Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group

Author: Valery V. Volchkov,Vitaly V. Volchkov

Publisher: Springer Science & Business Media

ISBN: 1848825331

Category: Mathematics

Page: 671

View: 2870

The theory of mean periodic functions is a subject which goes back to works of Littlewood, Delsarte, John and that has undergone a vigorous development in recent years. There has been much progress in a number of problems concerning local - pects of spectral analysis and spectral synthesis on homogeneous spaces. The study oftheseproblemsturnsouttobecloselyrelatedtoavarietyofquestionsinharmonic analysis, complex analysis, partial differential equations, integral geometry, appr- imation theory, and other branches of contemporary mathematics. The present book describes recent advances in this direction of research. Symmetric spaces and the Heisenberg group are an active ?eld of investigation at 2 the moment. The simplest examples of symmetric spaces, the classical 2-sphere S 2 and the hyperbolic plane H , play familiar roles in many areas in mathematics. The n Heisenberg groupH is a principal model for nilpotent groups, and results obtained n forH may suggest results that hold more generally for this important class of Lie groups. The purpose of this book is to develop harmonic analysis of mean periodic functions on the above spaces.
Release

Offbeat Integral Geometry on Symmetric Spaces

Author: Valery V. Volchkov,Vitaly V. Volchkov

Publisher: Springer Science & Business Media

ISBN: 3034805721

Category: Mathematics

Page: 592

View: 8689

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 The book includes many significant recent results, some of them hitherto unpublished, among which can be pointed out uniqueness theorems for various classes of functions, far-reaching generalizations of the two-radii problem, the modern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.
Release

The Geometry of Heisenberg Groups

With Applications in Signal Theory, Optics, Quantization, and Field Quantization

Author: Ernst Binz,Sonja Pods

Publisher: American Mathematical Soc.

ISBN: 0821844954

Category: Mathematics

Page: 299

View: 5588

The three-dimensional Heisenberg group, being the simplest non-commutative Lie group, appears prominently in various applications of mathematics. The goal of this book is to present basic geometric and algebraic properties of the Heisenberg group and its relation to other important mathematical structures (the skew field of quaternions, symplectic structures, and representations) and to describe some of its applications. In particular, the authors address such subjects as well as signal analysis and processing, geometric optics, and quantization. In each case, the authors present necessary details of the applied topic being considered. With no prerequisites beyond the standard mathematical curriculum, this book manages to encompass a large variety of topics being easily accessible in its fundamentals. It can be useful to students and researchers working in mathematics and in applied mathematics.
Release

Causal Symmetric Spaces

Author: Gestur Olafsson,Joachim Hilgert

Publisher: Academic Press

ISBN: 0080528724

Category: Mathematics

Page: 286

View: 5332

This book is intended to introduce researchers and graduate students to the concepts of causal symmetric spaces. To date, results of recent studies considered standard by specialists have not been widely published. This book seeks to bring this information to students and researchers in geometry and analysis on causal symmetric spaces. Includes the newest results in harmonic analysis including Spherical functions on ordered symmetric space and the holmorphic discrete series and Hardy spaces on compactly casual symmetric spaces Deals with the infinitesimal situation, coverings of symmetric spaces, classification of causal symmetric pairs and invariant cone fields Presents basic geometric properties of semi-simple symmetric spaces Includes appendices on Lie algebras and Lie groups, Bounded symmetric domains (Cayley transforms), Antiholomorphic Involutions on Bounded Domains and Para-Hermitian Symmetric Spaces
Release

Representation Theory, Complex Analysis, and Integral Geometry

Author: Bernhard Krötz,Omer Offen,Eitan Sayag

Publisher: Springer Science & Business Media

ISBN: 081764816X

Category: Mathematics

Page: 275

View: 9351

This book is an outgrowth of the special summer session held at the Max Planck Institute for Mathematics and the Hausdorff Research Institute for Mathematics. The contributions in the volume provide a window into a variety of subjects related to reductive groups, including real and complex analysis on homogeneous spaces, arithmetic aspects of moment geometry, geometry of flag varieties, restriction theory of representations, modern aspects of special functions, multiple Dirichlet series, and unfolding identities in the theory of automorphic forms.
Release

Cellular Automata and Groups

Author: Tullio Ceccherini-Silberstein,Michel Coornaert

Publisher: Springer Science & Business Media

ISBN: 9783642140341

Category: Computers

Page: 440

View: 9505

Cellular automata were introduced in the first half of the last century by John von Neumann who used them as theoretical models for self-reproducing machines. The authors present a self-contained exposition of the theory of cellular automata on groups and explore its deep connections with recent developments in geometric group theory, symbolic dynamics, and other branches of mathematics and theoretical computer science. The topics treated include in particular the Garden of Eden theorem for amenable groups, and the Gromov-Weiss surjunctivity theorem as well as the solution of the Kaplansky conjecture on the stable finiteness of group rings for sofic groups. The volume is entirely self-contained, with 10 appendices and more than 300 exercises, and appeals to a large audience including specialists as well as newcomers in the field. It provides a comprehensive account of recent progress in the theory of cellular automata based on the interplay between amenability, geometric and combinatorial group theory, symbolic dynamics and the algebraic theory of group rings which are treated here for the first time in book form.
Release

Quantum Theory, Groups and Representations

An Introduction

Author: Peter Woit

Publisher: Springer

ISBN: 3319646125

Category: Science

Page: 668

View: 7249

This text systematically presents the basics of quantum mechanics, emphasizing the role of Lie groups, Lie algebras, and their unitary representations. The mathematical structure of the subject is brought to the fore, intentionally avoiding significant overlap with material from standard physics courses in quantum mechanics and quantum field theory. The level of presentation is attractive to mathematics students looking to learn about both quantum mechanics and representation theory, while also appealing to physics students who would like to know more about the mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of the subject. The latter portions of the book focus on central mathematical objects that occur in the Standard Model of particle physics, underlining the deep and intimate connections between mathematics and the physical world. While an elementary physics course of some kind would be helpful to the reader, no specific background in physics is assumed, making this book accessible to students with a grounding in multivariable calculus and linear algebra. Many exercises are provided to develop the reader's understanding of and facility in quantum-theoretical concepts and calculations.
Release

Groups and Geometric Analysis

Integral Geometry, Invariant Differential Operators, and Spherical Functions

Author: Sigurdur Helgason

Publisher: American Mathematical Soc.

ISBN: 0821826735

Category: Mathematics

Page: 667

View: 8952

Group-theoretic methods have taken an increasingly prominent role in analysis. Some of this change has been due to the writings of Sigurdur Helgason. This book is an introduction to such methods on spaces with symmetry given by the action of a Lie group. The introductory chapter is a self-contained account of the analysis on surfaces of constant curvature. Later chapters cover general cases of the Radon transform, spherical functions, invariant operators, compact symmetric spaces and other topics. This book, together with its companion volume, Geometric Analysis on Symmetric Spaces (AMS Mathematical Surveys and Monographs series, vol. 39, 1994), has become the standard text for this approach to geometric analysis. Sigurdur Helgason was awarded the Steele Prize for outstanding mathematical exposition for Groups and Geometric Analysis and Differential Geometry, Lie Groups and Symmetric Spaces.
Release

The Geometry of Infinite-Dimensional Groups

Author: Boris Khesin,Robert Wendt

Publisher: Springer Science & Business Media

ISBN: 3540772634

Category: Mathematics

Page: 304

View: 3792

This monograph gives an overview of various classes of infinite-dimensional Lie groups and their applications in Hamiltonian mechanics, fluid dynamics, integrable systems, gauge theory, and complex geometry. The text includes many exercises and open questions.
Release

Introduction to Mechanics and Symmetry

A Basic Exposition of Classical Mechanical Systems

Author: J.E. Marsden,Tudor Ratiu

Publisher: Springer Science & Business Media

ISBN: 0387217924

Category: Science

Page: 586

View: 1387

A development of the basic theory and applications of mechanics with an emphasis on the role of symmetry. The book includes numerous specific applications, making it beneficial to physicists and engineers. Specific examples and applications show how the theory works, backed by up-to-date techniques, all of which make the text accessible to a wide variety of readers, especially senior undergraduates and graduates in mathematics, physics and engineering. This second edition has been rewritten and updated for clarity throughout, with a major revamping and expansion of the exercises. Internet supplements containing additional material are also available.
Release

Geometric Analysis on Symmetric Spaces

Author: Sigurdur Helgason

Publisher: American Mathematical Soc.

ISBN: 9780821815380

Category: Mathematics

Page: 611

View: 7783

Among Riemannian manifolds, symmetric spaces (in the sense of Cartan) provide an abundant supply of elegant examples, the structures of which are enhanced by the rich theory of semisimple Lie groups. On these spaces, global analysis, particularly integration theory and partial differential operators, arises in a natural way by the requirement of geometric invariance. In Euclidean space these two subjects are related by the Fourier transform. The Peter-Weyl theory for compact groups, and Cartan's refinement of it, provides a way to develop harmonic analysis on compact symmetric spaces. The noncompact symmetric spaces, however, present a multitude of new and natural problems. This book is devoted to geometric analysis on noncompact Riemannian spaces. The exposition in this book is accessible to readers with modest background in semisimple Lie group theory. In particular, familiarity with representation theory is not needed.
Release

Nonlinear Dispersive Equations

Existence and Stability of Solitary and Periodic Travelling Wave Solutions

Author: Jaime Angulo Pava

Publisher: American Mathematical Soc.

ISBN: 0821848976

Category: Mathematics

Page: 256

View: 3733

This book provides a self-contained presentation of classical and new methods for studying wave phenomena that are related to the existence and stability of solitary and periodic travelling wave solutions for nonlinear dispersive evolution equations. Simplicity, concrete examples, and applications are emphasized throughout in order to make the material easily accessible. The list of classical nonlinear dispersive equations studied includes Korteweg-de Vries, Benjamin-Ono, and Schrodinger equations. Many special Jacobian elliptic functions play a role in these examples. The author brings the reader to the forefront of knowledge about some aspects of the theory and motivates future developments in this fascinating and rapidly growing field. The book can be used as an instructive study guide as well as a reference by students and mature scientists interested in nonlinear wave phenomena.
Release

Extensions of Positive Definite Functions

Applications and Their Harmonic Analysis

Author: Palle Jorgensen,Steen Pedersen,Feng Tian

Publisher: Springer

ISBN: 331939780X

Category: Mathematics

Page: 231

View: 9181

This monograph deals with the mathematics of extending given partial data-sets obtained from experiments; Experimentalists frequently gather spectral data when the observed data is limited, e.g., by the precision of instruments; or by other limiting external factors. Here the limited information is a restriction, and the extensions take the form of full positive definite function on some prescribed group. It is therefore both an art and a science to produce solid conclusions from restricted or limited data. While the theory of is important in many areas of pure and applied mathematics, it is difficult for students and for the novice to the field, to find accessible presentations which cover all relevant points of view, as well as stressing common ideas and interconnections. We have aimed at filling this gap, and we have stressed hands-on-examples.
Release

Understanding Understanding

Essays on Cybernetics and Cognition

Author: Heinz von Foerster

Publisher: Springer Science & Business Media

ISBN: 0387217223

Category: Computers

Page: 362

View: 841

In these ground-breaking essays, Heinz von Foerster discusses some of the fundamental principles that govern how we know the world and how we process the information from which we derive that knowledge. The author was one of the founders of the science of cybernetics.
Release

Orthogonal Polynomials of Several Variables

Author: Charles F. Dunkl,Yuan Xu

Publisher: Cambridge University Press

ISBN: 9780521800433

Category: Mathematics

Page: 390

View: 7339

Orthogonal polynomials of several variables, approximation theory, symmetry-group methods.
Release

Spectral Theory and Mathematical Physics: Ergodic Schrödinger operators, singular spectrum, orthogonal polynomials, and inverse spectral theory

Author: Barry Simon

Publisher: American Mathematical Soc.

ISBN: 0821842498

Category: Mathematics

Page: 948

View: 7938

This Festschrift had its origins in a conference called SimonFest held at Caltech, March 27-31, 2006, to honor Barry Simon's 60th birthday. It is not a proceedings volume in the usual sense since the emphasis of the majority of the contributions is on reviews of the state of the art of certain fields, with particular focus on recent developments and open problems. The bulk of the articles in this Festschrift are of this survey form, and a few review Simon's contributions to a particular area. Part 1 contains surveys in the areas of Quantum Field Theory, Statistical Mechanics, Nonrelativistic Two-Body and $N$-Body Quantum Systems, Resonances, Quantum Mechanics with Electric and Magnetic Fields, and the Semiclassical Limit. Part 2 contains surveys in the areas of Random and Ergodic Schrodinger Operators, Singular Continuous Spectrum, Orthogonal Polynomials, and Inverse Spectral Theory. In several cases, this collection of surveys portrays both the history of a subject and its current state of the art. Exhaustive lists of references enhance the presentation offered in these surveys. A substantial part of the contributions to this Festschrift are survey articles on the state of the art of certain areas with special emphasis on open problems. This will benefit graduate students as well as researchers who want to get a quick, yet comprehensive introduction into an area covered in this volume.
Release

Factorization, Singular Operators and Related Problems

Author: Georgiĭ Semenovich Litvinchuk,Stefan Samko,Amarino Lebre,António F. dos Santos

Publisher: Taylor & Francis US

ISBN: 9781402014079

Category: Mathematics

Page: 333

View: 1550

Proceedings of the Conference in Honour of Professor Georgii Litvinchuk
Release

An Introduction to the Geometry of Stochastic Flows

Author: Fabrice Baudoin

Publisher: World Scientific

ISBN: 1860944817

Category: Mathematics

Page: 140

View: 9373

This book aims to provide a self-contained introduction to the local geometry of the stochastic flows associated with stochastic differential equations. It stresses the view that the local geometry of any stochastic flow is determined very precisely and explicitly by a universal formula referred to as the Chen-Strichartz formula. The natural geometry associated with the Chen-Strichartz formula is the sub-Riemannian geometry whose main tools are introduced throughout the text. By using the connection between stochastic flows and partial differential equations, we apply this point of view of the study of hypoelliptic operators written in Hormander's form.
Release