Copyright Year 2022
ISBN 9780815373315
June 27, 2022 Forthcoming by Chapman and Hall/CRC
328 Pages 22 B/W Illustrations
Linear Algebra is intended primarily as an undergraduate textbook but is written in such a way that it can be also a valuable resource for independent learning. The narrative of the book takes a matrix approach: the exposition is intertwined with matrices either as the main subject or as tools to explore the theory. Each chapter contains a description of its aims, a summary at the end of the chapter, exercises and solutions. The reader is carefully guided through the theory and techniques presented which are outlined throughout in "How to…" text boxes. Common mistakes and pitfalls are also pointed out as one goes along.
Written to be self-contained.
Ideal as a primary textbook for an undergraduate course in linear algebra.
Applications of the general theory which are of interest to disciplines outside of mathematics, such as engineering.
1. Matrices. 1.1. Real and Complex Matrices. 1.2. Matrix Calculus. 1.3. Matrix Inverses. 1.4. Elementary Matrices. 1.5. Exercises. 1.6. At a Glance. 2. Determinant. 2.1. Axiomatic Definition. 2.2. Leibniz's Formula. 2.3. Laplace's Formula. 2.4. Exercises. 2.5. At a Glance. 3. Vector Spaces. 3.1. Vector Spaces. 3.2. Linear Independence. 3.3. Bases and Dimension. 3.4. Null Space, Row Space and Column Space. 3.5. Sum and intersection of Subspaces. 3.6. Change of Basis. 3.7. Exercises. 3.8. At a Glance. 4. Eigenvalues and Eigenvectors. 4.1. Spectrum of a Matrix. 4.2. Spectral Properties. 4.3. Similarity and Diagonalisation. 4.4. Jordan Canonical Form. 4.5. Exercises. 4.6. At a Glance. 5. Linear Transformations. 5.1. Linear Transformations. 5.2. Matrix Representations. 5.3. Null Space and Image. 5.4. Isomorphisms and Rank-Nullity Theorem. 5.5. Composition and Invertibility. 5.6. Change of Basis. 5.7. Spectrum and Diagonalisation. 5.8. Exercises. 5.9. At a Glance. 6. Inner Product Spaces. 6.1. Real Inner Product Spaces. 6.2. Complex Inner Product Spaces. 6.3. Orthogonal Sets. 6.4. Orthogonal and Unitary Diagonalisation. 6.5. Singular Value decomposition. 6.6. Affine Subspaces of Rn. 6.7. Exercises. 6.8. At a Glance. 7 Special Matrices by Example. 7.1. Least Squares Solutions. 7.2. Markov Chains. 7.3. Population Dynamics. 7.4. Graphs. 7.5. Differential Equations. 7.6. Exercises. 7.7. At a Glance. 8. Appendix. 8.1. Uniqueness of Reduced Row Echelon Form. 8.2. Uniqueness of determinant. 8.3. Direct sum of Subspaces. 9. Solutions.
Copyright Year 2022
ISBN 9781032255385
May 26, 2022 Forthcoming by Chapman and Hall/CRC
174 Pages 28 B/W Illustrations
Format Hardback
In the 1960’s and 1970’s, mathematical biologists Sir Robert M. May, E.C. Pielou, and others utilized di?erence equations as models of ecological and epidemiological phenomena. Since then, with or without applications, the mathematics of di?erence equations has evolved into a ?eld unto itself. Di?erence equations with the maximum (or the minimum or the "rank-type") function were rigorously investigated from the mid 1990’s into the 2000’s, without any applications in mind. These equations often involved arguments varying from reciprocal terms with parameters in the numerators to other special functions.
Recently, the authors of Analysis of a Model for Epilepsy: Application of a Max-Type Difference Equation to Mesial Temporal Lobe Epilepsy book and their colleagues investigated the ?rst known application of a "max-type" di?erence equation. Their equation is a phenomenological model of epileptic seizures. In this book, the authors expand on that research and present a more comprehensive development of mathematical, numerical, and biological results. Additionally, they describe the first documented instance of a novel dynamical behavior that they call rippled almost periodic behavior, which can be described as an unpredictable pseudo-periodic behavior.
Suitable for researchers in mathematical neuroscience and potentially as supplementary reading for postgraduate students.
Thoroughly researched and replete with references.
1. Introduction: Epilepsy. 1.1. Brief Overview. 1.2. Mesial Temporal Lobe Epilepsy and Other Examples. 2. The Model. 2.1. Model Description. 2.2. Connection to a Simpler Model. 2.3. Connection Between the Model and Epileptic Seizures. 2.4. Open Problem: Seizure Threshold as a Function of Time. 3. Eventual Periodicity of the Model. 3.1. Bounded and Persistent Solutions. 3.2. Eventually Periodic Solutions with Periods Multiples of Six. 3.3. Eventually Periodic Solutions with Period 4. 3.4. Partially and Completely Seizure-Free States. 4. Rippled Almost Periodic Solutions. 4.1. Rippled Behaviour. 4.2. Rippled Almost Periodic Solutions. 4.3. Lyapunov Exponent. 4.4. The State of Status Epilepticus. 4.5. On Termination of Repetition. 5. Numerical Results and Biological Conclusions. 5.1. Bifurcation Diagrams. 5.2. Variability in Seizure Characteristics. 5.3. A Case of Variability in Region 1. 5.4. The Hyperexcitable State. 5.5. Impact of Individual Historical Differences. 6. Epilogue
Copyright Year 2022
ISBN 9781032281094
June 27, 2022 Forthcoming by Chapman and Hall/CRC
384 Pages 78 B/W Illustrations
Format Hardback
Through two previous editions, the third edition of this popular and intriguing text takes both an analytical/theoretical approach and a visual/intuitive approach to the local and global properties of curves and surfaces.
Requiring only multivariable calculus and linear algebra, it develops students’ geometric intuition through interactive graphics applets. Applets are presented in Maple workbook format, which readers can access using the free Maple Player.
The book explains the reasons for various definitions while the interactive applets offer motivation for definitions, allowing students to explore examples further, and give a visual explanation of complicated theorems. The ability to change parametric curves and parametrized surfaces in an applet lets students probe the concepts far beyond what static text permits. Investigative project ideas promote student research.
At users of the previous editions request, this third edition offers a broader list of exercises. More elementary exercises are added and some challenging problems are moved later in exercise sets to assure more graduated progress. The authors also add hints to motivate the more difficult exercises.
This student-friendly and readable approach offers additional examples, well-placed to assist student comprehension. In the presentation of the Gauss-Bonnet Theorem, the authors provide more intuition and steppingstones to help students grasp phenomenon behind it. Also, the concept of a homeomorphism is new to students even though it is a key theoretical component of the definition of a regular surface. Providing more examples show students how to prove certain functions are homeomorphisms.
Preface
1 Plane Curves: Local Properties
1.1 Parametrizations
1.2 Position, Velocity, and Acceleration
1.3 Curvature
1.4 Osculating Circles, Evolutes, Involutes
1.5 Natural Equations
2 Plane Curves: Global Properties
2.1 Basic Properties
2.2 Rotation Index
2.3 Isoperimetric Inequality
2.4 Curvature, Convexity, and the Four-Vertex Theorem
3 Curves in Space: Local Properties
3.1 Definitions, Examples, and Differentiation
3.2 Curvature, Torsion, and the Frenet Frame
3.3 Osculating Plane and Osculating Sphere
3.4 Natural Equations
4 Curves in Space: Global Properties
4.1 Basic Properties
4.2 Indicatrices and Total Curvature
4.3 Knots and Links
5 Regular Surfaces
5.1 Parametrized Surfaces
5.2 Tangent Planes; The Differential
5.3 Regular Surfaces
5.4 Change of Coordinates; Orientability
6 First and Second Fundamental Forms
6.1 The First Fundamental Form
6.2 Map Projections (Optional)
6.3 The Gauss Map
6.4 The Second Fundamental Form
6.5 Normal and Principal Curvatures
6.6 Gaussian and Mean Curvatures
6.7 Developable Surfaces; Minimal Surfaces
7 Fundamental Equations of Surfaces
7.1 Gauss’s Equations; Christoffel Symbols
7.2 Codazzi Equations; Theorema Egregium
7.3 Fundamental Theorem of Surface Theory
8 Gauss-Bonnet Theorem; Geodesics
8.1 Curvatures and Torsion
8.2 Gauss-Bonnet Theorem, Local Form
8.3 Gauss-Bonnet Theorem, Global Form
8.4 Geodesics
8.5 Geodesic Coordinates
8.6 Applications to Plane, Spherical, and Elliptic Geometry
8.7 Hyperbolic Geometry
9 Curves and Surfaces in n-dimensional Space
9.1 Curves in n-dimensional Euclidean Space
9.2 Surfaces in Euclidean n-Space
Appendix A: Tensor Notation
Index
Copyright Year 2023
ISBN 9781138606586
July 25, 2022 Forthcoming by Chapman and Hall/CRC
704 Pages 228 B/W Illustrations
Format Hardback
This book started as a collection of lecture notes for a course in differential equations taught by the Division of Applied Mathematics at Brown University. To some extent, it is a result of collective insights given by almost every instructor who taught such a course over the last 15 years. Therefore, the material and its presentation covered in this book were practically tested for many years.
This text is designed for a two-semester sophomore or junior level course in differential equations. It offers novel approaches in presentation and utilization of computer capabilities. This text intends to provide a solid background in differential equations for students majoring in a breadth of fields.
Differential equations are described in the context of applications. The author stresses differential equations constitute an essential part of modeling by showing their applications, including numerical algorithms and syntax of the four most popular software packages. Students learn how to formulate a mathematical model, how to solve differential equations (analytically or numerically), how to analyze them qualitatively, and how to interpret the results.
In writing this textbook, the author aims to assist instructors and students through:
?Showing a course in differential equations is essential for modeling real-life phenomena.
?Stressing the mastery of traditional solution techniques and presenting effective methods, including reliable numerical approximations.
?Providing qualitative analysis of ordinary differential equations. The reader should get an idea of how all solutions to the given problem behave, what are their validity intervals, whether there are oscillations, vertical or horizontal asymptotes, and what is their long-term behavior.
?The reader will learn various methods of solving, analysis, visualization, and approximation, exploiting the capabilities of computers.
?Introduces and employs Maple?, MathematicaR, MatLabR, and Maxima.
?This textbook facilitates the development of the student’s skills to model real-world problems.
Ordinary and partial differential equations is a classical subject that has been studied for about 300 years. The beauty and utility of differential equations and their application in mathematics, biology, chemistry, computer science, economics, engineering, geology, neuroscience, physics, the life sciences, and other fields reaffirm their inclusion in myriad curricula.
A great number of examples and exercises make this text well suited for self-study or for traditional use by a lecturer in class. Therefore, this textbook addresses the needs of two levels of audience, the beginning and the advanced.
Preface
1 Introduction
1.1 Motivation
1.2 Classification of Differential Equations
1.3 Solutions to Differential Equations
1.4 Particular and Singular Solutions
1.5 Direction Fields
1.6 Existence and Uniqueness
Review Questions for Chapter 1
2 First Order Equations
2.1 Separable Equations
2.1.1 Autonomous Equations
2.2 Equations Reducible to Separable Equations
2.2.1 Equations with Homogeneous Coefficients
2.2.2 Equations with Homogeneous Fractions
2.2.3 Equations with Linear Coefficients
2.3 Exact Differential Equations
2.4 Simple Integrating Factors
2.5 First-Order Linear Differential Equations
2.6 Special Classes of Equations
2.6.1 The Bernoulli Equation
2.6.2 The Riccati Equation
2.6.3 Equations with the Dependent or Independent Variable Missing
2.6.4 Equations Homogeneous with Respect to Their Dependent Variable
2.6.5 Equations Solvable for a Variable
2.7 Qualitative Analysis
2.7.1 Bifurcation Points
2.7.2 Validity Intervals of Autonomous Equations
Summary for Chapter 2
Review Questions for Chapter 2
3 Numerical Methods
3.1 Difference Equations
3.2 Euler’s Methods
3.3 The Polynomial Approximation
3.4 Error Estimates
3.5 The Runge?Kutta Methods
Summary for Chapter 3
Review Questions for Chapter 3
4 Second and Higher Order Linear Differential Equations
4.1 Second and Higher Order Differential Equations
4.1.1 Linear Operators
4.1.2 Exact Equations and Integrating Factors
4.1.3 Change of Variables
4.2 Linear Independence and Wronskians
4.3 The Fundamental Set of Solutions
4.4 Equations with Constant Coefficients
4.5 Complex Roots
4.6 Repeated Roots. Reduction of Order
4.6.1 Reduction of Order
4.6.2 Euler’s Equations
4.7 Nonhomogeneous Equations
4.7.1 The Annihilator
4.7.2 The Method of Undetermined Coefficients
4.8 Variation of Parameters
Summary for Chapter 4
Review Questions for Chapter 4
5 Laplace Transforms
5.1 The Laplace Transform
5.2 Properties of the Laplace Transform
5.3 Discontinuous and Impulse Functions
5.4 The Inverse Laplace Transform
5.4.1 Partial Fraction Decomposition
5.4.2 Convolution Theorem
5.4.3 The Residue Method
5.5 Homogeneous Differential Equations
5.5.1 Equations with Variable Coefficients
5.6 Nonhomogeneous Differential Equations
5.6.1 Differential Equations with Intermittent Forcing Functions
Summary for Chapter 5
Review Questions for Chapter 5
6 Series Solutions of Differential Equations 335
6.1 Power Series Solutions
6.2 Picard’s Iterations
6.3 Adomian Decomposition Method
6.4 Power Series Solutions to Equations with Analytic Coefficients
6.4.1 The Ordinary Point at Infinity
6.5 Euler Equations
6.6 Series Solutions Near a Regular Singular Point
6.6.1 Regular Singular Point at Infinity
6.6.2 Inhomogeneous Equations
6.7 Bessel Equations
6.7.1 Parametric Bessel Equation
6.7.2 Bessel Functions of Half-Integer Order
6.7.3 Related Differential Equations
Summary for Chapter 6
Review Questions for Chapter 6
7 Introduction to Systems of ODEs
7.1 Some ODE Models
7.1.1 RLC-circuits
7.1.2 Spring-Mass Systems
7.1.3 The Euler?Lagrange Equation
7.1.4 Pendulum
7.1.5 Laminated Material
7.1.6 Flow Problems
7.2 Matrices
7.3 Linear Systems of First Order ODEs
7.4 Reduction to a Single ODE
7.5 Existence and Uniqueness
Summary for Chapter 7
Review Questions for Chapter 7
8 Topics from Linear Algebra
8.1 The Calculus of Matrix Functions
8.2 Inverses and Determinants
8.2.1 Solving Linear Equations
8.3 Eigenvalues and Eigenvectors
8.4 Diagonalization
8.5 Sylvester’s Formula
8.6 The Resolvent Method
8.7 The Spectral Decomposition Method
Summary for Chapter 8
Review Questions for Chapter 8
9 Systems of Linear Differential Equations
9.1 Systems of Linear Equations
9.1.1 The Euler Vector Equations
9.2 Constant Coefficient Homogeneous Systems
9.2.1 Simple Real Eigenvalues
9.2.2 Complex Eigenvalues
9.2.3 Repeated Eigenvalues
9.2.4 Qualitative Analysis of Linear Systems
9.3 Variation of Parameters
9.3.1 Equations with Constant Coefficients
9.4 Method of Undetermined Coefficients
9.5 The Laplace Transformation
9.6 Second Order Linear Systems
Summary for Chapter 9
Review Questions for Chapter 9
10 Qualitative Theory of Differential Equations
10.1 Autonomous Systems
10.1.1 Two-Dimensional Autonomous Equations
10.2 Linearization
10.2.1 Two-Dimensional Autonomous Equations
10.2.2 Scalar Equations
10.3 Population Models
10.3.1 Competing Species
10.3.2 Predator-Prey Equations
10.3.3 Other Population Models
10.4 Conservative Systems
10.4.1 Hamiltonian Systems
10.5 Lyapunov’s Second Method
10.6 Periodic Solutions
10.6.1 Equations with Periodic Coefficients
Summary for Chapter 10
Review Questions for Chapter 10
11 Orthogonal Expansions
11.1 Sturm?Liouville Problems
11.2 Orthogonal Expansions
11.3 Fourier Series
11.3.1 Music as Motivation
11.3.2 Sturm?Liouville Periodic Problem
11.3.3 Fourier Series
11.4 Convergence of Fourier Series
11.4.1 Complex Fourier Series
11.4.2 The Gibbs Phenomenon
11.5 Even and Odd Functions
Summary for Chapter 11
Review Questions for Chapter 11
12 Partial Differential Equations
12.1 Separation of Variables for the Heat Equation
12.1.1 Two-Dimensional Heat Equation
12.2 Other Heat Conduction Problems
12.3 Wave Equation
12.3.1 Transverse Vibrations of Beams
12.4 Laplace Equation
12.4.1 Laplace Equation in Polar Coordinates
Summary for Chapter 12
Review Questions for Chapter 12
Bibliography
Index
Copyright Year 2022
ISBN 9781032290607
July 12, 2022 Forthcoming by Chapman and Hall/CRC
176 Pages 13 B/W Illustrations
Format Hardback
This book introduces multiplicative Frenet curves. We define multiplicative tangent, multiplicative normal, and multiplicative normal plane for a multiplicative Frenet curve. We investigate the local behaviours of a multiplicative parameterized curve around multiplicative biregular points, define multiplicative Bertrand curves and investigate some of their properties. A multiplicative rigid motion is introduced.
The book is addressed to instructors and graduate students, and also specialists in geometry, mathematical physics, differential equations, engineering, and specialists in applied sciences. The book is suitable as a textbook for graduate and under-graduate level courses in geometry and analysis. Many examples and problems are included.
The author introduces the main conceptions for multiplicative surfaces: multiplicative first fundamental form, the main multiplicative rules for differentiations on multiplicative surfaces, and the main multiplicative regularity conditions for multiplicative surfaces. An investigation of the main classes of multiplicative surfaces and second fundamental forms for multiplicative surfaces is also employed. Multiplicative differential forms and their properties, multiplicative manifolds, multiplicative Einstein manifolds and their properties, are investigated as well.
Many unique applications in mathematical physics, classical geometry, economic theory, and theory of time scale calculus are offered.
1. Elements of the Multiplicative Euclidean Geometry
2. Multiplicative Curves in Rn
3. Multiplicative Plane Curves
4. General Theory of Multiplicative Surfaces
5. Multiplicative Fundamental Equations of a Multiplicative Surface
6. Special Classes of Multiplicative Surfaces
7. Multiplicative Differential Forms
8. The Multiplicative Nature Connection
9. Multiplicative Riemannian Manifolds
10. The Multiplicative Curvature Tensor
Appendix A. The Multiplicative Lipschitz Condition
Appendix B. The Multiplicative Implicit Function Theorem
Copyright Year 2022
ISBN 9781032102757
June 3, 2022 Forthcoming by Chapman and Hall/CRC
272 Pages 36 B/W Illustrations
Format Hardback
This book has enjoyed considerable use and appreciation during its first four editions. With hundreds of students having learned out of early editions, the author continues to find ways to modernize and maintain a unique presentation.
What sets the book apart is the excellent writing style, exposition, and unique and through sets of exercises. This edition offers a more instructive Preface to assist instructors on developing the course they prefer. The prerequisites are more explicit and provide a roadmap for the course. Sample syllabi are included.
As would be expected in a fifth edition, the overall content and structure of the book is sound.
This new edition offers a more organized treatment of axiomatics. Throughout the book, there is a more careful and detailed treatment of the axioms of set theory. The rules of inference are more carefully elucidated.
An emphasis on the art of proof.
Enhanced number theory chapter presents some easily accessible but still-unsolved problems. These include the Goldbach conjecture, the twin-prime conjecture, and so forth.
The discussion of equivalence relations is revised to present reflexivity, symmetry, and transitivity before we define equivalence relations.
The discussion of the RSA cryptosystem in Chapter 10 is expanded.
The author introduces groups much earlier, as this is an incisive example of an axiomatic theory. Coverage of group theory, formerly in Chapter 11, has been moved up, this is an incisive example of an axiomatic theory.
Recognizing new ideas, the author has enhanced the overall presentation to create a Fifth Edition of this classic and widely-used textbook.
1. Basic Logic 2. Methods of Proof 3. Set Theory 4. Relations and Functions 5. Group Theory 6. Number Systems 7. More on the Real Number System 8. A Glimpse of Topology 9. Elementary Number Theory 10. Zero-Knowledge Proofs and Cryptography 11. An Example of an Axiomatic Theory