Spectral Problems In Geometry And Arithmetic

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Spectral Problems in Geometry and Arithmetic

Author : Thomas Branson
Publisher : American Mathematical Soc.
ISBN 13 : 0821809407
Total Pages : 190 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Spectral Problems in Geometry and Arithmetic by : Thomas Branson

Download or read book Spectral Problems in Geometry and Arithmetic written by Thomas Branson and published by American Mathematical Soc.. This book was released on 1999 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: These are the proceedings of the NSF-CBMS Conference on "Spectral Problems in Geometry and Arithmetic" held at the University of Iowa. The principal speaker was Peter Sarnak, who has been a central contributor to developments in this field. The volume approaches the topic from the geometric, physical, and number theoretic points of view. The remarkable new connections among seemingly disparate mathematical and scientific disciplines have surprised even veterans of the physical mathematics renaissance forged by gauge theory in the 1970s. Numerical experiments show that the local spacing between zeros of the Riemann zeta function is modelled by spectral phenomena: the eigenvalue distributions of random matrix theory, in particular the Gaussian unitary ensemble (GUE). Related phenomena are from the point of view of differential geometry and global harmonic analysis. Elliptic operators on manifolds have (through zeta function regularization) functional determinants, which are related to functional integrals in quantum theory. The search for critical points of this determinant brings about extremely subtle and delicate sharp inequalities of exponential type. This indicates that zeta functions are spectral objects-and even physical objects. This volume demonstrates that zeta functions are also dynamic, chaotic, and more.

Spectral Geometry

Author : Pierre H. Berard
Publisher : Springer
ISBN 13 : 3540409580
Total Pages : 284 pages
Book Rating : 4.5/5 (44 download)

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Book Synopsis Spectral Geometry by : Pierre H. Berard

Download or read book Spectral Geometry written by Pierre H. Berard and published by Springer. This book was released on 2006-11-14 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Spectral Theory in Riemannian Geometry

Author : Olivier Lablée
Publisher : Erich Schmidt Verlag GmbH & Co. KG
ISBN 13 : 9783037191514
Total Pages : 204 pages
Book Rating : 4.1/5 (915 download)

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Book Synopsis Spectral Theory in Riemannian Geometry by : Olivier Lablée

Download or read book Spectral Theory in Riemannian Geometry written by Olivier Lablée and published by Erich Schmidt Verlag GmbH & Co. KG. This book was released on 2015 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: Spectral theory is a diverse area of mathematics that derives its motivations, goals, and impetus from several sources. In particular, the spectral theory of the Laplacian on a compact Riemannian manifold is a central object in differential geometry. From a physical point a view, the Laplacian on a compact Riemannian manifold is a fundamental linear operator which describes numerous propagation phenomena: heat propagation, wave propagation, quantum dynamics, etc. Moreover, the spectrum of the Laplacian contains vast information about the geometry of the manifold. This book gives a self-contained introduction to spectral geometry on compact Riemannian manifolds. Starting with an overview of spectral theory on Hilbert spaces, the book proceeds to a description of the basic notions in Riemannian geometry. Then its makes its way to topics of main interests in spectral geometry. The topics presented include direct and inverse problems. Direct problems are concerned with computing or finding properties on the eigenvalues while the main issue in inverse problems is knowing the spectrum of the Laplacian, can we determine the geometry of the manifold? Addressed to students or young researchers, the present book is a first introduction to spectral theory applied to geometry. For readers interested in pursuing the subject further, this book will provide a basis for understanding principles, concepts, and developments of spectral geometry.

Progress in Inverse Spectral Geometry

Author : Stig I. Andersson
Publisher : Birkhäuser
ISBN 13 : 3034889380
Total Pages : 202 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Progress in Inverse Spectral Geometry by : Stig I. Andersson

Download or read book Progress in Inverse Spectral Geometry written by Stig I. Andersson and published by Birkhäuser. This book was released on 2012-12-06 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt: Most polynomial growth on every half-space Re (z) ::::: c. Moreover, Op(t) depends holomorphically on t for Re t> O. General references for much of the material on the derivation of spectral functions, asymptotic expansions and analytic properties of spectral functions are [A-P-S] and [Sh], especially Chapter 2. To study the spectral functions and their relation to the geometry and topology of X, one could, for example, take the natural associated parabolic problem as a starting point. That is, consider the 'heat equation': (%t + p) u(x, t) = 0 { u(x, O) = Uo(x), tP which is solved by means of the (heat) semi group V(t) = e- ; namely, u(·, t) = V(t)uoU· Assuming that V(t) is of trace class (which is guaranteed, for instance, if P has a positive principal symbol), it has a Schwartz kernel K E COO(X x X x Rt, E* ®E), locally given by 00 K(x, y; t) = L>-IAk(~k ® 'Pk)(X, y), k=O for a complete set of orthonormal eigensections 'Pk E COO(E). Taking the trace, we then obtain: 00 tA Op(t) = trace(V(t)) = 2::>- k. k=O Now, using, e. g., the Dunford calculus formula (where C is a suitable curve around a(P)) as a starting point and the standard for malism of pseudodifferential operators, one easily derives asymptotic expansions for the spectral functions, in this case for Op.

Geometry of Reflecting Rays and Inverse Spectral Problems

Author : Vesselin Petkov
Publisher : John Wiley & Sons
ISBN 13 :
Total Pages : 328 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Geometry of Reflecting Rays and Inverse Spectral Problems by : Vesselin Petkov

Download or read book Geometry of Reflecting Rays and Inverse Spectral Problems written by Vesselin Petkov and published by John Wiley & Sons. This book was released on 1992 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: The behaviour of reflecting rays plays an essential role in many problems of mathematical physics. This book studies different geometric properties of reflecting rays for manifolds with smooth boundary and their applications to different inverse spectral and scattering problems. This is a developing area in which the authors have made important contributions. Results concerning the particular problems studied and which arise in several important domains of modern physics are presented. Some chapters concerning the generic properties of reflecting rays can be used for courses for graduate students.

Spectral Action in Noncommutative Geometry

Author : Michał Eckstein
Publisher : Springer
ISBN 13 : 3319947885
Total Pages : 165 pages
Book Rating : 4.3/5 (199 download)

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Book Synopsis Spectral Action in Noncommutative Geometry by : Michał Eckstein

Download or read book Spectral Action in Noncommutative Geometry written by Michał Eckstein and published by Springer. This book was released on 2018-12-18 with total page 165 pages. Available in PDF, EPUB and Kindle. Book excerpt: What is spectral action, how to compute it and what are the known examples? This book offers a guided tour through the mathematical habitat of noncommutative geometry à la Connes, deliberately unveiling the answers to these questions. After a brief preface flashing the panorama of the spectral approach, a concise primer on spectral triples is given. Chapter 2 is designed to serve as a toolkit for computations. The third chapter offers an in-depth view into the subtle links between the asymptotic expansions of traces of heat operators and meromorphic extensions of the associated spectral zeta functions. Chapter 4 studies the behaviour of the spectral action under fluctuations by gauge potentials. A subjective list of open problems in the field is spelled out in the fifth Chapter. The book concludes with an appendix including some auxiliary tools from geometry and analysis, along with examples of spectral geometries. The book serves both as a compendium for researchers in the domain of noncommutative geometry and an invitation to mathematical physicists looking for new concepts.

An Introduction to Inverse Scattering and Inverse Spectral Problems

Author : Khosrow Chadan
Publisher : SIAM
ISBN 13 : 0898713870
Total Pages : 206 pages
Book Rating : 4.8/5 (987 download)

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Book Synopsis An Introduction to Inverse Scattering and Inverse Spectral Problems by : Khosrow Chadan

Download or read book An Introduction to Inverse Scattering and Inverse Spectral Problems written by Khosrow Chadan and published by SIAM. This book was released on 1997-01-01 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: Here is a clearly written introduction to three central areas of inverse problems: inverse problems in electromagnetic scattering theory, inverse spectral theory, and inverse problems in quantum scattering theory. Inverse problems, one of the most attractive parts of applied mathematics, attempt to obtain information about structures by nondestructive measurements. Based on a series of lectures presented by three of the authors, all experts in the field, the book provides a quick and easy way for readers to become familiar with the area through a survey of recent developments in inverse spectral and inverse scattering problems.

Numerical Analysis of Spectral Methods

Author : David Gottlieb
Publisher : SIAM
ISBN 13 : 0898710235
Total Pages : 167 pages
Book Rating : 4.8/5 (987 download)

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Book Synopsis Numerical Analysis of Spectral Methods by : David Gottlieb

Download or read book Numerical Analysis of Spectral Methods written by David Gottlieb and published by SIAM. This book was released on 1977-01-01 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt: A unified discussion of the formulation and analysis of special methods of mixed initial boundary-value problems. The focus is on the development of a new mathematical theory that explains why and how well spectral methods work. Included are interesting extensions of the classical numerical analysis.

Geometric and Computational Spectral Theory

Author : Alexandre Girouard
Publisher : American Mathematical Soc.
ISBN 13 : 147042665X
Total Pages : 298 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Geometric and Computational Spectral Theory by : Alexandre Girouard

Download or read book Geometric and Computational Spectral Theory written by Alexandre Girouard and published by American Mathematical Soc.. This book was released on 2017-10-30 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: A co-publication of the AMS and Centre de Recherches Mathématiques The book is a collection of lecture notes and survey papers based on the mini-courses given by leading experts at the 2015 Séminaire de Mathématiques Supérieures on Geometric and Computational Spectral Theory, held from June 15–26, 2015, at the Centre de Recherches Mathématiques, Université de Montréal, Montréal, Quebec, Canada. The volume covers a broad variety of topics in spectral theory, highlighting its connections to differential geometry, mathematical physics and numerical analysis, bringing together the theoretical and computational approaches to spectral theory, and emphasizing the interplay between the two.

Fractal Geometry, Complex Dimensions and Zeta Functions

Author : Michel L. Lapidus
Publisher : Springer Science & Business Media
ISBN 13 : 1461421764
Total Pages : 583 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Fractal Geometry, Complex Dimensions and Zeta Functions by : Michel L. Lapidus

Download or read book Fractal Geometry, Complex Dimensions and Zeta Functions written by Michel L. Lapidus and published by Springer Science & Business Media. This book was released on 2012-09-20 with total page 583 pages. Available in PDF, EPUB and Kindle. Book excerpt: Number theory, spectral geometry, and fractal geometry are interlinked in this in-depth study of the vibrations of fractal strings, that is, one-dimensional drums with fractal boundary. Throughout Geometry, Complex Dimensions and Zeta Functions, Second Edition, new results are examined and a new definition of fractality as the presence of nonreal complex dimensions with positive real parts is presented. The new final chapter discusses several new topics and results obtained since the publication of the first edition.

Spectral Analysis on Graph-like Spaces

Author : Olaf Post
Publisher : Springer Science & Business Media
ISBN 13 : 3642238394
Total Pages : 444 pages
Book Rating : 4.6/5 (422 download)

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Book Synopsis Spectral Analysis on Graph-like Spaces by : Olaf Post

Download or read book Spectral Analysis on Graph-like Spaces written by Olaf Post and published by Springer Science & Business Media. This book was released on 2012-01-06 with total page 444 pages. Available in PDF, EPUB and Kindle. Book excerpt: Small-radius tubular structures have attracted considerable attention in the last few years, and are frequently used in different areas such as Mathematical Physics, Spectral Geometry and Global Analysis. In this monograph, we analyse Laplace-like operators on thin tubular structures ("graph-like spaces''), and their natural limits on metric graphs. In particular, we explore norm resolvent convergence, convergence of the spectra and resonances. Since the underlying spaces in the thin radius limit change, and become singular in the limit, we develop new tools such as norm convergence of operators acting in different Hilbert spaces, an extension of the concept of boundary triples to partial differential operators, and an abstract definition of resonances via boundary triples. These tools are formulated in an abstract framework, independent of the original problem of graph-like spaces, so that they can be applied in many other situations where the spaces are perturbed.

Spectral Geometry of Partial Differential Operators

Author : Michael Ruzhansky
Publisher : Chapman & Hall/CRC
ISBN 13 : 9781138360716
Total Pages : 0 pages
Book Rating : 4.3/5 (67 download)

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Book Synopsis Spectral Geometry of Partial Differential Operators by : Michael Ruzhansky

Download or read book Spectral Geometry of Partial Differential Operators written by Michael Ruzhansky and published by Chapman & Hall/CRC. This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Access; Differential; Durvudkhan; Geometry; Makhmud; Michael; OA; Open; Operators; Partial; Ruzhansky; Sadybekov; Spectral; Suragan.

Modular Forms and Fermat’s Last Theorem

Author : Gary Cornell
Publisher : Springer Science & Business Media
ISBN 13 : 1461219744
Total Pages : 592 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Modular Forms and Fermat’s Last Theorem by : Gary Cornell

Download or read book Modular Forms and Fermat’s Last Theorem written by Gary Cornell and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 592 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the expanded lectures given at a conference on number theory and arithmetic geometry held at Boston University. It introduces and explains the many ideas and techniques used by Wiles, and to explain how his result can be combined with Ribets theorem and ideas of Frey and Serre to prove Fermats Last Theorem. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions and curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of the proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serres conjectures, Galois deformations, universal deformation rings, Hecke algebras, and complete intersections. The book concludes by looking both forward and backward, reflecting on the history of the problem, while placing Wiles'theorem into a more general Diophantine context suggesting future applications. Students and professional mathematicians alike will find this an indispensable resource.

Spectral Theory and Quantum Mechanics

Author : Valter Moretti
Publisher : Springer
ISBN 13 : 331970706X
Total Pages : 962 pages
Book Rating : 4.3/5 (197 download)

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Book Synopsis Spectral Theory and Quantum Mechanics by : Valter Moretti

Download or read book Spectral Theory and Quantum Mechanics written by Valter Moretti and published by Springer. This book was released on 2018-01-30 with total page 962 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book discusses the mathematical foundations of quantum theories. It offers an introductory text on linear functional analysis with a focus on Hilbert spaces, highlighting the spectral theory features that are relevant in physics. After exploring physical phenomenology, it then turns its attention to the formal and logical aspects of the theory. Further, this Second Edition collects in one volume a number of useful rigorous results on the mathematical structure of quantum mechanics focusing in particular on von Neumann algebras, Superselection rules, the various notions of Quantum Symmetry and Symmetry Groups, and including a number of fundamental results on the algebraic formulation of quantum theories. Intended for Master's and PhD students, both in physics and mathematics, the material is designed to be self-contained: it includes a summary of point-set topology and abstract measure theory, together with an appendix on differential geometry. The book also benefits established researchers by organizing and presenting the profusion of advanced material disseminated in the literature. Most chapters are accompanied by exercises, many of which are solved explicitly."

Fractal Geometry and Number Theory

Author : Michel L. Lapidus
Publisher : Springer Science & Business Media
ISBN 13 : 1461253144
Total Pages : 277 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Fractal Geometry and Number Theory by : Michel L. Lapidus

Download or read book Fractal Geometry and Number Theory written by Michel L. Lapidus and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 277 pages. Available in PDF, EPUB and Kindle. Book excerpt: A fractal drum is a bounded open subset of R. m with a fractal boundary. A difficult problem is to describe the relationship between the shape (geo metry) of the drum and its sound (its spectrum). In this book, we restrict ourselves to the one-dimensional case of fractal strings, and their higher dimensional analogues, fractal sprays. We develop a theory of complex di mensions of a fractal string, and we study how these complex dimensions relate the geometry with the spectrum of the fractal string. We refer the reader to [Berrl-2, Lapl-4, LapPol-3, LapMal-2, HeLapl-2] and the ref erences therein for further physical and mathematical motivations of this work. (Also see, in particular, Sections 7. 1, 10. 3 and 10. 4, along with Ap pendix B.) In Chapter 1, we introduce the basic object of our research, fractal strings (see [Lapl-3, LapPol-3, LapMal-2, HeLapl-2]). A 'standard fractal string' is a bounded open subset of the real line. Such a set is a disjoint union of open intervals, the lengths of which form a sequence which we assume to be infinite. Important information about the geometry of . c is contained in its geometric zeta function (c(8) = L lj. j=l 2 Introduction We assume throughout that this function has a suitable meromorphic ex tension. The central notion of this book, the complex dimensions of a fractal string . c, is defined as the poles of the meromorphic extension of (c.

A User's Guide to Spectral Sequences

Author : John McCleary
Publisher : Cambridge University Press
ISBN 13 : 0521567599
Total Pages : 579 pages
Book Rating : 4.5/5 (215 download)

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Book Synopsis A User's Guide to Spectral Sequences by : John McCleary

Download or read book A User's Guide to Spectral Sequences written by John McCleary and published by Cambridge University Press. This book was released on 2001 with total page 579 pages. Available in PDF, EPUB and Kindle. Book excerpt: Spectral sequences are among the most elegant and powerful methods of computation in mathematics. This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. The first part treats the algebraic foundations for this sort of homological algebra, starting from informal calculations. The heart of the text is an exposition of the classical examples from homotopy theory, with chapters on the Leray-Serre spectral sequence, the Eilenberg-Moore spectral sequence, the Adams spectral sequence, and, in this new edition, the Bockstein spectral sequence. The last part of the book treats applications throughout mathematics, including the theory of knots and links, algebraic geometry, differential geometry and algebra. This is an excellent reference for students and researchers in geometry, topology, and algebra.

Issues in Logic, Operations, and Computational Mathematics and Geometry: 2013 Edition

Author :
Publisher : ScholarlyEditions
ISBN 13 : 1490110119
Total Pages : 1187 pages
Book Rating : 4.4/5 (91 download)

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Book Synopsis Issues in Logic, Operations, and Computational Mathematics and Geometry: 2013 Edition by :

Download or read book Issues in Logic, Operations, and Computational Mathematics and Geometry: 2013 Edition written by and published by ScholarlyEditions. This book was released on 2013-05-01 with total page 1187 pages. Available in PDF, EPUB and Kindle. Book excerpt: Issues in Logic, Operations, and Computational Mathematics and Geometry: 2013 Edition is a ScholarlyEditions™ book that delivers timely, authoritative, and comprehensive information about Random Structures and Algorithms. The editors have built Issues in Logic, Operations, and Computational Mathematics and Geometry: 2013 Edition on the vast information databases of ScholarlyNews.™ You can expect the information about Random Structures and Algorithms in this book to be deeper than what you can access anywhere else, as well as consistently reliable, authoritative, informed, and relevant. The content of Issues in Logic, Operations, and Computational Mathematics and Geometry: 2013 Edition has been produced by the world’s leading scientists, engineers, analysts, research institutions, and companies. All of the content is from peer-reviewed sources, and all of it is written, assembled, and edited by the editors at ScholarlyEditions™ and available exclusively from us. You now have a source you can cite with authority, confidence, and credibility. More information is available at http://www.ScholarlyEditions.com/.