Series: Dolciani Mathematical Expositions
Hardback (ISBN-13: 9780883853467)
A Guide to Topology is an introduction to basic topology for graduate or advanced undergraduate students. It covers point-set topology, Moore-Smith convergence and function spaces. It treats continuity, compactness, the separation axioms, connectedness, completeness, the relative topology, the quotient topology, the product topology, and all the other fundamental ideas of the subject. The book is filled with examples and illustrations. Students studying for exams will find this book to be a concise, focused and informative resource. Professional mathematicians who need a quick review of the subject, or need a place to look up a key fact, will find this book to be a useful resource too.
* A short, sharp introduction that shows readers exactly what topology
is and why it is useful * Concludes with function spaces to acquaint students
with the applications of the basic ideas and techniques of topology * Includes
table of notation and glossary to help students get to grip with the material
quickly
Preface; Part I. Fundamentals: 1.1. What is topology*; 1.2. First definitions;
1.3 Mappings; 1.4. The separation axioms; 1.5. Compactness; 1.6. Homeomorphisms;
1.7. Connectedness; 1.8. Path-connectedness; 1.9. Continua; 1.10. Totally
disconnected spaces; 1.11. The Cantor set; 1.12. Metric spaces; 1.13. Metrizability;
1.14. Baire's theorem; 1.15. Lebesgue's lemma and Lebesgue numbers; Part
II. Advanced Properties: 2.1 Basis and subbasis; 2.2. Product spaces; 2.3.
Relative topology; 2.4. First countable and second countable; 2.5. Compactifications;
2.6. Quotient topologies; 2.7. Uniformities; 2.8. Morse theory; 2.9. Proper
mappings; 2.10. Paracompactness; Part III. Moore-Smith Convergence and
Nets: 3.1. Introductory remarks; 3.2. Nets; Part IV. Function Spaces: 4.1.
Preliminary ideas; 4.2. The topology of pointwise convergence; 4.3. The
compact-open topology; 4.4. Uniform convergence; 4.5. Equicontinuity and
the Ascoli-Arzela theorem; 4.6. The Weierstrass approximation theorem;
Table of notation; Glossary; Bibliography; Index.
Series: Mathematical Association of America Textbooks
Hardback (ISBN-13: 9780883857595)
This textbook is a complete introduction to Lie groups for undergraduate students. The only prerequisites are multi-variable calculus and linear algebra. The emphasis is placed on the algebraic ideas, with just enough analysis to define the tangent space and the differential and to make sense of the exponential map. This textbook works on the principle that students learn best when they are actively engaged. To this end nearly 200 problems are included in the text, ranging from the routine to the challenging level. Every chapter has a section called ‘Putting the pieces together’ in which all definitions and results are collected for reference and further reading is suggested.
* Assumes only a background in linear algebra and multi-variable calculus
* Contains nearly 200 problems to promote active learning * At the end
of each chapter the mathematical and historical context of the material
is given
1. Symmetries of vector spaces: 1.1. What is a symmetry*; 1.2. Distance
is fundamental; 1.3. Groups of symmetries; 1.4. Bilinear forms and symmetries
of spacetime; 1.5. Putting the pieces together; 1.6. A broader view: Lie
groups; 2. Complex numbers, quaternions and geometry: 2.1. Complex numbers;
2.2. Quaternions; 2.3. The geometry of rotations of R3; 2.4. Putting the
pieces together; 2.5. A broader view: octonions; 3. Linearization: 3.1.
Tangent spaces; 3.2. Group homomorphisms; 3.3. Differentials; 3.4. Putting
the pieces together; 3.5. A broader view: Hilbert's fifth problem; 4. One-parameter
subgroups and the exponential map: 4.1. One-parameter subgroups; 4.2. The
exponential map in dimension one; 4.3. Calculating the matrix exponential;
4.4. Properties of the matrix exponential; 4.5. Using exp to determine
L(G); 4.6. Differential equations; 4.7. Putting the pieces together; 4.8.
A broader view: Lie and differential equations; 4.9. Appendix on convergence;
5. Lie algebras: 5.1. Lie algebras; 5.2. Adjoint maps { big `A' and small
`a'; 5.3. Putting the pieces together; 5.4. A broader view: Lie theory;
6. Matrix groups over other fields: 6.1. What is a field*; 6.2. The unitary
group; 6.3. Matrix groups over finite fields; 6.4. Putting the pieces together;
6.5. A broader view of finite groups of Lie type and simple groups; Appendix
I. Linear algebra facts; Appendix II. Paper assignment used at Mount Holyoke
College; Appendix III. Opportunities for further study; Solutions to selected
problems; Bibliography.
Series: Classics in Applied Mathematics (No. 59)
Paperback (ISBN-13: 9780898716849)
Originally published in 1986, this valuable reference provides a detailed treatment of limit theorems and inequalities for empirical processes of real-valued random variables. It also includes applications of the theory to censored data, spacings, rank statistics, quantiles, and many functionals of empirical processes, including a treatment of bootstrap methods, and a summary of inequalities that are useful for proving limit theorems. At the end of the Errata section, the authors have supplied references to solutions for 11 of the 19 Open Questions provided in the book’s original edition.
* A classic originally published in 1986 * A valuable resource for researchers
in statistical theory, probability theory, biostatistics, econometrics
and computer science * Contains a solid treatment of Martingale approaches
to right-censored data
Preface for Classics Edition; Preface; 1. Introduction and survey of results;
2. Foundations, special spaces and special processes; 3. Convergence and
distributions of empirical processes; 4. Alternatives and processes of
residuals; 5. Integral test of fit and estimated empirical process; 6.
Martingale methods; 7. Censored data: the product-limit estimator; 8. Poisson
and exponential representations; 9. Some exact distributions; 10. Linear
and nearly linear bounds on the empirical distribution function Gn; 11.
Exponential inequalities and **/q* -metric convergence of Un and Vn; 12.
The Hungarian constructions of Kn, Un, and Vn; 13. Laws of the iterated
logarithm associated with Un and Vn; 14. Oscillations of the empirical
process; 15. The uniform empirical difference process Dn≡Un + Vn; 16.
The normalized uniform empirical process Zn and the normalized uniform
quantile process; 17. The uniform empirical process indexed by intervals
and functions; 18. The standardized quantile process Qn; 19. L-statistics;
20. Rank statistics; 21. Spacing; 22. Symmetry; 23. Further applications;
24. Large deviations; 25. Independent but not identically distributed random
variable; 26. Empirical measures and processes for general spaces; Appendix
A. Inequalities and miscellaneous; Appendix B. Counting processes Martingales;
References; Errata; Author index; Subject index.
Series: Cambridge Monographs on Applied and Computational Mathematics (No. 13)
Paperback (ISBN-13: 9780521183703)
Computational simulation of scientific phenomena and engineering problems often depends on solving linear systems with a large number of unknowns. This book gives insight into the construction of iterative methods for the solution of such systems and helps the reader to select the best solver for a given class of problems. The emphasis is on the main ideas and how they have led to efficient solvers such as CG, GMRES, and BI-CGSTAB. The author also explains the main concepts behind the construction of preconditioners. The reader is encouraged to gain experience by analysing numerous examples that illustrate how best to exploit the methods. The book also hints at many open problems and as such it will appeal to established researchers. There are many exercises that motivate the material and help students to understand the essential steps in the analysis and construction of algorithms.
* Now in paperback, it incorporates corrections from the author * Based
on extensive teaching experience * Contains numerous exercises and references
for further reading
Preface; 1. Introduction; 2. Mathematical preliminaries; 3. Basic iteration
methods; 4. Construction of approximate solutions; 5. The conjugate gradients
method; 6. GMRES and MINRES; 7. Bi-conjugate gradients; 8. How serious
is irregular convergence*; 9. BI-CGSTAB; 10. Solution of singular systems;
11. Solution of f (A)x = b with Krylov subspace information; 12. Miscellaneous;
13. Preconditioning; References; Index.
Series: Cambridge Monographs on Applied and Computational Mathematics (No. 16)
Paperback (ISBN-13: 9780521136099)
The emerging field of computational topology utilizes theory from topology and the power of computing to solve problems in diverse fields. Recent applications include computer graphics, computer-aided design (CAD), and structural biology, all of which involve understanding the intrinsic shape of some real or abstract space. A primary goal of this book is to present basic concepts from topology and Morse theory to enable a non-specialist to grasp and participate in current research in computational topology. The author gives a self-contained presentation of the mathematical concepts from a computer scientist’s point of view, combining point set topology, algebraic topology, group theory, differential manifolds, and Morse theory. He also presents some recent advances in the area, including topological persistence and hierarchical Morse complexes. Throughout, the focus is on computational challenges and on presenting algorithms and data structures when appropriate.
* Presents classical topological subject of Morse theory in a computer
science context * Material is widely used within computation geometry and
computer graphics
1. Introduction; Part I. Mathematics: 2. Spaces and filtrations; 3. Group theory; 4. Homology; 5. Morse theory; 6. New results; Part II. Algorithms: 7. The persistence algorithms; 8. Topological simplification; 9. The Morse-Smale algorithm; 10. The linking number algorithm; Part III. Applications: 11. Software; 12. Experiments; 13. Applications.