Series: Universitext
2009, Approx. 170 p. 30 illus., Softcover
ISBN: 978-1-84800-939-4
Due: December 2008
The study of formal languages and automata has proved to be a source of much interest and discussion amongst mathematicians in recent times. This book, written by Professor Ian Chiswell, attempts to provide a comprehensive textbook for undergraduate and postgraduate mathematicians with an interest in this developing field. The first three Chapters give a rigorous proof that various notions of recursively enumerable language are equivalent. Chapter Four covers the context-free languages, whereas Chapter Five clarifies the relationship between LR(k) languages and deterministic (context-free languages). Chiswell's book is unique in that it gives the reader a thorough introduction into the connections between group theory and formal languages. This information, contained within the final chapter, includes work on the Anisimov and Muller-Schupp theorems.
Preface.- Contents.- 1. Grammars and Machine Recognition.- 2. Recursive Functions.- 3. Recursively Enumerable Sets and Languages.- 4. Context-free language.- 5. Connections with Group Theory.- A. Results and Proofs Omitted in the Text.- B. The Halting Problem and Universal Turing Machines.- C. Cantor's Diagonal Argument.- D. Solutions to Selected Exercises.- References.- Index.
Series: Springer Monographs in Mathematics
2009, Approx. 385 p. 12 illus., Hardcover
ISBN: 978-0-387-85586-8
Due: January 2009
The purpose of this book is to present a modern account of mapping theory with emphasis on quasiconformal mapping and its generalizations. The modulus method was initiated by Arne Beurling and Lars Ahlfors to study conformal mappings, and later this method was extended and enhanced by several others. The techniques are geometric and they have turned out to be an indispensable tool in the study of quasiconformal and quasiregular mappings as well as their generalizations. The book is based on recent research papers and extends the modulus method beyond the classical applications of the modulus techniques presented in many monographs.
Introduction and notation.- Moduli and capacity.- Moduli and domains.- Q-Homeomorphisms.- Q-Homeomorphisms with Q in BMO.- More General Q-Homeomorphisms.- Ring Q-Homeomorphisms.- Mappings With Finite Length Distortions (FLD).- Lower Q-Homeomorphisms.- Mappings With Finite Area Distortions (FAD).- On Ring Solutions of the Beltrami Equation.- Homeomorphisms with Finite Means Dilations.- On the Mapping Theory in Metric Spaces.- Appendix.- Index.-
Series: Undergraduate Texts in Mathematics
2009, Approx. 170 p. 45 illus., 10 in color., Hardcover
ISBN: 978-0-387-85524-0
Due: January 2009
This is a textbook about classical elementary number theory and elliptic curves. The first part discusses elementary topics such as primes, factorization, continued fractions, and quadratic forms, in the context of cryptography, computation, and deep open research problems. The second part is about elliptic curves, their applications to algorithmic problems, and their connections with problems in number theory such as Fermat's Last Theorem, the Congruent Number Problem, and the Conjecture of Birch and Swinnerton-Dyer. The intended audience of this book is an undergraduate with some familiarity with basic abstract algebra, e.g. rings, fields, and finite abelian groups
Preface.- Prime Numbers.- The Ring of Integers Modulo n.- Public-Key Cryptography.- Quadratic Reciprocity.- Continued Fractions.- Elliptic Curves.- Answers and Hints.- References.-
Series: Graduate Texts in Mathematics , Vol. 251
2009, Approx. 220 p., Hardcover
ISBN: 978-1-84800-987-5
Due: February 15, 2009
The finite simple groups are the building blocks from which all the finite groups are made and as such are objects of fundamental importance throughout mathematics. They can be divided into the alternating, sporadic and Lie type groups, the latter further subdivided into classical and exceptional, or into untwisted and twisted types. This book is the first accessible introduction to all these families of finite simple groups at a level suitable for final year undergraduate and beginning graduate students.
The first five chapters provide a thorough grounding in the theory of the alternating and classical groups, followed by an overview of the exceptional groups (treated as automorphism groups of multilinear forms) and the sporadic groups. These chapters form the basis of a final year undergraduate course bringing their undergraduate studies to a fitting climax with seminal results from the late 20th century.
The final two chapters give an introduction to the theory of Lie algebras and Chevalley groups (which provides a unified approach to all the untwisted finite groups of Lie type) and to algebraic groups (which unites the twisted and untwisted types). These final chapters are ideal guides for undergraduate projects and prepare the students for further reading in more advanced texts on these important topics.
Contents. 1 Introduction. 1.1 Simple groups. 1.2 The Classification Theorem. 1.3 Facts about simple groups. 1.4 Prerequisites. 1.5 How to read this book. 2 The alternating groups. 2.1 Introduction. 2.2 Permutations. 2.3 Simplicity. 2.4 Outer automorphisms. 2.5 The outer automorphism of S6. 2.6 Subgroups of Sn. 2.7 The O’Nan-Scott Theorem. 2.8 The Schur double covers. 2.9 The exceptional triple covers. 2.10 Presentations. 2.11 Coxeter groups. 3 Classical groups. 3.1 Introduction. 3.2 Finite fields. 3.3 General linear groups. 3.4 Bilinear, sesquilinear and quadratic forms. 3.5 Symplectic groups. 3.6 Unitary groups. 3.7 Orthogonal groups in odd characteristic. 3.8 Orthogonal groups in characteristic 2. 3.9 Maximal subgroups of classical groups. 3.10 Clifford algebras and spin groups. 3.11 Exceptional covers and isomorphisms. 4 Exceptional groups. 4.1 Introduction. 4.2 Octonions and groups of type G2. 4.3 Triality. 4.4 Exceptional Jordan algebras and groups of type F4. 4.5 Trilinear forms and groups of type E6. 4.6 Groups of type E7 and E8. 4.7 Steinberg twisted groups. 4.8 An integral Jordan algebra. 4.9 The Suzuki groups. 4.10 The Ree groups. 4.11 Exceptional multipliers. 5 Sporadic groups. 5.1 Introduction. 5.2 The Mathieu groups. 5.3 The Leech lattice and the Conway groups. 5.4 The Fischer groups. 5.5 The Monster and Subgroups of the Monster. 5.6 Pariahs. 6 Conclusions.