Algebra Books

Book SynopsisAbout our authors Allen Angel received his BS and MS in mathematics from SUNY at New Paltz. He completed additional graduate work at Rutgers University. He taught at Sullivan County Community College and Monroe Community College, where he served as chairperson of the Mathematics Department. He served as Assistant Director of the National Science Foundation at Rutgers University for the summers of 1967 - 1970. He was President of The New York State Mathematics Association of Two-Year Colleges (NYSMATYC). He also served as Northeast Vice President of the American Mathematics Association of Two-Year Colleges (AMATYC). Allen lives in Palm Harbor, Florida but spends his summers in Penfield, New York. He enjoys playing tennis and watching sports. He also enjoys traveling with his wife Kathy. Dennis Runde received his BS and MS in mathematics from the University of Wisconsin - Platteville and Milwaukee, respectively. He has a PhD in MathemaTable of ContentsTable of Contents Basic Concepts 1.1 Study Skills for Success in Mathematics, and Using a Calculator 1.2 Sets and Other Basic Concepts 1.3 Properties of and Operations with Real Numbers 1.4 Order of Operations Mid-Chapter Test: Sections 1.1—1.4 1.5 Exponents 1.6 Scientific Notation Equations and Inequalities 2.1 Solving Linear Equations 2.2 Problem Solving and Using Formulas 2.3 Applications of Algebra Mid-Chapter Test: Sections 2.1—2.3 2.4 Additional Application Problems 2.5 Solving Linear Inequalities 2.6 Solving Equations and Inequalities Containing Absolute Values Graphs and Functions 3.1 Graphs 3.2 Functions 3.3 Linear Functions: Graphs and Applications 3.4 The Slope-Intercept Form of a Linear Equation Mid-Chapter Test: Sections 3.1—3.4 3.5 The Point-Slope Form of a Linear Equation 3.6 The Algebra of Functions 3.7 Graphing Linear Inequalities Systems of Equations and Inequalities 4.1 Solving Systems of Linear Equations in Two Variables 4.2 Solving Systems of Linear Equations in Three Variables 4.3 Systems of Linear Equations: Applications and Problem Solving Mid-Chapter Test: Sections 4.1—4.3 4.4 Solving Systems of Equations Using Matrices 4.5 Solving Systems of Equations Using Determinants and Cramer’s Rule 4.6 Solving Systems of Linear Inequalities Polynomials and Polynomial Functions 5.1 Addition and Subtraction of Polynomials 5.2 Multiplication of Polynomials 5.3 Division of Polynomials and Synthetic Division 5.4 Factoring a Monomial from a Polynomial and Factoring by Grouping Mid-Chapter Test: Sections 5.1—5.4 5.5 Factoring Trinomials 5.6 Special Factoring Formulas 5.7 A General Review of Factoring 5.8 Polynomial Equations Rational Expressions and Equations 6.1 The Domains of Rational Functions and Multiplication and Division of Rational Expressions 6.2 Addition and Subtraction of Rational Expressions 6.3 Complex Fractions 6.4 Solving Rational Equations Mid-Chapter Test: Sections 6.1—6.4 6.5 Rational Equations: Applications and Problem Solving 6.6 Variation Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.2 Rational Exponents 7.3 Simplifying Radicals 7.4 Adding, Subtracting, and Multiplying Radicals Mid-Chapter Test: Sections 7.1—7.4 7.5 Dividing Radicals 7.6 Solving Radical Equations 7.7 Complex Numbers Quadratic Functions 8.1 Solving Quadratic Equations by Completing the Square 8.2 Solving Quadratic Equations by the Quadratic Formula 8.3 Quadratic Equations: Applications and Problem Solving Mid-Chapter Test: Sections 8.1—8.3 8.4 Writing Equations in Quadratic Form 8.5 Graphing Quadratic Functions 8.6 Quadratic and Other Inequalities in One Variable Exponential and Logarithmic Functions 9.1 Composite and Inverse Functions 9.2 Exponential Functions 9.3 Logarithmic Functions 9.4 Properties of Logarithms Mid-Chapter Test: Sections 9.1—9.4 9.5 Common Logarithms 9.6 Exponential and Logarithmic Equations 9.7 Natural Exponential and Natural Logarithmic Functions Conic Sections 10.1 The Parabola and the Circle 10.2 The Ellipse Mid-Chapter Test: Sections 10.1—10.2 10.3 The Hyperbola 10.4 Nonlinear Systems of Equations and Their Applications Sequences, Series, and the Binomial Theorem 11.1 Sequences and Series 11.2 Arithmetic Sequences and Series 11.3 Geometric Sequences and Series Mid-Chapter Test: Sections 11.1—11.3 11.4 The Binomial Theorem Appendix: Geometric Formulas

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Book SynopsisThis book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations.Table of ContentsI. Basic Concepts.- 1. Definitions and first examples.- 1.1 The notion of Lie algebra.- 1.2 Linear Lie algebras.- 1.3 Lie algebras of derivations.- 1.4 Abstract Lie algebras.- 2. Ideals and homomorphisms.- 2.1 Ideals.- 2.2 Homomorphisms and representations.- 2.3 Automorphisms.- 3. Solvable and nilpotent Lie algebras.- 3.1 Solvability.- 3.2 Nilpotency.- 3.3 Proof of Engel’s Theorem.- II. Semisimple Lie Algebras.- 4. Theorems of Lie and Cartan.- 4.1 Lie’s Theorem.- 4.2 Jordan-Chevalley decomposition.- 4.3 Cartan’s Criterion.- 5. Killing form.- 5.1 Criterion for semisimplicity.- 5.2 Simple ideals of L.- 5.3 Inner derivations.- 5.4 Abstract Jordan decomposition.- 6. Complete reducibility of representations.- 6.1 Modules.- 6.2 Casimir element of a representation.- 6.3 Weyl’s Theorem.- 6.4 Preservation of Jordan decomposition.- 7. Representations of sl (2, F).- 7.1 Weights and maximal vectors.- 7.2 Classification of irreducible modules.- 8. Root space decomposition.- 8.1 Maximal toral subalgebras and roots.- 8.2 Centralizer of H.- 8.3 Orthogonality properties.- 8.4 Integrality properties.- 8.5 Rationality properties Summary.- III. Root Systems.- 9. Axiomatics.- 9.1 Reflections in a euclidean space.- 9.2 Root systems.- 9.3 Examples.- 9.4 Pairs of roots.- 10. Simple roots and Weyl group.- 10.1 Bases and Weyl chambers.- 10.2 Lemmas on simple roots.- 10.3 The Weyl group.- 10.4 Irreducible root systems.- 11. Classification.- 11.1 Cartan matrix of ?.- 11.2 Coxeter graphs and Dynkin diagrams.- 11.3 Irreducible components.- 11.4 Classification theorem.- 12. Construction of root systems and automorphisms.- 12.1 Construction of types A-G.- 12.2 Automorphisms of ?.- 13. Abstract theory of weights.- 13.1 Weights.- 13.2 Dominant weights.- 13.3 The weight ?.- 13.4 Saturated sets of weights.- IV. Isomorphism and Conjugacy Theorems.- 14. Isomorphism theorem.- 14.1 Reduction to the simple case.- 14.2 Isomorphism theorem.- 14.3 Automorphisms.- 15. Cartan subalgebras.- 15.1 Decomposition of L relative to ad x.- 15.2 Engel subalgebras.- 15.3 Cartan subalgebras.- 15.4 Functorial properties.- 16. Conjugacy theorems.- 16.1 The group g (L).- 16.2 Conjugacy of CSA’s (solvable case).- 16.3 Borel subalgebras.- 16.4 Conjugacy of Borel subalgebras.- 16.5 Automorphism groups.- V. Existence Theorem.- 17. Universal enveloping algebras.- 17.1 Tensor and symmetric algebras.- 17.2 Construction of U(L).- 17.3 PBW Theorem and consequences.- 17.4 Proof of PBW Theorem.- 17.5 Free Lie algebras.- 17. Generators and relations.- 17.1 Relations satisfied by L.- 17.2 Consequences of (S1)-(S3).- 17.3 Serre’s Theorem.- 17.4 Application: Existence and uniqueness theorems.- 18. The simple algebras.- 18.1 Criterion for semisimplicity.- 18.2 The classical algebras.- 18.3 The algebra G2.- VI. Representation Theory.- 20. Weights and maximal vectors.- 20.1 Weight spaces.- 20.2 Standard cyclic modules.- 20.3 Existence and uniqueness theorems.- 21. Finite dimensional modules.- 21.1 Necessary condition for finite dimension.- 21.2 Sufficient condition for finite dimension.- 21.3 Weight strings and weight diagrams.- 21.4 Generators and relations for V(?).- 22. Multiplicity formula.- 22.1 A universal Casimir element.- 22.2 Traces on weight spaces.- 22.3 Freudenthal’s formula.- 22.4 Examples.- 22.5 Formal characters.- 23. Characters.- 23.1 Invariant polynomial functions.- 23.2 Standard cyclic modules and characters.- 23.3 Harish-Chandra’s Theorem.- 24. Formulas of Weyl, Kostant, and Steinberg.- 24.1 Some functions on H*.- 24.2 Kostant’s multiplicity formula.- 24.3 Weyl’s formulas.- 24.4 Steinberg’s formula.- VII. Chevalley Algebras and Groups.- 25. Chevalley basis of L.- 25.1 Pairs of roots.- 25.2 Existence of a Chevalley basis.- 25.3 Uniqueness questions.- 25.4 Reduction modulo a prime.- 25.5 Construction of Chevalley groups (adjoint type).- 26. Kostant’s Theorem.- 26.1 A combinatorial lemma.- 26.2 Special case: sl (2, F).- 26.3 Lemmas on commutation.- 26.4 Proof of Kostant’s Theorem.- 27. Admissible lattices.- 27.1 Existence of admissible lattices.- 27.2 Stabilizer of an admissible lattice.- 27.3 Variation of admissible lattice.- 27.4 Passage to an arbitrary field.- 27.5 Survey of related results.- References.- Afterword (1994).- Index of Terminology.- Index of Symbols.

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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 SynopsisPresents the proceedings of the Joint Summer Research Conference on Algebraic $K$-theory held at the University of Washington in Seattle. This work includes an account of Voevodsky's proof of the Milnor conjecture relating the Milnor $K$-theory of fields to Galois cohomology.Table of ContentsConjectures de type local-global sur l'image des groupes de Chow dans la cohomologie etale by J.-L. Colliot-Thelene Algebraic theory of characteristic classes of bundles with connection by H. Esnault Polylogarithmic identities in cubical higher Chow groups by H. Gangl and S. Muller-Stach Topological cyclic homology of schemes by T. Geisser and L. Hesselholt Filtrations on higher algebraic $K$-theory by H. Gillet and C. Soule Motivic cohomology of smooth geometrically cellular varieties by B. Kahn Integral homology of $PGL_2$ over elliptic curves by K. P. Knudson Application of motivic complexes to negligible classes by E. Peyre Two-primary algebraic $K$-theory of spaces and related spaces of symmetries of manifolds by J. Rognes A mini-course on recent progress in algebraic $K$-theory and its relationship with topology and analysis by J. Rosenberg The Chow ring of a classifying space by B. Totaro Voevodsky's Seattle lectures: $K$-theory and motivic cohomology by V. Voevodsky Products in higher Chow groups and motivic cohomology by C. Weibel.

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Book SynopsisOrdered vector spaces and cones made their debut in mathematics at the beginning of the twentieth century. They were developed in parallel (but from a different perspective) with functional analysis and operator theory. This book offers a modern perspective on cones and ordered vector spaces.Table of ContentsCones Cones in topological vector spaces Yudin and pull-back cones Krein operators $\mathcal{K}$-lattices The order extension of $L'$ Piecewise affine functions Appendix: Linear topologies Bibliography Index.

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Book SynopsisOpen algebraic surfaces are a synonym for algebraic surfaces that are not necessarily complete. An open algebraic surface is understood as a Zariski open set of a projective algebraic surface. This book contains an account of the theory of open algebraic surfaces, as well as several applications, in particular to the study of affine surfaces.Trade ReviewAn indispensable reference work for experts in the field CMS NotesTable of ContentsComplete algebraic surfaces Open algebraic surfaces Affine algebraic surfaces Bibliography Index.

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Book SynopsisTable of ContentsTable of Contents: Parameters Local transfer Global stabilization The standard model A study of critical cases The local classification Local nontempered representations The global classification Inner forms Bibliography Index

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Book SynopsisAlgebraic Geometry has been at the centre of much of mathematics for hundreds of years. It is not an easy field to break into, despite its humble beginnings in the study of circles, ellipses, hyperbolas, and parabolas. This text consists of a series of exercises, plus some background information and explanations, starting with conics and ending with sheaves and cohomology.

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