Spectral Geometry Of Partial Differential Operators

Author : Michael Ruzhansky,Makhmud Sadybekov,Durvudkhan Suragan
Publisher : CRC Press
Page : 366 pages
File Size : 51,9 Mb
Release : 2020-02-07
Category : Mathematics
ISBN : 9780429780578

Get Book

Spectral Geometry of Partial Differential Operators by Michael Ruzhansky,Makhmud Sadybekov,Durvudkhan Suragan Pdf

The aim of Spectral Geometry of Partial Differential Operators is to provide a basic and self-contained introduction to the ideas underpinning spectral geometric inequalities arising in the theory of partial differential equations. Historically, one of the first inequalities of the spectral geometry was the minimization problem of the first eigenvalue of the Dirichlet Laplacian. Nowadays, this type of inequalities of spectral geometry have expanded to many other cases with number of applications in physics and other sciences. The main reason why the results are useful, beyond the intrinsic interest of geometric extremum problems, is that they produce a priori bounds for spectral invariants of (partial differential) operators on arbitrary domains. Features: Collects the ideas underpinning the inequalities of the spectral geometry, in both self-adjoint and non-self-adjoint operator theory, in a way accessible by anyone with a basic level of understanding of linear differential operators Aimed at theoretical as well as applied mathematicians, from a wide range of scientific fields, including acoustics, astronomy, MEMS, and other physical sciences Provides a step-by-step guide to the techniques of non-self-adjoint partial differential operators, and for the applications of such methods. Provides a self-contained coverage of the traditional and modern theories of linear partial differential operators, and does not require a previous background in operator theory.

Author : Pierre H. Berard
Publisher : Springer
Page : 284 pages
File Size : 42,7 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540409588

Get Book

Spectral Geometry by Pierre H. Berard Pdf

Author : Krzysztof P. Wojciechowski
Publisher : American Mathematical Soc.
Page : 338 pages
File Size : 42,5 Mb
Release : 2005
Category : Algèbres d'opérateurs - Congrès
ISBN : 9780821835364

Get Book

Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds by Krzysztof P. Wojciechowski Pdf

In recent years, increasingly complex methods have been brought into play in the treatment of geometric and topological problems for partial differential operators on manifolds. This collection of papers, resulting from a Workshop on Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds, provides a broad picture of these methods with new results. Subjects in the book cover a wide variety of topics, from recent advances in index theory and the more general theory of spectral invariants on closed manifolds and manifolds with boundary, to applications of those invariants in geometry, topology, and physics. Papers are grouped into four parts: Part I gives an overview of the subject from various points of view. Part II deals with spectral invariants, such as traces, indices, and determinants. Part III is concerned with general geometric and topological questions. Part IV deals specifically with problems on manifolds with singularities. The book is suitable for graduate students and researchers interested in spectral problems in geometry.

Author : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 54,9 Mb
Release : 1991
Category : Differential equations, Partial
ISBN : 0387546774

Get Book

Partial Differential Equations by Anonim Pdf

Author : Agostino Prastaro,Themistocles M. Rassias
Publisher : World Scientific
Page : 482 pages
File Size : 51,5 Mb
Release : 1994
Category : Mathematics
ISBN : 9810214073

Get Book

Geometry in Partial Differential Equations by Agostino Prastaro,Themistocles M. Rassias Pdf

This book emphasizes the interdisciplinary interaction in problems involving geometry and partial differential equations. It provides an attempt to follow certain threads that interconnect various approaches in the geometric applications and influence of partial differential equations. A few such approaches include: Morse-Palais-Smale theory in global variational calculus, general methods to obtain conservation laws for PDEs, structural investigation for the understanding of the meaning of quantum geometry in PDEs, extensions to super PDEs (formulated in the category of supermanifolds) of the geometrical methods just introduced for PDEs and the harmonic theory which proved to be very important especially after the appearance of the Atiyah-Singer index theorem, which provides a link between geometry and topology.

Author : M.A. Shubin
Publisher : Springer Science & Business Media
Page : 278 pages
File Size : 52,5 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783662067192

Get Book

Partial Differential Equations VII by M.A. Shubin Pdf

This EMS volume contains a survey of the principles and advanced techniques of the spectral theory of linear differential and pseudodifferential operators in finite-dimensional spaces. Also including a special section of Sunada's recent solution of Kac's celebrated problem of whether or not "one can hear the shape of a drum".

Author : M.A. Shubin
Publisher : Springer
Page : 274 pages
File Size : 51,6 Mb
Release : 2012-12-22
Category : Mathematics
ISBN : 366206720X

Get Book

Partial Differential Equations VII by M.A. Shubin Pdf

This EMS volume contains a survey of the principles and advanced techniques of the spectral theory of linear differential and pseudodifferential operators in finite-dimensional spaces. Also including a special section of Sunada's recent solution of Kac's celebrated problem of whether or not "one can hear the shape of a drum".

Author : Michael Demuth,Bert-Wolfgang Schulze
Publisher : Birkhäuser
Page : 346 pages
File Size : 42,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034882316

Get Book

Partial Differential Equations and Spectral Theory by Michael Demuth,Bert-Wolfgang Schulze Pdf

The intention of the international conference PDE2000 was to bring together specialists from different areas of modern analysis, mathematical physics and geometry, to discuss not only the recent progress in their own fields but also the interaction between these fields. The special topics of the conference were spectral and scattering theory, semiclassical and asymptotic analysis, pseudodifferential operators and their relation to geometry, as well as partial differential operators and their connection to stochastic analysis and to the theory of semigroups. The scientific advisory board of the conference in Clausthal consisted of M. Ben-Artzi (Jerusalem), Chen Hua (Peking), M. Demuth (Clausthal), T. Ichinose (Kanazawa), L. Rodino (Turin), B.-W. Schulze (Potsdam) and J. Sjöstrand (Paris). The book is aimed at researchers in mathematics and mathematical physics with interests in partial differential equations and all its related fields.

Author : David Borthwick
Publisher : Springer Nature
Page : 339 pages
File Size : 45,5 Mb
Release : 2020-03-12
Category : Mathematics
ISBN : 9783030380021

Get Book

Spectral Theory by David Borthwick Pdf

This textbook offers a concise introduction to spectral theory, designed for newcomers to functional analysis. Curating the content carefully, the author builds to a proof of the spectral theorem in the early part of the book. Subsequent chapters illustrate a variety of application areas, exploring key examples in detail. Readers looking to delve further into specialized topics will find ample references to classic and recent literature. Beginning with a brief introduction to functional analysis, the text focuses on unbounded operators and separable Hilbert spaces as the essential tools needed for the subsequent theory. A thorough discussion of the concepts of spectrum and resolvent follows, leading to a complete proof of the spectral theorem for unbounded self-adjoint operators. Applications of spectral theory to differential operators comprise the remaining four chapters. These chapters introduce the Dirichlet Laplacian operator, Schrödinger operators, operators on graphs, and the spectral theory of Riemannian manifolds. Spectral Theory offers a uniquely accessible introduction to ideas that invite further study in any number of different directions. A background in real and complex analysis is assumed; the author presents the requisite tools from functional analysis within the text. This introductory treatment would suit a functional analysis course intended as a pathway to linear PDE theory. Independent later chapters allow for flexibility in selecting applications to suit specific interests within a one-semester course.

Author : M.A. Shubin
Publisher : Springer Science & Business Media
Page : 296 pages
File Size : 54,7 Mb
Release : 2011-06-28
Category : Mathematics
ISBN : 9783642565793

Get Book

Pseudodifferential Operators and Spectral Theory by M.A. Shubin Pdf

I had mixed feelings when I thought how I should prepare the book for the second edition. It was clear to me that I had to correct all mistakes and misprints that were found in the book during the life of the first edition. This was easy to do because the mistakes were mostly minor and easy to correct, and the misprints were not many. It was more difficult to decide whether I should update the book (or at least its bibliography) somehow. I decided that it did not need much of an updating. The main value of any good mathematical book is that it teaches its reader some language and some skills. It can not exhaust any substantial topic no matter how hard the author tried. Pseudodifferential operators became a language and a tool of analysis of partial differential equations long ago. Therefore it is meaningless to try to exhaust this topic. Here is an easy proof. As of July 3, 2000, MathSciNet (the database of the American Mathematical Society) in a few seconds found 3695 sources, among them 363 books, during its search for "pseudodifferential operator". (The search also led to finding 963 sources for "pseudo-differential operator" but I was unable to check how much the results ofthese two searches intersected). This means that the corresponding words appear either in the title or in the review published in Mathematical Reviews.

Author : Dmitri Fursaev,Dmitri Vassilevich
Publisher : Springer Science & Business Media
Page : 294 pages
File Size : 52,9 Mb
Release : 2011-06-25
Category : Science
ISBN : 9789400702059

Get Book

Operators, Geometry and Quanta by Dmitri Fursaev,Dmitri Vassilevich Pdf

This book gives a detailed and self-contained introduction into the theory of spectral functions, with an emphasis on their applications to quantum field theory. All methods are illustrated with applications to specific physical problems from the forefront of current research, such as finite-temperature field theory, D-branes, quantum solitons and noncommutativity. In the first part of the book, necessary background information on differential geometry and quantization, including less standard material, is collected. The second part of the book contains a detailed description of main spectral functions and methods of their calculation. In the third part, the theory is applied to several examples (D-branes, quantum solitons, anomalies, noncommutativity). This book addresses advanced graduate students and researchers in mathematical physics with basic knowledge of quantum field theory and differential geometry. The aim is to prepare readers to use spectral functions in their own research, in particular in relation to heat kernels and zeta functions.

Author : Peter B. Gilkey
Publisher : CRC Press
Page : 315 pages
File Size : 49,5 Mb
Release : 2003-12-17
Category : Mathematics
ISBN : 9781135440749

Get Book

Asymptotic Formulae in Spectral Geometry by Peter B. Gilkey Pdf

A great deal of progress has been made recently in the field of asymptotic formulas that arise in the theory of Dirac and Laplace type operators. Asymptotic Formulae in Spectral Geometry collects these results and computations into one book. Written by a leading pioneer in the field, it focuses on the functorial and special cases methods of computing asymptotic heat trace and heat content coefficients in the heat equation. It incorporates the work of many authors into the presentation, and includes a complete bibliography that serves as a roadmap to the literature on the subject. Geometers, mathematical physicists, and analysts alike will undoubtedly find this book to be the definitive book on the subject

Author : Michael Levitin,Dan Mangoubi,Iosif Polterovich
Publisher : American Mathematical Society
Page : 346 pages
File Size : 42,9 Mb
Release : 2023-11-30
Category : Mathematics
ISBN : 9781470475253

Get Book

Topics in Spectral Geometry by Michael Levitin,Dan Mangoubi,Iosif Polterovich Pdf

It is remarkable that various distinct physical phenomena, such as wave propagation, heat diffusion, electron movement in quantum mechanics, oscillations of fluid in a container, can be described using the same differential operator, the Laplacian. Spectral data (i.e., eigenvalues and eigenfunctions) of the Laplacian depend in a subtle way on the geometry of the underlying object, e.g., a Euclidean domain or a Riemannian manifold, on which the operator is defined. This dependence, or, rather, the interplay between the geometry and the spectrum, is the main subject of spectral geometry. Its roots can be traced to Ernst Chladni's experiments with vibrating plates, Lord Rayleigh's theory of sound, and Mark Kac's celebrated question “Can one hear the shape of a drum?” In the second half of the twentieth century spectral geometry emerged as a separate branch of geometric analysis. Nowadays it is a rapidly developing area of mathematics, with close connections to other fields, such as differential geometry, mathematical physics, partial differential equations, number theory, dynamical systems, and numerical analysis. This book can be used for a graduate or an advanced undergraduate course on spectral geometry, starting from the basics but at the same time covering some of the exciting recent developments which can be explained without too many prerequisites.

Author : Bernhelm Booss,Krzysztof Wojciechowski,Krzysztof P. Wojciechowski
Publisher : American Mathematical Soc.
Page : 282 pages
File Size : 53,7 Mb
Release : 1999
Category : Geometry, Differential
ISBN : 9780821820612

Get Book

Geometric Aspects of Partial Differential Equations by Bernhelm Booss,Krzysztof Wojciechowski,Krzysztof P. Wojciechowski Pdf

This collection of papers by leading researchers gives a broad picture of current research directions in geometric aspects of partial differential equations. Based on lectures presented at a Minisymposium on Spectral Invariants - Heat Equation Approach, held in September 1998 at Roskilde University in Denmark, the book provides both a careful exposition of new perspectives in classical index theory and an introduction to currently active areas of the field. Presented here are new index theorems as well as new calculations of the eta-invariant, of the spectral flow, of the Maslov index, of Seiberg-Witten monopoles, heat kernels, determinants, non-commutative residues, and of the Ray-Singer torsion. New types of boundary value problems for operators of Dirac type and generalizations to manifolds with cuspidal ends, to non-compact and to infinite-dimensional manifolds are also discussed. Throughout the book, the use of advanced analysis methods for gaining geometric insight emerges as a central theme. Aimed at graduate students and researchers, this book would be suitable as a text for an advanced graduate topics course on geometric aspects of partial differential equations and spectral invariants.

Author : Jochen Brüning,Matthias Staudacher
Publisher : Walter de Gruyter GmbH & Co KG
Page : 517 pages
File Size : 51,6 Mb
Release : 2018-04-09
Category : Mathematics
ISBN : 9783110452150

Get Book

Space – Time – Matter by Jochen Brüning,Matthias Staudacher Pdf

This monograph describes some of the most interesting results obtained by the mathematicians and physicists collaborating in the CRC 647 "Space – Time – Matter", in the years 2005 - 2016. The work presented concerns the mathematical and physical foundations of string and quantum field theory as well as cosmology. Important topics are the spaces and metrics modelling the geometry of matter, and the evolution of these geometries. The partial differential equations governing such structures and their singularities, special solutions and stability properties are discussed in detail. Contents Introduction Algebraic K-theory, assembly maps, controlled algebra, and trace methods Lorentzian manifolds with special holonomy – Constructions and global properties Contributions to the spectral geometry of locally homogeneous spaces On conformally covariant differential operators and spectral theory of the holographic Laplacian Moduli and deformations Vector bundles in algebraic geometry and mathematical physics Dyson–Schwinger equations: Fix-point equations for quantum fields Hidden structure in the form factors ofN = 4 SYM On regulating the AdS superstring Constraints on CFT observables from the bootstrap program Simplifying amplitudes in Maxwell-Einstein and Yang-Mills-Einstein supergravities Yangian symmetry in maximally supersymmetric Yang-Mills theory Wave and Dirac equations on manifolds Geometric analysis on singular spaces Singularities and long-time behavior in nonlinear evolution equations and general relativity