Paperback (ISBN-13: 9780521091794)
Professor Caratheodory sets out the basic theory of conformal representations as simply as possible. In the early chapters on Mobius' and other elementary transformations and on non-Euclidean geometry, he deals with those elementary subjects that are necessary for an understanding of the general theory discussed in the remaining chapters.
1. Mobius Transformation; 2. Non-Euclidean Geometry; 3. Elementary Transformations; 4. Schwarz’s Lemma; 5. The Fundamental Theorems of Conformal Representation; 6. Transformation of the Frontier; 7. Transformation of Closed Surfaces; 8. The General Theorem of Uniformisation.
Paperback (ISBN-13: 9780521091909)
The purpose of this book is threefold: to be used for graduate courses on integral equations; to be a reference for researchers; and to describe methods of application of the theory. The author emphasizes the role of Volterra equations as a unifying tool in the study of functional equations, and investigates the relation between abstract Volterra equations and other types of functional-differential equations.
1. Introduction to the theory of integral equations; 2. Function spaces, operators, fixed points and monotone mappings; 3. Basic theory of Volterra equations: integral and abstract; 4. Some special classes of integral and integrodifferential equations; 5. Integral equations in abstract spaces; 6. Some applications of integral and integrodifferential
Series: Spectrum
Hardback (ISBN-13: 9780883855676)
For the majority of the twentieth century, philosophers of mathematics focused their attention on foundational questions. However, in the last quarter of the century they began to return to basics, and two new schools of thought were created: social constructivism and structuralism. The advent of the computer also led to proofs and development of mathematics assisted by computer, and to questions concerning the role of the computer in mathematics. This book of sixteen original essays is the first to explore this range of new developments in the philosophy of mathematics, in a language accessible to mathematicians. Approximately half the essays were written by mathematicians, and consider questions that philosophers have not yet discussed. The other half, written by philosophers of mathematics, summarise the discussion in that community during the last 35 years. A connection is made in each case to issues relevant to the teaching of mathematics.
Acknowledgments; Introduction; Part I. Proof and How it is Changing: 1. Proof: its nature and significance Michael Detlefsen; 2. Implications of experimental mathematics for the philosophy of mathematics Jonathan Borwein; 3. On the roles of proof in mathematics Joseph Auslander; Part II. Social Constructivist Views of Mathematics: 4. When is a problem solved* Philip J. Davis; 5. Mathematical practice as a scientific problem Reuben Hersh; 6. Mathematical domains: social constructs* Julian Cole; Part III. The Nature of Mathematical Objects and Mathematical Knowledge: 7. The existence of mathematical objects Charles Chihara; 8. Mathematical objects Stewart Shapiro; 9. Mathematical Platonism Mark Balaguer; 10. The nature of mathematical objects Oystein Linnebo; 11. When is one thing equal to some other thing* Barry Mazur; Part IV. The Nature of Mathematics and its Applications: 12. Extreme science: mathematics as the science of relations as such R. S. D. Thomas; 13. What is mathematics* A pedagogical answer to a philosophical question Guershon Harel; 14. What will count as mathematics in 2100* Keith Devlin; 15. Mathematics applied: the case of addition Mark Steiner; 16. Probability - a philosophical overview Alan Hajek; Glossary of common philosophical terms; About the editors.
Paperback (ISBN-13: 9780521091688)
This book provides an introduction to the theory of elliptic modular functions and forms, a subject of increasing interest because of its connexions with the theory of elliptic curves. Modular forms are generalizations of functions like theta functions. They can be expressed as Fourier series, and the Fourier coefficients frequently possess multiplicative properties which lead to a correspondence between modular forms and Dirichlet series having Euler products. The Fourier coefficients also arise in certain representational problems in the theory of numbers, for example in the study of the number of ways in which a positive integer may be expressed as a sum of a given number of squares. The treatment of the theory presented here is fuller than is customary in a textbook on automorphic or modular forms, since it is not confined solely to modular forms of integral weight (dimension). It will be of interest to professional mathematicians as well as senior undergraduate and graduate students in pure mathematics.
1. Groups of matrices and bilinear mappings; 2. Mapping properties; 3. Automorphic factors and multiplier systems; 4. General properties of modular forms; 5. Construction of modular forms; 6. Functions belonging to the full modular group; 7. Groups of level 2 and sums of squares; 8. Modular forms of level N; 9. Hecke operators and congruence groups; 10. Applications.
Hardback (ISBN-13: 9780521516440)
Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasizes algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The presentation alternates between theory and applications in order to motivate and illustrate the mathematics. The mathematical coverage includes the basics of number theory, abstract algebra and discrete probability theory. This edition now includes over 150 new exercises, ranging from the routine to the challenging, that flesh out the material presented in the body of the text, and which further develop the theory and present new applications. The material has also been reorganized to improve clarity of exposition and presentation. Ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.
* Now contains over 650 exercises, which present new applications to number theory and algebra * Minimal mathematics needed * An ideal textbook for an introductory graduate or advanced undergraduate course, geared towards computer science students
Preface; Preliminaries; 1. Basic properties of the integers; 2. Congruences; 3. Computing with large integers; 4. Euclid’s algorithm; 5. The distribution of primes; 6. Abelian groups; 7. Rings; 8. Finite and discrete probability distributions; 9. Probabilistic algorithms; 10. Probabilistic primality testing; 11. Finding generators and discrete logarithms in Z*p; 12. Quadratic reciprocity and computing modular square roots; 13. Modules and vector spaces; 14. Matrices; 15. Subexponential-time discrete logarithms and factoring; 16. More rings; 17. Polynomial arithmetic and applications; 18. Linearly generated sequences and applications; 19. Finite fields; 20. Algorithms for finite fields; 21. Deterministic primality testing; Appendix: Some useful facts; Bibliography; Index of notation; Index.
Paperback (ISBN-13: 9780521091800)
Primarily this book is intended for application of the operational calculus to mathematics, physics and technical problems. It gives the basic principles, ideas, and theorems clearly and extensively, but also many worked-out problems from mathematical and physical as well as from technical fields. The purely mathematical treatment is more advanced than is usual in books devoted primarily to practical applications, and the book will therefore be of value to those pure mathematicians who are interested in a rapid and simple derivation of complicated and unexpected relations between various mathematical functions, as well as to the engineer in search (for example) of a very simple treatment of transient phenomena in electrical networks.
1. General Introduction; 2. The Fourier Integral as Basis of the Operational Calculus; 3. Elementary Operational Images; 4. Elementary Rules; 5. The Delta or Impulse Function; 6. Questions Concerning the Convergence of the Definition Integral; 7. Asymptotic Relations and Operational Transposition of Series; 8. Linear Differential Equations with Constant Coefficients; 9. Simultaneous Linear Differential Equations with Constant Coefficients; Electric-Circuit Theory; 10. Linear Differential Equations with Variable Coefficients; 11. Operational Rules of more Complicated Character; 12. Step Functions and Other Discontinuous Functions; 13. Difference Equations; 14. Integral Equations; 15. Partial Differential Equations in the Operational Calculus of one Variable; 16. Simultaneous Operational Calculus.