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Research in recent years has highlighted the deep connections between the algebraic, geometric, and analytic structures of a discrete group. New methods and ideas have resulted in an exciting field, with many opportunities for new researchers. This book is an introduction to the area from a modern vantage point. It incorporates the main basics, such as Kesten's amenability criterion, Coulhon and Saloff-Coste inequality, random walk entropy and bounded harmonic functions, the Choquet-Deny Theorem, the Milnor-Wolf Theorem, and a complete proof of Gromov's Theorem on polynomial growth groups. The book is especially appropriate for young researchers, and those new to the field, accessible even to graduate students. An abundance of examples, exercises, and solutions encourage self-reflection and the internalization of the concepts introduced. The author also points to open problems and possibilities for further research.

Harmonic functions. --- Funcions harmòniques

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This text, the first of two volumes, provides a comprehensive and self-contained introduction to a wide range of fundamental results from ergodic theory and geometric measure theory. Topics covered include: finite and infinite abstract ergodic theory, Young's towers, measure-theoretic Kolmogorov-Sinai entropy, thermodynamics formalism, geometric function theory, various kinds of conformal measures, conformal graph directed Markov systems and iterated functions systems, semi-local dynamics of analytic functions, and nice sets. Many examples are included, along with detailed explanations of essential concepts and full proofs, in what is sure to be an indispensable reference for both researchers and graduate students.

Functions, Meromorphic. --- Funcions meromorfes

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Smoothness of functions. --- Smooth functions --- Functions --- Funcions --- Càlcul diferencial --- Càlcul --- Càlcul integral --- Funcions (Matemàtica) --- Funcions matemàtiques --- Anàlisi matemàtica --- Teoria de conjunts --- Aplicacions (Matemàtica) --- Constants matemàtiques --- Convergència (Matemàtica) --- Convolucions (Matemàtica) --- Funcions algebraiques --- Funcions contínues --- Funcions discontínues --- Funcions d'ona --- Funcions especials --- Teoria de l'aproximació --- Superfícies de Riemann

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Analytic continuation. --- Continuation, Analytic --- Functions of several complex variables --- Funcions de diverses variables complexes --- Variables complexes --- Funcions de variables complexes --- Aplicacions holomòrfiques --- Espais analítics --- Espais de Teichmüller --- Funcions automorfes --- Funcions holomorfes --- Teoria de mòduls

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This book provides the knowledge of the newly-established supertrigonometric and superhyperbolic functions with the special functions such as Mittag-Leffler, Wiman, Prabhakar, Miller-Ross, Rabotnov, Lorenzo-Hartley, Sonine, Wright and Kohlrausch-Williams-Watts functions, Gauss hypergeometric series and Clausen hypergeometric series. The special functions can be considered to represent a great many of the real-world phenomena in mathematical physics, engineering and other applied sciences. The audience benefits of new and original information and references in the areas of the special functions applied to model the complex problems with the power-law behaviors. The results are important and interesting for scientists and engineers to represent the complex phenomena arising in applied sciences therefore graduate students and researchers in mathematics, physics and engineering might find this book appealing.

Algebra --- Mathematics --- algebra --- functies (wiskunde) --- wiskunde --- Functions, Special. --- Funcions especials