Algebra Books

Book SynopsisThe theory of algebras, rings, and modules is one of the fundamental domains of modern mathematics. General algebra, more specifically non-commutative algebra, is poised for major advances in the twenty-first century (together with and in interaction with combinatorics), just as topology, analysis, and probability experienced in the twentieth century. This is the second volume of Algebras, Rings and Modules: Non-commutative Algebras and Rings by M. Hazewinkel and N. Gubarenis, a continuation stressing the more important recent results on advanced topics of the structural theory of associative algebras, rings and modules. Table of ContentsRings related to finite posets. Distributive and semidistributive rings. The group of extensions. Modules over semiperfect rings. Representations of primitive posets. Representations of quivers, species and finite dimensional algebras. Artinian rings of finite representation type. O-species and SPSD-rings of bounded representation type.

Read more less

Book SynopsisTable of ContentsR. Review of Basic Concepts. R.1 Sets R.2 Real Numbers and Their Properties R.3 Polynomials R.4 Factoring Polynomials R.5 Rational Expressions R.6 Rational Exponents R.7 Radical Expressions 1. Equations and Inequalities 1.1 Linear Equations 1.2 Applications and Modeling with Linear Equations 1.3 Complex Numbers 1.4 Quadratic Equations 1.5 Applications and Modeling with Quadratic Equations 1.6 Other Types of Equations and Applications 1.7 Inequalities 1.8 Absolute Value Equations and Inequalities 2. Graphs and Functions 2.1 Rectangular Coordinates and Graphs 2.2 Circles 2.3 Functions 2.4 Linear Functions 2.5 Equations of Lines and Linear Models 2.6 Graphs of Basic Functions 2.7 Graphing Techniques 2.8 Function Operations and Composition 3. Polynomial and Rational Functions. 3.1 Quadratic Functions and Models 3.2 Synthetic Division 3.3 Zeros of Polynomial Functions 3.4 Polynomial Functions: Graphs, Applications, and Models 3.5 Rational Functions: Graphs, Applications, and Models 3.6 Variation 4. Inverse, Exponential, and Logarithmic Functions. 4.1 Inverse Functions 4.2 Exponential Functions 4.3 Logarithmic Functions 4.4 Evaluating Logarithms and the Change-of-Base Theorem 4.5 Exponential and Logarithmic Equations 4.6 Applications and Models of Exponential Growth and Decay 5. Systems and Matrices 5.1 Systems of Linear Equations 5.2 Matrix Solution of Linear Systems 5.3 Determinant Solution of Linear Systems 5.4 Partial Fractions 5.5 Nonlinear Systems of Equations 5.6 Systems of Inequalities and Linear Programming 5.7 Properties of Matrices 5.8 Matrix Inverses Glossary Solutions to Selected Exercises Answers to Selected Exercises Index of Applications Index

Read more less

Book SynopsisBob Blitzer is a native of Manhattan and received aBachelor of Arts degree with dual majors in mathematics and psychology (minor:English literature) from the City College of New York. His unusual combinationof academic interests led him toward a Master of Arts in mathematics from theUniversity of Miami and a doctorate in behavioral sciences from NovaUniversity. Bob's love for teaching mathematics was nourished for nearly 30years at Miami Dade College, where he received numerous teaching awards,including Innovator of the Year from the League for Innovations in theCommunity College and an endowed chair based on excellence in the classroom. Inaddition to Algebra and Trigonometry, Bob has written textbookscovering developmental mathematics, introductory algebra, intermediate algebra,trigonometry, precalculus, and liberal arts mathematics, all published byPearson. When not secluded in his Northern California writer's cabin, Bob canbe found hikinTable of Contents P. Prerequisites: Fundamental Concepts of Algebra P.1 Algebraic Expressions, Mathematical Models, and Real Numbers P.2 Exponents and Scientific Notation P.3 Radicals and Rational Exponents P.4 Polynomials P.5 Factoring Polynomials P.6 Rational Expressions 1. Equations and Inequalities 1.1 Graphs and Graphing Utilities 1.2 Linear Equations and Rational Equations 1.3 Models and Applications 1.4 Complex Numbers 1.5 Quadratic Equations 1.6 Other Types of Equations 1.7 Linear Inequalities and Absolute Value Inequalities 2. Functions and Graphs 2.1 Basics of Functions and Their Graphs 2.2 More on Functions and Their Graphs 2.3 Linear Functions and Slope 2.4 More on Slope 2.5Transformations of Functions 2.6 Combinations of Functions; Composite Functions 2.7 Inverse Functions 2.8 Distance and Midpoint Formulas; Circles 3. Polynomial and Rational Functions 3.1 Quadratic Functions 3.2 Polynomial Functions and Their Graphs 3.3 Dividing Polynomials; Remainder and Factor Theorems 3.4 Zeros of Polynomial Functions 3.5 Rational Functions and Their Graphs 3.6 Polynomial and Rational Inequalities 3.7 Modeling Using Variation 4. Exponential and Logarithmic Functions 4.1 Exponential Functions 4.2 Logarithmic Functions 4.3 Properties of Logarithms 4.4 Exponential and Logarithmic Equations 4.5 Exponential Growth and Decay; Modeling Data 5. Trigonometric Functions 5.1 Angles and Radian Measure 5.2 Right Triangle Trigonometry 5.3 Trigonometric Functions of Any Angle 5.4 Trigonometric Functions of Real Numbers; Periodic Functions 5.5 Graphs of Sine and Cosine Functions 5.6 Graphs of Other Trigonometric Functions 5.7 Inverse Trigonometric Functions 5.8 Applications of Trigonometric Functions 6. Analytic Trigonometry 6.1 Verifying Trigonometric Identities 6.2 Sum and Difference Formulas 6.3Double-Angle, Power-Reducing, and Half-Angle Formulas 6.4 Product-to-Sumand Sum-to-Product Formulas 6.5 Trigonometric Equations 7. Additional Topics in Trigonometry 7.1 The Law of Sines 7.2 The Law of Cosines 7.3 Polar Coordinates 7.4 Graphs of Polar Equations 7.5 Complex Numbers in Polar Form; De Moivre's Theorem 7.6 Vectors 7.7 The Dot Product 8. Systems of Equations and Inequalities 8.1 Systems of Linear Equations in Two Variables 8.2 Systems of Linear Equations in Three Variables 8.3 Partial Fractions 8.4 Systems of Nonlinear Equations in Two Variables 8.5 Systems of Inequalities 8.6 Linear Programming 9. Matrices and Determinants 9.1 Matrix Solutions to Linear Systems 9.2 Inconsistent and Dependent Systems and Their Applications 9.3 Matrix Operations and Their Applications 9.4 Multiplicative Inverses of Matrices and Matrix Equations 9.5 Determinants and Cramer's Rule 10. Conic Sections and Analytic Geometry 10.1 The Ellipse 10.2 The Hyperbola 10.3 The Parabola 10.4 Rotation of Axes 10.5 Parametric Equations 10.6 Conic Sections in Polar Coordinates 11. Sequences, Induction, and Probability 11.1 Sequences and Summation Notation 11.2 Arithmetic Sequences 11.3 Geometric Sequences and Series 11.4 Mathematical Induction 11.5 The Binomial Theorem 11.6 Counting Principles, Permutations, and Combinations 11.7 Probability

Read more less

Book SynopsisMarge Lial (late) was always interested in math; it was her favorite subject in the first grade! Marge's intense desire to educate both her students and herself has inspired the writing of numerous best-selling textbooks. Marge, who received Bachelor's and Master's degrees from California State University at Sacramento, was affiliated with American River College. An avid reader and traveler, her travel experiences often found their way into her books as applications, exercise sets, and feature sets. Her interest in archeology led to trips to various digs and ruin sites, producing some fascinating problems for her textbooks involving such topics as the building of Mayan pyramids and the acoustics of ancient ball courts in the Yucatan. When John Hornsby enrolled as an undergraduate at Louisiana State University, he was uncertain whether he wanted to study mathematics education or journalism. His ultimate decision was to become a teacher, but afTable of ContentsR. Review of Basic Concepts R.1. Fractions, Decimals, and Percents R.2. Sets and Real Numbers R.3. Real Number Operations and Properties R.4. Integer and Rational Exponents R.5. Polynomials R.6. Factoring Polynomials R.7. Rational Expressions R.8. Radical Expressions 1. Equations and Inequalities 1.1. Linear Equations 1.2. Applications and Modeling with Linear Equations 1.3. Complex Numbers 1.4. Quadratic Equations 1.5. Applications and Modeling with Quadratic Equations 1.6. Other Types of Equations and Applications 1.7. Inequalities 1.8. Absolute Value Equations and Inequalities 2. Graphs and Functions 2.1. Rectangular Coordinates and Graphs 2.2. Circles 2.3. Functions 2.4. Linear Functions 2.5. Equations of Lines and Linear Models 2.6. Graphs of Basic Functions 2.7. Graphing Techniques 2.8. Function Operations and Composition 3. Polynomial and Rational Functions 3.1. Quadratic Functions and Models 3.2. Synthetic Division 3.3. Zeros of Polynomial Functions 3.4. Polynomial Functions: Graphs, Applications, and Models 3.5. Rational Functions: Graphs, Applications, and Models 3.6. Polynomial and Rational Inequalities 3.7. Variation 4. Inverse, Exponential, and Logarithmic Functions 4.1. Inverse Functions 4.2. Exponential Functions 4.3. Logarithmic Functions 4.4. Evaluating Logarithms and the Change-of-Base Theorem 4.5. Exponential and Logarithmic Equations 4.6. Applications of Models of Exponential Growth and Decay 5. Trigonometric Functions 5.1. Angles 5.2. Trigonometric Functions 5.3. Trigonometric Function Values and Angle Measures 5.4. Solutions and Applications of Right Triangles 6. The Circular Functions and Their Graphs 6.1. Radian Measure 6.2. The Unit Circle and Circular Functions 6.3. Graphs of the Sine and Cosine Functions 6.4. Translations of the Graphs of the Sine and Cosine Functions 6.5. Graphs of the Tangent and Cotangent Functions 6.6. Graphs of the Secant and Cosecant Functions 6.7. Harmonic Motion 7. Trigonometric Identities and Equations 7.1. Fundamental Identities 7.2. Verifying Trigonometric Identities 7.3. Sum and Difference Identities 7.4. Double-Angle and Half-Angle Identities 7.5. Inverse Circular Functions 7.6. Trigonometric Equations 7.7. Equations Involving Inverse Trigonometric Functions 8. Applications of Trigonometry 8.1. The Law of Sines 8.2. The Law of Cosines 8.3. Geometrically Defined Vectors and Applications 8.4. Algebraically Defined Vectors and the Dot Product 8.5. Trigonometric (Polar) Form of Complex Numbers; Products and Quotients 8.6. De Moivre's Theorem; Powers and Roots of Complex Numbers 8.7. Polar Equations and Graphs 8.8. Parametric Equations, Graphs, and Applications 9. Systems and Matrices 9.1. Systems of Linear Equations 9.2. Matrix Solution of Linear Systems 9.3. Determinant Solution of Linear Systems 9.4. Partial Fractions 9.5. Nonlinear Systems of Equations 9.6. Systems of Inequalities and Linear Programming 9.7. Properties of Matrices 9.8. Matrix Inverses 10. Analytic Geometry 10.1. Parabolas 10.2. Ellipses 10.3. Hyperbolas 10.4. Summary of the Conic Sections 11. Further Topics in Algebra 11.1. Sequences and Series 11.2. Arithmetic Sequences and Series 11.3. Geometric Sequences and Series 11.4. The Binomial Theorem 11.5. Mathematical Induction 11.6. Basics of Probability Appendix A Polar Form of Conic Sections Appendix B Rotation of Axes Appendix C Geometry Formulas

Read more less

Book SynopsisMarge Lial (late) was always interested in math; it was her favorite subject in the first grade! Marge's intense desire to educate both her students and herself has inspired the writing of numerous best-selling textbooks. Marge, who received Bachelor's and Master's degrees from California State University at Sacramento, was affiliated with American River College. An avid reader and traveler, her travel experiences often found their way into her books as applications, exercise sets, and feature sets. Her interest in archeology led to trips to various digs and ruin sites, producing some fascinating problems for her textbooks involving such topics as the building of Mayan pyramids and the acoustics of ancient ball courts in the Yucatan. When John Hornsby enrolled as an undergraduate at Louisiana State University, he was uncertain whether he wanted to study mathematics education or journalism. His ultimate decision was to become a teacher, but af

Read more less

Book Synopsis

Read more less

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

Read more less

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

Read more less

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.

Read more less

Book Synopsis

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

Book Synopsis

Read more less

Book Synopsis

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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

Read more less

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

Read more less

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.

Read more less

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.

Read more less

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

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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.

Read more less

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

Read more less

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

Read more less

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.

Read more less

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

Read more less