2007-2008 NEW Higher Mathematics Titles Higher Mathematics ~ Contents

2007 New Titles 

Abstract Algebra ............................................................. 68 Advanced Engineering Mathematics ............................... 64

ISBN-13: 978-0-07-312561-9 / MHID: 0-07-312561-X 

Advanced Geometry ....................................................... 68 Combinatorics ................................................................ 63

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BURTON The History of Mathematics: An Introduction, 6e ...........................................................................66 ISBN-13: 978-0-07-305189-5 / MHID: 0-07-305189-6

Differential Equations ..................................................... 58 

Graph Theory ................................................................. 65

DUMAS Transition to Higher Mathematics: Structure and Proof ...............................................................61 ISBN-13: 978-0-07-353353-7 / MHID: 0-07-353353-X

History of Mathematics ................................................... 66 Introductory Analysis ...................................................... 65

BURTON Elementary Number Theory, 6e...............................67 ISBN-13: 978-0-07-305188-8 / MHID: 0-07-305188-8

Complex Analysis ........................................................... 69

Dynamical System .......................................................... 64

BONA Introduction to Enumerative Combinatorics ...........63

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Linear Algebra ................................................................ 62

SIMMONS Differential Equations: Theory, Technique, and Practice ............................................................58 ISBN-13: 978-0-07-286315-4 / MHID: 0-07-286315-3

Logic .............................................................................. 64 Mathematical References ................................................ 70 Number Theory .............................................................. 67

2008 New Titles

Numerical Analysis......................................................... 67 Partial Differential Equations........................................... 60 Topology ........................................................................ 70 Transition to Higher Math/Foundations of Higher Math........................................................................... 61

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BROWN Frontier Series and Boundary Value Problems, 7e ............................................................................60 ISBN-13: 978-0-07-305193-2 / MHID: 0-07-305193-4

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Higher Mathematics Differential Equations

* Technology – Each chapter includes optional technology exercises designed to take advantage of computer algebra systems such as MAPLE or Mathematica for solving problems in differential equations.

CONTENTS

International Edition

NEW

DIFFERENTIAL EQUATIONS: THEORY, TECHNIQUE, AND PRACTICE

By George F. Simmons, Colorado College, and Steven G. Krantz, Washington University-St Louis 2007 (December 2005) / 768 pages / Hardcover ISBN-13: 978-0-07-286315-4 / MHID: 0-07-286315-3 ISBN-13: 978-0-07-125437-3 / MHID: 0-07-125437-4 [IE]

This traditional text is intended for mainstream one- or twosemester differential equations courses taken by undergraduates majoring in engineering, mathematics, and the sciences. Written by two of the world’s leading authorities on differential equations, Simmons/Krantz provides a cogent and accessible introduction to ordinary differential equations written in classical style. Its rich variety of modern applications in engineering, physics, and the applied sciences illuminate the concepts and techniques that students will use through practice to solve reallife problems in their careers. This text is part of the Walter Rudin Student Series in Advanced Mathematics. FEATURES * Award-winning Author Team – With Differential Equations: Theory, Technique and Practice, McGraw-Hill is proud to bring together the talents of two acclaimed authors and leading authorities in differential equations. George F. Simmons is renowned for his clear, direct, and engaging writing style over a 40-year career as author of highly-regarded textbooks for undergraduates in calculus and differential equations. Steven G. Krantz is one of the most visible and respected mathematicians in the community today, with over 45 books, 125 papers, and groundbreaking research in differential equations to his credit. * Applications – Simmons/Krantz features a wealth of modern applications of differential equations to bring the subject to life and demonstrate its utility to students. These applications show how differential equations is the centerpiece of many areas of engineering, physics, the life sciences, and mathematical modeling, and prepare students for more concrete applied work in future courses. * “Anatomy of an Application” – These sections, occurring at the end of each chapter, examine in detail a particular application of differential equations in engineering, physics, or the applied sciences. These applications apply topics learned in the chapter to practical problems students will face in these fields, and discuss in detail the differential equations used to solve those problems. * Historical context – The text underscores the historical context and human element behind the development of key concepts in ordinary differential equations with biographical sketches students will find thought-provoking and fun. Short Math Nuggets appear within sections to illuminate the lives, accomplishments, and tribulations of important historical figures behind individual concepts. Longer Historical Notes appearing at the end of certain chapters go into greater detail about giants in the field such as Leonhard Euler, Carl Friedrich Gauss, and Pierre Simon de Laplace, and how they helped develop key areas students now study in modern differential equations. * Exercises – Each chapter contains a variety of quality exercises, including Drill Exercises that test basic understanding, Challenge Problems that go into greater detail and feature a higher degree of complexity, and Problems for Discussion and Exploration that further develop students’ critical thinking skills. Hints are given when appropriate to assist students with more difficult problems. Available for separate purchase, McGraw-Hill offers a Student’s Solutions Manual containing worked solutions to all odd-numbered exercises. We also offer an Instructor’s Solutions Manual containing solutions to all even-numbered exercises available free to instructors who adopt the text.

Preface. 1 What is a Differential Equation? 1.1 Introductory Remarks 1.2 The Nature of Solutions 1.3 Separable Equations 1.4 First-Order Linear Equations 1.5 Exact Equations 1.6 Orthogonal Trajectories and Families of Curves 1.7 Homogeneous Equations 1.8 Integrating Factors 1.9 Reduction of Order 1.9.1 Dependent Variable Missing 1.9.2 Independent Variable Missing 1.10 The Hanging Chain and Pursuit Curves 1.10.1 The Hanging Chain 1.10.2 Pursuit Curves 1.11 Electrical Circuits Anatomy of an Application: The Design of a Dialysis Machine. Problems for Review and Discovery. 2 Second-Order Equations 2.1 Second-Order Linear Equations with Constant Coefficients 2.2 The Method of Undetermined Coefficients 2.3 The Method of Variation of Parameters 2.4 The Use of a Known Solution to Find Another 2.5 Vibrations and Oscillations 2.5.1 Undamped Simple Harmonic Motion 2.5.2 Damped Vibrations 2.5.3 Forced Vibrations 2.5.4 A Few Remarks About Electricity 2.6 Newton’s Law of Gravitation and Kepler’s Laws 2.6.1 Kepler’s Second Law 2.6.2 Kepler’s First Law 2.6.3 Kepler’s Third Law 2.7 Higher Order Equations. Anatomy of an Application: Bessel Functions and the Vibrating Membrane. Problems for Review and Discovery. 3 Qualitative Properties and Theoretical Aspects 3.0 Review of Linear Algebra 3.0.1 Vector Spaces 3.0.2 The Concept Linear Independence 3.0.3 Bases 3.0.4 Inner Product Spaces 3.0.5 Linear Transformations and Matrices 3.0.6 Eigenvalues and Eigenvectors 3.1 A Bit of Theory 3.2 Picard’s Existence and Uniqueness Theorem 3.2.1 The Form of a Differential Equation 3.2.2 Picard’s Iteration Technique 3.2.3 Some Illustrative Examples 3.2.4 Estimation of the Picard Iterates 3.3 Oscillations and the Sturm Separation Theorem 3.4 The Sturm Comparison Theorem. Anatomy of an Application: The Green’s Function. Problems for Review and Discovery. 4 Power Series Solutions and Special Functions 4.1 Introduction and Review of Power Series 4.1.1 Review of Power Series. 4.2 Series Solutions of First-Order Differential Equations. 4.3 Second-Order Linear Equations: Ordinary Points. 4.4 Regular Singular Points. 4.5 More on Regular Singular Points. 4.6 Gauss’s Hypergeometric Equation. Anatomy of an Application: Steady State Temperature in a Ball. Problems for Review and Discovery. 5 Fourier Series: Basic Concepts. 5.1 Fourier Coefficients. 5.2 Some Remarks about Convergence. 5.3 Even and Odd Functions: Cosine and Sine Series. 5.4 Fourier Series on Arbitrary Intervals. 5.5 Orthogonal Functions. Anatomy of an Application: Introduction to the Fourier Transform. Problems for Review and Discovery. 6 Partial Differential Equations and Boundary Value Problems. 6.1 Introduction and Historical Remarks. 6.2 Eigenvalues, Eigenfunctions, and the Vibrating String. 6.2.1 Boundary Value Problems. 6.2.2 Derivation of the Wave Equation. 6.2.3 Solution of the Wave Equation. 6.3 The Heat Equation. 6.4 The Dirichlet Problem for a Disc. 6.4.1 The Poisson Integral 6.5 Sturm-Liouville Problems. Anatomy of an Application: Some Ideas from Quantum Mechanics. Problems for Review and Discovery. 7 Laplace Transforms. 7.0 Introduction 7.1 Applications to Differential Equations 7.2 Derivatives and Integrals of Laplace Transforms 7.3 Convolutions 7.4 The Unit Step and Impulse Functions. Anatomy of an Application: Flow Initiated by an Impulsively-Started Flat Plate. Problems for Review and Discovery. 8 The Calculus of Variations 8.1 Introductory Remarks. 8.2 Euler’s Equation. 8.3 Isoperimetric Problems and the Like. 8.3.1 Lagrange Multipliers 8.3.2 Integral Side Conditions. 8.3.3 Finite Side Conditions. Anatomy of an Application: Hamilton’s Principle and its Implications. Problems for Review and Discovery. 9 Numerical Methods. 9.1 Introductory Remarks. 9.2 The Method of Euler. 9.3 The Error Term. 9.4 An Improved Euler Method 9.5 The Runge-Kutta Method. Anatomy of an Application: A Constant Perturbation Method for Linear, Second-Order Equations. Problems for Review and Discovery. 10 Systems of First-Order Equations 10.1 Introductory Remarks. 10.2 Linear Systems 10.3 Homogeneous Linear Systems with Constant Coefficients 10.4 Nonlinear Systems: Volterra’s Predator-Prey Equations. Anatomy of an Application: Solution of Systems with Matrices and Exponentials. Problems for Review and Discovery. 11 The Nonlinear Theory. 11.1 Some Motivating Examples 11.2 Specializing Down 11.3 Types of Critical Points: Stability 11.4 Critical Points and Stability for Linear Systems 11.5 Stability by Liapunov’s Direct Method 11.6 Simple Critical Points of Nonlinear Systems 11.7 Nonlinear Mechanics: Conservative Systems 11.8 Periodic Solutions: The Poincaré-Bendixson Theorem. Anatomy of an Application: Mechanical Analysis of a Block on a Spring. Problems for Review and Discovery. 12 Dynamical Systems 12.1 Flows 12.1.1 Dynamical Systems 12.1.2 Stable and Unstable Fixed Points 12.1.3 Linear Dynamics in the Plane 12.2 Some Ideas from Topology 12.2.1 Open and Closed Sets 12.2.2 The Idea of Connectedness 12.2.3 Closed Curves

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Higher Mathematics in the Plane 12.3 Planar Autonomous Systems 12.3.1 Ingredients of the Proof of Poincaré-Bendixson. Anatomy of an Application: Lagrange’s Equations. Problems for Review and Discovery. Bibliography

International Edition DIFFERENTIAL EQUATIONS: A MODELING APPROACH

SCHAUM’S OUTLINE OF DIFFERENTIAL EQUATIONS Third Edition

by Richard Bronson, Fairleigh Dickinson University-Madison and Gabriel Costa, US Military Academy 2006 (June 2006) / 384 pages ISBN-13: 978-0-07-145687-6 / MHID: 0-07-145687-2

A Schaum Publication Thoroughly updated, this third edition of Schaum’s Outline of Differential Equations offers you new, faster techniques for solving differential equations generated by the emergence of high-speed computers. Differential equations, a linchpin of modern math, are essential in engineering, the natural sciences, economics, and business. Includes:  563 fully solved problems  800-plus supplementary problems  New chapter on modeling

DIFFERENTIAL EQUATIONS

by Keng Cheng Ang 2005 (October 2005) ISBN-13: 978-0-07-125085-6 / MHID: 0-07-125085-9

An Asian Publication Many books on differential equations assume that the reader has a fairly sophisticated level of competence in calculus at the university level. Differential Equations: Models and Methods differs from them in that it enables a student with some basic knowledge of calculus to learn about differential equations and appreciate their applications. The focus of the book is on first order differential equations, their methods of solution and their use in mathematical models. Methods include analytic and graphical solutions, as well as numerical techniques. Readers will not only learn the necessary techniques of solving first order differential equations, but also how these equations can be applied in different fields. Examples have been carefully chosen to provide motivation for new concepts or techniques, and to illustrate the importance of differential equations. This book was written with student needs in mind; in particular, pre-university students taking the new GCE ‘A’ Level H3 Mathematics will find it useful in helping them through the course. CONTENTS Preface / 1. Basic Concepts 2. Analytic Solutions 3. Graphical Techniques 4. Numerical Methods 5. Mathematical Models 6. Further Applications / Further Reading / Appendix A: Table of Integrals / Appendix B: Method of Least Squares / Answers to Odd-numbered Problems / Index

by Glenn Ledder, University of Nebraska—Lincoln 2005 / 768 pages ISBN-13: 978-0-07-242229-0 / MHID: 0-07-242229-7 ISBN-13: 978-0-07-111151-5 / MHID: 0-07-111151-4 [IE]

Ledder’s innovative, student-centered approach reflects recent research on successful learning by emphasizing connections between new and familiar concepts and by engaging students in a dialogue with the material. Though streamlined, the text is also flexible enough to support a variety of teaching goals, in part through optional topics that give instructors considerable freedom in customizing their courses. Linear algebra is presented in self-contained sections to accommodate both courses that have a linear algebra prerequisite and those that do not. Throughout the text, a wide variety of examples from the physical, life and social sciences, among other areas, are employed to enhance student learning. In-depth Model Problems drawn from everyday experience highlight the key concepts or methods in each section. Other innovative features of the text include Instant Exercises that allow students to quickly test new skills and Case Studies that further explore the powerful problem-solving capability of differential equations. Readers will learn not only how to solve differential equations, but also how to apply their knowledge to areas in mathematics and beyond. CONTENTS 1 Introduction: 1.1 Natural Decay and Natural Growth. 1.2 Differential Equations and Solutions. 1.3 Mathematical Models and Mathematical Modeling. Case Study 1 Scientific Detection of Art Forgery. 2 Basic Concepts and Techniques: 2.1 A Collection of Mathematical Models. 2.2 Separable First-Order Equations. 2.3 Slope Fields. 2.4 Existence of Unique Solutions. 2.5 Euler’s Method. 2.6 Runge-Kutta Methods. Case Study 2 A Successful Volleyball Serve. 3 Homogeneous Linear Equations. 3.1 Linear Oscillators. 3.2 Systems of Linear Algebraic Equations. 3.3 Theory of Homogeneous Linear Equations. 3.4 Homogeneous Equations with Constant Coefficients. 3.5 Real Solutions from Complex Characteristic Values. 3.6 Multiple Solutions for Repeated Characteristic Values. 3.7 Some Other Homogeneous Linear Equations. Case Study 3 How Long Should Jellyfish Hold their Food? 4 Nonhomogeneous Linear Equations: 4.1 More on Linear Oscillator Models. 4.2 General Solutions for Nonhomogeneous Equations. 4.3 The Method of Undetermined Coefficients. 4.4 Forced Linear Oscillators. 4.5 Solving First-Order Linear Equations. 4.6 Particular Solutions for Second-Order Equations by Variation of Parameters. Case Study 4 A Tuning Circuit for a Radio. 5 Autonomous Equations and Systems: 5.1 Population Models. 5.2 The Phase Line. 5.3 The Phase Plane. 5.4 The Direction Field and Critical Points. 5.5 Qualitative Analysis. Case Study 5 A Self-Limiting Population. 6 Analytical Methods for Systems: 6.1 Compartment Models. 6.2 Eigenvalues and Eigenspaces. 6.3 Linear Trajectories. 6.4 Homogeneous Systems with Real Eigenvalues. 6.5 Homogeneous Systems with Complex Eigenvalues. 6.6 Additional Solutions for Deficient Matrices. 6.7 Qualitative Behavior of Nonlinear Systems. Case Study 6 Invasion by Disease. 7 The Laplace Transform: 7.1 Piecewise-Continuous Functions. 7.2 Definition and Properties of the Laplace Transform. 7.3 Solution of Initial-Value Problems with the Laplace Transform. 7.4 PiecewiseContinuous and Impulsive Forcing. 7.5 Convolution and the Impulse Response Function. Case Study 7 Growth of a Structured Population. 8 Vibrating Strings: A Focused Introduction to Partial Differential Equations: 8.1 Transverse Vibration of a String. 8.2 The General Solution of the Wave Equation. 8.3 Vibration Modes of a Finite String. 8.4 Motion of a Plucked String. 8.5 Fourier Series. Case Study 8 Stringed Instruments and Percussion. A Some Additional Topics: A.1 Using Integrating Factors to Solve First-Order Linear Equations (Chapter 2). A.2 Proof of the Existence and Uniqueness Theorem for First-Order Equations (Chapter 2). A.3 Error in Numerical Methods (Chapter 2). A.4 Power Series Solutions (Chapter 3). A.5 Matrix Functions (Chapter 6). A.6 Nonhomogeneous Linear Systems (Chapter 6). A.7 The One-Dimensional Heat Equation (Chapter 8). A.8 Laplace’s Equation (Chapter 8)

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Higher Mathematics International Edition DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES Second Edition by George F. Simmons, Colorado College 1991 / 640 pages ISBN-13: 978-0-07-057540-0 / MHID: 0-07-057540-1 ISBN-13: 978-0-07-112807-0 / MHID: 0-07-112807-7 [IE]

CONTENTS 1 The Nature of Differential Equations. 2 First Order Equations. 3 Second Order Linear Equations. 4 Qualitative Properties of Solutions. 5 Power Series Solutions and Special Functions. 6 Fourier Series and Orthogonal Functions. 7 Partial Differential Equations and Boundary Value Problems. 8 Some Special Functions of Mathematical Physics. 9 Laplace Transforms. 10 Systems of First Order Equations. 11 Nonlinear Equations. 12 The Calculus of Variations. 13 The Existence and Uniqueness of Solutions. 14 Numerical Methods.

SCHAUM’S OUTLINE OF DIFFERENTIAL EQUATIONS Second Edition

by Richard Bronson, Fairleigh Dickinson University 1994 / 368 pages ISBN-13: 978-0-07-008019-5 / MHID: 0-07-008019-4

A Schaum Publication CONTENTS Basic Concepts. Classification of First-Order Differential Equations. Separable First-Order Differential Equations. Exact First-Order Differential Equations. Linear First-Order Differential Equations. Applications of First-Order Differential Equations. Linear Differential Equations: Theory of Solutions. Second-Order Linear Homogeneous Differential Equations with Constant Coefficients. nTH-Order Linear Homogeneous Differential Equations with Constant Coefficients. The Method of Undetermined Coefficients. Variation of Parameters. Initial-Value Problems. Applications of Second-Order Linear Differential Equations. The Laplace Transform. The Inverse Laplace Transform. Convolutions and the Unit Step Function. Solutions of Linear Systems by Laplace Transform. Convolutions and the Unit Step Function. Solutions of Linear Differential Equations with Constant Coefficients by Laplace Transform. Solutions of Linear Differential Equations with Constant Coefficients by Laplace Transform. Solutions of Linear Systems by Laplace Transform. Matrices. eAt. Reduction of Linear Differential Equations to a First-Order System. Solutions of Linear Differential Equations with Constant Coefficients by Matrix Methods. Linear Differential Equations with Variable Coefficients. Regular Singular Points and the Method of Frobenius. Gamma and Bessel Functions. Graphical Methods for Solving First-Order Differential Equations. Numerical Methods for Solving First-Order Differential Equations. Numerical Methods for Systems. Second-Order Boundary-Value Problems. Eigenfunction Expansions. Appendix: Laplace Transforms. Answers to Supplementary Problems.

Partial Differential Equations International Edition

NEW

FOURIER SERIES AND BOUNDARY VALUE PROBLEMS Seventh Edition

by James Ward Brown, University of Michigan-Dearborn and Ruel Churchill (deceased) 2008 (August 2006) / 384 pages ISBN-13: 978-0-07-305193-2 / MHID: 0-07-305193-4 ISBN-13: 978-0-07-125917-0 / MHID: 0-07-125971-1 [IE]

Published by McGraw-Hill since its first edition in 1941, this classic text is an introduction to Fourier series and their applications to boundary value problems in partial differential equations of engineering and physics. It will primarily be used by students with a background in ordinary differential equations and advanced calculus. There are two main objectives of this text. The first is to introduce the concept of orthogonal sets of functions and representations of arbitrary functions in series of functions from such sets. The second is a clear presentation of the classical method of separation of variables used in solving boundary value problems with the aid of those representations. NEW TO THIS EDITION  Reorganization of Topics: Topics in the text have been realigned to allow for more focus on each section and to allow for more examples. The chapter on The Fourier Method has been moved earlier in the book (now Chapter 2). The former Fourier Series chapter has been split into two chapters (Chapter 3: Orthonormal Sets and Fourier Series and Chapter 4: Convergence of Fourier Series).  Problem Sets Revised: Problem sets have been broken up into more manageable segments to allow for each problem set to be very focused.  Examples Added: Additional examples have been added in each chapter to help illustrate important topics.

FEATURES  Primary Focus: The text’s primary focus is to find solutions to specific problems, rather than developing general theories.

CONTENTS Preface / 1 Fourier Series / 2 Convergence of Fourier Series / 3 Partial Differential Equations of Physics / 4 The Fourier Method / 5 Boundary Value Problems / 6 Fourier Integrals and Applications / 7 Orthonormal Sets / 8 Sturm-Liouville Problems and Applications / 9 Bessel Functions and Applications / 10 Legendre Polynomials and Applications / 11 Verification of Solutions and Uniqueness / Appendixes / Bibliography / Some Fourier Series Expansions / Solutions of Some Regular SturmLiouville Problems/ Index

INVITATION TO PUBLISH McGraw-Hill is interested in reviewing manuscript for publication. Please contact your local McGraw-Hill office or email to [email protected]

Visit McGraw-Hill Education (Asia) Website: www.mcgraw-hill.com.sg

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Higher Mathematics SCHAUM’S OUTLINE OF PARTIAL DIFFERENTIAL EQUATIONS

by Paul DuChateau, Colorado State University and D W Zachmann, Colorado State University 1986 / 256 pages ISBN-13: 978-0-07-017897-7 / MHID: 0-07-017897-6

A Schaum Publication CONTENTS Introduction. Classification and Characteristics. Qualitative Behavior of Solutions to Elliptic Equations. Qualitative Behavior of Solutions to Evolution Equations. First-Order Equations Eigenfunction Expansions and Integral Transforms: Theory. Eigenfunction Expansions and Integral Transforms: Applications. Green’s Functions. Difference Methods for Parabolic Equations. Difference and Characteristic Methods for Parabolic Equations. Difference Methods for Hyperbolic Equations. Difference Methods for Elliptic Equations. Variational Formulation of Boundary Value Problems. The Finite Element Method: An Introduction. Answers to Supplementary Problems.

FEATURES * Writing Style – Dumas/McCarthy presents the material in a friendly, conversational writing style designed to appeal to students, while maintaining the appropriate level of formality and rigor required for proper mastery of proof technique. The authors motivate understanding of formal definitions with discussions and remarks designed to help students grasp the basic direction of the arguments. * Attention to Detail – The authors take utmost care to present correct mathematical detail, structure, notation, and terminology at all times. Informal discussion and hints are clearly distinguished from formal definitions and proof with a “Discussion” label. * Exercises – Each chapter features numerous exercises of varying difficulty designed to direct students’ attention to the reading and compel them to think through the details of the proofs. * Flexible Topic Coverage – Dumas/McCarthy covers a wide and welldeveloped range of topics, and the chapters are self-contained enough to allow instructors to easily pick and choose the topics they wish to cover. The material can be structured to accommodate the time constraints of a quarter course as well as a semester-long course.

CONTENTS

International Edition ELEMENTS OF PARTIAL DIFFERENTIAL EQUATIONS by Sneddon 1985 / 344 pages ISBN-13: 978-0-07-085740-7 / MHID: 0-07-085740-7 [IE]

Transition to Higher Math/ Foundations of Higher Math International Edition

NEW

TRANSITION TO HIGHER MATHEMATICS Structure and Proof

By Bob A. Dumas, University Of Washington, and John E. McCarthy, Washington University-St Louis 2007 (February 2006) / 416 pages / Hardcover ISBN-13: 978-0-07-353353-7 / MHID: 0-07-353353-X ISBN-13: 978-0-07-110647-4 / MHID: 0-07-110647-2 [IE]

This text is intended for the Foundations of Higher Math bridge course taken by prospective math majors following completion of the mainstream Calculus sequence, and is designed to help students develop the abstract mathematical thinking skills necessary for success in later upper-level majors math courses. As lower-level courses such as Calculus rely more exclusively on computational problems to service students in the sciences and engineering, math majors increasingly need clearer guidance and more rigorous practice in proof technique to adequately prepare themselves for the advanced math curriculum. With their friendly writing style Bob Dumas and John McCarthy teach students how to organize and structure their mathematical thoughts, how to read and manipulate abstract definitions, and how to prove or refute proofs by effectively evaluating them. Its wealth of exercises give students the practice they need, and its rich array of topics give instructors the flexibility they desire to cater coverage to the needs of their school’s majors curriculum. This text is part of the Walter Rudin Student Series in Advanced Mathematics.

Chapter 0. Introduction. 0.1. Why this book is 0.2. What this book is 0.3. What this book is not 0.4. Advice to the Student 0.5. Advice to the Teacher 0.6. Acknowledgements Chapter 1. Preliminaries 1.1. “And” “Or” 1.2. Sets 1.3. Functions 1.4. Injections, Surjections, Bijections 1.5. Images and Inverses 1.6. Sequences 1.7. Russell’s Paradox 1.8. Exercises Chapter 2. Relations 2.1. Definitions 2.2. Orderings 2.3. Equivalence Relations 2.4. Constructing Bijections 2.5. Modular Arithmetic 2.6. Exercises Chapter 3. Proofs 3.1. Mathematics and Proofs 3.2. Propositional Logic 3.3. Formulas 3.4. Quantifiers 3.5. Proof Strategies 3.6. Exercises. Chapter 4. Principle of Induction 4.1. Well-orderings 4.2. Principle of Induction 4.3. Polynomials 4.4. Arithmetic-Geometric Inequality 4.5. Exercises Chapter 5. Limits 5.1. Limits 5.2. Continuity 5.3. Sequences of Functions 5.4. Exercises Chapter 6. Cardinality 6.1. Cardinality 6.2. Infinite Sets 6.3. Uncountable Sets 6.4. Countable Sets 6.5. Functions and Computability 6.6. Exercises. Chapter 7. Divisibility 7.1. Fundamental Theorem of Arithmetic 7.2. The Division Algorithm 7.3. Euclidean Algorithm 7.4. Fermat’s Little Theorem 7.5. Divisibility and Polynomials 7.6. Exercises Chapter 8. The Real Numbers. 8.1. The Natural Numbers 8.2. The Integers 8.3. The Rational Numbers 8.4. The Real Numbers 8.5. The Least Upper Bound Principle 8.6. Real Sequences 8.7. Ratio Test 8.8. Real Functions 8.9. Cardinality of the Real Numbers 8.10. Exercises Chapter 9. Complex Numbers 9.1. Cubics 9.2. Complex Numbers 9.3. Tartaglia-Cardano Revisited 9.4. Fundamental Theorem of Algebra 9.5. Application to Real Polynomials 9.6. Further remarks 9.7. Exercises Appendix A. The Greek Alphabet Appendix B. Axioms of Zermelo-Fraenkel with the Axiom of Choice Appendix C. Hints to get started on early exercises. Bibliography. Index

COMPLIMENTARY COPIES Complimentary desk copies are available for course adoption only. Kindly contact your local McGraw-Hill Representative or fax the Examination Copy Request Form available on the back pages of this catalog.

Visit McGraw-Hill Education Website: www.mheducation.com

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Higher Mathematics Linear Algebra LINEAR ALGEBRA DEMYSTIFIED

by David McMahon 2006 (October 2005) / 255 pages ISBN-13: 978-0-07-146579-3 / MHID: 0-07-146579-0

A Professional Reference Title Taught at junior level math courses at every university, Linear Algebra is essential for students in almost every technical and analytic discipline. CONTENTS PREFACE Chapter 1: Systems of Linear Equations Chapter 2: Matrix Algebra Chapter 3: Determinants Chapter 4: Vectors Chapter 5: Vector Spaces Chapter 6: Inner Product Spaces Chapter 7: Linear Transformations Chapter 8: The Eigenvalue Problem Chapter 9: Special Matrices Chapter 10: Matrix Decomposition / FINAL EXAM / HINTS AND SOLUTIONS / REFERENCES / INDEX

Cross Product 4.4 Matrix Transformations II 4.5 An Application to Computer Graphics Supplementary Exercises for Chapter 4 Chapter 5 The Vector Space Rn 5.1 Subspaces and Spanning 5.2 Independence and Dimension 5.3 Orthogonality 5.4 Rank of a Matrix 5.5 Similarity and Diagonalization 5.6 An Application to Correlation and Variance 5.7 An Application to Least Squares Approximation Supplementary Exercises for Chapter 5 Chapter 6 Vector Spaces 6.1 Examples and Basic Properties 6.2 Subspaces and Spanning Sets 6.3 Linear Independence and Dimension 6.4 Finite Dimensional Spaces 6.5 An Application to Polynomials 6.6 An Application to Differential Equations Supplementary Exercises for Chapter 6 Chapter 7 Linear Transformations 7.1 Examples and Elementary Properties 7.2 Kernel and Image of a Linear Transformation 7.3 Isomorphisms and Composition 7.4 More on Linear Recurrences Chapter 8 Orthogonality 8.1 Orthogonal Complements and Projections 8.2 Orthogonal Diagonalization 8.3 Positive Definite Matrices 8.4 QR-Factorization 8.5 Computing Eigenvalues 8.6 Complex Matrices 8.7 Best Approximation and Least Squares 8.8 Finite Fields and Linear Codes 8.9 An Application to Quadratic Forms 8.10 An Application to Systems of Differential Equations Chapter 9 Change of Basis 9.1 The Matrix of a Linear Transformation 9.2 Operators and Similarity 9.3 Invariant Subspaces and Direct Sums 9.4 Block Triangular Form *9.5 Jordan Canonical Form Chapter 10 Inner Product Spaces 10.1 Inner Products and Norms 10.2 Orthogonal Sets of Vectors 10.3 Orthogonal Diagonalization 10.4 Isometries 10.5 An Application to Fourier Approximation

International Edition LINEAR ALGEBRA WITH APPLICATIONS Fifth Edition

by Keith Nicholson, University of Calgary 2006 (January 2006) / 512 pages ISBN-13: 978-0-07-092277-8 / MHID: 0-07-092277-2 ISBN-13: 978-0-07-125353-6 / MHID: 0-07-125353-X [IE]

McGraw-Hill Canada Title W. Keith Nicholson’s Linear Algebra with Applications, Fifth Canadian Edition is written for first and second year students at both the college or university level. Its real world approach challenges students step-by-step, gradually bringing them to a higher level of understanding from abstract to more general concepts. Real world applications have been added to the new edition, including: Directed graphs Google PageRank Computer graphics Correlation and Variance Finite Fields and Linear Codes In addition to the new applications, the author offers several new exercises and examples throughout each chapter. Some new examples include: motivating matrix multiplication (Chapter 2) a new way to expand a linearly independent set to a basis using an existing basis While some instructors will use the text for one semester, ending at Chapter 5 The Vector Space Rn others will continue with more abstract concepts being introduced. Chapter 5 prepares students for the transition, acting as the “bridging” chapter, allowing challenging concepts like subspaces, spanning, independence and dimension to be assimilated first in the concrete context of Rn. This “bridging” concept eases students into the introduction of vector spaces in Chapter 6.

International Edition ELEMENTARY LINEAR ALGEBRA Second Edition

by Keith Nicholson, University of Calgary 2004 / 608 pages / softcover ISBN-13: 978-0-07-091142-0 / MHID: 0-07-091142-8 ISBN-13: 978-0-07-123439-9 / MHID: 0-07-123439-X [IE]

McGraw-Hill Canada Title CONTENTS Chapter 1 Linear Equations and Matrices: Matrices. Linear Equations. Homogeneous Systems. Matrix Multiplication. Matrix Inverses. Elementary Matrices. Lu-Factorization. Application ot Markov Chains. Chapter 2 Determinants and Eigenvalues: Cofactor Expansions. Determinants and Inversees. Diagonalization and Eigenvalues. Linear Dynamical Systems. Complex Eignevalues. Linear Recurrences. Polynomial Interpolation. Systems of Differential Equations. Chapter 3 Vector Geometry: Geometric Vectors. Dot Product and Projections. Lines and Planes. Matrix Transformation of R^2. The Cross Product: Optional. Chapter 4 The Vector Space R^n. Subspaces and Spanning. Linear Independence. Dimension. Rank. Orthogonality. Projections and Approximation. Orthogonal Diagonalization. Quadratic Forms. Linear Transformations. Complex Matrices. Singular Value Decomposition. Chapter 5 Vector Spaces: Examples and Basic Properties. Independence and Dimension. Linear Transformations. Isomorphisms and Matrices. Linear Operations and Similarity. Invariant Subspaces. General Inner Products. Appendix: A.1 Basic Trigonometry. A.2 Induction. A.3 Polynomials

CONTENTS Chapter 1 Systems of Linear Equations 1.1 Solutions and Elementary Operations 1.2 Gaussian Elimination 1.3 Homogeneous Equations 1.4 An Application to Network Flow 1.5 An Application to Electrical Networks 1.6 An Application to Chemical Reactions Supplementary Exercises for Chapter 1 Chapter 2 Matrix Algebra 2.1 Matrix Addition, Scalar Multiplication, and Transposition 2.2 Matrix Multiplication 2.3 Matrix Inverses 2.4 Elementary Matrices 2.5 Matrix Transformations 2.6 LU-Factorization 2.7 An Application to Input-Output Economic Models 2.8 An Application to Markov Chains Supplementary Exercises for Chapter 2 Chapter 3 Determinants and Diagonalization 3.1 The Cofactor Expansion 3.2 Determinants and Matrix Inverses 3.3 Diagonalization and Eigenvalues 3.5 An Application to Linear Recurrences 3.6 An Application to Population Growth 3.7 Proof of the Cofactor Expansion Supplementary Exercises for Chapter 3 Chapter 4 Vector Geometry 4.1 Vectors and Lines 4.2 Projections and Planes 4.3 The

INVITATION TO PUBLISH McGraw-Hill is interested in reviewing manuscript for publication. Please contact your local McGraw-Hill office or email to [email protected]

Visit McGraw-Hill Education (Asia) Website: www.mcgraw-hill.com.sg

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Higher Mathematics Combinatorics

SCHAUM’S EASY OUTLINES: LINEAR ALGEBRA by Seymour Lipschutz, Temple University - Philadelphia Marc Lipson, University of Georgia 2003 ISBN-13: 978-0-07-139880-0 / MHID: 0-07-139880-5

A Schaum Publication What could be better than the bestselling Schaum’s Outline series? For students looking for a quick nuts-and-bolts overview, it would have to be Schaum’s Easy Outline series. Every book in this series is a pared-down, simplified, and tightly focused version of its predecessor. With an emphasis on clarity and brevity, each new title features a streamlined and updated format and the absolute essence of the subject, presented in a concise and readily understandable form. Graphic elements such as sidebars, reader-alert icons, and boxed highlights stress selected points from the text, illuminate keys to learning, and give students quick pointers to the essentials.

International Edition SCHAUM’S OUTLINE OF LINEAR ALGEBRA Third Edition

by Seymour Lipschutz, Temple University, and Marc Lipson, University of Georgia 2001 / 480 pages ISBN-13: 978-0-07-136200-9 / MHID: 0-07-136200-2 ISBN-13: 978-0-07-118947-7 / MHID: 0-07-118947-5 [IE]

A Schaum Publication (International Edition is not for sale in Japan) This third edition of the successful outline in linear algebra - which sold more than 400,000 copies in its past two editions—has been thoroughly updated to increase its applicability to the fields in which linear algebra is now essential: computer science, engineering, mathematics, physics, and quantitative analysis. Revised coverage includes new problems relevant to computer science and a revised chapter on linear equations. More than 100,000 students enroll in beginning and advanced Linear Algebra courses each year. This outline is appropriate for both first- and second-level linear algebra courses.

SCHAUM’S 3,000 SOLVED PROBLEMS IN LINEAR ALGEBRA by Seymour Lipschultz, Temple University 1989 / 496 pages ISBN-13: 978-0-07-038023-3 / MHID: 0-07-038023-6

A Schaum Publication CONTENTS Vectors in R and C. Matrix Algebra. Systems of Linear Equations. Square Matrices. Determinants. Algebraic Structures. Vector Spaces and Subspaces. Linear Dependence, Basis, Dimension. Mappings. Linear Mappings. Spaces of Linear Mappings. Matrices and Linear Mappings. Change of Basis, Similarity. Inner Product Spaces, Orthogonality. Polynomials Over A Field. Eigenvalues and Eigenvectors. Diagonalization. Canonical Forms. Linear Functional and the Dual Space. Bilinear, Quadratic, and Hermitian Forms. Linear Operators on Inner Product Spaces. Applications to Geometry and Calculus.

International Edition

NEW

INTRODUCTION TO ENUMERATIVE COMBINATORICS

By Miklos Bona, University Of Florida @ Gainesville 2007 (September 2005) / 533 pages / Hardcover ISBN-13: 978-0-07-312561-9 / MHID: 0-07-312561-X ISBN-13: 978-0-07-125415-1 / MHID: 0-07-125415-3 [IE]

Written by one of the leading authors and researchers in the field, this comprehensive modern text is written for one- or two-semester undergraduate courses in General Combinatorics or Enumerative Combinatorics taken by math and computer science majors. Introduction to Enumerative Combinatorics features a strongly-developed focus on enumeration, a vitally important area in introductory combinatorics crucial for further study in the field. Miklós Bóna’s text is one of the very first enumerative combinatorics books written specifically for the needs of an undergraduate audience, with a lively and engaging style that is ideal for presenting the material to sophomores or juniors. This text is part of the Walter Rudin Student Series in Advanced Mathematics. FEATURES * The Author--Miklós Bóna is one of the country’s leading combinatorics researchers, and the author of two successful combinatorics textbooks. McGraw-Hill is privileged to bring his clear, insightful style and attention to mathematical detail to the Walter Rudin Student Series in Advanced Mathematics. * Enumeration Focus – Bóna’s more detailed focus on enumeration develops this key area of combinatorics more deeply and more effectively to undergrads than general combinatorics texts that sacrifice depth of enumeration coverage in order to cast a wider net to more peripheral topics. Bóna serves the growing need in undergraduate courses for more dedicated coverage of enumerative combinatorics, while still offering a comprehensive-enough selection of additional topics to suit most general combinatorics courses. ...from the Foreword, by Richard Stanley (Massachusetts Institute of Technology): “Miklós Bóna has done a masterful job of bringing an overview of all of enumerative combinatorics within reach of undergraduates.” * Writing Style – The clarity, detail, rigor, and mathematical accuracy in Bóna’s presentation are balanced with an informal writing style that proves ideal for explaining difficult material to weaker students while allowing stronger students to move ahead without getting bogged down. Bóna explains and motivates key concepts with accurate and understandable definitions, and presents them in concrete examples before placing them in a larger mathematical framework. * Flexibile Topic Coverage – Introduction to Enumerative Combinatorics gives instructors flexibility in choosing topics to suit the nature of their course. The text can be used in a one-semester course by using the most general topics, in a two-semester course by utilizing the full text and optional extra topics in the exercise sets, or in a second-semester Enumerative Combinatorics course that follows a first-semester General Combinatorics course by using the material in the last 6 chapters. * Flexible Level--Bóna maintains a clear, straightforward progression within individual topics from basic to advanced material, allowing instructors the freedom to skip over parts that would prove too difficult for some students. Exercises cover a wide range of difficulty levels, and are numerous enough to allow instructors choices to suit every ability level.

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Higher Mathematics Logic

CONTENTS Foreword. Preface. Acknowledgments. I How: Methods. 1 Basic Methods. 1.1 When We Add and When We Subtract 1.1.1 When We Add 1.1.2 When We Subtract 1.2 When We Multiply 1.2.1 The Product Principle 1.2.2 Using Several Counting Principles 1.2.3 When Repetitions Are Not Allowed 1.3 When We Divide 1.3.1 The Division Principle 1.3.2 Subsets 1.4 Applications of Basic Counting Principles 1.4.1 Bijective Proofs 1.4.2 Properties of Binomial Coefficients 1.4.3 Permutations With Repetition.1.5 The Pigeonhole Principle 1.6 Notes 1.7 Chapter Review 1.8 Exercises 1.9 Solutions to Exercises 1.10 Supplementary Exercises. 2 Direct Applications of Basic Methods 2.1 Multisets and Compositions 2.1.1 Weak Compositions 2.1.2 Compositions 2.2 Set Partitions 2.2.1 Stirling Numbers of the Second Kind 2.2.2 Recurrence Relations for Stirling Numbers of the Second Kind 2.2.3 When the Number of Blocks Is Not Fixed 2.3 Partitions of Integers 2.3.1 Nonincreasing Finite Sequences of Integers 2.3.2 Ferrers Shapes and Their Applications 2.3.3 Excursion: Euler’s Pentagonal Number Theorem 2.4 The Inclusion-Exclusion Principle 2.4.1 Two Intersecting Sets 2.4.2 Three Intersecting Sets 2.4.3 Any Number of Intersecting Sets 2.5 The Twelvefold Way 2.6 Notes 2.7 Chapter Review 2.8 Exercises 2.9 Solutions to Exercises 2.10 Supplementary Exercises 3 Generating Functions 3.1 Power Series 3.1.1 Generalized Binomial Coefficients 3.1.2 Formal Power Series 3.2 Warming Up: Solving Recursions 3.2.1 Ordinary Generating Functions 3.2.2 Exponential Generating Functions 3.3 Products of Generating Functions 3.3.1 Ordinary Generating Functions 3.3.2 Exponential Generating Functions 3.4 Excursion: Composition of Two Generating Functions 3.4.1 Ordinary Generating Functions 3.4.2 Exponential Generating Functions 3.5 Excursion: A Different Type of Generating Function 3.6 Notes 3.7 Chapter Review 3.8 Exercises 3.9 Solutions to Exercises 3.10 Supplementary Exercises. II What: Topics. 4 Counting Permutations 4.1 Eulerian Numbers 4.2 The Cycle Structure of Permutations 4.2.1 Stirling Numbers of the First Kind 4.2.2 Permutations of a Given Type 4.3 Cycle Structure and Exponential Generating Functions 4.4 Inversions 4.4.1 Counting Permutations with Respect to Inversions 4.5 Notes 4.6 Chapter Review 4.7 Exercises 4.8 Solutions to Exercises 4.9 Supplementary Exercises 5 Counting Graphs 5.1 Counting Trees and Forests 5.1.1 Counting Trees 5.2 The Notion of Graph Isomorphisms 5.3 Counting Trees on Labeled Vertices 5.3.1 Counting Forests 5.4 Graphs and Functions 5.4.1 Acyclic Functions 5.4.2 Parking Functions 5.5 When the Vertices Are Not Freely Labeled 5.5.1 Rooted Plane Trees 5.5.2 Binary Plane Trees 5.6 Excursion: Graphs on Colored Vertices 5.6.1 Chromatic Polynomials 5.6.2 Counting k-colored Graphs 5.7 Graphs and Generating Functions 5.7.1 Generating Functions of Trees 5.7.2 Counting Connected Graphs 5.7.3 Counting Eulerian Graphs 5.8 Notes 5.9 Chapter Review 5.10 Exercises 5.11 Solutions to Exercises 5.12 Supplementary Exercises 6 Extremal Combinatorics 6.1 Extremal Graph Theory 6.1.1 Bipartite Graphs 6.1.2 Tur´an’s Theorem 6.1.3 Graphs Excluding Cycles 6.1.4 Graphs Excluding Complete Bipartite Graphs 6.2 Hypergraphs 6.2.1 Hypergraphs with Pairwise Intersecting Edges 6.2.2 Hypergraphs with Pairwise Incomparable Edges 6.3 Something Is More Than Nothing: Existence Proofs 6.3.1 Property B 6.3.2 Excluding Monochromatic Arithmetic Progressions 6.3.3 Codes Over Finite Alphabets 6.4 Notes 6.5 Chapter Review 6.6 Exercises 6.7 Solutions to Exercises 6.8 Supplementary Exercises. III What Else: Special Topics. 7 Symmetric Structures 7.1 Hypergraphs with Symmetries 7.2 Finite Projective Planes 7.2.1 Excursion: Finite Projective Planes of Prime Power Order 7.3 Error-Correcting Codes 7.3.1 Words Far Apart 7.3.2 Codes from Hypergraphs 7.3.3 Perfect Codes 7.4 Counting Symmetric Structures 7.5 Notes 7.6 Chapter Review 7.7 Exercises 7.8 Solutions to Exercises 7.9 Supplementary Exercises 8 Sequences in Combinatorics 8.1 Unimodality 8.2 Log-Concavity 8.2.1 Log-Concavity Implies Unimodality 8.2.2 The Product Property 8.2.3 Injective Proofs 8.3 The Real Zeros Property 8.4 Notes 8.5 Chapter Review 8.6 Exercises 8.7 Solutions to Exercises 8.8 Supplementary Exercises 9 Counting Magic Squares and Magic Cubes 9.1 An Interesting Distribution Problem 9.2 Magic Squares of Fixed Size 9.2.1 The Case of n = 3 9.2.2 The Function Hn(r) for Fixed n 9.3 Magic Squares of Fixed Line Sum 9.4 Why Magic Cubes Are Different 9.5 Notes 9.6 Chapter Review 9.7 Exercises 9.8 Supplementary Exercises. A The Method of Mathematical Induction. A.1 Weak Induction A.2 Strong Induction References. Index. List of Frequently Used Notation

SCHAUM’S EASY OUTLINE OF LOGIC

by John Nolt, University of Tennessee, Dennis Rohatyn, University of San Diego and Achille Varzi, Columbia University-New York 2006 (September 2005) / 160pages ISBN-13: 978-0-07-145535-0 / MHID: 0-07-145535-3

A Schaum Publication Pared-down, simplified, and tightly focused, Schaum’s Easy Outline of Logic is perfect for anyone turned off by dense text. Cartoons, sidebars, icons, and other graphic pointers get the material across fast, and concise text focuses on the essence of logic. This is the ideal book for last-minute test preparation.

Advanced Engineering Mathematics International Edition SCHAUM’S OUTLINE OF ADVANCED MATHEMATICS FOR ENGINEERS AND SCIENTISTS, SI METRIC by Murray R Spiegel, Rensselaer Polytechnic Institute 1971 / 416 pages ISBN-13: 978-0-07-060216-8 / MHID: 0-07-060216-6 (Non SI Metric) ISBN-13: 978-0-07-099064-7 / MHID: 0-07-099064-6 [IE, SI Metric]

A Schaum Publication (International Edition is not for sale in Japan.)

Dynamical System International Edition SCHAUM’S OUTLINE OF VECTOR ANALYSIS by Murray R Spiegel, deceased 1968 / 240 pages ISBN-13: 978-0-07-060228-1 / MHID: 0-07-060228-X ISBN-13: 978-0-07-099009-8 / MHID: 0-07-099009-3 [IE]

A Schaum Publication (International Edition is not for sale in Japan.)

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Higher Mathematics Graph Theory International Edition INTRODUCTION TO GRAPH THEORY

by Gary Chartrand, Western Michigan University—Kalamazoo and Ping Zhang, Western Michigan University—Kalamazoo 2005 (May 2004) / 464 pages ISBN-13: 978-0-07-320416-1 / MHID: 0-07-320416-1 ISBN-13: 978-0-07-123822-9 / MHID: 0-07-123822-0 [IE]

Written by one of the leading authors in the field, this text provides a student-friendly approach to graph theory for undergraduates. Much care has been given to present the material at the most effective level for students taking a first course in graph theory. Gary Chartrand and Ping Zhang’s lively and engaging style, historical emphasis, unique examples and clearly-written proof techniques make it a sound yet accessible text that stimulates interest in an evolving subject and exploration in its many applications. This text is part of the Walter Rudin Student Series in Advanced Mathematics. CONTENTS 1. Introduction: Graphs and Graph Models. Connected Graphs. Common Classes of Graphs. 2. Degrees: The Degree of a Vertex. Regular Graphs. Degree Sequences. Excursion: Graphs and Matrices. Exploration: Irregular Graphs. 3. Isomorphic Graphs: The Definition of Isomorphisms. Isomorphism as a Relation. Excursion: Recognition, Reconstruction, Solvability. Excursion: Graphs and Groups. 4. Trees: Bridges. Trees. The Minimum Spanning Tree Problem. Excursion: The Number of Spanning Trees. Exploration: Comparing Trees. 5. Connectivity: Cut-Vertices. Blocks. Connectivity. Menger’s Theorem. Exploration: Geodetic Sets. 6. Traversability: Eulerian Graphs. Hamiltonian Graphs. Exploration: Hamiltonian Walks and Numbers. Excursion: The Early Books of Graph Theory. 7. Digraphs: Strong Digraphs. Tournaments. Excursion: How to Make Decisions. Exploration: Wine Bottle Problems. 8. Matchings and Factorization: Matchings. Factorizations. Decompositions and Graceful Labelings. Excursion: Instant Insanity. Excursion: The Petersen Graph. Exploration: -Labeling of Graphs. 9. Planarity: Planar Graphs. Embedding Graphs on Surfaces. Excursion: Graphs Minors. Exploration: Embedding Graphs in Graphs. 10. Coloring Graphs: The Four Color Problem. Vertex Coloring. Edge Coloring. Excursion: The Heawood Map-Coloring Theorem. Exploration: Local Coloring. 11. Ramsey Numbers: The Ramsey Number of Graphs. Turan’s Theorem. Exploration: Rainbow Ramsey Numbers. Excursion: Erd?umbers. 12. Distance: The Center of a Graph. Distant Vertices. Excursion: Locating Numbers. Excursion: Detour Distance and Directed Distance. Exploration: The Channel Assignment Problem. Exploration: Distance Between Graphs. 13. Domination: The Domination Number of a Graph. Exploration: Stratification. Exploration: Lights Out. Excursion: And Still It Grows More Colorful. Appendix 1. Sets and Logic. Appendix 2. Equivalence Relations and Functions. Appendix 3. Methods of Proof. Answers and Hints to Odd-Numbered Exercises. References. Index of Symbols. Index of Mathematical Terms

SCHAUM’S OUTLINE OF COMBINATORICS by V K Balakrishnan, University of Maine 1995 / 320 pages ISBN-13: 978-0-07-003575-1 / MHID: 0-07-003575-X

A Schaum Publication Combinatorial and graph-theoretic principles are used in many areas of pure and applied mathematics and also in such fields as electric circuit theory (graph theory, in fact, grew out of Kirchoff's Laws) and quantum physics. Finite element methods, now important in civil engineering, are in part graph-theoretic. Dr. Balakrishnan's book will treat, via its compendium of solved problems, some of the major (programmable) algorithms of graph theory, and, in a separable chapter, will deal with applications of the very powerful Polya Counting Theorem.

International Edition APPLIED AND ALGORITHMIC GRAPH THEORY

by Gary Chartrand, Western Michigan University, and Ortrud Oellermann, University of Natal, South Africa 1993 / 432 pages ISBN-13: 978-0-07-557101-8 / MHID: 0-07-557101-3 (Out-of-Print) ISBN-13: 978-0-07-112575-8 / MHID: 0-07-112575-2 [IE]

CONTENTS 1 An Introduction to Graphs/2 An Introduction to Algorithms/3 Trees/4 Paths and Distance and Graphs/5 Networks/6 Matchings and Factorizations/7 Eulerian Graphs/8 Hamiltonian Graphs/9 Planar Graphs/10 Coloring Graphs/11 Digigraphs/12 Extremal Graph Theory

Introductory Analysis International Edition INTRODUCTION TO MATHEMATICAL ANALYSIS

by William Parzynski, Philip Zipse both of Montclair State College 1982 / 352 pages ISBN-13: 978-0-07-048845-8 / MHID: 0-07-048845-2 (Out-of-Print) ISBN-13: 978-0-07-066467-8 / MHID: 0-07-066467-6 [IE]

International Edition SCHAUM’S OUTLINE OF GRAPH THEORY: INCLUDING HUNDREDS OF SOLVED PROBLEMS by V K Balakrishnan, University of Maine 1997 / 288 pages ISBN-13: 978-0-07-005489-9 / MHID: 0-07-005489-4

PRINCIPLES OF MATHEMATICAL ANALYSIS Third Edition

by Walter Rudin, University of Wisconsin-Madison 1976 / 325 pages ISBN-13: 978-0-07-054235-8 / MHID: 0-07-054235-X ISBN-13: 978-0-07-085613-4 / MHID: 0-07-085613-3 [IE]

A Schaum Publication

CONTENTS

CONTENTS

CHAPTER 1: The Real Numbers: Section 1.1 Sets. Section 1.2 Functions. Section 1.3 Algebraic and order properties. Section 1.4 The positive integers. Section 1.5 The least upper bound axiom. CHAPTER 2: Sequences: Section 2.1 Sequences and limits. Section 2.2 Limit theorems. Section 2.3 Monotonic sequences. Section 2.4 Sequences defined inductively. Section 2.5 Sequences, Cauchy sequences. Section 2.6 Infinite limits. CHAPTER 3: Functions and Continuity: Section 3.1 Limit of a function. Section 3.2 Limit theorems. Section 3.3 Other

Graphs and Digraphs. Connectivity. Eulerian and Hamiltonian Graphs. Optimization Involving Trees. Shortest Path Problems. Flow and Connectivity. Planarity and Duality. Graph Colorings. Additional Topics. List of Technical Terms and Symbols Used.

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Higher Mathematics limits. Section 3.4 Continuity. Section 3.5 Intermediate values, extreme values. Section 3.6 Uniform continuity. CHAPTER 4: The Derivative: Section 4.1 Definition of the derivative. Section 4.2 Rules for differentiation. Section 4.3 The Mean Value Theorem. Section 4.4 Inverse functions. CHAPTER 5: The Integral: Section 5.1 The definition of the integral. Section 5.2 Properties of the integral. Section 5.3 Existence theory. Section 5.4 The Fundamental Theorem of Calculus. Section 5.5 Improper integrals. CHAPTER 6: Infinite Series: Section 6.1 Basic theory. Section 6.2 Absolute convergence. Section 6.3 Power series. Section 6.4 Taylor series. CHAPTER 7: Sequences and Series of Functions: Section 7.1 Uniform convergence. Section 7.2 Consequences of uniform convergence. Section 7.3 Two examples. Solutions and Hints for Selected Problems. Index

promises in depth of coverage to cram in more topics, Burton weaves topics together seamlessly with his engaging narrative. The text flows smoothly from topic to topic, providing a complete and accurate picture of the richness of math history without getting bogged down in detail or skipping over important developments. More than a robust historical resource, Burton’s The History of Mathematics: An Introduction is also a good read. * Insightful Exercises--Students work through assorted problems of varying difficulty from a particular historical period and solve them by applying the procedures of the day to achieve an intuitive understanding of how these concepts were discovered and developed through time. * Flexible Organization--The text offers enough material to suit a twosemester course, but is flexible enough in its organization to be used in the more common one-semester course.

CONTENTS

History Of Mathematics International Edition

NEW

THE HISTORY OF MATHEMATICS AN INTRODUCTION Sixth Edition

By David M. Burton, University Of New Hampshire 2007 (November 2005) / 752 pages / Hardcover ISBN-13: 978-0-07-305189-5 / MHID: 0-07-305189-6 ISBN-13: 978-0-07-125389-5 / MHID: 0-07-125389-0 [IE]

The History of Mathematics: An Introduction, Sixth Edition, is written for the one- or two-semester math history course taken by juniors or seniors, and covers the history behind the topics typically covered in an undergraduate math curriculum or in elementary schools or high schools. Elegantly written in David Burton’s imitable prose, this classic text provides rich historical context to the mathematics that undergrad math and math education majors encounter every day. Burton illuminates the people, stories, and social context behind mathematics’ greatest historical advances while maintaining appropriate focus on the mathematical concepts themselves. Its wealth of information, mathematical and historical accuracy, and renowned presentation make The History of Mathematics: An Introduction, Sixth Edition a valuable resource that teachers and students will want as part of a permanent library. NEW TO THIS EDITION * Contemporary Coverage--The Sixth Edition features new expanded coverage of Chinese and Islamic histories, reflecting a more global outlook in current math history courses on the cultures that shaped modern mathematics. * An Evolving Classic--The Sixth Edition delves further into early 20th century American history to include the achievements of George Birkhoff and Norbert Wiener, expands coverage of Henri Poincaré’s career and the role of number theorists P.G. Lejeune Dirichlet and Carl Gustav Jacobi, and pays increased attention to several individuals touched upon too lightly in previous editions. These changes, implemented with the aid of user and market feedback, make Burton the classic text that changes with the times. * Revamped Table of Contents--The Sixth Edition features a broadened Table of Contents that more effectively conveys the material in each chapter, making it easier to pinpoint coverage of a particular period, topic, or individual.

FEATURES * Renowned Writing Style--Whereas other texts are often too technical to appeal to anyone but the most serious mathematician, or are too encyclopedic to weave its topics together, or make too many com-

Preface. 1 Early Number Systems and Symbols 1.1 Primitive Counting. A Sense of Number. Notches as Tally Marks. The Peruvian Quipus: Knots as Numbers. 1.2 Number Recording of the Egyptians and Greeks. The History of Herodotus. Hieroglyphic Representation of Numbers. Egyptian Hieratic Numeration. The Greek Alphabetic Numeral System. 1.3 Number Recording of the Babylonians. Babylonian Cuneiform Script. Deciphering Cuneiform: Grotefend and Rawlinson. The Babylonian Positional Number System. Writing in Ancient China. 2 Mathematics in Early Civilizations 2.1 The Rhind Papyrus. Egyptian Mathematical Papyri. A Key To Deciphering: The Rosetta Stone 2.2 Egyptian Arithmetic. Early Egyptian Multiplication. The Unit Fraction Table. Representing Rational Numbers 2.3 Four Problems from the Rhind Papyrus. The Method of False Position. A Curious Problem. Egyptian Mathematics as Applied Arithmetic. 2.4 Egyptian Geometry. Approximating the Area of a Circle. The Volume of a Truncated Pyramid. Speculations About the Great Pyramid 2.5 Babylonian Mathematics. A Tablet of Reciprocals. The Babylonian Treatment of Quadratic Equations. Two Characteristic Babylonian Problems. 2.6 Plimpton. A Tablet Concerning Number Triples. Babylonian Use of the Pythagorean Theorem. The Cairo Mathematical Papyrus. 3 The Beginnings of Greek Mathematics 3.1 The Geometric Discoveries of Thales. Greece and the Aegean Area. The Dawn of Demonstrative Geometry: Thales of Miletos. Measurements Using Geometry. 3.2 Pythagorean Mathematics. Pythagoras and His Followers. Nichomachus’ Introductio Arithmeticae. The Theory of Figurative Numbers. Zeno’s Paradox 3.3 The Pythagorean Problem. Geometric Proofs of the Pythagorean Theorem. Early Solutions of the Pythagorean Equation. The Crisis of Incommensurable Quantities. Theon’s Side and Diagonal Numbers Eudoxus of Cnidos. 3.4 Three Construction Problems of Antiquity. Hippocrates and the Quadrature of the Circle. The Duplication of the Cube. The Trisection of an Angle. 3.5 The Quadratrix of Hippias. Rise of the Sophists. Hippias of Elis. The Grove of Academia: Plato’s Academy. 4 The Alexandrian School: Euclid. 4.1 Euclid and the Elements. A Center of Learning: The Museum. Euclid’s Life and Writings. 4.2 Euclidean Geometry. Euclid’s Foundation for Geometry. Book I of the Elements. Euclid’s Proof of the Pythagorean Theorem. Book II on Geometric Algebra. Construction of the Regular Pentagon. 4.3 Euclid’s Number Theory. Euclidean Divisibility Properties. The Algorithm of Euclid. The Fundamental Theorem of Arithmetic. An Infinity of Primes. 4.4 Eratosthenes, the Wise Man of Alexandria. The Sieve of Eratosthenes. Measurement of the Earth. The Almagest of Claudius Ptolemy. Ptolemy’s Geographical Dictionary. 4.5 Archimedes. The Ancient World’s Genius. Estimating the Value of. The Sand-Reckoner Quadrature of a Parabolic Segment. Apollonius of Perga: the Conics. 5 The Twilight of Greek Mathematics: Diophantus. 5.1 The Decline of Alexandrian Mathematics. The Waning of the Golden Age. The Spread of Christianity. Constantinople, A Refuge for Greek Learning. 5.2 The Arithmetica. Diophantus’s Number Theory. Problems from the Arithmetica. 5.3 Diophantine Equations in Greece, India and China. The Cattle Problem of Archimedes. Early Mathematics in India. The Chinese Hundred Fowls Problem. 5.4 The Later Commentators. The Mathematical Collection of Pappus. Hypatia, the First Woman Mathematician. Roman Mathematics: Boethius and Cassiodorus. 5.5 Mathematics in the Near and Far East. The Algebra of al-Khowârizmî. Abû Kamil and Thâbit ibn Qurra. Omar Khayyam The Astronomers al-Tusi and al-Karashi. The Ancient Chinese Nine Chapters. Later Chinese Mathematical Works. 6 The First Awakening: Fibonacci. 6.1 The Decline and Revival of Learning. The Carolingian Pre-Renaissance. Transmission of Arabic Learning to the West. The Pioneer Translators: Gerard and Adelard. 6.2 The Liber Abaci and Liber Quadratorum. The Hindu-Arabic Numerals. Libonacci’s Liver Quadratorum.

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Higher Mathematics The Works of Jordanus de Nemore. 6.3 The Fibonacci Sequence. The Liber Abaci’s Rabbit Problem. Some Properties of Fibonacci Numbers. 6.4 Fibonacci and the Pythagorean Problem. Pythagorean Number Triples. Fibonacci’s Tournament Problem. 7 The Renaissance of Mathematics: Cardan and Tartaglia. 7.1 Europe in the Fourteenth and Fifteenth Centuries. The Italian Renaissance. Artificial Writing: The Invention of Printing. Founding of the Great Universities A Thirst for Classical Learning. 7.2 The Battle of the Scholars. Restoring the Algebraic Tradition: Robert Recorde. The Italian Algebraists: Pacioli, del Ferro and Tartaglia. Cardan, A Scoundrel Mathematician 7.3 Cardan’s Ars Magna. Cardan’s Solution of the Cubic Equation. Bombelli and Imaginary Roots of the Cubic. 7.4 Ferrari’s Solution of the Quartic Equation. The Resolvant Cubic. The Story of the Quintic Equation: Ruffini, Abel and Galois. 8 The Age of Descartes and Newton. 8.1 The Dawn of Modern Mathematics. The 17th Century Spread of Knowledge. Galileo’s Telescopic Observations. The Beginning of Modern Notation: Francois Vièta. The Decimal Fractions of Simon Steven. Napier’s Invention of Logarithms. The Astronomical Discoveries of Brahe and Kepler. 8.2 Descartes: The Discours de la Méthod. The Writings of Descartes. Inventing Cartesian Geometry. The Algebraic Aspect of La Géometrie. Descartes’ Principia Philosophia. Perspective Geometry: Desargues and Poncelet. 8.3 Newton: The Principia Mathematica. The Textbooks of Oughtred and Harriot. Wallis’ Arithmetica Infinitorum. The Lucasian Professorship: Barrow and Newton. Newton’s Golden Years. The Laws of Motion. Later Years: Appointment to the Mint. 8.4 Gottfried Leibniz: The Calculus Controversy. The Early Work of Leibniz. Leibniz’s Creation of the Calculus. Newton’s Fluxional Calculus. The Dispute over Priority. Maria Agnesi and Emilie du Châtelet. 9 The Development of Probability Theory: Pascal, Bernoulli, and Laplace. 9.1 The Origins of Probability Theory. Graunt’s Bills of Mortality. James of Chance: Dice and Cards. The Precocity of the Young Pascal. Pascal and the Cycloid. De Méré’s Problem of Points. 9.2 Pascal’s Arithmetic Triangle. The Traité du Triangle Arithmétique. Mathematical Induction. Francesco Maurolico’s Use of Induction. 9.3 The Bernoullis and Laplace. Christiaan Huygens’s Pamphlet on Probability. The Bernoulli Brothers: John and James. De Moivre’s Doctrine of Chances The Mathematics of Celestial Phenomena: Laplace. Mary Fairfax Somerville. Laplace’s Research on Probability Theory. Daniel Bernoulli, Poisson and Chebyshev. 10 The Revival of Number Theory: Fermat, Euler, and Gauss. 10.1 Martin Mersenne and the Search for Perfect Numbers. Scientific Societies Marin Mersenne’s Mathematical Gathering. Numbers, Perfect and Not So Perfect. 10.2 From Fermat to Euler. Fermat’s Arithmetica. The Famous Last Theorem of Fermat. The Eighteenth Century Enlightenment Maclaurin’s Treatise on Fluxions. Euler’s Life and Contributions. 10.3 The Prince of Mathematicians: Carl Friedrich Gauss. The Period of the French Revolution: Lagrange and Monge. Gauss’s Disquisitiones Arithmeticae. The Legacy of Gauss: Congruence Theory. Dirichlet and Jacobi. 11 Nineteenth-Century Contributions: Lobachevsky to Hilbert. 11.1 Attempts to Prove the Parallel Postulate. The Efforts of Proclus, Playfair and Wallis. Saccheri Quadrilaterals. The Accomplishments of Legendre. Legendre’s Eléments de géometrie. 11.2 The Founders of Non-Euclidean Geometry. Gauss’s Attempt at a New Geometry. The Struggle of John Bolyai. Creation of Non-Euclidean Geometry: Lobachevsky. Models of the New Geometry: Riemann, Beltrami and Klein. Grace Chisholm Young 11.3 The Age of Rigor. D’Alembert and Cauchy on Limits. Fourier’s Series. The Father of Modern Analysis, Weierstrass. Sonya Kovalevsky. The Axiomatic Movement: Pasch and Hilbert 11.4 Arithmetic Generalized. Babbage and the Analytical Engine. Peacock’s Treatise on Algebra. The Representations of Complex Numbers. Hamilton’s Discovery of Quaternions. Matrix Algebra: Cayley and Sylvester. Boole’s Algebra of Logic 12 Transition to the Twenthieth Century 12.1 The Emergence of American Mathematics. Ascendency of the German Universities. American Mathematics Takes Root: 1800-1900. The Twentieth Century Consolidation 12.2 Counting the Infinite. The Last Universalist: Poincaré. Cantor’s Theory of Infinite Sets. Kronecker’s View of Set Theory. Countable and Uncountable Sets. Transcendental Numbers. The Continuum Hypothesis 12.3 The Paradoxes of Set Theory. The Early Paradoxes. Zermelo and the Axiom of Choice. The Logistic School: Frege, Peano and Russell. Hilbert’s Formalistic Approach: Brouwer’s Intuitionism. 13 Extensions and Generalizations: Hardy, Hausdorff, and Noether. 13.1 Hardy and Ramanujan. The Tripos Examination. The Rejuvenation of English Mathematics. A Unique Collaboration: Hardy and Littlewood. India’s Prodigy, Ramanujan 13.2 The Beginnings of Point-Set Topology. Frechet’s Metric Spaces. The Neighborhood Spaces of Hausdorff. Banach and Normed Linear Spaces. 13.3 Some Twentieth-Century Developments. Emmy Noether’s Theory of Rings. Von Neumann and the Computer. Women in Modern Mathematics. A Few Recent Advances. General Bibliography. Additional Reading. The Greek Alphabet Solutions to Selected Problems. Index

Numerical Analysis SCHAUM’S OUTLINE OF NUMERICAL ANALYSIS Second Edition by Francis Scheid, Boston University 1988 / 471 pages ISBN-13: 978-0-07-055221-0 / MHID: 0-07-055221-5

A Schaum Publication CONTENTS What Is Numerical Analysis? The Collocation Polynomial. Finite Differences. Factorial Polynomials. Summation. The Newton Formula. Operators and Collocation Polynomials. Unequally-Spaced Arguments. Splines. Osculating Polynomials. The Taylor Polynomial. Interpolation. Numerical Differentiation. Numerical Integration. Gaussian Integration. Singular Integrals. Sums and Series. Difference Equations. Differential Equations. Differential Problems of Higher Order. Least-Squares Polynomial Approximation. Min-Max Polynomial Approximation. Approximation By Rational Functions. Trigonometric Approximation. Nonlinear Algebra. Linear Systems. Linear Programming. Overdetermined Systems. Boundary Value Problems. Monte Carlo Methods.

International Edition ELEMENTARY NUMERICAL ANALYSIS An Algorithmic Approach, Third Edition

by Samuel D. Conte, Purdue University, Carl de Boor, University of Wisconsin-Madison 1980 / 408 pages ISBN-13: 978-0-07-012447-9 / MHID: 0-07-012447-7 (Out-of-Print) ISBN-13: 978-0-07-066228-5 / MHID: 0-07-066228-2 [IE]

Number Theory International Edition

NEW

ELEMENTARY NUMBER THEORY Sixth Edition

By David M. Burton, University Of New Hampshire 2007 (October 2005) / 528 pages / Hardcover ISBN-13: 978-0-07-305188-8 / MHID: 0-07-305188-8 ISBN-13: 978-0-07-124425-1 / MHID: 0-07-124425-5 [IE]

Elementary Number Theory, Sixth Edition, is written for the onesemester undergraduate number theory course taken by math majors, secondary education majors, and computer science students. This contemporary text provides a simple account of classical number theory, set against a historical background that shows the subject’s evolution from antiquity to recent research. Written in David Burton’s engaging style, Elementary Number Theory reveals the attraction that has drawn leading mathematicians and amateurs alike to number theory over the course of history. NEW TO THIS EDITION * Expanded Cryptography Coverage--Coverage on cryptosystems has been considerably expanded in the Sixth Edition into a new chapter, “Introduction to Cryptography,” reflecting the profound recent influence of fast computers on on number theory and its applications.

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Higher Mathematics * Recent Achievements and Ongoing Research--The Sixth Edition expands the chapter on recent developments to include new techniques for factorization of large numbers, and the resolution of such challenging conjectures as the confirmation of the Catalan Conjecture and the composite nature of Fermat number F31. The text also updates the progress of several still-unsolved problems that occupy number theorists to this day. These changes reflect the current vitality of number theory as an area of research mathematics. * Flexible Organization – The Sixth Edition has been streamlined to allow greater flexibility to instructors in choosing topics. Later chapters are designed to be independent of one another and certain sections can be omitted without affecting continuity, allowing instructors to omit certain topics in shorter quarter-term and summer courses or pick and choose the material they find most essential. These options are described in detail in the book’s Preface.

FEATURES * Historical Emphasis – Burton carefully places the topics of number theory within a human context, weaving historical and biographical remarks into the text to show how number theory was constructed pieceby-piece by individual contributors throughout history. This historical focus underscores the timeless appeal of number theory as a research subject, and encourages students who may be daunted by complex problems by highlighting the straightforward creative processes that mathematicians used to tackle them. * Exercises – The book provides more than 750 exercises, including computational exercises that test basic understanding and develop important techniques, more theoretical exercises to give practice in constructing proofs, and optional exercises that convey additional topics and ideas. Exercises cover a wide range of difficulty levels, with hints included when appropriate. * Ancillaries – McGraw-Hill offers a Student’s Solutions Manual providing worked solutions to all odd-numbered exercises in the text, and an Instructor’s Solutions Manual providing worked solutions to all exercises in the text.

CONTENTS Preface. New To This Edition. 1 Preliminaries 1.1 Mathematical Induction 1.2 The Binomial Theorem 2 Divisibility Theory in the Integers 2.1 Early Number Theory 2.1 The Division Algorithm 2.2 The Greatest Common Divisor 2.3 The Euclidean Algorithm 2.4 The Diophantine Equation ax + by = c 3 Primes and Their Distribution 3.1 The Fundamental Theorem of Arithmetic 3.2 The Sieve of Eratosthenes 3.3 The Goldbach Conjecture 4 The Theory of Congruences 4.1 Carl Friedrich Gauss 4.2 Basic Properties of Congruence 4.3 Binary and Decimal Representations of Integers 4.4 Linear Congruences and the Chinese Remainder Theorem 5 Fermat’s Theorem 5.1 Pierre de Fermat 5.2 Fermat’s Little Theorem and Pseudoprimes 5.3 Wilson’s Theorem 5.4 The Fermat-Kraitchik Factorization Method 6 Number-Theoretic Functions 6.1 The Sum and Number of Divisors 6.2 The Möbius Inversion Formula 6.3 The Greatest Integer Function 6.4 An Application to the Calendar 7 Euler’s Generalization of Fermat’s Theorem 7.1 Leonhard Euler 7.2 Euler’s Phi-Function 7.3 Euler’s Theorem 7.4 Some Properties of the Phi-Function. 8 Primitive Roots and Indices 8.1 The Order of an Integer Modulo n 8.2 Primitive Roots for Primes 8.3 Composite Numbers Having Primitive Roots 8.4 The Theory of Indices 9 The Quadratic Reciprocity Law 9.1 Euler’s Criterion 9.2 The Legendre Symbol and Its Properties 9.3 Quadratic Reciprocity 9.4 Quadratic Congruences with Composite Moduli 10 Introduction to Cryptography 10.1 From Caesar Cipher to Public Key Cryptography 10.2 The Knapsack Cryptosystem 10.3 An Application of Primitive Roots to Cryptography 11 Numbers of Special Form 11.1 Marin Mersenne 11.2 Perfect Numbers 11.3 Mersenne Primes and Amicable Numbers 11.4 Fermat Numbers 12 Certain Nonlinear Diophantine Equations 12.1 The Equation x2 + y2 = z2 12.2 Fermat’s Last Theorem 13 Representation of Integers as Sums of Squares 13.1 Joseph Louis Lagrange 13.2 Sums of Two Squares 13.3 Sums of More than Two Squares 14 Fibonacci Numbers 14.1 Fibonacci 14.2 The Fibonacci Sequence 14.3 Certain Identities Involving Fibonacci Numbers 15 Continued Fractions 15.1 Srinivasa Ramanujan 15.2 Finite Continued Fractions 15.3 Infinite Continued Fractions 15.4 Pell’s Equation 16 Some Twentieth-Century Developments. 16.1 Hardy, Dickson, and Erdös 16.2 Primality Testing and Factorization 16.3 An Application to Factoring: Remote Coin Flipping 16.4 The Prime Number Theorem and Zeta Function. Miscellaneous

Problems. Appendixes. General References. Suggested Further Reading Tables. Answers to Selected Problems. Index.

International Edition ELEMENTARY NUMBER THEORY Second Edition

by Charles Vanden Eynden, Illinois State University 2001 / 288 pages ISBN-13: 978-0-07-232571-3 / MHID: 0-07-232571-2 (Out of Print) ISBN-13: 978-0-07-118193-8 / MHID: 0-07-118193-8 [IE]

CONTENTS 0 What is Number Theory? 1 Divisibility. 2 Prime Numbers. 3 Numerical Functions. 4 The Algebra of Congruence Classes. 5 Congruences of Higher Degree. 6 The Number Theory of the Reals. 7 Diophantine Equations.

Abstract Algebra SCHAUM’S OUTLINE OF MODERN ABSTRACT ALGEBRA by Frank Ayres (deceased) 1965 / 256 pages ISBN-13: 978-0-07-002655-1 / MHID: 0-07-002655-6

A Schaum Publication CONTENTS

Sets. Relations and Operations. The Natural Numbers. The Integers. Some Properties of Integers. The Rational Numbers. The Real Numbers. The Complex Numbers. Groups. Rings. Integral Domains. Division Rings. Fields. Polynomials. Vector Spaces. Matrices. Matrix Polynomials. Linear Algebra. Boolean Algebra.

Advanced Geometry SCHAUM’S OUTLINE OF DIFFERENTIAL GEOMETRY by Martin M. Lipschutz, Hahnemann Medical College 1969 / 288 pages ISBN-13: 978-0-07-037985-5 / MHID: 0-07-037985-8

A Schaum Publication CONTENTS

Vectors. Vector Functions of Real Variable. Concept of Curve. Curvature and Torsion. Theory of Curves. Elementary Topology in Euclidean Spaces. Vector Functions of Vector Variable. Concept of Curve. First and Second Fundamental Forms. Theory of Surfaces. Tensor Analysis. Intrinsic Geometry. Appendix. Existence Theorem for Curves. Existence Theorem for Surfaces.

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Higher Mathematics Complex Analysis

International Edition REAL AND COMPLEX ANALYSIS Third Edition

International Edition COMPLEX VARIABLES AND APPLICATIONS Seventh Edition

by James W Brown, University of Michigan and Ruel V Churchill (deceased) 2004 / 480 pages ISBN-13: 978-0-07-287252-1 / MHID: 0-07-287252-7 ISBN-13: 978-0-07-123365-1 / MHID: 0-07-123365-2 [IE]

CONTENTS 1 Complex Numbers: Sums and Products. Basic Algebraic Properties. Further Properties. Moduli. Complex Conjugates. Exponential Form. Products and Quotients in Exponential Form. Roots of Complex Numbers. Examples. Regions in the Complex Plane. 2 Analytic Functions: Functions of a Complex Variable. Mappings. Mappings by the Exponential Function. Limits. Theorems on Limits. Limits Involving the Point at Infinity. Continuity. Derivatives. Differentiation Formulas. Cauchy Riemann Equations. Sufficient Conditions for Differentiability. Polar Coordinates. Analytic Functions. Examples. Harmonic Functions. Uniquely Determined Analytic Functions. Reflection Principle. 3 Elementary Functions: The Exponential Function. The Logarithmic Function. Branches and Derivatives of Logarithms. Some Identities Involving Logarithms. Complex Exponents. Trigonometric Functions. Hyperbolic Functions. Inverse Trigonometric and Hyperbolic Functions. 4 Integrals: Derivatives of Functions w(t). Definite Integrals of Functions w(t). Contours. Contour Integrals. Examples. Upper Bounds for Moduli of Contour Integrals. Antiderivatives. Examples. Cauchy Goursat Theorem. Proof of the Theorem. Simply and Multiply Connected Domains. Cauchy Integral Formula. Derivatives of Analytic Functions. Liouville’s Theorem and the Fundamental Theorem of Algebra. Maximum Modulus Principle. 5 Series: Convergence of Sequences. Convergence of Series. Taylor Series. Examples. Laurent Series. Examples. Absolute and Uniform Convergence of Power Series. Continuity of Sums of Power Series. Integration and Differentiation of Power Series. Uniqueness of Series Representations. Multiplication and Division of Power Series. 6 Residues and Poles: Residues. Cauchy’s Residue Theorem. Using a Single Residue. The Three Types of Isolated Singular Points. Residues at Poles. Examples. Zeros of Analytic Functions. Zeros and Poles. Behavior of f Near Isolated Singular Points. 7 Applications of Residues: Evaluation of Improper Integrals. Example. Improper Integrals from Fourier Analysis. Jordan’s Lemma. Indented Paths. An Indentation Around a Branch Point. Integration Along a Branch Cut. Definite Integrals Involving Sines and Cosines. Argument Principle. Rouch?squo;s Theorem. Inverse Laplace Transforms. Examples. 8 Mapping by Elementary Functions: Linear Transformations. The Transformation w = 1/z. Mappings by 1/z. Linear Fractional Transformations. An Implicit Form. Mappings of the Upper Half Plane. The Transformation w = sin z. Mappings by z2 and Branches of z1/2. Square Roots of Polynomials. Riemann Surfaces. Surfaces for Related Functions. 9 Conformal Mapping. Preservation of Angles. Scale Factors. Local Inverses. Harmonic Conjugates. Transformations of Harmonic Functions. Transformations of Boundary Conditions. 10 Applications of Conformal Mapping. Steady Temperatures. Steady Temperatures in a Half Plane. A Related Problem. Temperatures in a Quadrant. Electrostatic Potential. Potential in a Cylindrical Space. TwoDimensional Fluid Flow. The Stream Function. Flows Around a Corner and Around a Cylinder. 11 The Schwarz Christoffel Transformation: Mapping the Real Axis onto a Polygon. Schwarz Christoffel Transformation. Triangles and Rectangles. Degenerate Polygons. Fluid Flow in a Channel Through a Slit. Flow in a Channel with an Offset. Electrostatic Potential about an Edge of a Conducting Plate. 12 Integral Formulas of the Poisson Type: Poisson Integral Formula. Dirichlet Problem for a Disk. Related Boundary Value Problems. Schwarz Integral Formula. Dirichlet Problem for a Half Plane. Neumann Problems. Appendixes. Bibliography. Table of Transformations of Regions. Index

by Walter Rudin, University of Wisconsin 1987 / 483 pages ISBN-13: 978-0-07-054234-1 / MHID: 0-07-054234-1 ISBN-13: 978-0-07-100276-9 / MHID: 0-07-100276-6 [IE]

CONTENTS Preface. Prologue: The Exponential Function. Chapter 1: Abstract Integration: Set-theoretic notations and terminology. The concept of measurability. Simple functions. Elementary properties of measures. Arithmetic in [0, infinity]. Integration of positive functions. Integration of complex functions. The role played by sets of measure zero. Exercises. Chapter 2: Positive Borel Measures: Vector spaces. Topological preliminaries. The Riesz representation theorem. Regularity properties of Borel measures. Lebesgue measure. Continuity properties of measurable functions. Exercises. Chapter 3: L^p-Spaces: Convex functions and inequalities. The L^p-spaces. Approximation by continuous functions. Exercises. Chapter 4: Elementary Hilbert Space Theory: Inner products and linear functionals. Orthonormal sets. Trigonometric series. Exercises. Chapter 5: Examples of Banach Space Techniques: Banach spaces. Consequences of Baire’s theorem. Fourier series of continuous functions. Fourier coefficients of L -functions. The Hahn-Banach theorem. An abstract approach to the Poisson integral. Exercises. Chapter 6: Complex Measures: Total variation. Absolute continuity. Consequences of the Radon-Nikodym theorem. Bounded linear functionals on L^p. The Riesz representation theorem. Exercises. Chapter 7: Differentiation: Derivatives of measures. The fundamental theorem of Calculus. Differentiable transformations. Exercises. Chapter 8: Integration on Product Spaces: Measurability on cartesian products. Product measures. The Fubini theorem. Completion of product measures. Convolutions. Distribution functions. Exercises. Chapter 9: Fourier Transforms: Formal properties. The inversion theorem. The Plancherel theorem. The Banach algebra L. Exercises. Chapter 10: Elementary Properties of Holomorphic Functions: Complex differentiation. Integration over paths. The local Cauchy theorem. The power series representation. The open mapping theorem. The global Cauchy theorem. The calculus of residues. Exercises. Chapter 11: Harmonic Functions: The Cauchy-Riemann equations. The Poisson integral. The mean value property. Boundary behavior of Poisson integrals. Representation theorems. Exercises. Chapter 12: The Maximum Modulus Principle: Introduction. The Schwarz lemma. The PhragmenLindel’s Method. An interpolation theorem. A converse of the maximum modulus theorem. Exercises. Chapter 13: Approximation by Rational Functions: Preparation. Runge’s theorem. The Mittag-Leffler theorem. Simply connected regions. Exercises. Chapter 14: Conformal Mapping: Preservation of angles. Linear fractional transformations. Normal families. The Riemann mapping theorem. The class. Continuity at the boundary. Conformal mapping of an annulus. Exercises. Chapter 15: Zeros of Holomorphic Functions: Infinite Products. The Weierstrass factorization theorem. An interpolation problem. Jensen’s formula. Blaschke products. The M’zas theorem. Exercises. Chapter 16: Analytic Continuation: Regular points and singular points. Continuation along curves. The monodromy theorem. Construction of a modular function. The Picard theorem. Exercises. Chapter 17: H^p-Spaces: Subharmonic functions . The spaces H^p and N. The theorem of F. and M. Riesz. Factorization theorems. The shift operator. Conjugate functions. Exercises. Chapter 18: Elementary Theory of Banach Algebras: Introduction. The invertible elements. Ideals and homomorphisms. Applications. Exercises. Chapter 19: Holomorphic Fourier Transforms: Introduction. Two theorems of Paley and Wiener. Quasi-analytic classes. The Denjoy-Carleman theorem. Exercises. Chapter 20: Uniform Approximation by Polynomials: Introduction. Some lemmas. Mergelyan’s theorem. Exercises. Appendix: Hausdorff’s Maximality Theorem. Notes and Comments. Bibliography. List of Special Symbols. Index

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Higher Mathematics International Edition COMPLEX ANALYSIS Third Edition

by Lars Ahlfors, Harvard University 1979 / 336 pages ISBN-13: 978-0-07-000657-7 / MHID: 0-07-000657-1 ISBN-13: 978-0-07-085008-8 / MHID: 0-07-085008-9 [IE]

CONTENTS Chapter 1: Complex Numbers: 1 The Algebra of Complex Numbers. 2 The Geometric Representation of Complex Numbers. Chapter 2: Complex Functions: 1 Introduction to the Concept of Analytic Function. 2 Elementary Theory of Power Series. 3 The Exponential and Trigonometric Functions. Chapter 3: Analytic Functions as Mappings: 1 Elementary Point Set Topology. 2 Conformality. 3 Linear Transformations. 4 Elementary Conformal Mappings. Chapter 4: Complex Integration: 1 Fundamental Theorems. 2 Cauchy’s Theorem for a Rectangle. 3 Local Properties of Analytical Functions. 4 The General Form of Cauchy’s Theorem. 5 The Calculus of Residues. 6 Harmonic Functions. Chapter 5: Series and Product Developments: 1 Power Series Expansions. 2 Partial Fractions and Factorization. 3 Entire Functions. 4 The Riemann Zeta Function. 5 Normal Families. Chapter 6: Conformal Mapping, Dirichlet’s Problem: 1 The Riemann Mapping Theorem. 2 Conformal Mapping of Polygons. 3 A Closer Look at Harmonic Functions. 4 The Dirichlet Problem. 5 Canonical Mappings of Multiply Connected Regions. Chapter 7: Elliptic Functions: 1 Simply Periodic Functions. 2 Doubly Periodic Functions. 3 The Weierstrass Theory. Chapter 8: Global Analytic Functions: 1 Analytic Continuation. 2 Algebraic Functions. 3 Picard’s Theorem. 4 Linear Differential Equations. Index

International Edition SCHAUM’S OUTLINE OF COMPLEX VARIABLES

by Murray R Spiegel, formerly of Rensselaer Polytechnic Institute 1968 / 320 pages ISBN-13: 978-0-07-060230-4 / MHID: 0-07-060230-1 ISBN-13: 978-0-07-099010-4 / MHID: 0-07-099010-7 [IE, SI Metric]

A Schaum Publication (International Edition is not for sale in Japan.) CONTENTS Complex Numbers. Functions. Limits and Continuity. Complex Differentiation and the Cauchy Riemann Equations. Complex Integration and Cauchy’s Theorem. Cauchy’s Integral Formulas and Related Theorems. Infinite Series. Taylor’s and Laurent Series. The Residue Theorem: Evaluation of Integrals and Series. Conformal Mappings. Physical Applications of Conformal Mapping. Special Topics.

Topology International Edition TOPOLOGY

by Sheldon W Davis, Miami University—Oxford 2005 / 448 page ISBN-13: 978-0-07-291006-3 / MHID: 0-07-291006-2 ISBN-13: 978-0-07-124339-1 / MHID: 0-07-124339-9 [IE]

A volume in the Walter Rudin Student Series. CONTENTS 1 Sets, Functions, Notation: Cantor-Bernstein Theorem. Countable Set. 2 Metric Spaces: Topology Generated by a Metric. Complete

Metric Space. Cantor Intersection Theorem. Baire Category Theorem. 3 Continuity: Banach Fixed Point Theorem. 4 Topological Spaces: Subspace Topology. Continuous Function. Base. Sorgenfrey Line. Lindel? Theorem. 5 Basic Constructions: Products. Product Topology. 6 Separation Axioms: Hausdorff. Regular Normal. Urysohn’s Lemma. Tietze Extension Theorem. 7 Compactness: Heine-Borel Theorem. Tychonoff Theorem. Lebesgue Number. 8 Local Compactness: OnePoint Compactification. 9 Connectivity: Intermediate Value Theorem. Connected Subspaces. Products of Connected Spaces. Components. 10 Other Types of Connectivity: Pathwise Connected. Locally Pathwise Connected. Locally Connected. 11 Continua: Irreducible: Cut Point. Moore’s Characterization of [0, 1]. 12 Homotopy: Contractible Space. Fundamental Group.

SCHAUM’S OUTLINE OF GENERAL TOPOLOGY by Seymour Lipschutz, Temple University 1986 / 256 pages ISBN-13: 978-0-07-037988-6 / MHID: 0-07-037988-2

A Schaum Publication CONTENTS Sets and Relations. Functions. Cardinality, Order. Topology of the Line and Plane. Topological Spaces. Definitions. Bases and Subbases. Continuity and Topological Equivalence. Metric and Normed Spaces. Countability. Separation Axioms. Compactness. Product Spaces. Connectedness. Complete Metric Spaces. Function Spaces. Appendix. Properties of the Real Numbers.

Mathematical References GREAT JOBS FOR MATH MAJORS Second Edition

by Stephen Lambert and Ruth DeCotis 2006 (September 2005) / 208 pages ISBN-13: 978-0-07-144859-8 / MHID: 0-07-144859-4

A Professional Reference Answers the question “What can I do with a major in math?” It isn’t always obvious what a math major can offer to the workplace. But it provides you with valuable skills and training that can be applied to a wide range of careers. Great Jobs for Math Majors helps you explore these possibilities.

GETTING STARTED WITH THE T1-84 PLUS GRAPHING CALCULATOR by Wee Leng Ng 2006 (October 2005) / 84 pages ISBN-13: 978-0-07-125247-8 / MHID: 0-07-125247-9

An Asian Publication With the recent introduction of the TI-84 Plus graphing calculator into the A-level Mathematics curriculum, students can now reduce the time spent on tedious computations. Getting Started with the TI-84 Plus Graphing Calculator is an invaluable guide to the basic skills required to utilize the graphing calculator, and to help students get the most out of their new tool. Filled with comprehensive key press instructions, screen-shots and useful tips at almost every step, students as well as teachers are bound to find this example-based book a rich reference source and a handy companion to their TI-84 Plus.

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Higher Mathematics CONTENTS How to use this book 1. Basic Calculations / The Keys of the TI84+ / Entering and Editing Mathematical Expressions / Accessing Menus / Basic Numeric Calculations / 2. Basic Features of Function Graphing / Entering and Graphing Functions / Changing the Viewing Window / 3. The Equation Solver / Solving Equations Without Parameters / Solving Equations With Parameters / 4. Advanced Graphing Features / Defining Functions in Terms of Other Functions / Entering and Graphing a Function with Parameters / Graphing a Family of Functions / Restricting the Domain of a Function / Shading Above/Below a Function / Entering and Plotting a Graph Defined Parametrically / Entering and Graphing a Polar Graph / 5. Calculus / Numerical Derivative / Numerical Integral / Turning Points / Drawing Tangent Lines / 6. Matrices / The

Matrix Menu / Operations on Matrices / 7. Complex Numbers / Selecting the Display Format/ Rectangular Complex Mode / Polar Complex Mode / Entering expressions involving Complex Numbers / Finding the Argument and Modulus / 8. Vectors / Performing Vector Operations / Finding the Magnitude of a Vector / Finding the Scalar Product / Finding the Vector Product / 9. Sequences and Series / Sequences on the Home Screen / Defining Sequences Using the Editor

MCGRAW-HILL DICTIONARY OF MATHEMATICS Second Edition by McGraw-Hill 2003 / 336 pages ISBN-13: 978-0-07-141049-6 / MHID: 0-07-141049-X

Derived from the content of the respected McGraw-Hill Dictionary of Scientific and Technical Terms Sixth Edition, each title provides thousands of definitions of words and phrases encountered in a specific discipline. All include: *Pronunciation guide for every term *Acronyms, cross-references, and abbreviations * Append-ices with conversion tables; listings of scientific, technical, and mathematical notation; tables of relevant data; and more * A convenient, quick-find format

SCHAUM’S EASY OUTLINES: MATHEMATICAL HANDBOOK OF FORMULAS AND TABLES

by Murray R Spiegel, Rensselaer Polytechnic Institute, and John Liu, Temple University 2001 / 144 pages ISBN-13: 978-0-07-136974-9 / MHID: 0-07-136974-0

MATH PROOFS DEMYSTIFIED

By Stan Gibilisco 2005 / 290 pages / Softcover ISBN-13: 978-0-07-144576-4 / MHID: 0-07-144576-5

CONTENTS Part One: The Rules of Reason. Chapter 1: The Basics of Propositional Logic. Chapter 2: How Sentences are Put Together. Chapter 3: Formalities and Techniques. Chapter 4: Vagaries of Logic. Test: Part One. Part Two: Proofs in Action. Chapter 5: Some Theoretical Geometry. Chapter 6: Sets and Numbers. Chapter 7: A Few Historic Tidbits. Test: Part Two. Final Exam. Answers to Quiz, Test and Exam Questions. Suggested Additional References. Index

A Schaum Publication Concise, easy-to-use versions of the biggest and best-selling classics in the Scaum's Outline series. Format will be streamlined and updated for a contemporary look and feel. Each book will extract the absolute essence of the subject, presenting it in concise and readily understand-able form, emphasizing clarity and brevity. Graphic elements like sidebars, reader-alert icons, and boxed highlights will feature selected points from the text, highlighting keys to learning and giving students quick pointers to the essentials.

International Edition PRE-CALCULUS DEMYSTIFIED

By Rhonda Huettenmueller 2005 / 468 pages / Softcover ISBN-13: 978-0-07-143927-5 / MHID: 0-07-143927-7

CONTENTS Preface Chapter 1: The Slope and Equation of a Line Chapter 2: Introduction to Functions Chapter 3: Functions and Their Graphs Chapter 4: Combinations of Functions and Inverse Functions Chapter 5: Translations and Special Functions Chapter 6: Quadratic Functions Chapter 7: Polynomial Functions Chapter 8: Rational Functions Chapter 9: Exponents and Logarithms Chapter 10: Systems of Equations and Inequalities Chapter 11: Matrices Chapter 12: Conic Sections Chapter 13: Trigonometry Chapter 14: Sequences and Series Appendix. Final Exam

DIFFERENTIAL EQUATIONS DEMYSTIFIED

SCHAUM’S OUTLINE OF MATHEMATICAL HANDBOOK OF FORMULAS AND TABLES Second Edition

by Murray R Spiegel, Rensselaer Polytechnic Institute, and John Liu, Temple University 1999 / 278 pages ISBN-13: 978-0-07-038203-9 / MHID: 0-07-038203-4 ISBN-13: 978-0-07-116765-9 / MHID: 0-07-116765-X [IE]

(International Edition is not for sale in Japan.) CONTENTS Section I: Elementary Constants, Products, Formulas. Section II: Geometry. Geometric Formulas. Section III: Elementary Transcendental Functions. Section IV: Calculus. Derivatives. Section V: Differential Equations and Vector Analysis. Section VI: Series. Section VII: Special Functions and Polynomials. Section VIII: Laplace and Fourier Transforms. Section IX: Elliptic and Miscellaneous Special Functions. Section X: Inequalities and Infinite Products. Section XI: Probability and Statistics. Section XII: Numerical Methods.

By Steven G Krantz, Washington University-St Louis 2005 / 323 pages / Softcover ISBN-13: 978-0-07-144025-7 / MHID: 0-07-144025-9

CONTENTS Preface. Chapter 1: What Is a Differential Equation? Chapter 2: Second-Order Equations Chapter 3: Power Series Solutions and Special Functions Chapter 4: Fourier Series: Basic Concepts Chapter 5: Partial Differential Equations and Boundary Value Problems Chapter 6: Laplace Transforms Chapter 7: Numerical Methods Chapter 8: Systems of First-Order Equations. Final Exam. Solutions to Exercises. Bibliography Index.

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