Copyright 2024
Hardback
ISBN 9781032662527
456 Pages 166 B/W Illustrations
June 14, 2024
Boundary value problems play a significant role in modeling systems characterized by established conditions at their boundaries. On the other hand, initial value problems hold paramount importance in comprehending dynamic processes and foreseeing future behaviors. The fusion of these two types of problems yields profound insights into the intricacies of the conduct exhibited by many physical and mathematical systems regulated by linear partial differential equations.
Boundary Value Problems for Linear Partial Differential Equations provides students with the opportunity to understand and exercise the benefits of this fusion, equipping them with realistic, practical tools to study solvable linear models of electromagnetism, fluid dynamics, geophysics, optics, thermodynamics and specifically, quantum mechanics. Emphasis is devoted to motivating the use of these methods by means of concrete examples taken from physical models.
Features
No prerequisites apart from knowledge of differential and integral calculus and ordinary differential equations.
Provides students with practical tools and applications
Contains numerous examples and exercises to help readers understand the concepts discussed in the book.
1. Introduction. 1.1. Partial Differential Equations. 1.2. Boundary and Initial Conditions. 1.3. Local Solvability. 1.4. Characteristics. 1.5. General Solutions. 1.6. Remarkable Lives and Achievements. 2. Linear PDEs. 2.1. Linear Differential Operators. 2.2. 1D Eigenvalue Problems. 2.3. Linear BVPs in Physics. 2.4. Exercises. 3. Separation of Variables Method. 3.1. Separable Homogeneous Linear PDEs. 3.2. Separable Homogeneous BVP. 3.3. Equations of Mathematical Physics. 3.4. Helmholtz Equation. 3.5. Remarkable Lives and Achievements. 3.6. Exercises. 4. Symmetric Differential Operators. 4.1. Hilbert Spaces. 4.2. Orthogonal Sets of Functions. 4.3. Green Functions. 4.4. Symmetric Differential Operators. 4.5. Sturm?Liouville Differential Operators. 4.6. Remarkable Lives and Achievements. 4.7. Exercises. 5. Fourier Analysis. 5.1. Fourier Trigonometric Bases. 5.2. Fourier Series. 5.3. Convergence of Fourier Series. 5.4. Fourier Transform. 5.5. Remarkable Lives and Achievements. 5.6. Exercises. 6. Eigenfunction Expansion Method. 6.1. Preliminary Discussion: Restricted Inhomogeneities. 6.2. Application to Evolution Equations. 6.3. General Discussion: Full Inhomogeneties. 6.4. Unbounded Domains and Fourier Transform. 7. Special Functions. 7.1. Frobenius Series. 7.2. Ordinary Points. 7.3. Regular Singular Points. 7.4. Bessel Equation. 7.5. Euler Gamma Function. 7.6. Remarkable Lives and Achievements. 7.7. Exercises. 8. Cylindrical and Spherical BVPs. 8.1. Cylindrical BVPs. 8.2. Spherical BVPs. 8.3. Beyond Cylindrical and Spherical. 8.4. Exercises.
Copyright 2024
Hardback
ISBN 9781032003665
564 Pages 33 B/W Illustrations
May 29, 2024
Vector Partitions, Visible Points and Ramanujan Functions offers a novel theory of Vector Partitions, though very much grounded in the long-established work of others, that could be developed as an extension to the existing theory of Integer Partitions. The book is suitable for graduate students in physics, applied mathematics, number theory and computational mathematics. It takes the reader up to research level, presenting new results alongside known classical results from integer partitions and areas of vector and multipartite partition theory. It also sets forth new directions for research for the more advanced reader.
Above all, the intention of the book is to bring new inspiration to others who study mathematics and related areas. It is hoped that some new ideas will be launched to add value and insight into many of the classical and new theories surrounding partitions. The book is an appreciation of the many gifted authors of research into partitions over the past century and before, in the hope that more may come of this for future generations.
Provides a step-by-step guide through the known literature on Integer and Vector Partitions, and a focus on the not so well-known Visible Point Vector identities
Serves as a reference for graduate students and researchers in physics, applied mathematics, number theory and computational mathematics
Offers a variety of practical examples as well as sets of exercises suitable for students and researchers
Geoffrey B. Campbell completed his PhD at Australian National University in 1998 under the esteemed physicist Professor Rodney Baxter. His affiliation with the Australian National University Mathematical Sciences Institute has continued for over 30 years. Within that time frame, Geoffrey also served eight years as an Honorary Research Fellow at LaTrobe University Mathematics and Statistics Department in Melbourne. Currently he writes ongoing articles for the Australian Mathematical Society Gazette. Within the international scope, Geoffrey currently serves as a PhD external committee member for a mathematics graduate student at Washington State University in America.
Geoffrey has built a career within Australian Commonwealth and State government departments, including as an Advisor at the Department of Prime Minister and Cabinet; as Analyst Researcher for a Royal Commission. Geoffrey specializes in complex data, machine learning including data analytics. He is also a published poet in Australian anthologies and literary magazines.
Section I. Background and History. 1. History timeline partitions. Section
II. Integer Partition Theory. 3. Integer partition generating functions.
4. Continued fraction partition identities. 5. Partition congruences. 6.
Ferrers diagrams. 7.Durfee Squares. 8. Gaussian polynomials. 9. Plane Partitions
from MacMahon to Andrews. 10. Asymptotics for Partition Functions. 11.
Rogers-Ramanujan identities in Statistical Mechanics. Section III. Vector
Partition Theory. 12. Vector partitions and their generating function identities.
13. Integer Partitions generalized to Vector Partitions. 14. Weighted Vector
Partitions as hybrid n-space variations. 15. Functional Equations for n-space
Vector Partitions. 16. Binary Partitions and their Vector Generalizations.
17. n-ary Partitions and their Vector Generalizations. 18. Some Binary
and n-ary Partition Analytic Formulas. Section IV. 19. Features of the
Visible Lattice Points. 20. Visible Point Vector Identities in the first
Hyperquadrant. 21. Visible Point Vector Identities in Hyperpyramid lattices.
22. Polylogarithms, and Parametric Euler Sum identities. 23. Visible Point
Vector identities from particular Euler sum values. 24. Visible Point Vector
Identities in Skewed Hyperpyramid lattices. 25. Harmonic Sums applied to
VPV Identities. 26. The Ramanujan trigonometric function and visible point
identities. 27. Other non-weighted n-space Vector Partition Theorems. 28.
VPV Identity cases related to some exponential relations. Section V. Models,
Interpretations and some Useful Tools. 29. 2D and 3D Stepping Stones, Forests,
Orchards and Light Diffusions. 30. Euler Products over Primes and new VPV
Formulas. 31. Determinants, Bell Polynomial Expansions for Vector Partitions.
32. Glossary.
Copyright 2024
Hardback
ISBN 9781032617978
584 Pages 29 B/W Illustrations
August 6, 2024
This book introduces Probability Theory with R software and explains abstract concepts in a simple and easy-to-understand way by combining theory and computation. It discusses conceptual and computational examples in detail, to provide a thorough understanding of basic techniques and develop an enjoyable read for students seeking suitable material for self-study. It illustrates fundamental concepts including fields, sigma-fields, random variables and their expectations, various modes of convergence of a sequence of random variables, laws of large numbers and the central limit theorem.
? Computational exercises based on R software are included in each Chapter
? Includes a brief introduction to the basic functions of R software for beginners in R and serves as a ready reference.
? Includes Numerical computations, simulation studies, and visualizations using R software as easy tools to explain abstract concepts
? Provides multiple-choice questions for practice
? Incorporates self-explanatory R codes in every chapter
This textbook is for advanced students, professionals, and academic researchers of Statistics, Biostatistics, Economics and Mathematics.
1. Sigma Field, Borel Field and Probability Measure 2. Random Variables and Random Vectors 3. Distribution Function 4. Expectation and Characteristic Function 5. Independence 6. Almost Sure Convergence and Borel Zero-One Law 7. Convergence in Probability, in Law and in r-th Mean 8. Convergence of a Sequence of Expectations 9. Laws of Large Numbers 10. Central Limit Theorem 11. Solutions to Conceptual Exercises 12. Bibliography 13. Index
Copyright 2025
Paperback
ISBN 9781032811802
287 Pages 73 B/W Illustrations
August 2, 2024
This classic text, originally from the noted logician Elliot Mendelson, is intended to be an easy-to-read introduction to the basic ideas and techniques of game theory. It can be used as a class textbook or for self-study.
Introducing Game Theory and its Applications presents an easy-to-read introduction to the basic ideas and techniques of game theory. After a brief introduction, the authors begin with a chapter devoted to combinatorial games--a topic neglected or treated minimally in most other texts. The focus then shifts to two-person zero-sum games and their solution.
Here the authors present the simplex method, based on linear programming, for solving these games and develop within this presentation the required background. The final chapter presents some of the fundamental ideas and tools of non-zero-sum games and games with more than two players, including an introduction to cooperative game theory.
The book is suitable for a first undergraduate course in game theory, or a graduate course for students with limited previous exposure. It is useful for students who need to learn some game theory for a related subject (e.g., microeconomics) and have a limited mathematical background. It also prepares its readers for more advanced study of game theory's applications in economics, business, and the physical, biological, and social sciences.
The authors hope this book breeds curiosity into the subject and is designed to satisfy this to some extent by reading this book. The book will prepare readers for deeper study of game theory applications in many fields of study.
Preface
Introduction
1 Combinatorial games
1.1 Definition of combinatorial games
1.2 Fundamental theorem of combinatorial games
1.3 Nim
1.4 Hex and other games
1.5 Tree games
1.6 Grundy functions
1.7 Bogus Nim-sums
1.8 Chapter summary
2 Two-person zero-sum games
2.1 Games in normal form
2.2 Saddle points and equilibrium pairs
2.3 Maximin and minimax
2.4 Mixed strategies
2.5 2-by-2 matrix games
2.6 2-by-n, m-by-2 and 3-by-3 matrix games
2.7 Linear programming
2.8 Chapter summary
3 Solving two-person zero-sum games using LP
3.1 Perfect canonical linear programming problems
3.2 The simplex method
3.3 Pivoting
3.4 The perfect phase of the simplex method
3.5 The Big M method
3.6 Bland's rules to prevent cycling
3.7 Duality and the simplex method
3.8 Solution of game matrices
3.9 Chapter summary
4 Non-zero-sum games and k-person games
4.1 The general setting
4.2 Nash equilibria
4.3 Graphical method for 2 A? 2 matrix games
4.4 Inadequacies of Nash equilibria & cooperative games
4.5 The Nash arbitration procedure
4.6 Games with two or more players
4.7 Coalitions
4.8 Games in coalition form
4.9 The Shapley value
4.10 The Banzhaf power index
4.11 Imputations
4.12 Strategic equivalence
4.13 Stable sets
4.14 Chapter summary
5 Imperfect Information Games
5.1 The general setting
5.2 Complete information games in extensive form
5.3 Imperfect information games in extensive form
5.4 Games with random effects
5.5 Chapter summary
6 Computer solutions to games
6.1 Zero-sum games - invertible matrices
6.2 Zero sum games - linear program problem (LP)
6.3 Special Linear Programming Capabilities
6.4 Non-zero sum games - linear complementarity problem (LCP)
6.5 Special game packages
6.6 Chapter summary
Appendices
Appendix A Utility theory
Appendix B Nash's theorem
Appendix C Finite probability theory
Appendix D Calculus & Differentiation
Appendix E Linear Algebra
Appendix F Linear Programming
Appendix G Named Games and Game Data
Answers to selected exercises
Bibliography
Index