Non-relativistic quantum theory

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Christian Kassel

This book provides an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and on Drinfeld's recent fundamental contributions. The first part presents in detail the quantum groups attached to SL2 as well as the basic concepts of the theory of Hopf algebras. Part Two focuses on Hopf algebras that produce solutions of the Yang-Baxter equation, and on Drinfeld's quantum double construction. In the following part we construct isotopy invariants of knots and links in the three-dimensional Euclidean space, using the language of tensor categories. The last part is an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations, culminating in the construction of Kontsevich's universal knot invariant.

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Jukka Saranen, Gennadi Vainikko

Classical boundary integral equations arising from the potential theory and acoustics (Laplace and Helmholtz equations) are derived. Using the parametrization of the boundary these equations take a form of periodic pseudodifferential equations. A general theory of periodic pseudodifferential equations and methods of solving are developed, including trigonometric Galerkin and collocation methods, their fully discrete versions with fast solvers, quadrature and spline based methods. The theory of periodic pseudodifferential operators is presented in details, with preliminaries (Fredholm operators, periodic distributions, periodic Sobolev spaces) and full proofs. This self-contained monograph can be used as a textbook by graduate/postgraduate students. It also contains a lot of carefully chosen exercises.

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P.K. Jain, O.P. Ahuja, Khalil Ahmad

The book is intended to serve as a textbook for an introductory course in functional analysis for the senior undergraduate and graduate students. It can also be useful for the senior students of applied mathematics, statistics, operations research, engineering and theoretical physics. The text starts with a chapter on Preliminaries discussing basic concepts and results which would be taken for granted later in the book. This is followed by chapters on Normed and Banach Spaces, Bounded Linear Operators, Bounded Linear Functionals, The Concept and Specific Geometry of Hilbert Spaces, Functionals and Operators on Hilbert Spaces and Introduction to Spectral Theory. An appendix have been given on Schauder Bases. The salient features of the book are presentation of the subject in a natural way, description of the concepts with justification, clear and precise exposition avoiding pendantry, various examples and counter examples, and graded problems throughout each chapter. Notes and remarks within the text enhances the utility of the book for the students.

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Guoqing Chen, Mingsheng Ying, Kai-Yuan Cai

Fuzzy Logic and Soft Computing contains contributions from world-leading experts from both the academic and industrial communities. The first part of the volume consists of invited papers by international authors describing possibilistic logic in decision analysis, fuzzy dynamic programming in optimization, linguistic modifiers for word computation, and theoretical treatments and applications of fuzzy reasoning. The second part is composed of eleven contributions from Chinese authors focusing on some of the key issues in the fields: stable adaptive fuzzy control systems, partial evaluations and fuzzy reasoning, fuzzy wavelet neural networks, analysis and applications of genetic algorithms, partial repeatability, rough set reduction for data enriching, limits of agents in process calculus, medium logic and its evolution, and factor spaces canes. These contributions are not only theoretically sound and well-formulated, but are also coupled with applicability implications and/or implementation treatments. The domains of applications realized or implied are: decision analysis, word computation, databases and knowledge discovery, power systems, control systems, and multi-destinational routing. Furthermore, the articles contain materials that are an outgrowth of recently conducted research, addressing fundamental and important issues of fuzzy logic and soft computing.