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This book studies the modules arising in Fourier expansions of automorphic forms, namely Fourier term modules on SU(2,1), the smallest rank one Lie group with a non-abelian unipotent subgroup. It considers the “abelian” Fourier term modules connected to characters of the maximal unipotent subgroups of SU(2,1), and also the “non-abelian” modules, described via theta functions. A complete description of the submodule structure of all Fourier term modules is given, with a discussion of the consequences for Fourier expansions of automorphic forms, automorphic forms with exponential growth included. These results can be applied to prove a completeness result for Poincaré series in spaces of square integrable automorphic forms. Aimed at researchers and graduate students interested in automorphic forms, harmonic analysis on Lie groups, and number-theoretic topics related to Poincaré series, the book will also serve as a basic reference on spectral expansion with Fourier-Jacobi coefficients. Only a background in Lie groups and their representations is assumed.

Number theory. --- Fourier analysis. --- Topological groups. --- Lie groups. --- Number Theory. --- Fourier Analysis. --- Topological Groups and Lie Groups. --- Teoria de nombres --- Anàlisi de Fourier

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This book presents a novel theory of multibody dynamics with distinct features, including unified continuum theory, multiscale modeling technology of multibody system, and motion formalism implementation. All these features together with the introductions of fundamental concepts of vector, dual vector, tensor, dual tensor, recursive descriptions of joints, and the higher-order implicit solvers formulate the scope of the book’s content. In this book, a multibody system is defined as a set consisted of flexible and rigid bodies which are connected by any kinds of joints or constraints to achieve the desired motion. Generally, the motion of multibody system includes the translation and rotation; it is more efficient to describe the motion by using the dual vector or dual tensor directly instead of defining two types of variables, the translation and rotation separately. Furthermore, this book addresses the detail of motion formalism and its finite element implementation of the solid, shell-like, and beam-like structures. It also introduces the fundamental concepts of mechanics, such as the definition of vector, dual vector, tensor, and dual tensor, briefly. Without following the Einstein summation convention, the first- and second-order tensor operations in this book are depicted by linear algebraic operation symbols of row array, column array, and two-dimensional matrix, making these operations easier to understand. In addition, for the integral of governing equations of motion, a set of ordinary differential equations for the finite element-based discrete system, the book discussed the implementation of implicit solvers in detail and introduced the well-developed RADAU IIA algorithms based on post-error estimation to make the contents of the book complete. The intended readers of this book are senior engineers and graduate students in related engineering fields.

Multibody systems. --- Vibration. --- Mechanics, Applied. --- Solids. --- Topological groups. --- Lie groups. --- Multibody Systems and Mechanical Vibrations. --- Solid Mechanics. --- Topological Groups and Lie Groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Groups, Topological --- Continuous groups --- Solid state physics --- Transparent solids --- Applied mechanics --- Engineering, Mechanical --- Engineering mathematics --- Cycles --- Mechanics --- Sound --- Multi-body systems --- Systems, Multibody --- Mechanics, Analytic

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This book explores generalized Lorenz–Mie theories when the illuminating beam is an electromagnetic arbitrary shaped beam relying on the method of separation of variables. Although it particularly focuses on the homogeneous sphere, the book also considers other regular particles. It discusses in detail the methods available for evaluating beam shape coefficients describing the illuminating beam. In addition it features applications used in many fields such as optical particle sizing and, more generally, optical particle characterization, morphology-dependent resonances and the mechanical effects of light for optical trapping, optical tweezers and optical stretchers. Furthermore, it provides various computer programs relevant to the content. In the last years many new developments took place so that a new edition became necessary. This new book now incorporates solutions for many more particle shapes and morphologies, various kinds of illuminating beams, and also to mechanical effects of light, whispering-gallery modes and resonances, and optical particle characterization techniques. In addition, the new book considers localized approximations, on the renewal of the finite series technique, on a new categorization of optical forces, and the study of Bessel beams, Mathieu beams, Laguerre-Gauss beams, frozen waves.

Topological groups. --- Lie groups. --- Fluid mechanics. --- Electrodynamics. --- Telecommunication. --- Topological Groups and Lie Groups. --- Engineering Fluid Dynamics. --- Classical Electrodynamics. --- Microwaves, RF Engineering and Optical Communications. --- Electric communication --- Mass communication --- Telecom --- Telecommunication industry --- Telecommunications --- Communication --- Information theory --- Telecommuting --- Dynamics --- Hydromechanics --- Continuum mechanics --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Groups, Topological --- Continuous groups --- Electrodinàmica --- Mecànica de fluids --- Equacions de Maxwell

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This book is based on the author's mini course delivered at Tokyo University of Marine Science and Technology in March 2019. The shuffle approach to Drinfeld-Jimbo quantum groups of finite type (embedding their "positive" subalgebras into q-deformed shuffle algebras) was first developed independently in the 1990s by J. Green, M. Rosso, and P. Schauenburg. Motivated by similar ideas, B. Feigin and A. Odesskii proposed a shuffle approach to elliptic quantum groups around the same time. The shuffle algebras in the present book can be viewed as trigonometric degenerations of the Feigin-Odesskii elliptic shuffle algebras. They provide combinatorial models for the "positive" subalgebras of quantum affine algebras in their loop realizations. These algebras appeared first in that context in the work of B. Enriquez. Over the last decade, the shuffle approach has been applied to various problems in combinatorics (combinatorics of Macdonald polynomials and Dyck paths, generalization to wreath Macdonald polynomials and operators), geometric representation theory (especially the study of quantum algebras' actions on the equivariant K-theories of various moduli spaces such as affine Laumon spaces, Nakajima quiver varieties, nested Hilbert schemes), and mathematical physics (the Bethe ansatz, quantum Q-systems, and quantized Coulomb branches of quiver gauge theories, to name just a few). While this area is still under active investigation, the present book focuses on quantum affine/toroidal algebras of type A and their shuffle realization, which have already illustrated a broad spectrum of techniques. The basic results and structures discussed in the book are of crucial importance for studying intrinsic properties of quantum affinized algebras and are instrumental to the aforementioned applications.

Ordered algebraic structures --- Algebra --- Topological groups. Lie groups --- Mathematical physics --- algebra --- wiskunde --- fysica --- topologie

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This volume presents lectures given at the Wisła 20-21 Winter School and Workshop: Groups, Invariants, Integrals, and Mathematical Physics, organized by the Baltic Institute of Mathematics. The lectures were dedicated to differential invariants – with a focus on Lie groups, pseudogroups, and their orbit spaces – and Poisson structures in algebra and geometry and are included here as lecture notes comprising the first two chapters. Following this, chapters combine theoretical and applied perspectives to explore topics at the intersection of differential geometry, differential equations, and category theory. Specific topics covered include: The multisymplectic and variational nature of Monge-Ampère equations in dimension four Integrability of fifth-order equations admitting a Lie symmetry algebra Applications of the van Kampen theorem for groupoids to computation of homotopy types of striped surfaces A geometric framework to compare classical systems of PDEs in the category of smooth manifolds Groups, Invariants, Integrals, and Mathematical Physics is ideal for graduate students and researchers working in these areas. A basic understanding of differential geometry and category theory is assumed.

Mathematical physics. --- Group theory. --- Topological groups. --- Lie groups. --- Differential equations. --- Algebra, Homological. --- Mathematical Physics. --- Group Theory and Generalizations. --- Topological Groups and Lie Groups. --- Differential Equations. --- Category Theory, Homological Algebra. --- Àlgebra diferencial --- Física matemàtica --- Physics --- Science