Starting from simple generalizations of factorials and binomial coefficients, this book gives a friendly and accessible introduction to q-analysis, a subject consisting primarily of identities between certain kinds of series and products. Many applications of these identities to combinatorics and number theory are developed in detail. There are numerous exercises to help students appreciate the beauty and power of the ideas, and the history of the subject is kept consistently in view.
The book has few prerequisites beyond calculus. It is well suited to a capstone course, or for self-study in combinatorics or classical analysis. Ph.D. students and research mathematicians will also find it useful as a reference.
Undergraduate students interested in q-analysis, combinatorics, and number theory.
2020; 519 pp; Softcover
MSC: Primary 05; 11; 33;
Print ISBN: 978-1-4704-5623-8
PUBLICATION PLANNED FOR: January 2021
AVAILABILITY: Not yet published - available from January 2021
FORMAT: Hardback ISBN: 9781108479097
Address vector and matrix methods necessary in numerical methods and optimization of linear systems in engineering with this unified text. Treats the mathematical models that describe and predict the evolution of our processes and systems, and the numerical methods required to obtain approximate solutions. Explores the dynamical systems theory used to describe and characterize system behaviour, alongside the techniques used to optimize their performance. Integrates and unifies matrix and eigenfunction methods with their applications in numerical and optimization methods. Consolidating, generalizing, and unifying these topics into a single coherent subject, this practical resource is suitable for advanced undergraduate students and graduate students in engineering, physical sciences, and applied mathematics.
Unifies topics in matrix, numerical, and optimization methods along with dynamical systems
Illustrates the connections between linear algebraic and differential equations
Features end-of-chapter exercises and online solutions
Part I. Matrix Methods:
1. Vector and matrix algebra
2. Algebraic eigenproblems and their applications
3. Differential eigenproblems and their applications
4. Vector and matrix calculus
5. Analysis of discrete dynamical systems
Part II. Numerical Methods:
6. Computational linear algebra
7. Numerical methods for differential equations
8. Finite-difference methods for boundary-value problems
9. Finite-difference methods for initial-value problems
Part III. Least Squares and Optimization:
10. Least-squares methods
11. Data analysis ? curve fitting and interpolation
12. Optimization and root finding of algebraic systems
13. Data-driven methods and reduced-order modeling.
Part of Cambridge Studies in Advanced Mathematics
PUBLICATION PLANNED FOR: March 2021
AVAILABILITY: Not yet published - available from March 2021
FORMAT: HardbackISBN: 9781108836838
This introduction to the singularly perturbed methods in the nonlinear elliptic partial differential equations emphasises the existence and local uniqueness of solutions exhibiting concentration property. The authors avoid using sophisticated estimates and explain the main techniques by thoroughly investigating two relatively simple but typical non-compact elliptic problems. Each chapter then progresses to other related problems to help the reader learn more about the general theories developed from singularly perturbed methods. Designed for PhD students and junior mathematicians intending to do their research in the area of elliptic differential equations, the text covers three main topics. The first is the compactness of the minimization sequences, or the Palais-Smale sequences, or a sequence of approximate solutions; the second is the construction of peak or bubbling solutions by using the Lyapunov-Schmidt reduction method; and the third is the local uniqueness of these solutions.
Provides self-contained materials for PhD students and junior mathematicians who wish to acquaint themselves with singularly perturbed methods
Makes the techniques understandable without involving too many sophisticated estimates
Discusses the general theories developed from the singularly perturbed methods
1. Non-Compact Elliptic Problems
2. Perturbation Methods
3. Local Uniqueness of Solutions
4. Construction of Infinitely Many Solutions
5. A Compactness Theorem and Application
6. The Appendix.
Martin Gardner's Mathematical Games columns in Scientific American inspired and entertained several generations of mathematicians and scientists. Gardner in his crystal-clear prose illuminated corners of mathematics, especially recreational mathematics, that most people had no idea existed. His playful spirit and inquisitive nature invite the reader into an exploration of beautiful mathematical ideas along with him. These columns were both a revelation and a gift when he wrote them; no one?before Gardner?had written about mathematics like this. They continue to be a marvel.
This is the original 1971 edition and contains columns published in the magazine from 1963?1965.
Martin Gardner's Mathematical Games Volume: 5; 2020; 262 pp;
MSC: Primary 00;
ISBN: 978-1-4704-6356-4
Martin Gardner's Mathematical Games columns in Scientific American inspired and entertained several generations of mathematicians and scientists. Gardner in his crystal-clear prose illuminated corners of mathematics, especially recreational mathematics, that most people had no idea existed. His playful spirit and inquisitive nature invite the reader into an exploration of beautiful mathematical ideas along with him. These columns were both a revelation and a gift when he wrote them; no one?before Gardner?had written about mathematics like this. They continue to be a marvel.
This collection of fifteen e-books contains every column Gardner wrote for Scientific American in the years 1956?1986. In each book the columns were updated and corrected as necessary.
Martin Gardner's Mathematical Games
2020; 4408 pp;
MSC: Primary 00;
Print ISBN: 978-1-4704-6369-4