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These lecturers provide a clear introduction to Lie group methods for determining and using symmetries of differential equations, a variety of their applications in gas dynamics and other nonlinear models as well as the author's remarkable contribution to this classical subject. It contains material that is useful for students and teachers but cannot be found in modern texts. For example, the theory of partially invariant solutions developed by Ovsyannikov provides a powerful tool for solving systems of nonlinear differential equations and investigating complicated mathematical models. Readers

Differential equations. --- Lie groups. --- Differential algebraic groups. --- Algebraic groups, Differential --- Differential algebra --- Group theory --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- 517.91 Differential equations --- Differential equations

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The book demonstrates the development of integral geometry on domains of homogeneous spaces since 1990. It covers a wide range of topics, including analysis on multidimensional Euclidean domains and Riemannian symmetric spaces of arbitrary ranks as well as recent work on phase space and the Heisenberg group. The book includes many significant recent results, some of them hitherto unpublished, among which can be pointed out uniqueness theorems for various classes of functions, far-reaching generalizations of the two-radii problem, the modern versions of the Pompeiu problem, and explicit reconstruction formulae in problems of integral geometry. These results are intriguing and useful in various fields of contemporary mathematics. The proofs given are “minimal” in the sense that they involve only those concepts and facts which are indispensable for the essence of the subject. Each chapter provides a historical perspective on the results presented and includes many interesting open problems. Readers will find this book relevant to harmonic analysis on homogeneous spaces, invariant spaces theory, integral transforms on symmetric spaces and the Heisenberg group, integral equations, special functions, and transmutation operators theory.

Algebra --- Algebraic geometry --- Differential geometry. Global analysis --- Harmonic analysis. Fourier analysis --- Mathematical analysis --- Mathematics --- algebra --- analyse (wiskunde) --- differentiaal geometrie --- Fourierreeksen --- functies (wiskunde) --- mathematische modellen --- wiskunde --- geometrie --- Symmetric spaces. --- Integral geometry.

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The book demonstrates the development of integral geometry on domains of homogeneous spaces since 1990. It covers a wide range of topics, including analysis on multidimensional Euclidean domains and Riemannian symmetric spaces of arbitrary ranks as well as recent work on phase space and the Heisenberg group. The book includes many significant recent results, some of them hitherto unpublished, among which can be pointed out uniqueness theorems for various classes of functions, far-reaching generalizations of the two-radii problem, the modern versions of the Pompeiu problem, and explicit reconstruction formulae in problems of integral geometry. These results are intriguing and useful in various fields of contemporary mathematics. The proofs given are “minimal” in the sense that they involve only those concepts and facts which are indispensable for the essence of the subject. Each chapter provides a historical perspective on the results presented and includes many interesting open problems. Readers will find this book relevant to harmonic analysis on homogeneous spaces, invariant spaces theory, integral transforms on symmetric spaces and the Heisenberg group, integral equations, special functions, and transmutation operators theory.

Geometry, Differential. --- Harmonic analysis. --- Integral geometry. --- Symmetric spaces. --- Symmetric spaces --- Integral geometry --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Geometry, Integral --- Spaces, Symmetric --- Mathematics. --- Integral transforms. --- Operational calculus. --- Special functions. --- Differential geometry. --- Special Functions. --- Abstract Harmonic Analysis. --- Integral Transforms, Operational Calculus. --- Differential Geometry. --- Geometry, Differential --- Functions, special. --- Integral Transforms. --- Global differential geometry. --- Transform calculus --- Integral equations --- Transformations (Mathematics) --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Mathematical analysis --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Special functions --- Differential geometry --- Operational calculus --- Differential equations --- Electric circuits

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An Introduction to Quasisymmetric Schur Functions is aimed at researchers and graduate students in algebraic combinatorics. The goal of this monograph is twofold. The first goal is to provide a reference text for the basic theory of Hopf algebras, in particular the Hopf algebras of symmetric, quasisymmetric and noncommutative symmetric functions and connections between them. The second goal is to give a survey of results with respect to an exciting new basis of the Hopf algebra of quasisymmetric functions, whose combinatorics is analogous to that of the renowned Schur functions.

Combinatorial analysis. --- Combinatorics. --- Quasisymmetric groups. --- Schur functions. --- Schur functions --- Quasisymmetric groups --- Combinatorial analysis --- Engineering & Applied Sciences --- Computer Science --- Hopf algebras. --- Combinatorics --- Algebras, Hopf --- S-functions --- Schur's functions --- Mathematics. --- Topological groups. --- Lie groups. --- Applied mathematics. --- Engineering mathematics. --- Algorithms. --- Topological Groups, Lie Groups. --- Applications of Mathematics. --- Algebra --- Mathematical analysis --- Algebraic topology --- Holomorphic functions --- Topological Groups. --- Math --- Science --- Groups, Topological --- Continuous groups --- Algorism --- Arithmetic --- Foundations --- Engineering --- Engineering analysis --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Mathematics

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This monograph lays down the foundations of the theory of complex Kleinian groups, a “newborn” area of mathematics whose origin can be traced back to the work of Riemann, Poincaré, Picard and many others. Kleinian groups are, classically, discrete groups of conformal automorphisms of the Riemann sphere, and these can themselves be regarded as groups of holomorphic automorphisms of the complex projective line CP1. When we go into higher dimensions, there is a dichotomy: Should we look at conformal automorphisms of the n-sphere? or should we look at holomorphic automorphisms of higher dimensional complex projective spaces? These two theories differ in higher dimensions. In the first case we are talking about groups of isometries of real hyperbolic spaces, an area of mathematics with a long-standing tradition; in the second, about an area of mathematics that is still in its infancy, and this is the focus of study in this monograph. It brings together several important areas of mathematics, e.g. classical Kleinian group actions, complex hyperbolic geometry, crystallographic groups and the uniformization problem for complex manifolds.

Kleinian groups. --- Differentiable dynamical systems. --- Topological Groups. --- Differential equations, partial. --- Dynamical Systems and Ergodic Theory. --- Topological Groups, Lie Groups. --- Several Complex Variables and Analytic Spaces. --- Partial differential equations --- Groups, Topological --- Continuous groups --- Differential dynamical systems --- Dynamical systems, Differentiable --- Dynamics, Differentiable --- Differential equations --- Global analysis (Mathematics) --- Topological dynamics --- Dynamics. --- Ergodic theory. --- Topological groups. --- Lie groups. --- Functions of complex variables. --- Ergodic transformations --- Mathematical physics --- Measure theory --- Transformations (Mathematics) --- Dynamical systems --- Kinetics --- Mathematics --- Mechanics, Analytic --- Force and energy --- Mechanics --- Physics --- Statics --- Complex variables --- Elliptic functions --- Functions of real variables --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups