Series: Cambridge Studies in Advanced Mathematics (No. 47)
Paperback (ISBN-13: 9780521108478)
In this introduction to commutative algebra, the author leads the beginning student through the essential ideas, without getting embroiled in technicalities. The route chosen takes the reader quickly to the fundamental concepts for understanding complex projective geometry, the only prerequisites being a basic knowledge of linear and multilinear algebra and some elementary group theory. In the first part, the general theory of Noetherian rings and modules is developed. A certain amount of homological algebra is included, and rings and modules of fractions are emphasised, as preparation for working with sheaves. In the second part, the central objects are polynomial rings in several variables with coefficients in the field of complex numbers. After Noether’s normalisation lemma and Hilbert’s Nullstellensatz, affine complex schemes and their morphisms are introduced; Zariski’s main theorem and Chevalley’s semi-continuity theorem are then proved. Finally, a detailed study of Weil and Cartier divisors provides a solid background for modern intersection theory. This is an excellent textbook for those who seek an efficient and rapid introduction to the geometric applications of commutative algebra.
* Excellent author * No comparable books at this level
1. Rings, homomorphisms, ideals; 2. Modules; 3. Noetherian rings and modules; 4. Artinian rings and modules; 5. Finitely generated modules over Noetherian rings; 6. A first contact with homological algebra; 7. Fractions; 8. Integral extensions of rings; 9. Algebraic extensions of rings; 10. Noether’s normalisation lemma; 11. Affine schemes; 12. Morphisms of affine schemes; 13. Zariski’s main theorem; 14. Integrally closed Noetherian rings; 15. Weil divisors; 16. Cartier divisors; Subject index; Symbols index.
Paperback (ISBN-13: 9780521542616)
Mathematical methods are essential tools for all physical scientists. This second edition provides a comprehensive tour of the mathematical knowledge and techniques that are needed by students in this area. In contrast to more traditional textbooks, all the material is presented in the form of problems. Within these problems the basic mathematical theory and its physical applications are well integrated. The mathematical insights that the student acquires are therefore driven by their physical insight. Topics that are covered include vector calculus, linear algebra, Fourier analysis, scale analysis, complex integration, Green’s functions, normal modes, tensor calculus and perturbation theory. The second edition contains new chapters on dimensional analysis, variational calculus, and the asymptotic evaluation of integrals. This book can be used by undergraduates and lower-level graduate students in the physical sciences. It can serve as a stand-alone text, or as a source of problems and examples to complement other textbooks.
* All the material is presented in the form of problems * Mathematical
insights are gained by getting the reader to develop answers themselves
* Many applications of the mathematics are given
Contents
1. Introduction; 2. Dimensional analysis; 3. Power series; 4. Spherical and cylindrical co-ordinates; 5. The gradient; 6. The divergence of a vector field; 7. The curl of a vector field; 8. The theorem of Gauss; 9. The theorem of Stokes; 10. The Laplacian; 11. Conservation laws; 12. Scale analysis; 13. Linear algebra; 14. The Dirac delta function; 15. Fourier analysis; 16. Analytic functions; 17. Complex integration; 18. Green’s functions: principles; 19. Green’s functions: examples; 20. Normal modes; 21. Potential theory; 22. Cartesian tensors; 23. Perturbation theory; 24. Asymptotic evaluation of integrals; 25. Variational calculus; 26. Epilogue, on power and knowledge; References.
Paperback (ISBN-13: 9780898716689)
The absence of derivatives, often combined with the presence of noise or lack of smoothness, is a major challenge for optimization. This book explains how sampling and model techniques are used in derivative-free methods and how these methods are designed to efficiently and rigorously solve optimization problems. Although readily accessible to readers with a modest background in computational mathematics, it is also intended to be of interest to researchers in the field. Introduction to Derivative-Free Optimization is the first contemporary comprehensive treatment of optimization without derivatives. This book covers most of the relevant classes of algorithms from direct search to model-based approaches. It contains a comprehensive description of the sampling and modeling tools needed for derivative-free optimization; these tools allow the reader to better analyze the convergent properties of the algorithms and identify their differences and similarities.
* Intended for anyone interested in using optimization on problems where
derivatives are difficult or impossible to obtain * Includes a comprehensive
description of the sampling and modeling tools needed for derivative-free
optimization * Contains analysis of convergence for modified Nelder*Mead
and implicit-filtering methods as well as for model-based methods
Preface; 1. Introduction; Part I. Sampling and Modeling: 2. Sampling and linear models; 3. Interpolating nonlinear models; 4. Regression nonlinear models; 5. Underdetermined interpolating models; 6. Ensuring well poisedness and suitable derivative-free models; Part II. Frameworks and Algorithms: 7. Directional direct-search methods; 8. Simplicial direct-search methods; 9. Line-search methods based on simplex derivatives; 10. Trust-region methods based on derivative-free models; 11. Trust-region interpolation-based methods; Part III. Review of Other Topics: 12. Review of surrogate model management; 13. Review of constrained and other extensions to derivative-free optimization; Appendix: software for derivative-free optimization; Bibliography; Index.
Series: Cambridge Tracts in Mathematics (No. 116)
Paperback (ISBN-13: 9780521105828)
The sign-solvability of a linear system implies that the signs of the entries of the solution are determined solely on the basis of the signs of the coefficients of the system. That it might be worthwhile and possible to investigate such linear systems was recognised by Samuelson in his classic book Foundations of Economic Analysis. Sign-solvability is part of a larger study which seeks to understand the special circumstances under which an algebraic, analytic or geometric property of a matrix can be determined from the combinatorial arrangement of the positive, negative and zero elements of the matrix. The large and diffuse body of literature connected with sign-solvability is presented as a coherent whole for the first time in this book, displaying it as a beautiful interplay between combinatorics and linear algebra. One of the features of this book is that algorithms that are implicit in many of the proofs have been explicitly described and their complexity has been commented on.
Preface; 1. Sign-solvability; Bibliography; 2. L-matrices; Bibliography; 3. Sign-solvability and digraphs; Bibliography; 4. S*-matrices; Bibliography; 5. Beyond S*-matrices; Bibliography; 6. SNS-matrices; Bibliography; 7. S2NS-matrices; Bibliography; 8. Extremal properties of L-matrices; Bibliography; 9. The inverse sign pattern graph; Bibliography; 10. Sign stability; Bibliography; 11. Related Topics; Bibliography; Master Bibliography; Index
Series: Classroom Resource Materials
Hardback (ISBN-13: 9780883857571)
This text approaches the learning of group theory visually. It allows the student to see groups, experiment with groups and understand their significance. It brings groups, subgroups, homomorphisms, products, and quotients into clear view. Every topic and theorem is accompanied with a visual demonstration of its meaning and import, from the basics of groups and subgroups through advanced structural concepts such as semidirect products and Sylow theory. Opening chapters anchor the reader's intuitions with puzzles and symmetrical objects, defining groups as collections of actions. This approach gives early access to Cayley diagrams, the visualization technique central to the book, due to its unique ability to make group structure visually evident. This book is ideal as a supplement for a first course in group theory or alternatively as recreational reading.
* Includes over 300 full-colour illustrations * Moves from the basics of
group theory to Sylow theory and Galois theory * An applications chapter
shows how group theory describes the symmetry in crystals, dancing, art,
architecture, and more
Preface; Overview; 1. What is a group?; 2. What do groups look like?; 3.
Why study groups?; 4. Algebra at last; 5. Five families; 6. Subgroups;
7. Products and quotients; 8. The power of homomorphisms; 9. Sylow Theory;
10. Galois theory.