Series: Lecture Notes in Logic
Hardback (ISBN-13: 9780521760652)
The Annual European Meeting of the Association for Symbolic Logic, also known as the Logic Colloquium, is among the most prestigious annual meetings in the field. The current volume, Logic Colloquium 2007, with contributions from plenary speakers and selected special session speakers, contains both expository and research papers by some of the best logicians in the world. This volume covers many areas of contemporary logic: model theory, proof theory, set theory, and computer science, as well as philosophical logic, including tutorials on cardinal arithmetic, on Pillay’s conjecture, and on automatic structures. This volume will be invaluable for experts as well as those interested in an overview of central contemporary themes in mathematical logic.
* Fully refereed proceedings of the Logic Colloquium 2007 made after the conference * Contains surveys based on tutorial talks * Contains full scientific articles
1. Decorated linear order types and the theory of concatenation Vedran Cacic, Pavel Pudlak, Greg Restall, Alasdair Urquhart and Albert Visser; 2. Cardinal preserving elementary embeddings Andres Eduardo Caicedo; 3. Proof interpretations and majorizability Fernando Ferreira; 4. Proof mining in practice Philipp Gerhardy; 5. Cardinal structure under AD Steve Jackson; 6. Three lectures on automatic structures Bakhadyr Khoussainov and Mia Minnes; 7. Pillay's conjecture and its solution - a survey Ya'acov Peterzil; 8. Proof theory and meaning: on the context of deducibility Greg Restall; 9. Bounded super real closed rings Marcus Tressl; 10. Analytic combinatorics of the transfinite: a unifying Tauberian perspective Andreas Weiermann.
Series: Encyclopedia of Mathematics and its Applications (No. 135)
Hardback (ISBN-13: 9780521515979)
This collaborative volume presents recent trends arising from the fruitful interaction between the themes of combinatorics on words, automata and formal language theory, and number theory. Presenting several important tools and concepts, the authors also reveal some of the exciting and important relationships that exist between these different fields. Topics include numeration systems, word complexity function, morphic words, Rauzy tilings and substitutive dynamical systems, Bratelli diagrams, frequencies and ergodicity, Diophantine approximation and transcendence, asymptotic properties of digital functions, decidability issues for D0L systems, matrix products and joint spectral radius. Topics are presented in a way that links them to the three main themes, but also extends them to dynamical systems and ergodic theory, fractals, tilings and spectral properties of matrices. Graduate students, research mathematicians and computer scientists working in combinatorics, theory of computation, number theory, symbolic dynamics, fractals, tilings and stringology will find much of interest in this book.
* Provides a useful overview of the most recent results * Written for readers with either a mathematical or computer science background * Chapters are self-contained so readers do not have to consult extra material
Introduction Valerie Berthe and Michel Rigo; 1. Preliminaries; 2. Number representation and finite automata Ch. Frougny and J. Sakarovitch; 3. Abstract numeration systems P. Lecomte and M. Rigo; 4. Factor complexity J. Cassaigne and F. Nicolas; 5. Substitutions, Rauzy fractals, and tilings V. Berthe, A. Siegel and J. Thuswaldner; 6. Combinatorics on Bratelli diagrams and dynamical systems F. Durand; 7. Infinite words with uniform frequencies, and invariant measures S. Ferenczi and T. Monteil; 8. Transcendence and Diophantine approximation B. Adamczewski and Y. Bugeaud; 9. Analysis of digital functions and applications M. Drmota and P. Grabner; 10. The equality problem for purely substitutive words J. Honkala; 11. Long products of matrices V. Blondel and R. Jungers; References; Notation index; General index.
Series: Cambridge Tracts in Mathematics (No. 182)
Hardback (ISBN-13: 9780521111843)
A nonlinear Markov evolution is a dynamical system generated by a measure-valued ordinary differential equation with the specific feature of preserving positivity. This feature distinguishes it from general vector-valued differential equations and yields a natural link with probability, both in interpreting results and in the tools of analysis. This brilliant book, the first devoted to the area, develops this interplay between probability and analysis. After systematically presenting both analytic and probabilistic techniques, the author uses probability to obtain deeper insight into nonlinear dynamics, and analysis to tackle difficult problems in the description of random and chaotic behavior. The book addresses the most fundamental questions in the theory of nonlinear Markov processes: existence, uniqueness, constructions, approximation schemes, regularity, law of large numbers and probabilistic interpretations. Its careful exposition makes the book accessible to researchers and graduate students in stochastic and functional analysis with applications to mathematical physics and systems biology.
* Presents a multidimensional view of the subject by exploring different methods * End-of-chapter exercises enable the reader to test their understanding * Applications of these processes include non-equilibrium statistical mechanics, evolutionary biology, population and disease dynamics, and dynamics of economic and social systems
Preface; Basic notations; 1. Introduction; Part I. Tools From Markov Processes: 2. Probability and analysis; 3. Probabilistic constructions; 4. Analytic constructions; 5. Unbounded coefficients; Part II. Nonlinear Markov Processes and Semigroups: 6. Integral generators; 7. Generators of Levy*Khintchine type; 8. Smoothness with respect to initial data; Part III. Applications to Interacting Particles: 9. The dynamic law of large numbers; 10. The dynamic central limit theorem; 11. Developments and comments; 12. Appendices; References; Index.
Series: London Mathematical Society Lecture Note Series (No. 375)
Paperback (ISBN-13: 9780521744317)
Over the last few decades triangulated categories have become increasingly important, to the extent that they can now be viewed as a unifying theory underlying major parts of modern mathematics. This collection of survey articles, written by leading experts, covers fundamental aspects of triangulated categories, as well as applications in algebraic geometry, representation theory, commutative algebra, microlocal analysis and algebraic topology. These self-contained articles are a useful introduction for graduate students entering the field and a valuable reference for experts.
* Ideal reference for mathematicians interested in modern aspects of triangulated categories * Explores applications from very different areas of mathematics * A suitable introduction to the subject for graduate students
Preface; Introduction; 1. Cohomology over complete intersections via exterior algebras Luchezar Avramov and Srikanth Iyengar; 2. Cluster algebras, quiver representations and triangulated categories Bernhard Keller; 3. Localization for triangulated categories Henning Krause; 4. Homological algebra in bivariant K-theory and other triangulated categories Ralf Meyer and Ryszard Nest; 5. Derived categories and Grothendieck duality Amnon Neeman; 6. Algebraic versus topological triangulated categories Stefan Schwede; 7. Derived categories and algebraic geometry Raphael Rouquier; 8. Triangulated categories for the analysts Pierre Schapira; 9. Derived categories of coherent sheaves on algebraic varieties Yukinobu Toda; 10. Rigid dualizing complexes via differential graded algebras Amnon Yekutieli.
Series: Encyclopedia of Mathematics and its Applications (No. 136)
Hardback (ISBN-13: 9780521117821)
This comprehensive introduction to the calculus of variations and its main principles also presents their real-life applications in various contexts: mathematical physics, differential geometry, and optimization in economics. Based on the authors' original work, it provides an overview of the field with examples and exercises suitable for graduate students entering research. The method of presentation will appeal to readers with diverse backgrounds in functional analysis, differential geometry and partial differential equations. Each chapter includes detailed heuristic arguments, providing thorough motivation for the material developed later in the text. Since much of the material has a strong geometric flavor, the authors have supplemented the text with many figures to illustrate the abstract concepts. Its extensive bibliography, glossary and index also make this a valuable reference for researchers working in a variety of fields who are interested in partial differential equations and functional analysis.
* Rich with examples, exercises, figures and historical comments and includes a rich index and a comprehensive reference list * Provides theoretical methods that allow the reader to develop research in basic fields connected with applications * Contains new, previously unpublished material
Foreword Jean Mawhin; Preface; Part I. Variational Principles in Mathematical Physics: 1. Variational principles; 2. Variational inequalities; 3. Nonlinear eigenvalue problems; 4. Elliptic systems of gradient type; 5. Systems with arbitrary growth nonlinearities; 6. Scalar field systems; 7. Competition phenomena in Dirichlet problems; 8. Problems to Part I; Part II. Variational Principles in Geometry: 9. Sublinear problems on Riemannian manifolds; 10. Asymptotically critical problems on spheres; 11. Equations with critical exponent; 12. Problems to Part II; Part III. Variational Principles in Economics: 13. Mathematical preliminaries; 14. Minimization of cost-functions on manifolds; 15. Best approximation problems on manifolds; 16. A variational approach to Nash equilibria; 17. Problems to Part III; Appendix A. Elements of convex analysis; Appendix B. Function spaces; Appendix C. Category and genus; Appendix D. Clarke and Degiovanni gradients; Appendix E. Elements of set-valued analysis; References; Index.
Series: Encyclopedia of Mathematics and its Applications (No. 132)
Hardback (ISBN-13: 9780521762687)
Relational mathematics is to operations research and informatics what numerical mathematics is to engineering: it is intended to help modelling, reasoning, and computing. Its applications are therefore diverse, ranging from psychology, linguistics, decision aid, and ranking to machine learning and spatial reasoning. Although many developments have been made in recent years, they have rarely been shared amongst this broad community of researchers. This first comprehensive overview begins with an easy introduction to the topic, assuming a minimum of prerequisites; but it is nevertheless theoretically sound and up to date. It is suitable for applied scientists, explaining all the necessary mathematics from scratch using a multitude of visualised examples, via matrices and graphs. It ends with tangible results on the research level. The author illustrates the theory and demonstrates practical tasks in operations research, social sciences and the humanities.
* A thorough reference for researchers working in a wide variety of fields * The theory is illustrated by a wealth of exercises and practical examples * Assumes only basic prerequisites with mathematics explained from scratch
Preface; 1. Introduction; Part I. Representations of Relations: 2. Sets, subsets and elements; 3. Relations; Part II. Operations and Constructions: 4. Algebraic operations on relations; 5. Order and function: the standard view; 6. Relations and vectors; 7. Domain construction; Part III. Algebra: 8. Relation algebra; 9. Orders and lattices; 10. Rectangles, fringes, inverses; 11. Concept analysis; Part IV. Applications: 12. Orderings: an advanced view; 13. Preference and indifference; 14. Aggregating preferences; 15. Relational graph theory; 16. Standard Galois mechanisms; Part V. Advanced Topics: 17. Mathematical applications; 18. Implication structures; 19. Power operations; Appendix A. Notations; Appendix B. Postponed proofs of Part II; Appendix C. Algebraic visualization; Appendix D. Historical annotations; Table of symbols; References; Index.