Part of London Mathematical Society Lecture Note Series
Publication planned for: August 2015
format: Paperback
isbn: 9781107462502
This volume contains nine survey articles based on the invited lectures given at the 25th British Combinatorial Conference, held at the University of Warwick in July 2015. This biennial conference is a well-established international event, with speakers from around the world. The volume provides an up-to-date overview of current research in several areas of combinatorics, including graph theory, Ramsey theory, combinatorial geometry and curves over finite fields. Each article is clearly written and assumes little prior knowledge on the part of the reader. The authors are some of the world's foremost researchers in their fields, and here they summarise existing results and give a unique preview of cutting-edge developments. The book provides a valuable survey of the present state of knowledge in combinatorics, and will be useful to researchers and advanced graduate students, primarily in mathematics but also in computer science and statistics.
Preface Artur Czumaj, Agelos Georgakopolous, Daniel Kral', Vadim Lozin and Oleg Pikhurko
1. Ramsey classes: examples and constructions Manuel Bodirsky
2. Recent developments in graph Ramsey theory David Conlon, Jacob Fox and Benny Sudakov
3. Controllability and matchings in random bipartite graphs Paul Balister and Stefanie Gerke
4. Some old and new problems in combinatorial geometry I: around Borsuk's problem Gil Kalai
5. Randomly generated groups Tomasz ?uczak
6. Curves over finite fields and linear recurring sequences Omran Ahmadi and Gary McGuire
7. New tools and results in graph minor structure theory Sergey Norin
8. Well quasi-order in combinatorics: embeddings and homomorphisms Sophie Huczynska and Nik Ru?kuc
9. Constructions of block codes from algebraic curves over finite fields Liming Ma and Chaoping Xing.
Part of Cambridge Studies in Advanced Mathematics
Publication planned for: April 2016
isbn: 9781107108745
The study of model spaces, the closed invariant subspaces of the backward shift operator, is a vast area of research with connections to complex analysis, operator theory and functional analysis. This self-contained text is the ideal introduction for newcomers to the field. It sets out the basic ideas and quickly takes the reader through the history of the subject before ending up at the frontier of mathematical analysis. Open questions point to potential areas of future research, offering plenty of inspiration to graduate students wishing to advance further.
Preface
Notation
1. Preliminaries
2. Inner functions
3. Hardy spaces
4. Operators on the Hardy space
5. Model spaces
6. Operators between model spaces
7. Boundary behavior
8. Conjugation
9. The compressed shift
10. The commutant lifting theorem
11. Clark measures
12. Riesz bases
13. Truncated Toeplitz operators
References
Index.
Publication planned for: December 2015
format: Hardback
isbn: 9781107118508
From social networks such as Facebook, the World Wide Web and the Internet, to the complex interactions between proteins in the cells of our bodies, we constantly face the challenge of understanding the structure and development of networks. The theory of random graphs provides a framework for this understanding, and in this book the authors give a gentle introduction to the basic tools for understanding and applying the theory. Part I includes sufficient material, including exercises, for a one semester course at the advanced undergraduate or beginning graduate level. The reader is then well prepared for the more advanced topics in Parts II and III. A final part provides a quick introduction to the background material needed. All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.
Preface
Part I. Basic Models:
1. Random graphs
2. Evolution
3. Vertex degrees
4. Connectivity
5. Small subgraphs
6. Spanning subgraphs
7. Extreme characteristics
8. Extremal properties
Part II. Basic Model Extensions:
9. Inhomogeneous graphs
10. Fixed degree sequence
11. Intersection graphs
12. Digraphs
13. Hypergraphs
Part III. Other Models:
14. Trees
15. Mappings
16. k-out
17. Real-world networks
18. Weighted graphs
19. Brief notes on uncovered topics
Part IV. Tools and Methods:
20. Moments
21. Inequalities
22. Differential equations method
23. Branching processes
24. Entropy
References
Author index
Main index.
Part of Cambridge Studies in Advanced Mathematics
Publication planned for: October 2015
format: Hardback
isbn: 9781107126961
Rich with examples and applications, this textbook provides a coherent and self-contained introduction to ergodic theory, suitable for a variety of one- or two-semester courses. The authors' clear and fluent exposition helps the reader to grasp quickly the most important ideas of the theory, and their use of concrete examples illustrates these ideas and puts the results into perspective. The book requires few prerequisites, with background material supplied in the appendix. The first four chapters cover elementary material suitable for undergraduate students ? invariance, recurrence and ergodicity ? as well as some of the main examples. The authors then gradually build up to more sophisticated topics, including correlations, equivalent systems, entropy, the variational principle and thermodynamical formalism. The 400 exercises increase in difficulty through the text and test the reader's understanding of the whole theory. Hints and solutions are provided at the end of the book.
Preface
1. Recurrence
2. Existence of invariant measures
3. Ergodic theorems
4. Ergodicity
5. Ergodic decomposition
6. Unique ergodicity
7. Correlations
8. Equivalent systems
9. Entropy
10. Variational principle
11. Expanding maps
12. Thermodynamical formalism
Appendix. Topics of measure theory, topology and analysis
Hints or solutions for selected exercises
References
Index.
Part of Cambridge Monographs on Applied and Computational Mathematics
Publication planned for: December 2015
format: Hardback
isbn: 9781107001008
This book is concerned with digital image processing techniques that use partial differential equations (PDEs) for the task of image 'inpainting', an artistic term for virtual image restoration or interpolation, whereby missing or occluded parts in images are completed based on information provided by intact parts. Computer graphic designers, artists and photographers have long used manual inpainting to restore damaged paintings or manipulate photographs. Today, mathematicians apply powerful methods based on PDEs to automate this task. This book introduces the mathematical concept of PDEs for virtual image restoration. It gives the full picture, from the first modelling steps originating in Gestalt theory and arts restoration to the analysis of resulting PDE models, numerical realisation and real-world application. This broad approach also gives insight into functional analysis, variational calculus, optimisation and numerical analysis and will appeal to researchers and graduate students in mathematics with an interest in image processing and mathematical analysis.
1. Introduction
2. Overview of mathematical inpainting methods
3. The principle of good continuation
4. Second-order diffusion equations for inpainting
5. Higher-order PDE inpainting
6. Transport inpainting
7. The Mumford?Shah image for inpainting
8. Inpainting mechanisms of transport and diffusion
9. Applications.
Part of Cambridge Studies in Advanced Mathematics
Publication planned for: November 2015
format: Hardback
isbn: 9781107097605
Diophantine number theory is an active area that has seen tremendous growth over the past century, and in this theory unit equations play a central role. This comprehensive treatment is the first volume devoted to these equations. The authors gather together all the most important results and look at many different aspects, including effective results on unit equations over number fields, estimates on the number of solutions, analogues for function fields and effective results for unit equations over finitely generated domains. They also present a variety of applications. Introductory chapters provide the necessary background in algebraic number theory and function field theory, as well as an account of the required tools from Diophantine approximation and transcendence theory. This makes the book suitable for young researchers as well as experts who are looking for an up-to-date overview of the field.
Preface
Summary
Glossary of frequently used notation
Part I. Preliminaries:
1. Basic algebraic number theory
2. Algebraic function fields
3. Tools from Diophantine approximation and transcendence theory
Part II. Unit equations and applications:
4. Effective results for unit equations in two unknowns over number fields
5. Algorithmic resolution of unit equations in two unknowns
6. Unit equations in several unknowns
7. Analogues over function fields
8. Effective results for unit equations over finitely generated domains
9. Decomposable form equations
10. Further applications
References
Index.