Algebra Books

Book SynopsisThis book introduces students to projective geometry from an analytic perspective, mixing recent results from the past 100 years with the history of the field in one of the most comprehensive surveys of the subject. The subject is taught conceptually, with worked examples and diagrams to aid in understanding.Trade Review'This book provides a lively and lovely perspective on real projective spaces, combining art, history, groups and elegant proofs.' William M. Kantor'This book is a celebration of the projective viewpoint of geometry. It gradually introduces the reader to the subject, and the arguments are presented in a way that highlights the power of projective thinking in geometry. The reader surprisingly discovers not only that Euclidean and related theorems can be realized as derivatives of projective results, but there are also unnoticed connections between results from ancient times. The treatise also contains a large number of exercises and is dotted with worked examples, which help the reader to appreciate and deeply understand the arguments they refer to. In my opinion this is a book that will definitely change the way we look at the Euclidean and projective analytic geometry.' Alessandro Siciliano, Università degli Studi della BasilicataTable of ContentsPreface; Part I. The Real Projective Plane: 1. Fundamental aspects of the real projective plane; 2. Collineations; 3. Polarities and conics; 4. Cross-ratio; 5. The group of the conic; 6. Involution; 7. Affine plane geometry viewed projectively; 8. Euclidean plane geometry viewed projectively; 9. Transformation geometry: Klein's point of view; 10. The power of projective thinking; 11. From perspective to projective; 12. Remarks on the history of projective geometry; Part II. Two Real Projective 3-Space: 13. Fundamental aspects of real projective space; 14. Triangles and tetrahedra; 15. Reguli and quadrics; 16. Line geometry; 17. Projections; 18. A glance at inversive geometry; Part III. Higher Dimensions: 19. Generalising to higher dimensions; 20. The Klein quadric and Veronese surface; Appendix: Group actions; References; Index.

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Book SynopsisThis textbook teaches the fundamentals of algebra, keeping points clear, succinct and focused, with plenty of diagrams and practice but relatively few words. It assumes a basic knowledge but revises the key prerequisites before moving on. Definitions are highlighted for easy understanding and reference, and worked examples illustrate the explanations. Chapters are interwoven with exercises, whilst each chapter also ends with a comprehensive set of exercises, with answers in the back of the book. Introductory paragraphs describe the real-world application of each topic, and also include briefly where relevant any interesting historical facts about the development of the mathematical subject. This text is intended for undergraduate students in engineering taking a course in algebra. It works for the Foundation and 1st year levels.Table of Contents1. Prerequisite 2. Functions 3. Quadratic equations 4. Exponential and logarithmic functions 5. Trigonometry 1 6. Trigonometry 2 7. Polynomial factorisation 8. Numerical methods 9 Binomial theorem 10. Matrices 11. Complex numbers 12. Sequence 13. Series 14. Partial fractions 15. Vector Algebra Answers to Exercise Questions.

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Book SynopsisThis English edition of Yuri I. Manin's well-received lecture notes provides a concise but extremely lucid exposition of the basics of algebraic geometry and sheaf theory. The lectures were originally held in Moscow in the late 1960s, and the corresponding preprints were widely circulated among Russian mathematicians. This book will be of interest to students majoring in algebraic geometry and theoretical physics (high energy physics, solid body, astrophysics) as well as to researchers and scholars in these areas."This is an excellent introduction to the basics of Grothendieck's theory of schemes; the very best first reading about the subject that I am aware of. I would heartily recommend every grad student who wants to study algebraic geometry to read it prior to reading more advanced textbooks."- Alexander BeilinsonTrade Review“This slim volume is still a valuable introduction to schemes and nicely complements the textbooks on this topic which have appeared in the meantime.” (C. Baxa, Monatshefte für Mathematik, Vol. 201 (4), August, 2023)“Throughout the text there are many instructive examples, remarks, and clarifying footnotes. The style of exposition is rather concise, very elegant, extremely lucid and enlightening, versatile, and – despite its venerable age of fifty years – absolutely modern and timely. … an excellent source for students, instructors, and mathematical physicists. No doubt, with this textbook, the mathematical community has another general standard reference in algebraic geometry at its disposal.” (Werner Kleinert, zbMATH 1390.14002, 2018)Table of ContentsEditor's Preface.- Author's Preface.- 1 Affine Schemes.- 2 Sheaves, Schemes, and Projective Spaces.- References.- Index.

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Book SynopsisThis book reviews the theory of 'generalized B*-algebras' (GB*-algebras), a class of complete locally convex *-algebras which includes all C*-algebras and some of their extensions. A functional calculus and a spectral theory for GB*-algebras is presented, together with results such as Ogasawara's commutativity condition, Gelfand–Naimark type theorems, a Vidav–Palmer type theorem, an unbounded representation theory, and miscellaneous applications. Numerous contributions to the subject have been made since its initiation by G.R. Allan in 1967, including the notable early work of his student P.G. Dixon. Providing an exposition of existing research in the field, the book aims to make this growing theory as familiar as possible to postgraduate students interested in functional analysis, (unbounded) operator theory and its relationship to mathematical physics. It also addresses researchers interested in extensions of the celebrated theory of C*-algebras.Trade Review“This book deals with the theory of locally convex algebras, in general, and of generalized B_-algebras (GB_-algebras in short) in particular. It is well written and self-contained.” (Lahbib Oubbi, Mathematical Reviews, November, 2023)“The book has been written by specialists that are actively working in the field. The choice of the presented material has been done with great care. The bibliography contains all classical monographs, all important papers, and most recent ones. The book leads the reader 'smoothly' ... . It will therefore serve as an excellent introduction to this theory for graduate students. It should also provide a valuable reference source for researchers in the field.” (Andrzej Sołtysiak, zbMATH 1498.46001, 2022)Table of Contents1. Introduction.- 2. A Spectral Theory for Locally Convex Algebras.- 3. Generalized B*-Algebras: Functional Representation Theory.- 4. Commutative Generalized B*-Algebras: Functional Calculus and Equivalent Topologies.- 5. Extended C*-Algebras and Extended W*-Algebras.- 6. Generalized B*-Algebras: Unbounded *-Representation Theory.- 7. Applications I: Miscellanea.- 8. Applications II: Tensor Products.

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Book SynopsisThis study guide is designed for students taking courses in differential equations. The textbook includes examples, questions, and exercises that will help engineering students to review and sharpen their knowledge of the subject and enhance their performance in the classroom. Offering detailed solutions, multiple methods for solving problems, and clear explanations of concepts, this hands-on guide will improve student’s problem-solving skills and basic and advanced understanding of the topics covered in electric circuit analysis courses. Trade Review“Both problems and solutions are presented very clearly. The large format of the book, together with its generous spacing, means that each page has a good deal of text with plenty of additional room for notes. … this book is more likely to be of interest for courses directed at engineering students.” (Bill Satzer, MAA Reviews, October 30, 2022)Table of Contents1) Problems: First-Order Differential Equations2) Solutions of Problems: First-Order Differential Equations3) Problems: Second-Order Differential Equations4) Solutions of Problems: Second-Order Differential Equations5) Problems: Series and their Applications in Solving Differential Equations6) Solutions of Problems: Series and their Applications in Solving Differential Equations7) Problems: Laplace Transform and its Applications in Solving Differential Equations8) Solutions of Problems: Laplace Transform and its Applications in Solving Differential Equations

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Book SynopsisThis open access book offers a self-contained introduction to the homotopy theory of simplicial and dendroidal sets and spaces. These are essential for the study of categories, operads, and algebraic structure up to coherent homotopy. The dendroidal theory combines the combinatorics of trees with the theory of Quillen model categories. Dendroidal sets are a natural generalization of simplicial sets from the point of view of operads. In this book, the simplicial approach to higher category theory is generalized to a dendroidal approach to higher operad theory. This dendroidal theory of higher operads is carefully developed in this book. The book also provides an original account of the more established simplicial approach to infinity-categories, which is developed in parallel to the dendroidal theory to emphasize the similarities and differences. Simplicial and Dendroidal Homotopy Theory is a complete introduction, carefully written with the beginning researcher in mind and ideally suited for seminars and courses. It can also be used as a standalone introduction to simplicial homotopy theory and to the theory of infinity-categories, or a standalone introduction to the theory of Quillen model categories and Bousfield localization.Trade Review“This book is a readable and carefully organized account of dendroidal sets by two of the main figures in the field. It gives a self-contained, detailed description of dendroidal sets and spaces … . Each chapter is also accompanied by a short section of historical notes giving background, references, and historical perspectives on the ideas presented.” (Ben C Walter, Mathematical Reviews, December, 2023)Table of ContentsPart I The Elementary Theory of Simplicial and Dendroidal Sets.- 1 Operads.- 2 Simplicial Sets.- 3 Dendroidal Sets.- 4 Tensor Products of Dendroidal Sets.- 5 Kan Conditions for Simplicial Sets.- 6 Kan Conditions for Dendroidal Sets.- Part II The Homotopy Theory of Simplicial and Dendroidal Sets.- 7 Model Categories.- 8 Model Structures on the Category of Simplicial Sets.- 9 Three Model Structures on the Category of Dendroidal Sets.- Part III The Homotopy Theory of Simplicial and Dendroidal Spaces.- 10 Reedy Categories and Diagrams of Spaces.- 11 Mapping Spaces and Bousfield Localizations.- 12 Dendroidal Spaces and ∞-Operads.- 13 Left Fibrations and the Covariant Model Structure.- 14 Simplicial Operads and ∞-Operads.- Epilogue.- References.- Index.

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Book SynopsisThe 1963 Göttingen notes of T. A. Springer are well known in the field but have been unavailable for some time. This book is a translation of those notes, completely updated and revised. The part of the book dealing with the algebraic structures is on a fairly elementary level, presupposing basic results from algebra. Trade ReviewFrom the reviews of the first edition: "This book is an updated and revised English version of the German notes on octaves, Jordan algebras and exceptional groups which appeared as mimeographed lecture notes of Göttingen University in 1963. It is still an excellent reference on the subject … ." (Huberta Lausch, Zentralblatt MATH, Vol. 1087, 2006)Table of Contents1. Composition Algebras.- 2. The Automorphism Group of an Octonion Algebra.- 3. Triality.- 4. Twisted Composition Algebras.- 5. J-algebras and Albert Algebras.- 6. Proper J-algebras and Twisted Composition Algebras.- 7. Exceptional Groups.- 8. Cohomological Invariants.

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Book Synopsisby a more general quadratic algebra (possibly obtained by deformation) and then to derive Rq [G] by requiring it to possess the latter as a comodule. A third principle is to focus attention on the tensor structure of the cat­ egory of (!; modules. This means of course just defining an algebra structure on Rq[G]; but this is to be done in a very specific manner. Concretely the category is required to be braided and this forces (9.4.2) the existence of an "R-matrix" satisfying in particular the quantum Yang-Baxter equation and from which the algebra structure of Rq[G] can be written down (9.4.5). Finally there was a search for a perfectly self-dual model for Rq[G] which would then be isomorphic to Uq(g). Apparently this failed; but V. G. Drinfeld found that it could be essentially made to work for the "Borel part" of Uq(g) denoted U (b) and further found a general construction (the Drinfeld double) q mirroring a Lie bialgebra. This gives Uq(g) up to passage to a quotient. One of the most remarkable aspects of the above superficially different ap­ proaches is their extraordinary intercoherence. In particular they essentially all lead for G semisimple to the same and hence "canonical", objects Rq[G] and Uq(g), though this epithet may as yet be premature.Table of ContentsI. Hopf Algebras.- 1.1 Axioms of a Hopf Algebra.- 1.2 Group Algebras and Enveloping Algebras.- 1.3 Adjoint Action.- 1.4 The Hopf Dual.- 1.5 Comments and Complements.- 2. Excerpts from the Classical Theory.- 2.1 Lie Algebras.- 2.2 Algebraic Lie Algebras.- 2.3 Algebraic Groups.- 2.4 Lie Algebras of Algebraic Groups.- 2.5 Comments and Complements.- 3. Encoding the Cartan Matrix.- 3.1 Quantum Weyl Algebras.- 3.2 The Drinfeld Double.- 3.3 The Rosso Form and the Casimir Invariant.- 3.4 The Classical Limit and the Shapovalev Form.- 3.5 Comments and Complements.- 4. Highest Weight Modules.- 4.1 The Jantzen Filtration and Sum Formula.- 4.2 Kac-Moody Lie Algebras.- 4.3 Integrable Modules for Uq(gc).- 4.4 Demazure Modules and Product Formulae.- 4.5 Comments and Complements.- 5. The Crystal Basis.- 5.1 Operators in the Crystal Limit.- 5.2 Crystals.- 5.3 Ad-invariant Filtrations, Twisted Actions and the Crystal Basis for Uq(n-).- 5.4 The Grand Loop.- 5.5 Comments and Complements.- 6. The Global Bases.- 6.1 The ? Operation and the Embedding Theorem.- 6.2 Globalization.- 6.3 The Demazure Property.- 6.4 Littelmann’s Path Crystals.- 6.5 Comments and Complements.- 7. Structure Theorems for Uq(g).- 7.1 Local Finiteness for the Adjoint Action.- 7.2 Positivity of the Rosso Form.- 7.3 The Separation Theorem.- 7.4 Noetherianity.- 7.5 Comments and Complements.- 8. The Primitive Spectrum of Uq(g).- 8.1 The Poincaré Series of the Harmonic Space.- 8.2 Factorization of the Quantum PRV Determinants.- 8.3 Verma Module Annihilators.- 8.4 Equivalence of Categories.- 8.5 Comments and Complements.- 9. Structure Theorems for Rq[G].- 9.1 Commutativity Relations.- 9.2 Surjectivity and Injectivity Theorems.- 9.3 The Adjoint Action.- 9.4 The R-Matrix.- 9.5 Comments and Complements.- 10. The Prime Spectrum of Rq[G].- 10.1 Highest Weight Modules.- 10.2 The Quantum Weyl Group.- 10.3 Prime and Primitive Ideals of Rq[G].- 10.4 Hopf Algebra Automorphisms.- 10.5 Comments and Complements.- A.2 Excerpts from Ring Theory.- A.3 Combinatorial Identities and Dimension Theory.- A.4 Remarks on Constructions of Quantum Groups.- A.5 Comments and Complements.- Index of Notation.

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Book SynopsisTable of Contents1. Linear Equations 1.1 Introduction to Linear Systems 1.2 Matrices, Vectors, and Gauss-Jordan Elimination 1.3 On the Solutions of Linear Systems; Matrix Algebra 2. Linear Transformations 2.1 Introduction to Linear Transformations and Their Inverses 2.2 Linear Transformations in Geometry 2.3 Matrix Products 2.4 The Inverse of a Linear Transformation 3. Subspaces of Rn and Their Dimensions 3.1 Image and Kernel of a Linear Transformation 3.2 Subspace of Rn; Bases and Linear Independence 3.3 The Dimension of a Subspace of Rn 3.4 Coordinates 4. Linear Spaces 4.1 Introduction to Linear Spaces 4.2 Linear Transformations and Isomorphisms 4.3 The Matrix of a Linear Transformation 5. Orthogonality and Least Squares 5.1 Orthogonal Projections and Orthonormal Bases 5.2 Gram-Schmidt Process and QR Factorization 5.3 Orthogonal Transformations and Orthogonal Matrices 5.4 Least Squares and Data Fitting 5.5 Inner Product Spaces 6. Determinants 6.1 Introduction to Determinants 6.2 Properties of the Determinant 6.3 Geometrical Interpretations of the Determinant; Cramer's Rule 7. Eigenvalues and Eigenvectors 7.1 Diagonalization 7.2 Finding the Eigenvalues of a Matrix 7.3 Finding the Eigenvectors of a Matrix 7.4 More on Dynamical Systems 7.5 Complex Eigenvalues 7.6 Stability 8. Symmetric Matrices and Quadratic Forms 8.1 Symmetric Matrices 8.2 Quadratic Forms 8.3 Singular Values 9. Linear Differential Equations 9.1 An Introduction to Continuous Dynamical Systems 9.2 The Complex Case: Euler's Formula 9.3 Linear Differential Operators and Linear Differential Equations Appendix A. Vectors Appendix B: Techniques of Proof Answers to Odd-numbered Exercises Subject Index Name Index

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Book SynopsisAbout our author Jay Lehmann has taught for the past 25 years at College of San Mateo, where he received the shiny apple award for excellence in teaching. ?He has presented at over 80 conferences including AMATYC and ICTCM over the past 16 years. Jay is currently the newsletter editor for California Mathematics Council, Community Colleges (CMC3). ?Still young at heart, he plays in a rock band appropriately named the Procrastinistas. Jay has authored several algebra textbooks published by Pearson and has also recently completed a Prestatistics textbook. In the words of the author: Before writing my algebra series, it was painfully apparent that my students couldn't relate to the applications in the course. I was plagued with the question, What is this good for? To try to bridge that gap, I wrote some labs, which facilitated my students in collecting data, finding models via curve fitting, and using the models to make estimates and predictions.Table of ContentsTable of Contents Introduction to Modeling 1.1 Variables and Constants 1.2 Scatterplots 1.3 Exact Linear Relationships 1.4 Approximate Linear Relationships Taking It to the Lab: Climate Change Lab • Volume Lab • Linear Graphing Lab: Topic of Your Choice Operations and Expressions 2.1 Expressions 2.2 Operations with Fractions 2.3 Absolute Value and Adding Real Numbers 2.4 Change in a Quantity and Subtracting Real Numbers 2.5 Ratios, Percents, and Multiplying and Dividing Real Numbers 2.6 Exponents and Order of Operations Taking It to the Lab: Climate Change Lab (continued from Chapter 1) • Stocks Lab Using Slope to Graph Linear Equations 3.1 Graphing Equations of the Forms y = mx + b and x = a 3.2 Graphing Linear Models; Unit Analysis 3.3 Slope of a Line 3.4 Using Slope to Graph Linear Equations 3.5 Rate of Change Taking It to the Lab: Climate Change Lab (continued from Chapter 2) • Workout Lab • Balloon Lab Simplifying Expressions and Solving Equations 4.1 Simplifying Expressions 4.2 Simplifying More Expressions 4.3 Solving Linear Equations in One Variable 4.4 Solving More Linear Equations in One Variable 4.5 Comparing Expressions and Equations 4.6 Formulas Linear Functions and Linear Inequalities in One Variable 5.1 Graphing Linear Equations 5.2 Functions 5.3 Function Notation 5.4 Finding Linear Equations 5.5 Finding Equations of Linear Models 5.6 Using Function Notation with Linear Models to Make Estimates and Predictions 5.7 Solving Linear Inequalities in One Variable Taking It to the Lab: Climate Change Lab (continued from Chapter 3) • Golf Ball Lab • Rope Lab • Shadow Lab • Linear Lab: Topic of Your Choice Systems of Linear Equations and Systems of Linear Inequalities 6.1 Using Graphs and Tables to Solve Systems 6.2 Using Substitution to Solve Systems 6.3 Using Elimination to Solve Systems 6.4 Using Systems to Model Data 6.5 Perimeter, Value, Interest, and Mixture Problems 6.6 Linear Inequalities in Two Variables; Systems of Linear Inequalities in Two Variables Taking It to the Lab: Climate Change Lab (continued from Chapter 5) • Sports Lab • Truck Lab Polynomial Functions and Properties of Exponents 7.1 Adding and Subtracting Polynomial Expressions and Functions 7.2 Multiplying Polynomial Expressions and Functions 7.3 Powers of Polynomials; Product of Binomial Conjugates 7.4 Properties of Exponents 7.5 Dividing Polynomials: Long Division and Synthetic Division Taking It to the Lab: Climate Change Lab (continued from Chapter 6) • Projectile Lab Factoring Polynomials and Solving Polynomial Equations 8.1 Factoring Trinomials of the Form x2 + bx + c and Differences of Two Squares 8.2 Factoring Out the GCF; Factoring by Grouping 8.3 Factoring Trinomials of the Form ax2 + bx + c 8.4 Sums and Differences of Cubes; A Factoring Strategy 8.5 Using Factoring to Solve Polynomial Equations 8.6 Using Factoring to Make Predictions with Quadratic Models Quadratic Functions 9.1 Graphing Quadratic Functions in Vertex Form 9.2 Graphing Quadratic Functions in Standard Form 9.3 Simplifying Radical Expressions 9.4 Using the Square Root Property to Solve Quadratic Equations 9.5 Solving Quadratic Equations by Completing the Square 9.6 Using the Quadratic Formula to Solve Quadratic Equations 9.7 Solving Systems of Linear Equations in Three Variables; Finding Quadratic Functions 9.8 Finding Quadratic Models 9.9 Modeling with Quadratic Functions Taking It to the Lab: Climate Change Lab (continued from Chapter 7) • Projectile Lab (continued from Chapter 7) • Projectile Lab (Using a CBR or CBL) • Water Flow Lab • Quadratic Lab: Topic of Your Choice Exponential Functions 10.1 Integer Exponents 10.2 Rational Exponents 10.3 Graphing Exponential Functions 10.4 Finding Equations of Exponential Functions 10.5 Using Exponential Functions to Model Data Taking It to the Lab: Stringed Instrument Lab • Cooling Water Lab • Exponential Lab: Topic of Your Choice Logarithmic Functions 11.1 Composite Functions 11.2 Inverse Functions 11.3 Logarithmic Functions 11.4 Properties of Logarithms 11.5 Using the Power Property with Exponential Models to Make Predictions 11.6 More Properties of Logarithms 11.7 Natural Logarithms Taking It to the Lab: China and India Populations Lab • Folding Paper Lab • Exponential/Logarithmic Lab: Topic of Your Choice Rational Functions 12.1 Finding the Domains of Rational Functions and Simplifying Rational Expressions 12.2 Multiplying and Dividing Rational Expressions; Converting Units 12.3 Adding and Subtracting Rational Expressions 12.4 Simplifying Complex Rational Expressions 12.5 Solving Rational Equations 12.6 Modeling with Rational Functions 12.7 Proportions; Similar Triangles 12.8 Variation Taking It to the Lab: Climate Change Lab (continued from Chapter 9) • Illumination Lab • Boyle’s Law Lab • Estimating 5 Lab Radical Functions 13.1 Simplifying Radical Expressions 13.2 Adding, Subtracting, and Multiplying Radical Expressions 13.3 Rationalizing Denominators and Simplifying Quotients of Radical Expressions 13.4 Graphing and Combining Square Root Functions 13.5 Solving Radical Equations 13.6 Modeling with Square Root Functions Taking It to the Lab: Pendulum Lab Sequences and Series 14.1 Arithmetic Sequences 14.2 Geometric Sequences 14.3 Arithmetic Series 14.4 Geometric Series Taking It to the Lab: Bouncing Ball Lab • Stacked Cups Lab Additional Topics 15.1 Absolute Value: Equations and Inequalities 15.2 Performing Operations with Complex Numbers 15.3 Pythagorean Theorem, Distance Formula, and Circles 15.4 Ellipses and Hyperbolas 15.5 Solving Nonlinear Systems of Equations Appendix A. Using a TI-83 or TI-84 Graphing Calculator Appendix B. Using StatCrunch

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Book SynopsisConsidered a classic by many, A First Course in Abstract Algebra is an in-depth introduction to abstract algebra. Focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures.Table of Contents(*) Not required for the remainder of the text. (**) This section is required only for Chapters 17 and 36.). 0. Sets and Relations. I. GROUPS AND SUBGROUPS. 1. Introduction and Examples. 2. Binary Operations. 3. Isomorphic Binary Structures. 4. Groups. 5. Subgroups. 6. Cyclic Groups. 7. Generators and Cayley Digraphs. II. PERMUTATIONS, COSETS, AND DIRECT PRODUCTS. 8. Groups of Permutations. 9. Orbits, Cycles, and the Alternating Groups. 10. Cosets and the Theorem of Lagrange. 11. Direct Products and Finitely Generated Abelian Groups. 12. *Plane Isometries. III. HOMOMORPHISMS AND FACTOR GROUPS. 13. Homomorphisms. 14. Factor Groups. 15. Factor-Group Computations and Simple Groups. 16. **Group Action on a Set. 17. *Applications of G-Sets to Counting. IV. RINGS AND FIELDS. 18. Rings and Fields. 19. Integral Domains. 20. Fermat's and Euler's Theorems. 21. The Field of Quotients of an Integral Domain. 22. Rings of Polynomials. 23. Factorization of Polynomials over a Field. 24. *Noncommutative Examples. 25. *Ordered Rings and Fields. V. IDEALS AND FACTOR RINGS. 26. Homomorphisms and Factor Rings. 27. Prime and Maximal Ideas. 28. *Gröbner Bases for Ideals. VI. EXTENSION FIELDS. 29. Introduction to Extension Fields. 30. Vector Spaces. 31. Algebraic Extensions. 32. *Geometric Constructions. 33. Finite Fields. VII. ADVANCED GROUP THEORY. 34. Isomorphism Theorems. 35. Series of Groups. 36. Sylow Theorems. 37. Applications of the Sylow Theory. 38. Free Abelian Groups. 39. Free Groups. 40. Group Presentations. VIII. *GROUPS IN TOPOLOGY. 41. Simplicial Complexes and Homology Groups. 42. Computations of Homology Groups. 43. More Homology Computations and Applications. 44. Homological Algebra. IX. Factorization. 45. Unique Factorization Domains. 46. Euclidean Domains. 47. Gaussian Integers and Multiplicative Norms. X. AUTOMORPHISMS AND GALOIS THEORY. 48. Automorphisms of Fields. 49. The Isomorphism Extension Theorem. 50. Splitting Fields. 51. Separable Extensions. 52. *Totally Inseparable Extensions. 53. Galois Theory. 54. Illustrations of Galois Theory. 55. Cyclotomic Extensions. 56. Insolvability of the Quintic. Appendix: Matrix Algebra. Notations. Answers to odd-numbered exercises not asking for definitions or proofs. Index.

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Book SynopsisProvides clear presentations of more than sixty important unsolved problems in mathematical systems and control theory. This book consists of ten parts representing various problem areas, and each chapter sets forth a different problem presented by a researcher in the particular area and in the same way.Trade Review"This is an extremely important book that presents, in a clear way, many important and stimulating mathematical problems in systems and control. It will be an important reference for both researchers and people outside the field."—William W. Hager, University of Florida"This book covers a wide range of systems from linear to nonlinear, deterministic to stochastic, finite dimensional to infinite dimensional, and so on. It includes at least some set of problems that will interest any researcher in the field."—Kemin Zhou, Louisiana State UniversityTable of ContentsFrontmatter Preface xiii Associate Editors xv Website xvii PART 1. LINEAR SYSTEMS 1 Problem 1.1. Stability and composition of transfer functions Guillermo Fernandez-Anaya, Juan Carlos Martinez-Garcia 3 Problem 1.2. The realization problem for Herglotz-Nevanlinna functions Seppo Hassi, Henk de Snoo, Eduard Tsekanovskii 8 Problem 1.3. Does any analytic contractive operator function on the polydisk have a dissipative scattering nD realization? Dmitry S. Kalyuzhniy-Verbovetzky 14 Problem 1.4. Partial disturbance decoupling with stability Juan Carlos Martinez-Garcia, Michel Malabre, Vladimir Kucera 18 Problem 1.5. Is Monopoli's model reference adaptive controller correct? A. S. Morse 22 Problem 1.6. Model reduction of delay systems Jonathan R. Partington 29 Problem 1.7. Schur extremal problems Lev Sakhnovich 33 Problem 1.8. The elusive iff test for time-controllability of behaviors Amol J. Sasane 36 Problem 1.9. A Farkas lemma for behavioral inequalities A.A. (Tonny) ten Dam, J.W. (Hans) Nieuwenhuis 40 Problem 1.10. Regular feedback implementability of linear differential behaviors H. L. Trentelman 44 Problem 1.11. Riccati stability Erik I. Verriest 49 Problem 1.12. State and first order representations Jan C. Willems 54 Problem 1.13. Projection of state space realizations Antoine Vandendorpe, Paul Van Dooren 58 PART 2. STOCHASTIC SYSTEMS 65 Problem 2.1. On error of estimation and minimum of cost for wide band noise driven systems Agamirza E. Bashirov 67 Problem 2.2. On the stability of random matrices Giuseppe C. Calafiore, Fabrizio Dabbene 71 Problem 2.3. Aspects of Fisher geometry for stochastic linear systems Bernard Hanzon, Ralf Peeters 76 Problem 2.4. On the convergence of normal forms for analytic control systems Wei Kang, Arthur J. Krener 82 PART 3. NONLINEAR SYSTEMS 87 Problem 3.1. Minimum time control of the Kepler equation Jean-Baptiste Caillau, Joseph Gergaud, Joseph Noailles 89 Problem 3.2. Linearization of linearly controllable systems R. Devanathan 93 Problem 3.3. Bases for Lie algebras and a continuous CBH formula Matthias Kawski 97 Problem 3.4. An extended gradient conjecture Luis Carlos Martins Jr., Geraldo Nunes Silva 103 Problem 3.5. Optimal transaction costs from a Stackelberg perspective Geert Jan Olsder 107 Problem 3.6. Does cheap control solve a singular nonlinear quadratic problem? Yuri V. Orlov 111 Problem 3.7. Delta-Sigma modulator synthesis Anders Rantzer 114 Problem 3.8. Determining of various asymptotics of solutions of nonlinear time-optimal problems via right ideals in the moment algebra G. M. Sklyar, S. Yu. Ignatovich 117 Problem 3.9. Dynamics of principal and minor component flows U. Helmke, S. Yoshizawa, R. Evans, J.H. Manton, and I.M.Y. Mareels 122 PART 4. DISCRETE EVENT, HYBRID SYSTEMS 129 Problem 4.1. L2-induced gains of switched linear systems Joao P. Hespanha 131 Problem 4.2. The state partitioning problem of quantized systems Jan Lunze 134 Problem 4.3. Feedback control in flowshops S.P. Sethi and Q. Zhang 140 Problem 4.4. Decentralized control with communication between controllers Jan H. van Schuppen 144 PART 5. DISTRIBUTED PARAMETER SYSTEMS 151 Problem 5.1. Infinite dimensional backstepping for nonlinear parabolic PDEs Andras Balogh, Miroslav Krstic 153 Problem 5.2. The dynamical Lame system with boundary control: on the structure of reachable sets M.I. Belishev 160 Problem 5.3. Null-controllability of the heat equation in unbounded domains Sorin Micu, Enrique Zuazua 163 Problem 5.4. Is the conservative wave equation regular? George Weiss 169 Problem 5.5. Exact controllability of the semilinear wave equation Xu Zhang, Enrique Zuazua 173 Problem 5.6. Some control problems in electromagnetics and fluid dynamics Lorella Fatone, Maria Cristina Recchioni, Francesco Zirilli 179 PART 6. STABILITY, STABILIZATION 187 Problem 6.1. Copositive Lyapunov functions M. K. Camlibel, J. M. Schumacher 189 Problem 6.2. The strong stabilization problem for linear time-varying systems Avraham Feintuch 194 Problem 6.3. Robustness of transient behavior Diederich Hinrichsen, Elmar Plischke, Fabian Wirth 197 Problem 6.4. Lie algebras and stability of switched nonlinear systems Daniel Liberzon 203 Problem 6.5. Robust stability test for interval fractional order linear systems Ivo Petrs, YangQuan Chen, Blas M. Vinagre 208 Problem 6.6. Delay-independent and delay-dependent Aizerman problem Vladimir Rasvan 212 Problem 6.7. Open problems in control of linear discrete multidimensional systems Li Xu, Zhiping Lin, Jiang-Qian Ying, Osami Saito, Yoshihisa Anazawa 221 Problem 6.8. An open problem in adaptative nonlinear control theory Leonid S. Zhiteckij 229 Problem 6.9. Generalized Lyapunov theory and its omega-transformable regions Sheng-Guo Wang 233 Problem 6.10. Smooth Lyapunov characterization of measurement to error stability Brian P. Ingalls, Eduardo D. Sontag 239 PART 7. CONTROLLABILITY, OBSERVABILITY 245 Problem 7.1. Time for local controllability of a 1-D tank containing a fluid modeled by the shallow water equations Jean-Michel Coron 247 Problem 7.2. A Hautus test for infinite-dimensional systems Birgit Jacob, Hans Zwart 251 Problem 7.3. Three problems in the field of observability Philippe Jouan 256 Problem 7.4. Control of the KdV equation Lionel Rosier 260 PART 8. ROBUSTNESS, ROBUST CONTROL 265 Problem 8.1. H[infinity]-norm approximation A.C. Antoulas, A. Astolfi 267 Problem 8.2. Noniterative computation of optimal value in H[infinity] control Ben M. Chen 271 Problem 8.3. Determining the least upper bound on the achievable delay margin Daniel E. Davison, Daniel E. Miller 276 Problem 8.4. Stable controller coefficient perturbation in floating point implementation Jun Wu, Sheng Chen 280 PART 9. IDENTIFICATION, SIGNAL PROCESSING 285 Problem 9.1. A conjecture on Lyapunov equations and principal angles in sub-space identification Katrien De Cock, Bart De Moor 287 Problem 9.2. Stability of a nonlinear adaptive system for filtering and parameter estimation Masoud Karimi-Ghartemani, Alireza K. Ziarani 293 PART 10. ALGORITHMS, COMPUTATION 297 Problem 10.1. Root-clustering for multivariate polynomials and robust stability analysis Pierre-Alexandre Bliman 299 Problem 10.2. When is a pair of matrices stable? Vincent D. Blondel, Jacques Theys, John N. Tsitsiklis 304 Problem 10.3. Freeness of multiplicative matrix semigroups Vincent D. Blondel, Julien Cassaigne, Juhani Karhumaki 309 Problem 10.4. Vector-valued quadratic forms in control theory Francesco Bullo, Jorge Cortes, Andrew D. Lewis, Sonia Martinez 315 Problem 10.5. Nilpotent bases of distributions Henry G. Hermes, Matthias Kawski 321 Problem 10.6. What is the characteristic polynomial of a signal flow graph? Andrew D. Lewis 326 Problem 10.7. Open problems in randomized [mu] analysis Onur Toker 330

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Book SynopsisThis book covers the basic topics in arithmetic and algebra with which every college student should be thoroughly familiar. It is written with the student in mind, in a style and at a level appropriate for student understanding.

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Book SynopsisTable of ContentsA REVIEW OF ALGEBRA FUNDAMENTALS. Real Numbers and Mathematical Operations. Solving Linear Equations. Percents. Scientific Notation. 1. MATHEMATICAL MODEL FUNDAMENTALS. Mathematical Models. Formulas. Ratio and Proportion. Word Problem Strategies. 2. GEOMETRIC MODELS. Models and Patterns in Plane Geometry. Models and Patterns in Triangles. Models and Patterns in Art and Architecture: Perspective and Symmetry. Models and Patterns in Art and Architecture and Nature: Scale and Proportion. Models and Patterns in Music. 3. GRAPHING. Rectangular Coordinate System. Graphing Linear Equations. Slope. Writing Equations of Lines. Applications and Uses of Graphs. 4. FUNCTIONS. Functions. Using Function Notation. Linear Functions as Models. Direct and Inverse Variation. Quadratic Functions as Models. Exponential Functions as Models. 5. MATHEMATICAL MODELS IN CONSUMER MATH. Mathematical Models in the Business World. Mathematical Models in Banking. Mathematical Models in Consumer Credit. Mathematical Models in Purchasing an Automobile. Mathematical Models in Purchasing a Home. Mathematical Models in Insurance Options and Rates. Mathematical Models in Stocks, Mutual Funds, and Bonds. Mathematical Models in Personal Income. 6. MODELING WITH SYSTEMS OF EQUATIONS. Solving Systems by Graphing. Solving Systems Algebraically. Applications of Linear Systems. Systems of Non-Linear Functions. 7. PROBABILITY MODELS. Sets and Set Theory. What is Probability? Theoretical Probability. Odds. Tree Diagrams and the Counting Principle. Probabilities Involving "Or". Probabilities Involving "And". Permutations and Combinations. 8. MODELING WITH STATISTICS. Introduction to Statistics and Surveys. Frequency Tables and Histograms. Reading and Interpreting Graphical Information. Descriptive Statistics. Variation. Normal Curve. Scatter Diagrams and Linear Regression.

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Book SynopsisStraightedge and compass.- Euclid's approach to geometry.- Coordinates.- Vectors and Euclidean spaces.- Perspective.- Projective planes.- Transformations.- Non-Euclidean geometry.Trade ReviewFrom the reviews:"This is an introductory book on geometry, easy to read, written in an engaging style. The author’s goal is … to increase one’s overall understanding and appreciation of the subject. … Along the way, he presents elegant proofs of well-known theorems … . The advantage of the author’s approach is clear: in a short space he gives a brief introduction to many sides of geometry and includes many beautiful results, each explained from a perspective that makes it easy to understand." (Robin Hartshorne, SIAM Review, Vol. 48 (2), 2006)"The pillars of the title are … Euclidean construction and axioms, coordinates and vectors, projective geometry, and transformations and non-Euclidean geometry. … The writing style is both student-friendly and deeply informed. The pleasing brevity of the book … makes the book especially suitable as an instruction to geometry for the large and critically important population of undergraduate mathematics majors … . Each chapter concludes with a well-written discussion section that combines history with glances at further results. There is a good selection of thought-provoking exercises." (R. J. Bumcrot, Mathematical Reviews, Issue 2006 e)"The author acts on the assumption of four approaches to geometry: The axiomatic way, using linear Algebra, projective geometry and transformation groups. … Each of the chapters closes with a discussion giving hints on further aspects and historical remarks. … The book can be recommended to be used in undergraduate courses on geometry … ." (F. Manhart, Internationale Mathematische Nachrichten, Issue 203, 2006)"Any new mathematics textbook by John Stillwell is worth a serious look. Stillwell is the prolific author of more than half a dozen textbooks … . I would not hesitate to recommend this text to any professor teaching a course in geometry who is more interested in providing a rapid survey of topics rather than an in-depth, semester-long, examination of any particular one." (Mark Hunacek, The Mathematical Gazette, Vol. 91 (521), 2007)"The title refers to four different approaches to elementary geometry which according to the author only together show this field in all its splendor: via straightedge and compass constructions, linear algebra, projective geometry and transformation groups. … the book can be recommended warmly to undergraduates to get in touch with geometric thinking." (G. Kowol, Monatshefte für Mathematik, Vol. 150 (3), 2007)"This book presents a tour on various approaches to a notion of geometry and the relationship between these approaches. … The book shows clearly how useful it is to use various tools in a description of basic geometrical questions to find the simplest and the most intuitive arguments for different problems. The book is a very useful source of ideas for high school teachers." (EMS Newsletter, March, 2007)“The four pillars of geometry approaches geometry in four different ways, devoting two chapters to each, the first chapter being concrete and introductory, the second more abstract. … The content is quite elementary and is based on lectures given by the author at the University of San Francisco in 2004. … The book of Stillwell is a very good first introduction to geometry especially for the axiomatic and the projective point of view.” (Yves Félix, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)Table of ContentsStraightedge and compass.- Euclid’s approach to geometry.- Coordinates.- Vectors and Euclidean spaces.- Perspective.- Projective planes.- Transformations.- Non-Euclidean geometry.

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Book Synopsis'Probably its most significant distinguishing feature is that this book is more algebraically oriented than most undergraduate number theory texts.'MAA ReviewsIntroduction to Number Theory is dedicated to concrete questions about integers, to place an emphasis on problem solving by students. When undertaking a first course in number theory, students enjoy actively engaging with the properties and relationships of numbers.The book begins with introductory material, including uniqueness of factorization of integers and polynomials. Subsequent topics explore quadratic reciprocity, Hensel's Lemma, p-adic powers series such as exp(px) and log(1+px), the Euclidean property of some quadratic rings, representation of integers as norms from quadratic rings, and Pell's equation via continued fractions.Throughout the five chapters and more than 100 exercises and solutions, readers gain the advantage of a number theory book that focuses on doing calculations. This textbook is a valuable resource for undergraduates or those with a background in university level mathematics.

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Book Synopsis

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Book Synopsis

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