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The authors consider the Hodge Laplacian Delta on the Heisenberg group H_n, endowed with a left-invariant and U(n)-invariant Riemannian metric. For 0le kle 2n+1, let Delta_k denote the Hodge Laplacian restricted to k-forms. In this paper they address three main, related questions: (1) whether the L^2 and L^p-Hodge decompositions, 1 , hold on H_n; (2) whether the Riesz transforms dDelta_k^{-frac 12} are L^p-bounded, for 1.

Harmonic analysis.

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The book targets undergraduate and postgraduate mathematics students and helps them develop a deep understanding of mathematical analysis. Designed as a first course in real analysis, it helps students learn how abstract mathematical analysis solves mathematical problems that relate to the real world. As well as providing a valuable source of inspiration for contemporary research in mathematics, the book helps students read, understand and construct mathematical proofs, develop their problem-solving abilities and comprehend the importance and frontiers of computer facilities and much more. It offers comprehensive material for both seminars and independent study for readers with a basic knowledge of calculus and linear algebra. The first nine chapters followed by the appendix on the Stieltjes integral are recommended for graduate students studying probability and statistics, while the first eight chapters followed by the appendix on dynamical systems will be of use to students of biology and environmental sciences. Chapter 10 and the appendixes are of interest to those pursuing further studies at specialized advanced levels. Exercises at the end of each section, as well as commentaries at the end of each chapter, further aid readers’ understanding. The ultimate goal of the book is to raise awareness of the fine architecture of analysis and its relationship with the other fields of mathematics.

Algebra --- Harmonic analysis. Fourier analysis --- Mathematics --- Fourieranalyse --- algebra --- wiskunde

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This textbook is designed for a one year course covering the fundamentals of partial differential equations, geared towards advanced undergraduates and beginning graduate students in mathematics, science, engineering, and elsewhere.  The exposition carefully balances solution techniques, mathematical rigor, and significant applications, all illustrated by numerous examples.  Extensive exercise sets appear at the end of almost every subsection, and include straightforward computational problems to develop and reinforce new techniques and results, details on theoretical developments and proofs, challenging projects both computational and conceptual, and supplementary material that motivates the student to delve further into the subject. No previous experience with the subject of partial differential equations or Fourier theory is assumed, the main prerequisites being undergraduate calculus, both one- and multi-variable, ordinary differential equations, and basic linear algebra.  While the classical topics of separation of variables, Fourier analysis, boundary value problems, Green's functions, and special functions continue to form the core of an introductory course, the inclusion of nonlinear equations, shock wave dynamics, symmetry and similarity, the Maximum Principle, financial models, dispersion and solitons, Huygens' Principle, quantum mechanical systems, and more make this text well attuned to recent developments and trends in this active field of contemporary research.  Numerical approximation schemes are an important component of any introductory course, and the text covers the two most basic approaches: finite differences and finite elements. Peter J. Olver is professor of mathematics at the University of Minnesota.  His wide-ranging research interests are centered on the development of symmetry-based methods for differential equations and their manifold applications.  He is the author of over 130 papers published in major scientific research journals as well as 4 other books, including the definitive Springer graduate text, Applications of Lie Groups to Differential Equations, and another undergraduate text, Applied Linear Algebra. A Solutions Manual for instrucors is available by clicking on "Selected Solutions Manual" under the Additional Information section on the right-hand side of this page.   .

Harmonic analysis. Fourier analysis --- Partial differential equations --- Differential equations --- Mathematics --- Fourieranalyse --- differentiaalvergelijkingen --- wiskunde

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Time series, or longitudinal, data are ubiquitous in the social sciences. Unfortunately, analysts often treat the time series properties of their data as a nuisance rather than a substantively meaningful dynamic process to be modeled and interpreted. Time Series Analysis for the Social Sciences provides accessible, up-to-date instruction and examples of the core methods in time series econometrics. Janet M. Box-Steffensmeier, John R. Freeman, Jon C. Pevehouse and Matthew P. Hitt cover a wide range of topics including ARIMA models, time series regression, unit-root diagnosis, vector autoregressive models, error-correction models, intervention models, fractional integration, ARCH models, structural breaks, and forecasting. This book is aimed at researchers and graduate students who have taken at least one course in multivariate regression. Examples are drawn from several areas of social science, including political behavior, elections, international conflict, criminology, and comparative political economy.

Time-series analysis --- Time-series analysis. --- Analysis of time series --- Autocorrelation (Statistics) --- Harmonic analysis --- Mathematical statistics --- Probabilities --- Mathematical models.

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This text is addressed to graduate students in mathematics and to interested researchers who wish to acquire an in depth understanding of Euclidean Harmonic analysis. The text covers modern topics and techniques in function spaces, atomic decompositions, singular integrals of nonconvolution type, and the boundedness and convergence of Fourier series and integrals. The exposition and style are designed to stimulate further study and promote research. Historical information and references are included at the end of each chapter. This third edition includes a new chapter entitled "Multilinear Harmonic Analysis" which focuses on topics related to multilinear operators and their applications. Sections 1.1 and 1.2 are also new in this edition. Numerous corrections have been made to the text from the previous editions and several improvements have been incorporated, such as the adoption of clear and elegant statements. A few more exercises have been added with relevant hints when necessary. Reviews from the Second Edition: “The books cover a large amount of mathematics. They are certainly a valuable and useful addition to the existing literature and can serve as textbooks or as reference books. Students will especially appreciate the extensive collection of exercises.” —Andreas Seeger, Mathematical Reviews “The exercises at the end of each section supplement the material of the section nicely and provide a good chance to develop additional intuition and deeper comprehension. The historical notes in each chapter are intended to provide an account of past research as well as to suggest directions for further investigation. The volume is mainly addressed to graduate students who wish to study harmonic analysis.” —Leonid Golinskii, zbMATH.

Mathematics --- Functional analysis --- Harmonic analysis. Fourier analysis --- Mathematical analysis --- Fourieranalyse --- analyse (wiskunde) --- Fourierreeksen --- functies (wiskunde) --- mathematische modellen --- wiskunde