Series: London Mathematical Society Student Texts (No. 76)
Hardback (ISBN-13: 9781107002531)
Paperback (ISBN-13: 9780521175623)
275 exercises
Page extent: 310 pages
Size: 228 x 152 mm
Joseph Liouville is recognised as one of the great mathematicians of the nineteenth century, and one of his greatest achievements was the introduction of a powerful new method into elementary number theory. This book provides a gentle introduction to this method, explaining it in a clear and straightforward manner. The many applications provided include applications to sums of squares, sums of triangular numbers, recurrence relations for divisor functions, convolution sums involving the divisor functions, and many others. All of the topics discussed have a rich history dating back to Euler, Jacobi, Dirichlet, Ramanujan and others, and they continue to be the subject of current mathematical research. Williams places the results in their historical and contemporary contexts, making the connection between Liouville's ideas and modern theory. This is the only book in English entirely devoted to the subject and is thus an extremely valuable resource for both students and researchers alike.
* Demonstrates that some analytic formulae in number theory can be proved in an elementary arithmetic manner * Motivates students to do their own research * Includes an extensive bibliography
Preface; 1. Joseph Liouville (1809*1888); 2. Liouville's ideas in number
theory; 3. The arithmetic functions Šk(n), Šk*(n), dk,m(n) and Fk(n);
4. The equation i2 + jk = n; 5. An identity of Liouville; 6. A recurrence
relation for Š*(n); 7. The Girard*Fermat theorem; 8. A second identity
of Liouville; 9. Sums of two, four and six squares; 10. A third identity
of Liouville; 11. Jacobi's four squares formula; 12. Besge's formula; 13.
An identity of Huard, Ou, Spearman and Williams; 14. Four elementary arithmetic
formulae; 15. Some twisted convolution sums; 16. Sums of two, four, six
and eight triangular numbers; 17. Sums of integers of the form x2+xy+y2;
18. Representations by x2+y2+z2+2t2, x2+y2+2z2+2t2 and x2+2y2+2z2+2t2;
19. Sums of eight and twelve squares; 20. Concluding remarks; References;
Index.
Hardback (ISBN-13: 9780521872003)
Paperback (ISBN-13: 9780521692908)
10 b/w illus.
Page extent: 120 pages
Size: 228 x 152 mm
V. I. Arnold reveals some unexpected connections between such apparently unrelated theories as Galois fields, dynamical systems, ergodic theory, statistics, chaos and the geometry of projective structures on finite sets. The author blends experimental results with examples and geometrical explorations to make these findings accessible to a broad range of mathematicians, from undergraduate students to experienced researchers.
* Written by one of the greatest mathematicians of our age * Provides a
general overview suitable for mathematicians at all levels * Examples and
explanations may be used in all applications of Galois field theory
Preface; 1. What is a Galois field*; 2. The organisation and tabulation
of Galois fields; 3. Chaos and randomness in Galois field tables; 4. Equipartition
of geometric progressions along a finite one-dimensional torus; 5. Adiabatic
study of the distribution of geometric progressions of residues; 6. Projective
structures generated by a Galois field; 7. Projective structures: example
calculations; 8. Cubic field tables; Index.
Hardback (ISBN-13: 9780521515320)
Paperback (ISBN-13: 9780521735872)
80 b/w illus.
Page extent: 735 pages
Size: 247 x 174 mm
The partial differential equations that govern scalar and vector fields are the very language used to model a variety of phenomena in solid mechanics, fluid flow, acoustics, heat transfer, electromagnetism and many others. A knowledge of the main equations and of the methods for analyzing them is therefore essential to every working physical scientist and engineer. Andrea Prosperetti draws on many yearsf research experience to produce a guide to a wide variety of methods, ranging from classical Fourier-type series through to the theory of distributions and basic functional analysis. Theorems are stated precisely and their meaning explained, though proofs are mostly only sketched, with comments and examples being given more prominence. The book structure does not require sequential reading: each chapter is self-contained and users can fashion their own path through the material. Topics are first introduced in the context of applications, and later complemented by a more thorough presentation.
* Its modular structure makes the book suitable for a variety of uses and
users * Homework sets are available from www.cambridge.org/9780521735872
* Appendix provides material that is usually covered in mathematical analysis
courses (e.g. the Lebesgue integral) but is often unfamiliar to the applied
mathematician
Preface; To the reader; List of tables; Part I. General Remarks and Basic Concepts: 1. The classical field equations; 2. Some simple preliminaries; Part II. Applications: 3. Fourier series: applications; 4. Fourier transform: applications; 5. Laplace transform: applications; 6. Cylindrical systems; 7. Spherical systems; Part III. Essential Tools: 8. Sequences and series; 9. Fourier series: theory; 10. The Fourier and Hankel transforms; 11. The Laplace transform; 12. The Bessel equation; 13. The Legendre equation; 14. Spherical harmonics; 15. Green's functions: ordinary differential equations; 16. Green's functions: partial differential equations; 17. Analytic functions; 18. Matrices and finite-dimensional linear spaces; Part IV. Some Advanced Tools: 19. Infinite-dimensional spaces; 20. Theory of distributions; 21. Linear operators in infinite-dimensional spaces; Appendix; References; Index.
Series: London Mathematical Society Lecture Note Series (No. 382)
Paperback (ISBN-13: 9780521154338)
Page extent: 280 pages
Size: 228 x 152 mm
This book introduces a new approach to building models of bounded arithmetic, with techniques drawn from recent results in computational complexity. Propositional proof systems and bounded arithmetics are closely related. In particular, proving lower bounds on the lengths of proofs in propositional proof systems is equivalent to constructing certain extensions of models of bounded arithmetic. This offers a clean and coherent framework for thinking about lower bounds for proof lengths, and it has proved quite successful in the past. This book outlines a brand new method for constructing models of bounded arithmetic, thus for proving independence results and establishing lower bounds for proof lengths. The models are built from random variables defined on a sample space which is a non-standard finite set and sampled by functions of some restricted computational complexity. It will appeal to anyone interested in logical approaches to fundamental problems in complexity theory.
* A brand new approach to problems of complexity theory * Self-contained
so readers do not need to study other material * Presents some of the most
recent developments in proof complexity
Preface; Acknowledgements; Introduction; Part I. Basics: 1. The definition
of the models; 2. Measure on Ą; 3. Witnessing quantifiers; 4. The truth
in N and the validity in K(F); Part II. Second Order Structures: 5. Structures
K(F,G); Part III. AC0 World: 6. Theories I¢0, I¢0(R) and V10; 7. Shallow
Boolean decision tree model; 8. Open comprehension and open induction;
9. Comprehension and induction via quantifier elimination: a general reduction;
10. Skolem functions, switching lemma, and the tree model; 11. Quantifier
elimination in K(Ftree,Gtree); 12. Witnessing, independence and definability
in V10; Part IV. AC0(2) World: 13. Theory Q2V10; 14. Algebraic model; 15.
Quantifier elimination and the interpretation of Q2; 16. Witnessing and
independence in Q2V10; Part V. Towards Proof Complexity: 17. Propositional
proof systems; 18. An approach to lengths-of-proofs lower bounds; 19. PHP
principle; Part VI. Proof Complexity of Fd and Fd(+): 20. A shallow PHP
model; 21. Model K(Fphp,Gphp) of V10; 22. Algebraic PHP model*; Part VII.
Polynomial-Time and Higher Worlds: 23. Relevant theories; 24. Witnessing
and conditional independence results; 25. Pseudorandom sets and a Lowenheim*Skolem
phenomenon; 26. Sampling with oracles; Part VIII. Proof Complexity of EF
and Beyond: 27. Fundamental problems in proof complexity; 28. Theories
for EF and stronger proof systems; 29. Proof complexity generators: definitions
and facts; 30. Proof complexity generators: conjectures; 31. The local
witness model; Appendix. Non-standard models and the ultrapower construction;
Standard notation, conventions and list of symbols; References; Name index;
Subject index.
Series: Cambridge Monographs on Applied and Computational Mathematics (No. 11)
Paperback (ISBN-13: 9780521173278)
Page extent: 365 pages
Size: 229 x 152 mm
Numerical simulation of compressible, inviscid time-dependent flow is a major branch of computational fluid dynamics. Its primary goal is to obtain accurate representation of the time evolution of complex flow patterns, involving interactions of shocks, interfaces, and rarefaction waves. The Generalized Riemann Problem (GRP) algorithm, developed by the authors for this purpose, provides a unifying eshellf which comprises some of the most commonly used numerical schemes of this process. This 2003 monograph gives a systematic presentation of the GRP methodology, starting from the underlying mathematical principles, through basic scheme analysis and scheme extensions (such as reacting flow or two-dimensional flows involving moving or stationary boundaries). An array of instructive examples illustrates the range of applications, extending from (simple) scalar equations to computational fluid dynamics. Background material from mathematical analysis and fluid dynamics is provided, making the book accessible to both researchers and graduate students of applied mathematics, science and engineering.
* Presents the relevant mathematical background along with systematic analysis
of the GRP methods * Introduces basic mathematical concepts first in simpler
scalar conservation laws, repeats the process for more general settings
* Includes detailed econstructionf tables, allowing for the actual writing
of suitable computer codes
Preface; List of figures; 1. Introduction; Part I. Basic Theory: 2. Scalar conservation laws; Appendix A - entropy conditions for scalar conservation laws; 3. The GRP method for scalar conservation laws; Appendix B - convergence of the Godunov scheme; 4. Systems of conservation laws; Appendix C - Riemann solver for a y-law gas; 5. The generalized Riemann problem (GRP) for compressible fluid dynamics; Appendix D - the MUSCL scheme; 6. Analytical and numerical treatment of fluid dynamical problems; Part II. Numerical Implementation: 7. From the GRP algorithm to scientific computing; 8. Geometric extensions; 9. A physical extension: reacting flow; 10. Wave interaction in a duct - a comparative study; Bibliography; Glossary; Index.