Microlocal Analysis for Differential Operators

An Introduction

Author: Alain Grigis,Johannes Sjöstrand

Publisher: Cambridge University Press

ISBN: 9780521449861

Category: Mathematics

Page: 151

View: 4633

This book corresponds to a graduate course given many times by the authors, and should prove to be useful to mathematicians and theoretical physicists.

Spectral Asymptotics in the Semi-Classical Limit

Author: M. Dimassi,J. Sjostrand

Publisher: Cambridge University Press

ISBN: 9780521665445

Category: Mathematics

Page: 227

View: 4832

This book presents the basic methods and applications in semiclassical approximation in the light of developments.

An Introduction to Semiclassical and Microlocal Analysis

Author: André Bach

Publisher: Springer Science & Business Media

ISBN: 1475744951

Category: Mathematics

Page: 191

View: 9723

This book presents the techniques used in the microlocal treatment of semiclassical problems coming from quantum physics in a pedagogical, way and is mainly addressed to non-specialists in the subject. It is based on lectures taught by the author over several years, and includes many exercises providing outlines of useful applications of the semi-classical theory.

Lectures on Nonlinear Hyperbolic Differential Equations

Author: Lars Hörmander

Publisher: Springer Science & Business Media

ISBN: 9783540629214

Category: Mathematics

Page: 289

View: 6789

In this introductory textbook, a revised and extended version of well-known lectures by L. Hörmander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Four chapters are devoted to microanalysis of the singularities of the solutions. This part assumes some familiarity with pseudodifferential operators which are standard in the theory of linear differential operators, but the extension to the more exotic classes of opertors needed in the nonlinear theory is presented in complete detail.

Semiclassical Analysis

Author: Maciej Zworski

Publisher: American Mathematical Soc.

ISBN: 0821883208

Category: Mathematics

Page: 431

View: 1475

This book is an excellent, comprehensive introduction to semiclassical analysis. I believe it will become a standard reference for the subject. --Alejandro Uribe, University of Michigan Semiclassical analysis provides PDE techniques based on the classical-quantum (particle-wave) correspondence. These techniques include such well-known tools as geometric optics and the Wentzel-Kramers-Brillouin approximation. Examples of problems studied in this subject are high energy eigenvalue asymptotics and effective dynamics for solutions of evolution equations. From the mathematical point of view, semiclassical analysis is a branch of microlocal analysis which, broadly speaking, applies harmonic analysis and symplectic geometry to the study of linear and nonlinear PDE. The book is intended to be a graduate level text introducing readers to semiclassical and microlocal methods in PDE. It is augmented in later chapters with many specialized advanced topics which provide a link to current research literature.

Geometric Scattering Theory

Author: Richard B. Melrose

Publisher: Cambridge University Press

ISBN: 9780521498104

Category: Mathematics

Page: 116

View: 8223

This book is an overview of scattering theory. The author shows how this theory provides a parametrization of the continuous spectrum of an elliptic operator on a complete manifold with uniform structure at infinity. In the first two lectures the author describes the simple and fundamental case of the Laplacian on Euclidean space to introduce the theory's basic framework. In the next three lectures, he outlines various results on Euclidean scattering, and the methods used to prove them. In the last three lectures he extends these ideas to non-Euclidean settings.

Geometric Analysis

Author: Hubert L. Bray,Greg Galloway,Rafe Mazzeo,Natasa Sesum

Publisher: American Mathematical Soc.

ISBN: 1470423138

Category: Geometric analysis

Page: 456

View: 7277

This volume includes expanded versions of the lectures delivered in the Graduate Minicourse portion of the 2013 Park City Mathematics Institute session on Geometric Analysis. The papers give excellent high-level introductions, suitable for graduate students wishing to enter the field and experienced researchers alike, to a range of the most important areas of geometric analysis. These include: the general issue of geometric evolution, with more detailed lectures on Ricci flow and Kähler-Ricci flow, new progress on the analytic aspects of the Willmore equation as well as an introduction to the recent proof of the Willmore conjecture and new directions in min-max theory for geometric variational problems, the current state of the art regarding minimal surfaces in R3, the role of critical metrics in Riemannian geometry, and the modern perspective on the study of eigenfunctions and eigenvalues for Laplace–Beltrami operators.

Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators

Author: Nicolas Lerner

Publisher: Springer Science & Business Media

ISBN: 9783764385101

Category: Mathematics

Page: 397

View: 7755

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.

Asymptotic Analysis in General Relativity

Author: Thierry Daudé,Dietrich Häfner,Jean-Philippe Nicolas

Publisher: Cambridge University Press

ISBN: 1316649407

Category: Mathematics

Page: N.A

View: 2315

Introduction to modern methods for classical and quantum fields in general relativity / Thierry Daudé, Dietrich Häfner, and Jean-Philippe Nicolas -- Geometry of black hole spacetimes / Lars Andersson, Thomas B. Ackdahl, and Pieter Blue -- An introduction to Quantum Field Theory on curved space-times / Christian Gerard -- A minicourse on microlocal analysis for wave propagation / Andras Vasy -- An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity / Sean N. Curry and A. Rod Gover

Nonlinear Optical and Atomic Systems

At the Interface of Physics and Mathematics

Author: Christophe Besse,Jean-Claude Garreau

Publisher: Springer

ISBN: 3319190156

Category: Science

Page: 338

View: 1815

Focusing on the interface between mathematics and physics, this book offers an introduction to the physics, the mathematics, and the numerical simulation of nonlinear systems in optics and atomic physics. The text covers a wide spectrum of current research on the subject, which is an extremely active field in physics and mathematical physics, with a very broad range of implications, both for fundamental science and technological applications: light propagation in microstructured optical fibers, Bose-Einstein condensates, disordered systems, and the newly emerging field of nonlinear quantum mechanics. Accessible to PhD students, this book will also be of interest to post-doctoral researchers and seasoned academics.

Eigenfunctions of the Laplacian on a Riemannian Manifold

Author: Steve Zelditch

Publisher: American Mathematical Soc.

ISBN: 1470410370

Category: Eigenfunctions

Page: 394

View: 1209

Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The local analysis of eigenfunctions pertains to the behavior of the eigenfunctions on wavelength scale balls. After re-scaling to a unit ball, the eigenfunctions resemble almost-harmonic functions. Global analysis refers to the use of wave equation methods to relate properties of eigenfunctions to properties of the geodesic flow. The emphasis is on the global methods and the use of Fourier integral operator methods to analyze norms and nodal sets of eigenfunctions. A somewhat unusual topic is the analytic continuation of eigenfunctions to Grauert tubes in the real analytic case, and the study of nodal sets in the complex domain. The book, which grew out of lectures given by the author at a CBMS conference in 2011, provides complete proofs of some model results, but more often it gives informal and intuitive explanations of proofs of fairly recent results. It conveys inter-related themes and results and offers an up-to-date comprehensive treatment of this important active area of research.

Mathematical Models, Methods and Applications

Author: Abul Hasan Siddiqi,Pammy Manchanda,Rashmi Bhardwaj

Publisher: Springer

ISBN: 9812879730

Category: Mathematics

Page: 298

View: 5029

The present volume contains invited talks of 11th biennial conference on “Emerging Mathematical Methods, Models and Algorithms for Science and Technology”. The main message of the book is that mathematics has a great potential to analyse and understand the challenging problems of nanotechnology, biotechnology, medical science, oil industry and financial technology. The book highlights all the features and main theme discussed in the conference. All contributing authors are eminent academicians, scientists, researchers and scholars in their respective fields, hailing from around the world.

A Course in Convexity

Author: Alexander Barvinok

Publisher: American Mathematical Soc.

ISBN: 0821829688

Category: Mathematics

Page: 366

View: 451

Convexity is a simple idea that manifests itself in a surprising variety of places. This fertile field has an immensely rich structure and numerous applications. Barvinok demonstrates that simplicity, intuitive appeal, and the universality of applications make teaching (and learning) convexity a gratifying experience. The book will benefit both teacher and student: It is easy to understand, entertaining to the reader, and includes many exercises that vary in degree of difficulty. Overall, the author demonstrates the power of a few simple unifying principles in a variety of pure and applied problems. The prerequisites are minimal amounts of linear algebra, analysis, and elementary topology, plus basic computational skills. Portions of the book could be used by advanced undergraduates. As a whole, it is designed for graduate students interested in mathematical methods, computer science, electrical engineering, and operations research. The book will also be of interest to research mathematicians, who will find some results that are recent, some that are new, and many known results that are discussed from a new perspective.

Advances in Elliptic Curve Cryptography

Author: Ian F. Blake,Gadiel Seroussi,Nigel P. Smart

Publisher: Cambridge University Press

ISBN: 9781139441223

Category: Mathematics

Page: N.A

View: 8920

Since the appearance of the authors' first volume on elliptic curve cryptography in 1999 there has been tremendous progress in the field. In some topics, particularly point counting, the progress has been spectacular. Other topics such as the Weil and Tate pairings have been applied in new and important ways to cryptographic protocols that hold great promise. Notions such as provable security, side channel analysis and the Weil descent technique have also grown in importance. This second volume addresses these advances and brings the reader up to date. Prominent contributors to the research literature in these areas have provided articles that reflect the current state of these important topics. They are divided into the areas of protocols, implementation techniques, mathematical foundations and pairing based cryptography. Each of the topics is presented in an accessible, coherent and consistent manner for a wide audience that will include mathematicians, computer scientists and engineers.

Handbook of Mathematical Methods in Imaging

Author: Otmar Scherzer

Publisher: Springer Science & Business Media

ISBN: 0387929193

Category: Mathematics

Page: 1512

View: 5404

The Handbook of Mathematical Methods in Imaging provides a comprehensive treatment of the mathematical techniques used in imaging science. The material is grouped into two central themes, namely, Inverse Problems (Algorithmic Reconstruction) and Signal and Image Processing. Each section within the themes covers applications (modeling), mathematics, numerical methods (using a case example) and open questions. Written by experts in the area, the presentation is mathematically rigorous. The entries are cross-referenced for easy navigation through connected topics. Available in both print and electronic forms, the handbook is enhanced by more than 150 illustrations and an extended bibliography. It will benefit students, scientists and researchers in applied mathematics. Engineers and computer scientists working in imaging will also find this handbook useful.

Algorithms for Synthetic Aperture Radar Imagery

Author: Edmund G. Zelnio,Frederick D. Garber

Publisher: Society of Photo Optical

ISBN: 9780819466907

Category: Computer algorithms

Page: N.A

View: 8426

Evolution equations

existence, regularity and singularities

Author: Bogdan Bojarski,Dariusz Wrzosek,Wojciech M. Zajączkowski,Stefan Banach International Mathematical Center

Publisher: N.A


Category: Evolution equations

Page: 226

View: 4742

Lectures on the Mathematics of Quantum Mechanics II: Selected Topics

Author: Gianfausto Dell'Antonio

Publisher: Springer

ISBN: 9462391157

Category: Science

Page: 381

View: 6319

The first volume (General Theory) differs from most textbooks as it emphasizes the mathematical structure and mathematical rigor, while being adapted to the teaching the first semester of an advanced course in Quantum Mechanics (the content of the book are the lectures of courses actually delivered.). It differs also from the very few texts in Quantum Mechanics that give emphasis to the mathematical aspects because this book, being written as Lecture Notes, has the structure of lectures delivered in a course, namely introduction of the problem, outline of the relevant points, mathematical tools needed, theorems, proofs. This makes this book particularly useful for self-study and for instructors in the preparation of a second course in Quantum Mechanics (after a first basic course). With some minor additions it can be used also as a basis of a first course in Quantum Mechanics for students in mathematics curricula. The second part (Selected Topics) are lecture notes of a more advanced course aimed at giving the basic notions necessary to do research in several areas of mathematical physics connected with quantum mechanics, from solid state to singular interactions, many body theory, semi-classical analysis, quantum statistical mechanics. The structure of this book is suitable for a second-semester course, in which the lectures are meant to provide, in addition to theorems and proofs, an overview of a more specific subject and hints to the direction of research. In this respect and for the width of subjects this second volume differs from other monographs on Quantum Mechanics. The second volume can be useful for students who want to have a basic preparation for doing research and for instructors who may want to use it as a basis for the presentation of selected topics.

Spectral Theory of Infinite-Area Hyperbolic Surfaces

Author: David Borthwick

Publisher: Birkhäuser

ISBN: 3319338773

Category: Mathematics

Page: 463

View: 5910

This text introduces geometric spectral theory in the context of infinite-area Riemann surfaces, providing a comprehensive account of the most recent developments in the field. For the second edition the context has been extended to general surfaces with hyperbolic ends, which provides a natural setting for development of the spectral theory while still keeping technical difficulties to a minimum. All of the material from the first edition is included and updated, and new sections have been added. Topics covered include an introduction to the geometry of hyperbolic surfaces, analysis of the resolvent of the Laplacian, scattering theory, resonances and scattering poles, the Selberg zeta function, the Poisson formula, distribution of resonances, the inverse scattering problem, Patterson-Sullivan theory, and the dynamical approach to the zeta function. The new sections cover the latest developments in the field, including the spectral gap, resonance asymptotics near the critical line, and sharp geometric constants for resonance bounds. A new chapter introduces recently developed techniques for resonance calculation that illuminate the existing results and conjectures on resonance distribution. The spectral theory of hyperbolic surfaces is a point of intersection for a great variety of areas, including quantum physics, discrete groups, differential geometry, number theory, complex analysis, and ergodic theory. This book will serve as a valuable resource for graduate students and researchers from these and other related fields. Review of the first edition: "The exposition is very clear and thorough, and essentially self-contained; the proofs are detailed...The book gathers together some material which is not always easily available in the literature...To conclude, the book is certainly at a level accessible to graduate students and researchers from a rather large range of fields. Clearly, the reader...would certainly benefit greatly from it." (Colin Guillarmou, Mathematical Reviews, Issue 2008 h)