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This monograph presents the basic concepts of hyperbolic Lobachevsky geometry and their possible applications to modern nonlinear applied problems in mathematics and physics, summarizing the findings of roughly the last hundred years. The central sections cover the classical building blocks of hyperbolic Lobachevsky geometry, pseudo spherical surfaces theory, net geometrical investigative techniques of nonlinear differential equations in partial derivatives, and their applications to the analysis of the physical models. As the sine-Gordon equation appears to have profound “geometrical roots” and numerous applications to modern nonlinear problems, it is treated as a universal “object” of investigation, connecting many of the problems discussed. The aim of this book is to form a general geometrical view on the different problems of modern mathematics, physics and natural science in general in the context of non-Euclidean hyperbolic geometry.

Geometry, Hyperbolic. --- Geometry, Algebraic. --- Differential equations, Partial. --- Partial differential equations --- Algebraic geometry --- Geometry --- Hyperbolic geometry --- Lobachevski geometry --- Lobatschevski geometry --- Geometry, Non-Euclidean --- Geometry, algebraic. --- Differential equations, partial. --- Algebraic Geometry. --- Partial Differential Equations. --- Mathematical Physics. --- Algebraic geometry. --- Partial differential equations. --- Mathematical physics. --- Physical mathematics --- Physics --- Mathematics

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Group theory --- Representations of groups. --- 512.547 --- Representations of groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Linear representations of abstract groups. Group characters --- 512.547 Linear representations of abstract groups. Group characters --- Groupes finis --- Lie, Groupes de --- Représentations de groupes --- Groupes topologiques --- Groupes compacts

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Series of scalars, vectors, or functions are among the fundamental objects of mathematical analysis. When the arrangement of the terms is fixed, investigating a series amounts to investigating the sequence of its partial sums. In this case the theory of series is a part of the theory of sequences, which deals with their convergence, asymptotic behavior, etc. The specific character of the theory of series manifests itself when one considers rearrangements (permutations) of the terms of a series, which brings combinatorial considerations into the problems studied. The phenomenon that a numerical series can change its sum when the order of its terms is changed is one of the most impressive facts encountered in a university analysis course. The present book is devoted precisely to this aspect of the theory of series whose terms are elements of Banach (as well as other topological linear) spaces. The exposition focuses on two complementary problems. The first is to char­ acterize those series in a given space that remain convergent (and have the same sum) for any rearrangement of their terms; such series are usually called uncon­ ditionally convergent. The second problem is, when a series converges only for certain rearrangements of its terms (in other words, converges conditionally), to describe its sum range, i.e., the set of sums of all its convergent rearrangements.

Banach spaces. --- Convergence. --- Banach spaces --- Convergence --- Functional analysis. --- Mathematical analysis. --- Analysis (Mathematics). --- Geometry. --- Functional Analysis. --- Analysis. --- Mathematics --- Euclid's Elements --- 517.1 Mathematical analysis --- Mathematical analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Analyse fonctionnelle --- Functional analysis --- Series. --- Algebra --- Processes, Infinite --- Sequences (Mathematics) --- Functions --- Functions of complex variables --- Generalized spaces --- Topology --- Séries (mathématiques) --- Series (mathematique) --- Sommation

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Differential equations, Linear --- Selfadjoint operators --- Wiener-Hopf equations