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An organised step-by-step introduction to the theory of compact quantum groups, starting with examples coming from quantum physics, which stems from the basic undergraduate mathematics curriculum. Introducing more abstract concepts along the way when needed, the reader is led from the fundamentals of the theory to recent research results. The emphasis is put on the combinatorics underlying compact quantum groups, which is very elementary to describe but leads to profound results. This book includes many exercises to help students work through new concepts and ideas and consolidate their understanding. The theory itself is illustrated by an array of examples, some related to other fields of Mathematics such as free probability theory or graph theory. The book is intended for graduate students, motivated undergraduate students and researchers.

Quantum groups. --- Representations of quantum groups. --- Combinatorial analysis. --- Grups quàntics --- Anàlisi combinatòria

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In 1938, at the Institute for Advanced Study, E Hecke gave a series of lectures on his theory of correspondence between modular forms and Dirichlet series. Since then, the Hecke correspondence has remained an active feature of number theory and, indeed, it is more important today than it was in 1936 when Hecke published his original papers. This book is an amplified and up-to-date version of the former author's lectures at the University of Illinois at Urbana-Champaign, based on Hecke's notes. Providing many details omitted from Hecke's notes, it includes various new and important developments

Dirichlet series. --- Forms (Mathematics) --- Modular functions. --- Hecke operators. --- Operators, Hecke --- Forms, Modular --- Operator theory --- Functions, Modular --- Elliptic functions --- Group theory --- Number theory --- Quantics --- Algebra --- Mathematics --- Series, Dirichlet --- Series

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Mathematical analysis --- Forms (Mathematics) --- Numerical analysis --- Polynomials --- Waring's problem --- Waring problem --- Partitions (Mathematics) --- Algebra --- Quantics --- Mathematics --- Waring, Problème de --- Polynômes --- Analyse numérique --- Formes (mathématiques) --- Waring, Problème de. --- Polynômes. --- Analyse numérique.

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Automorphic forms are an important complex analytic tool in number theory and modern arithmetic geometry. They played for example a vital role in Andrew Wiles's proof of Fermat's Last Theorem. This text provides a concise introduction to the world of automorphic forms using two approaches: the classic elementary theory and the modern point of view of adeles and representation theory. The reader will learn the important aims and results of the theory by focussing on its essential aspects and restricting it to the 'base field' of rational numbers. Students interested for example in arithmetic geometry or number theory will find that this book provides an optimal and easily accessible introduction into this topic.

Automorphic forms. --- Forms (Mathematics) --- Quantics --- Mathematics. --- Algebra. --- Group theory. --- Number theory. --- Mathematics, general. --- Number Theory. --- Group Theory and Generalizations. --- Algebra --- Mathematics --- Automorphic functions --- Mathematical analysis --- Groups, Theory of --- Substitutions (Mathematics) --- Number study --- Numbers, Theory of --- Math --- Science

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Mathematics. --- Física matemàtica --- Math --- Science --- Mecànica --- Acústica --- Anàlisi de sistemes --- Anàlisi dimensional --- Grups quàntics --- Elasticitat --- Equació de Yang-Baxter --- Matemàtica en l'electrònica --- Problemes de contorn --- Teoria del potencial (Física) --- Teoria ergòdica --- Teories no lineals --- Rutes aleatòries (Matemàtica)