Download or read book Pseudodifferential Methods in Number Theory written by André Unterberger and published by Birkhäuser. This book was released on 2018-07-16 with total page 175 pages. Available in PDF, EPUB and Kindle. Book excerpt: Classically developed as a tool for partial differential equations, the analysis of operators known as pseudodifferential analysis is here regarded as a possible help in questions of arithmetic. The operators which make up the main subject of the book can be characterized in terms of congruence arithmetic. They enjoy a Eulerian structure, and are applied to the search for new conditions equivalent to the Riemann hypothesis. These consist in the validity of certain parameter-dependent estimates for a class of Hermitian forms of finite rank. The Littlewood criterion, involving sums of Möbius coefficients, and the Weil so-called explicit formula, which leads to his positivity criterion, fit within this scheme, using in the first case Weyl's pseudodifferential calculus, in the second case Fuchs'. The book should be of interest to people looking for new possible approaches to the Riemann hypothesis, also to new perspectives on pseudodifferential analysis and on the way it combines with modular form theory. Analysts will have no difficulty with the arithmetic aspects, with which, save for very few exceptions, no previous acquaintance is necessary.

Download or read book Functional Calculus of Pseudodifferential Boundary Problems written by Gerd Grubb and published by Springer Science & Business Media. This book was released on 1996-01-26 with total page 544 pages. Available in PDF, EPUB and Kindle. Book excerpt: Pseudodifferential methods are central to the study of partial differential equations, because they permit an "algebraization." The main purpose of this book is to set up an operational calculus for operators defined from differential and pseudodifferential boundary values problems via a resolvent construction. A secondary purposed is to give a complete treatment of the properties of the calculus of pseudodifferential boundary problems with transmission, both the first version by Boutet de Monvel (brought completely up to date in this edition) and in version containing a parameter running in an unbounded set. And finally, the book presents some applications to evolution problems, index theory, fractional powers, spectral theory and singular perturbation theory. Thus the book’s improved proofs and modern points of view will be useful to research mathematicians and to graduate students studying partial differential equations and pseudodifferential operators.

Download or read book Elementary Introduction to the Theory of Pseudodifferential Operators written by Xavier Saint Raymond and published by Routledge. This book was released on 2018-02-06 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the 19th century, the Fourier transformation was introduced to study various problems of partial differential equations. Since 1960, this old tool has been developed into a well-organized theory called microlocal analysis that is based on the concept of the pseudo-differential operator. This book provides the fundamental knowledge non-specialists need in order to use microlocal analysis. It is strictly mathematical in the sense that it contains precise definitions, statements of theorems and complete proofs, and follows the usual method of pure mathematics. The book explains the origin of the theory (i.e., Fourier transformation), presents an elementary construcion of distribution theory, and features a careful exposition of standard pseudodifferential theory. Exercises, historical notes, and bibliographical references are included to round out this essential book for mathematics students; engineers, physicists, and mathematicians who use partial differential equations; and advanced mathematics instructors.

Download or read book Functional Calculus of Pseudodifferential Boundary Problems written by Gerd Grubb and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 536 pages. Available in PDF, EPUB and Kindle. Book excerpt: Pseudodifferential methods are central to the study of partial differential equations, because they permit an "algebraization." The main purpose of this book is to set up an operational calculus for operators defined from differential and pseudodifferential boundary values problems via a resolvent construction. A secondary purposed is to give a complete treatment of the properties of the calculus of pseudodifferential boundary problems with transmission, both the first version by Boutet de Monvel (brought completely up to date in this edition) and in version containing a parameter running in an unbounded set. And finally, the book presents some applications to evolution problems, index theory, fractional powers, spectral theory and singular perturbation theory. Thus the book’s improved proofs and modern points of view will be useful to research mathematicians and to graduate students studying partial differential equations and pseudodifferential operators.

Download or read book Pseudodifferential Operators with Automorphic Symbols written by André Unterberger and published by Birkhäuser. This book was released on 2015-06-22 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main results of this book combine pseudo differential analysis with modular form theory. The methods rely for the most part on explicit spectral theory and the extended use of special functions. The starting point is a notion of modular distribution in the plane, which will be new to most readers and relates under the Radon transformation to the classical one of modular form of the non-holomorphic type. Modular forms of the holomorphic type are addressed too in a more concise way, within a general scheme dealing with quantization theory and elementary, but novel, representation-theoretic concepts.

Download or read book The Technique of Pseudodifferential Operators written by Heinz Otto Cordes and published by Cambridge University Press. This book was released on 1995-02-23 with total page 398 pages. Available in PDF, EPUB and Kindle. Book excerpt: Pseudodifferential operators arise naturally in a solution of boundary problems for partial differential equations. The formalism of these operators serves to make the Fourier-Laplace method applicable for nonconstant coefficient equations. This book presents the technique of pseudodifferential operators and its applications, especially to the Dirac theory of quantum mechanics. The treatment uses 'Leibniz formulas' with integral remainders or as asymptotic series. While a pseudodifferential operator is commonly defined by an integral formula, it also may be described by invariance under action of a Lie group. The author discusses connections to the theory of C*-algebras, invariant algebras of pseudodifferential operators under hyperbolic evolution, and the relation of the hyperbolic theory to the propagation of maximal ideals. The Technique of Pseudodifferential Operators will be of particular interest to researchers in partial differential equations and mathematical physics.

Download or read book Pseudodifferential Equations Over Non Archimedean Spaces written by W. A. Zúñiga-Galindo and published by Springer. This book was released on 2017-01-08 with total page 186 pages. Available in PDF, EPUB and Kindle. Book excerpt: Focusing on p-adic and adelic analogues of pseudodifferential equations, this monograph presents a very general theory of parabolic-type equations and their Markov processes motivated by their connection with models of complex hierarchic systems. The Gelfand-Shilov method for constructing fundamental solutions using local zeta functions is developed in a p-adic setting and several particular equations are studied, such as the p-adic analogues of the Klein-Gordon equation. Pseudodifferential equations for complex-valued functions on non-Archimedean local fields are central to contemporary harmonic analysis and mathematical physics and their theory reveals a deep connection with probability and number theory. The results of this book extend and complement the material presented by Vladimirov, Volovich and Zelenov (1994) and Kochubei (2001), which emphasize spectral theory and evolution equations in a single variable, and Albeverio, Khrennikov and Shelkovich (2010), which deals mainly with the theory and applications of p-adic wavelets.

Download or read book Analytic Methods In The Theory Of Differential And Pseudo Differential Equations Of Parabolic Type written by Samuil D. Eidelman and published by Birkhäuser. This book was released on 2012-12-06 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to new classes of parabolic differential and pseudo-differential equations extensively studied in the last decades, such as parabolic systems of a quasi-homogeneous structure, degenerate equations of the Kolmogorov type, pseudo-differential parabolic equations, and fractional diffusion equations. It will appeal to mathematicians interested in new classes of partial differential equations, and physicists specializing in diffusion processes.

Download or read book Singular Ordinary Differential Operators and Pseudodifferential Equations written by Johannes Elschner and published by Springer. This book was released on 2006-11-14 with total page 199 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Periodic Integral and Pseudodifferential Equations with Numerical Approximation written by Jukka Saranen and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 461 pages. Available in PDF, EPUB and Kindle. Book excerpt: An attractive book on the intersection of analysis and numerical analysis, deriving classical boundary integral equations arising from the potential theory and acoustics. This self-contained monograph can be used as a textbook by graduate/postgraduate students. It also contains a lot of carefully chosen exercises.

Download or read book Pseudodifferential Operators and Spectral Theory written by M.A. Shubin and published by Springer Science & Business Media. This book was released on 2001-07-03 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the second edition of Shubin's classical book. It provides an introduction to the theory of pseudodifferential operators and Fourier integral operators from the very basics. The applications discussed include complex powers of elliptic operators, Hörmander asymptotics of the spectral function and eigenvalues, and methods of approximate spectral projection. Exercises and problems are included to help the reader master the essential techniques. The book is written for a wide audience of mathematicians, be they interested students or researchers.

Download or read book Metrics on the Phase Space and Non Selfadjoint Pseudo Differential Operators written by Nicolas Lerner and published by Springer Science & Business Media. This book was released on 2011-01-30 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the study of pseudo-di?erential operators, with special emphasis on non-selfadjoint operators, a priori estimates and localization in the phase space. We have tried here to expose the most recent developments of the theory with its applications to local solvability and semi-classical estimates for non-selfadjoint operators. The?rstchapter,Basic Notions of Phase Space Analysis,isintroductoryand gives a presentation of very classical classes of pseudo-di?erential operators, along with some basic properties. As an illustration of the power of these methods, we give a proof of propagation of singularities for real-principal type operators (using aprioriestimates,andnotFourierintegraloperators),andweintroducethereader to local solvability problems. That chapter should be useful for a reader, say at the graduate level in analysis, eager to learn some basics on pseudo-di?erential operators. The second chapter, Metrics on the Phase Space begins with a review of symplectic algebra, Wigner functions, quantization formulas, metaplectic group and is intended to set the basic study of the phase space. We move forward to the more general setting of metrics on the phase space, following essentially the basic assumptions of L. H ̈ ormander (Chapter 18 in the book [73]) on this topic.

Download or read book Pseudodifferential Analysis on Symmetric Cones written by Andre Unterberger and published by CRC Press. This book was released on 1995-12-13 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: Symmetric cones, possibly disguised under non-linear changes of coordinates, are the building blocks of manifolds with edges, corners, or conical points of a very general nature. Besides being a canonical open set of some Euclidean space, a symmetric cone L has an intrinsic Riemannian structure of its own, turning it into a symmetric space. These two structures make it possible to define on L a pseudodifferential analysis (the Fuchs calculus). The considerable interest in pseudodifferential problems on manifolds with non-smooth boundaries makes the precise analyses presented in this book both interesting and important. Much of the material in this book has never been previously published. The methods used throughout the text rely heavily on the use of tools from quantum mechanics, such as representation theory and coherent states. Classes of operators defined by their symbols are given intrinsic characterizations. Harmonic analysis is discussed via the automorphism group of the complex tube over L. The basic definitions governing the Fuchs calculus are provided, and a thorough exposition of the fundamental facts concerning the geometry of symmetric cones is given. The relationship with Jordan algebras is outlined and the general theory is illustrated by numerous examples. The book offers the reader the technical tools for proving the main properties of the Fuchs calculus, with an emphasis on using the non-Euclidean Riemannian structure of the underlying cone. The fundamental results of pseudodifferential analysis are presented. The authors also develop the relationship to complex analysis and group representation. This book benefits researchers interested in analysis on non-smooth domains or anyone working in pseudodifferential analysis. People interested in the geometry or harmonic analysis of symmetric cones will find in this valuable reference a new range of applications of complex analysis on tube-type symmetric domains and of the theory of Jordan algebras.

Download or read book Symplectic Methods in Harmonic Analysis and in Mathematical Physics written by Maurice A. de Gosson and published by Springer Science & Business Media. This book was released on 2011-07-30 with total page 351 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to give a rigorous and complete treatment of various topics from harmonic analysis with a strong emphasis on symplectic invariance properties, which are often ignored or underestimated in the time-frequency literature. The topics that are addressed include (but are not limited to) the theory of the Wigner transform, the uncertainty principle (from the point of view of symplectic topology), Weyl calculus and its symplectic covariance, Shubin’s global theory of pseudo-differential operators, and Feichtinger’s theory of modulation spaces. Several applications to time-frequency analysis and quantum mechanics are given, many of them concurrent with ongoing research. For instance, a non-standard pseudo-differential calculus on phase space where the main role is played by “Bopp operators” (also called “Landau operators” in the literature) is introduced and studied. This calculus is closely related to both the Landau problem and to the deformation quantization theory of Flato and Sternheimer, of which it gives a simple pseudo-differential formulation where Feichtinger’s modulation spaces are key actors. This book is primarily directed towards students or researchers in harmonic analysis (in the broad sense) and towards mathematical physicists working in quantum mechanics. It can also be read with profit by researchers in time-frequency analysis, providing a valuable complement to the existing literature on the topic. A certain familiarity with Fourier analysis (in the broad sense) and introductory functional analysis (e.g. the elementary theory of distributions) is assumed. Otherwise, the book is largely self-contained and includes an extensive list of references.

Download or read book The Technique of Pseudodifferential Operators written by Heinz Otto Cordes and published by . This book was released on 2014-05-14 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: A technique used in the theory of partial differential equations with applications to quantum mechanics.

Download or read book Periodic Integral and Pseudodifferential Equations with Numerical Approximation written by Jukka Saranen and published by Springer. This book was released on 2014-03-12 with total page 452 pages. Available in PDF, EPUB and Kindle. Book excerpt: An attractive book on the intersection of analysis and numerical analysis, deriving classical boundary integral equations arising from the potential theory and acoustics. This self-contained monograph can be used as a textbook by graduate/postgraduate students. It also contains a lot of carefully chosen exercises.

Download or read book Distributions and Operators written by Gerd Grubb and published by Springer Science & Business Media. This book was released on 2008-10-14 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives an introduction to distribution theory, based on the work of Schwartz and of many other people. It is the first book to present distribution theory as a standard text. Each chapter has been enhanced with many exercises and examples.