Series: Cambridge Series in Statistical and Probabilistic Mathematics (No. 13)
Paperback (ISBN-13: 9780521121057)
Page extent: 256 pages
This book was first published in 2003. Derived from extensive teaching experience in Paris, this book presents around 100 exercises in probability. The exercises cover measure theory and probability, independence and conditioning, Gaussian variables, distributional computations, convergence of random variables, and random processes. For each exercise the authors have provided detailed solutions as well as references for preliminary and further reading. There are also many insightful notes to motivate the student and set the exercises in context. Students will find these exercises extremely useful for easing the transition between simple and complex probabilistic frameworks. Indeed, many of the exercises here will lead the student on to frontier research topics in probability. Along the way, attention is drawn to a number of traps into which students of probability often fall. This book is ideal for independent study or as the companion to a course in advanced probability theory.
* Class tested at the prestigious Paris school * Draws attention to a number of traps that must be avoided by students of probability * Includes detailed solutions to all exercises as well as references to the literature and contextual notes
1. Measure theory and probability; 2. Independence and conditioning; 3. Gaussian variables; 4. Distributional computations; 5. Convergence of random variables; 6. Random processes.
Series: New Mathematical Library (No. 44)
Paperback (ISBN-13: 9780883856482)
Page extent: 294 pages
The familiar plane geometry of secondary school - figures composed of lines and circles - takes on a new life when viewed as the study of properties that are preserved by special groups of transformations. No longer is there a single, universal geometry: different sets of transformations of the plane correspond to intriguing, disparate geometries. This book is the concluding Part IV of Geometric Transformations, but it can be studied independently of Parts I, II, and III. The present Part IV develops the geometry of transformations of the plane that map circles to circles (conformal or anallagmatic geometry). The notion of inversion, or reflection in a circle, is the key tool employed. Applications include ruler-and-compass constructions and the Poincare model of hyperbolic geometry. The straightforward, direct presentation assumes only some background in elementary geometry and trigonometry.
* Numerous exercises lead the reader to a mastery of the methods and concepts * The second half of the book contains detailed solutions of all the problems * Accessible to undergraduates - assumes only familiarity with elementary geometry and trigonometry
1. Reflection in a circle (inversion); Notes to Section 1; 2. Application of inversions to the solution of construction; Problems: constructions with compass alone; Problems involving the construction of circles; Notes to Section 2; 3. Pencils of circles. The radical axis of two circles; Notes to Section 3; 4. Inversion (concluding section); Notes to Section 4; 5. Axial circular transformations; A. Dilatation; B. Axial inversion; Notes to Section 5; Supplement I; Non-euclidean geometry of Lobachevskii-Bolyai, or hyperbolic geometry; Notes to Supplement I; Solutions; Section 1; Section 2; Section 3; Circular transformations; Section 4; Section 5; Supplement II; Notes to Supplement II.
Hardback (ISBN-13: 9780898716856)
Page extent: 554 pages
Full of features and applications, this acclaimed textbook for upper undergraduate level and graduate level students includes all the major topics of computational linear algebra, including solution of a system of linear equations, least-squares solutions of linear systems, computation of eigenvalues, eigenvectors, and singular value problems. Drawing from numerous disciplines of science and engineering, the author covers a variety of motivating applications. When a physical problem is posed, the scientific and engineering significance of the solution is clearly stated. Each chapter contains a summary of the important concepts developed in that chapter, suggestions for further reading, and numerous exercises, both theoretical and MATLABR and MATCOM based. The author also provides a list of key words for quick reference. The MATLAB toolkit available online, ‘MATCOM', contains implementations of the major algorithms in the book and will enable students to study different algorithms for the same problem, comparing efficiency, stability, and accuracy.
* Online content includes appendices containing MATLAB codes and the MATCOM toolkit solutions to selected problems, as well as an extra chapter on special topics * The important topics of generalized and quadratic eigenvalue problems that arise in practical engineering applications are described in great detail * To help stimulate the creativity of students, the algorithms are presented in a way so that they are readily usable in a computational setting
Preface; 1. Linear algebra problems, their importance, and computational difficulties; 2. A review of some required concepts from core linear algebra; 3. Floating point numbers and errors in computations; 4. Stability of algorithms and conditioning of problems; 5. Gaussian elimination and LU factorization; 6. Numerical solutions of linear systems; 7. QR factorization, singular value decomposition, and projections; 8. Least-squares solutions to linear systems; 9. Numerical matrix eigenvalue problems; 10. Numerical symmetric eigenvalue problem and singular value decomposition; 11. Generalized and quadratic eigenvalue problems; 12. Iterative methods for large and sparse problems: an overview; 13. Key terms in numerical linear algebra; Bibliography; Index; Online materials; 14. Special topics; Appendix A. Some software for matrix computations; Appendix B. A brief introduction to MATLABR; Appendix C. MATCOM and selected MATCOM commands; Appendix D. Partial solutions and answers to selected problems.
Series: Lecture Notes in Logic
Hardback (ISBN-13: 9780521115148)
Page extent: 375 pages
Kurt Godel (1906*1978) did groundbreaking work that transformed logic and other important aspects of our understanding of mathematics, especially his proof of the incompleteness of formalized arithmetic. This book on different aspects of his work and on subjects in which his ideas have contemporary resonance includes papers from a May 2006 symposium celebrating Godel’s centennial as well as papers from a 2004 symposium. Proof theory, set theory, philosophy of mathematics, and the editing of Godel’s writings are among the topics covered. Several chapters discuss his intellectual development and his relation to predecessors and contemporaries such as Hilbert, Carnap, and Herbrand. Others consider his views on justification in set theory in light of more recent work and contemporary echoes of his incompleteness theorems and the concept of constructible set.
* The only existing collection of essays devoted to Godel's work with a broad focus * It combines mathematical work, historical analysis, and philosophy
Part I. General: 1. The Godel editorial project: a synopsis Solomon Feferman; 2. Future tasks for Godel scholars John W. Dawson, Jr., and Cheryl A. Dawson; Part II. Proof Theory: 3. Kurt Godel and the metamathematical tradition Jeremy Avigad; 4. Only two letters: the correspondence between Herbrand and Godel Wilfried Sieg; 5. Godel's reformulation of Gentzen's first consistency proof for arithmetic: the no-counter-example interpretation W. W. Tait; 6. Godel on intuition and on Hilbert's finitism W. W. Tait; 7. The Godel hierarchy and reverse mathematics Stephen G. Simpson; 8. On the outside looking in: a caution about conservativeness John P. Burgess; Part III. Set Theory: 9. Godel and set theory Akihiro Kanamori; 10. Generalizations of Godel's universe of constructible sets Sy-David Friedman; 11. On the question of absolute undecidability Peter Koellner; Part IV: Philosophy of Mathematics: 12. What did Godel believe and when did he believe it* Martin Davis; 13. On Godel's way in: the influence of Rudolf Carnap Warren Goldfarb; 14. Godel and Carnap Steve Awodey and A. W. Carus; 15. On the philosophical development of Kurt Godel Mark van Atten and Juliette Kennedy; 16. Platonism and mathematical intuition in Kurt Godel's thought Charles Parsons; 17. Godel's conceptual realism Donald A. Martin.
Paperback (ISBN-13: 9780521131384)
50 exercises
Page extent: 300 pages
Aimed at advanced undergraduates, this self-contained textbook covers the key ideas of special and general relativity together with their applications. The textbook introduces students to basic geometric concepts, such as metrics, connections and curvature, before examining general relativity in more detail. It shows the observational evidence supporting the theory, and the description general relativity provides of black holes and cosmological space-times. The textbook is in full colour, with numerous worked examples and exercises with solutions. Key points and equations are highlighted for easy identification, and each chapter ends with a summary list of important concepts and results. This textbook provides the essential background for an up-to-date discussion of modern observational cosmology. Each chapter builds on the previous one as concepts are developed, making it ideal for self-study.
* Provides the essential background for an up-to-date discussion of modern observational cosmology * Each chapter builds on the previous one as concepts are developed, making it ideal for self-study * Contains worked examples, exercises with solutions, and summary lists of important concepts and results
1. Special relativity and spacetime; 2. Special relativity and physical laws; 3. Geometry and curved spacetime; 4. General relativity; 5. The Schwarzschild solution and black holes; 6. Testing general relativity; 7. Cosmological solutions; 8. Our Universe; Index.
Series: Lecture Notes in Logic
Hardback (ISBN-13: 9780521119696)
Page extent: 176 pages
This book presents a unifying framework for using priority arguments to prove theorems in computability. Priority arguments provide the most powerful theorem-proving technique in the field, but most of the applications of this technique are ad hoc, masking the unifying principles used in the proofs. The proposed framework presented isolates many of these unifying combinatorial principles and uses them to give shorter and easier-to-follow proofs of computability-theoretic theorems. Standard theorems of priority levels 1, 2, and 3 are chosen to demonstrate the framework’s use, with all proofs following the same pattern. The last section features a new example requiring priority at all finite levels. The book will serve as a resource and reference for researchers in logic and computability, helping them to prove theorems in a shorter and more transparent manner.
* Presents a new approach to priority argument proofs in computability theory using a framework * Present isolation of the general combinatorial properties used in priority arguments from the techniques special to particular theorems * Offers a presentation of shorter proofs of standard computability-theoretic theorems at lower levels, all following the same pattern4. A new and greatly simplified proof of a theorem whose proof uses priority arguments at all finite levels
1. Introduction; 2. Systems of trees of strategies; 3. Σ1 constructions; 4. Δ2 constructions; 5. μ2 constructions; 6. Δ3 constructions; 7. Σ3 constructions; 8. Paths and links; 9. Backtracking; 10. Higher level constructions; 11. Infinite systems of trees.