PIECEWISE POLYNOMIAL INTERPOLATION 181 8.1 The Vandermonde System 181 8.2 The Lagrange Form of the Interpolation Polynomial 181 8.3 The Newton Form of the Interpolation Polynomial 185 8.3.1 Divided Differences 187 8.4 The Error in Polynomial Interpolation 190 8.5 Interpolation at Chebyshev Points and chebfun 192 8.6 Piecewise Polynomial Interpolation 197 8.6.1 Piecewise Cubic Hermite Interpolation 200 8.6.2 Cubic Spline Interpolation 201 8.7 Some Applications 204 8.8 Chapter 8 Exercises 206 Chapter 9: NUMERICAL DIFFERENTIATION AND RICHARDSON EXTRAPOLATION 212 9.1 Numerical Differentiation 213 9.2 Richardson Extrapolation 221 9.3 Chapter 9 Exercises 225 Chapter 10: NUMERICAL INTEGRATION 227 10.1 Newton-Cotes Formulas 227 10.2 Formulas Based on Piecewise Polynomial Interpolation 232 10.3 Gauss Quadrature 234 10.3.1 Orthogonal Polynomials 236 10.4 Clenshaw-Curtis Quadrature 240 10.5 Romberg Integration 242 10.6 Periodic Functions and the Euler-Maclaurin Formula 243 10.7 Singularities 247 10.8 Chapter 10 Exercises 248 Chapter 11: NUMERICAL SOLUTION OF THE INITIAL VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATIONS 251 11.1 Existence and Uniqueness of Solutions 253 11.2 One-Step Methods 257 11.2.1 Euler's Method 257 11.2.2 Higher-Order Methods Based on Taylor Series 262 11.2.3 Midpoint Method 262 11.2.4 Methods Based on Quadrature Formulas 264 11.2.5 Classical Fourth-Order Runge-Kutta and Runge-Kutta-Fehlberg Methods 265 11.2.6 An Example Using MATLAB's ODE Solver 267 11.2.7 Analysis of One-Step Methods 270 11.2.8 Practical Implementation Considerations 272 11.2.9 Systems of Equations 274 11.3 Multistep Methods 275 11.3.1 Adams-Bashforth and Adams-Moulton Methods 275 11.3.2 General Linear m-Step Methods 277 11.3.3 Linear Difference Equations 280 11.3.4 The Dahlquist Equivalence Theorem 283 11.4 Stiff Equations 284 11.4.1 Absolute Stability 285 11.4.2 Backward Differentiation Formulas (BDF Methods) 289 11.4.3 Implicit Runge-Kutta (IRK) Methods 290 11.5 Solving Systems of Nonlinear Equations in Implicit Methods 291 11.5.1 Fixed Point Iteration 292 11.5.2 Newton's Method 293 11.6 Chapter 11 Exercises 295 Chapter 12: MORE NUMERICAL LINEAR ALGEBRA: EIGENVALUES AND ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS 300 12.1 Eigenvalue Problems 300 12.1.1 The Power Method for Computing the Largest Eigenpair 310 12.1.2 Inverse Iteration 313 12.1.3 Rayleigh Quotient Iteration 315 12.1.4 The QR Algorithm 316 12.1.5 Google's PageRank 320 12.2 Iterative Methods for Solving Linear Systems 327 12.2.1 Basic Iterative Methods for Solving Linear Systems 327 12.2.2 Simple Iteration 328 12.2.3 Analysis of Convergence 332 12.2.4 The Conjugate Gradient Algorithm 336 12.2.5 Methods for Nonsymmetric Linear Systems 334 12.3 Chapter 12 Exercises 345 Chapter 13: NUMERICAL SOLUTION OF TWO-POINT BOUNDARY VALUE PROBLEMS 350 13.1 An Application: Steady-State Temperature Distribution 350 13.2 Finite Difference Methods 352 13.2.1 Accuracy 354 13.2.2 More General Equations and Boundary Conditions 360 13.3 Finite Element Methods 365 13.3.1 Accuracy 372 13.4 Spectral Methods 374 13.5 Chapter 13 Exercises 376 Chapter 14: NUMERICAL SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS 379 14.1 Elliptic Equations 381 14.1.1 Finite Difference Methods 381 14.1.2 Finite Element Methods 386 14.2 Parabolic Equations 388 14.2.1 Semidiscretization and the Method of Lines 389 14.2.2 Discretization in Time 389 14.3 Separation of Variables 396 14.3.1 Separation of Variables for Difference Equations 400 14.4 Hyperbolic Equations 402 14.4.1 Characteristics 402 14.4.2 Systems of Hyperbolic Equations 403 14.4.3 Boundary Conditions 404 14.4.4 Finite Difference Methods 404 14.5 Fast Methods for Poisson's Equation 409 14.5.1 The Fast Fourier Transform 411 14.6 Multigrid Methods 414 14.7 Chapter 14 Exercises 418 APPENDIX A REVIEW OF LINEAR ALGEBRA 421 A.1 Vectors and Vector Spaces 421 A.2 Linear Independence and Dependence 422 A.3 Span of a Set of Vectors; Bases and Coordinates; Dimension of a Vector Space 423 A.4 The Dot Product; Orthogonal and Orthonormal Sets; the Gram-Schmidt Algorithm 423 A.5 Matrices and Linear Equations 425 A.6 Existence and Uniqueness of Solutions; the Inverse; Conditions for Invertibility 427 A.7 Linear Transformations; the Matrix of a Linear Transformation 431 A.8 Similarity Transformations; Eigenvalues and Eigenvectors 432 APPENDIX B TAYLOR'S THEOREM IN MULTIDIMENSIONS 436 References 439 Index 445

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Book SynopsisThe hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula.Table of Contents*FrontMatter, pg. i*Contents, pg. vii*Acknowledgments, pg. xi*Introduction, pg. 1*Chapter One. Clifford and Heisenberg algebras, pg. 12*Chapter Two. The hypoelliptic Laplacian on X = G/K, pg. 22*Chapter Three. The displacement function and the return map, pg. 48*Chapter Four. Elliptic and hypoelliptic orbital integrals, pg. 76*Chapter Five. Evaluation of supertraces for a model operator, pg. 92*Chapter Six. A formula for semisimple orbital integrals, pg. 113*Chapter Seven. An application to local index theory, pg. 120*Chapter Eight. The case where [k (gamma); p0] = 0, pg. 138*Chapter Nine. A proof of the main identity, pg. 142*Chapter Ten. The action functional and the harmonic oscillator, pg. 161*Chapter Eleven. The analysis of the hypoelliptic Laplacian, pg. 187*Chapter Twelve. Rough estimates on the scalar heat kernel, pg. 212*Chapter Thirteen. Refined estimates on the scalar heat kernel for bounded b, pg. 248*Chapter Fourteen. The heat kernel qXb;t for bounded b, pg. 262*Chapter Fifteen. The heat kernel qXb;t for b large, pg. 290*Bibliography, pg. 317*Subject Index, pg. 323*Index of Notation, pg. 325

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Book SynopsisTrade Review"Lane explores secondary, or hidden, mathematical gems that a player might discover upon mature reflection. . . . Just as most car drivers prefer not to inquire how the internal combustion engine works, most video-type users prefer not to ask how computer magic works. For the few who do ask questions, Lane assures us and as his book testifies, 'there's a lot of mathematics under the surface'."---Andrew James Simoson, MathSciNet"Lane explains some pretty technical concepts in an accessible way. . . . A fun survey of interesting maths related through the lens of video games."---Paul Taylor, Aperiodical"The examples [in Power-Up] were carefully chosen from very popular games, so even the most casual player will have heard of the vast majority of the games discussed. In general, Lane's writing is easy to digest, and the use of color and high-quality paper gives the book a nice look and feel." * Choice *"Power­Up is a very readable book based on examples taken from popular video games. . . . It is a pity that too many people are deprived of the pleasure of finding things out via the intellectual game of mathematics. Hopefully, the effort of the likes of Matthew Lane will someday solve the severe marketing problem of mathematics." * Computing Reviews *"Overall the book is excellent. Lane has written a high readable text with colorful illustrations. You won’t regret reading it and maybe Power-Up will add a new level of insight to your computer gaming." * MAA Reviews *"Matthew Lane explores the mathematical underpinning many popular video games in this well-written and very enjoyable book that is pitched at a very broad audience"---Dominic Thorrington, Mathematics TodayTable of ContentsAcknowledgments xi Introduction 1 1. Let's Get Physical 7 1.1 Platforming Perils 7 1.2 Platforming in Three Dimensions 10 1.3 LittleBigPlanet: Exploring Physics through Gameplay 12 1.4 From 2D to 3D: Bending Laws in Portal 14 1.5 Exploring Reality with A Slower Speed of Light 18 1.6 Exploring Alternative Realities 21 1.7 Beyond Physics: Minecraft or Mine Field? 26 1.8 Closing Remarks 27 1.9 Addendum: Describing Distortion 29 2. Repeat Offenders 34 2.1 Let's Play the Feud! 34 2.2 Game Shows and Birthdays 36 2.3 Beyond the First Duplicate 39 2.4 The Draw Something Debacle 41 2.5 Delayed Repetition: Increasing N 46 2.6 Delayed Repetition:Weight Lifting 48 2.7 The Completionist's Dilemma 53 2.8 Closing Remarks 55 2.9 Addendum: In Search of a Minimal k 55 3. Get Out the Voting System 58 3.1 Everybody Votes, but Not for Everything 58 3.2 Plurality Voting: An Example 60 3.3 Ranked-Choice Voting Systems and Arrow's Impossibility Theorem 61 3.4 An Escape from Impossibility? 66 3.5 Is There a "Best" System? 68 3.6 What Game Developers Know that Politicians Don't 71 3.7 The Best of the Rest 76 3.8 Closing Remarks 82 3.9 Addendum: TheWilson Score Confidence Interval 83 4. Knowing the Score 86 4.1 Ranking Players 86 4.2 Orisinal Original 87 4.3 What's in a Score? 91 4.4 Threes! Company 98 4.5 A Mathematical Model of Threes! 100 4.6 Invalid Scores 105 4.7 Lowest of the Low 109 4.8 Highest of the High 116 4.9 Closing Remarks 121 5. The Thrill of the Chase 122 5.1 I'ma GonnaWin! 122 5.2 Shell Games 123 5.3 Green-Shelled Monsters 125 5.4 Generalizations and Limitations 129 5.5 Seeing Red 131 5.6 Apollonius Circle Pursuit 134 5.7 Overview of aWinning Strategy 136 5.8 Pinpointing the Intersections 141 5.9 Blast Radius 145 5.10 The Pursuer and Pursued in Ms. Pac-Man 148 5.11 Concluding Remarks 153 5.12 Addendum: The Pursuit Curve for Red Shells and a Refined Inequality 153 6. Gaming Complexity 158 6.1 From Russia with Fun 158 6.2 P, NP, and Kevin Bacon 160 6.3 Desktop Diversions 165 6.4 Platforming Problems 169 6.5 Fetch Quests: An Overview 170 6.6 Fetch Quests and Traveling Salesmen 175 6.7 Closing Remarks 183 7. The Friendship Realm 184 7.1 Taking It to the Next Level 184 7.2 Friendship as Gameplay: The Sims and Beyond 186 7.3 A Game-Inspired Friendship Model 190 7.4 Approximations to the Model 193 7.5 The Cost of Maintaining a Friendship 195 7.6 From Virtual Friends to Realistic Romance 198 7.7 Modeling Different Personalities 200 7.8 Improving the Model (Again!) 203 7.9 Concluding Remarks 209 8. Order in Chaos 210 8.1 The Essence of Chaos 210 8.2 Love in the Time of Chaos 211 8.3 Shell Games Revisited 216 8.4 How's theWeather? 223 8.5 Concluding Remarks 225 9. The Value of Games 227 9.1 More Important Than Math 227 9.2 Why Games? 230 9.3 What Next? 242 Notes 244 Bibliography 269 Index 273

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