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Author : Pierre H. Berard
Publisher : Springer
ISBN 13 : 3540409580
Total Pages : 284 pages
Book Rating : 4.5/5 (44 download)
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:
Author : Jing Hua
Publisher : Academic Press
ISBN 13 : 0128138424
Total Pages : 152 pages
Book Rating : 4.1/5 (281 download)
Download or read book Spectral Geometry of Shapes written by Jing Hua and published by Academic Press. This book was released on 2020-01-15 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: Spectral Geometry of Shapes presents unique shape analysis approaches based on shape spectrum in differential geometry. It provides insights on how to develop geometry-based methods for 3D shape analysis. The book is an ideal learning resource for graduate students and researchers in computer science, computer engineering and applied mathematics who have an interest in 3D shape analysis, shape motion analysis, image analysis, medical image analysis, computer vision and computer graphics. Due to the rapid advancement of 3D acquisition technologies there has been a big increase in 3D shape data that requires a variety of shape analysis methods, hence the need for this comprehensive resource. Presents the latest advances in spectral geometric processing for 3D shape analysis applications, such as shape classification, shape matching, medical imaging, etc. Provides intuitive links between fundamental geometric theories and real-world applications, thus bridging the gap between theory and practice Describes new theoretical breakthroughs in applying spectral methods for non-isometric motion analysis Gives insights for developing spectral geometry-based approaches for 3D shape analysis and deep learning of shape 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)
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.
Author : Giampiero Esposito
Publisher : Cambridge University Press
ISBN 13 : 0521648629
Total Pages : 227 pages
Book Rating : 4.5/5 (216 download)
Download or read book Dirac Operators and Spectral Geometry written by Giampiero Esposito and published by Cambridge University Press. This book was released on 1998-08-20 with total page 227 pages. Available in PDF, EPUB and Kindle. Book excerpt: A clear, concise and up-to-date introduction to the theory of the Dirac operator and its wide range of applications in theoretical physics for graduate students and researchers.
Author : M.-E. Craioveanu
Publisher : Springer Science & Business Media
ISBN 13 : 940172475X
Total Pages : 447 pages
Book Rating : 4.4/5 (17 download)
Download or read book Old and New Aspects in Spectral Geometry written by M.-E. Craioveanu and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 447 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is known that to any Riemannian manifold (M, g ) , with or without boundary, one can associate certain fundamental objects. Among them are the Laplace-Beltrami opera tor and the Hodge-de Rham operators, which are natural [that is, they commute with the isometries of (M,g)], elliptic, self-adjoint second order differential operators acting on the space of real valued smooth functions on M and the spaces of smooth differential forms on M, respectively. If M is closed, the spectrum of each such operator is an infinite divergent sequence of real numbers, each eigenvalue being repeated according to its finite multiplicity. Spectral Geometry is concerned with the spectra of these operators, also the extent to which these spectra determine the geometry of (M, g) and the topology of M. This problem has been translated by several authors (most notably M. Kac). into the col loquial question "Can one hear the shape of a manifold?" because of its analogy with the wave equation. This terminology was inspired from earlier results of H. Weyl. It is known that the above spectra cannot completely determine either the geometry of (M , g) or the topology of M. For instance, there are examples of pairs of closed Riemannian manifolds with the same spectra corresponding to the Laplace-Beltrami operators, but which differ substantially in their geometry and which are even not homotopically equiva lent.
Author : Stig I. Andersson
Publisher : Birkhäuser
ISBN 13 : 3034889380
Total Pages : 202 pages
Book Rating : 4.0/5 (348 download)
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.
Author : Dmitri Fursaev
Publisher : Springer Science & Business Media
ISBN 13 : 9400702051
Total Pages : 294 pages
Book Rating : 4.4/5 (7 download)
Download or read book Operators, Geometry and Quanta written by Dmitri Fursaev and published by Springer Science & Business Media. This book was released on 2011-06-25 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Urakawa Hajime
Publisher : World Scientific
ISBN 13 : 9813109106
Total Pages : 312 pages
Book Rating : 4.8/5 (131 download)
Download or read book Spectral Geometry Of The Laplacian: Spectral Analysis And Differential Geometry Of The Laplacian written by Urakawa Hajime and published by World Scientific. This book was released on 2017-06-02 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: The totality of the eigenvalues of the Laplacian of a compact Riemannian manifold is called the spectrum. We describe how the spectrum determines a Riemannian manifold. The continuity of the eigenvalue of the Laplacian, Cheeger and Yau's estimate of the first eigenvalue, the Lichnerowicz–Obata's theorem on the first eigenvalue, the Cheng's estimates of the kth eigenvalues, and Payne–Pólya–Weinberger's inequality of the Dirichlet eigenvalue of the Laplacian are also described. Then, the theorem of Colin de Verdière, that is, the spectrum determines the totality of all the lengths of closed geodesics is described. We give the V Guillemin and D Kazhdan's theorem which determines the Riemannian manifold of negative curvature.
Author : Vladimir G. Berkovich
Publisher : American Mathematical Soc.
ISBN 13 : 0821890204
Total Pages : 169 pages
Book Rating : 4.8/5 (218 download)
Download or read book Spectral Theory and Analytic Geometry over Non-Archimedean Fields written by Vladimir G. Berkovich and published by American Mathematical Soc.. This book was released on 2012-08-02 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: The purpose of this book is to introduce a new notion of analytic space over a non-Archimedean field. Despite the total disconnectedness of the ground field, these analytic spaces have the usual topological properties of a complex analytic space, such as local compactness and local arcwise connectedness. This makes it possible to apply the usual notions of homotopy and singular homology. The book includes a homotopic characterization of the analytic spaces associated with certain classes of algebraic varieties and an interpretation of Bruhat-Tits buildings in terms of these analytic spaces. The author also studies the connection with the earlier notion of a rigid analytic space. Geometrical considerations are used to obtain some applications, and the analytic spaces are used to construct the foundations of a non-Archimedean spectral theory of bounded linear operators. This book requires a background at the level of basic graduate courses in algebra and topology, as well as some familiarity with algebraic geometry. It would be of interest to research mathematicians and graduate students working in algebraic geometry, number theory, and -adic analysis.
Author : Peter B. Gilkey
Publisher : CRC Press
ISBN 13 : 9780849382772
Total Pages : 294 pages
Book Rating : 4.3/5 (827 download)
Download or read book Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture written by Peter B. Gilkey and published by CRC Press. This book was released on 1999-07-27 with total page 294 pages. Available in PDF, EPUB and Kindle. Book excerpt: This cutting-edge, standard-setting text explores the spectral geometry of Riemannian submersions. Working for the most part with the form valued Laplacian in the class of smooth compact manifolds without boundary, the authors study the relationship-if any-between the spectrum of Dp on Y and Dp on Z, given that Dp is the p form valued Laplacian and pi: Z ® Y is a Riemannian submersion. After providing the necessary background, including basic differential geometry and a discussion of Laplace type operators, the authors address rigidity theorems. They establish conditions that ensure that the pull back of every eigenform on Y is an eigenform on Z so the eigenvalues do not change, then show that if a single eigensection is preserved, the eigenvalues do not change for the scalar or Bochner Laplacians. For the form valued Laplacian, they show that if an eigenform is preserved, then the corresponding eigenvalue can only increase. They generalize these results to the complex setting as well. However, the spinor setting is quite different. For a manifold with non-trivial boundary and imposed Neumann boundary conditions, the result is surprising-the eigenvalues can change. Although this is a relatively rare phenomenon, the authors give examples-a circle bundle or, more generally, a principal bundle with structure group G where the first cohomology group H1(G;R) is non trivial. They show similar results in the complex setting, show that eigenvalues can decrease in the spinor setting, and offer a list of unsolved problems in this area. Moving to some related topics involving questions of positive curvature, for the first time in mathematical literature the authors establish a link between the spectral geometry of Riemannian submersions and the Gromov-Lawson conjecture. Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture addresses a hot research area and promises to set a standard for the field. Researchers and applied mathematicians interested in mathematical physics and relativity will find this work both fascinating and important.
Author : Alex Barnett
Publisher : American Mathematical Soc.
ISBN 13 : 0821853198
Total Pages : 339 pages
Book Rating : 4.8/5 (218 download)
Download or read book Spectral Geometry written by Alex Barnett and published by American Mathematical Soc.. This book was released on 2012 with total page 339 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the International Conference on Spectral Geometry, held July 19-23, 2010, at Dartmouth College, Dartmouth, New Hampshire. Eigenvalue problems involving the Laplace operator on manifolds have proven to be a consistently fertile area of geometric analysis with deep connections to number theory, physics, and applied mathematics. Key questions include the measures to which eigenfunctions of the Laplacian on a Riemannian manifold condense in the limit of large eigenvalue, and the extent to which the eigenvalues and eigenfunctions of a manifold encode its geometry. In this volume, research and expository articles, including those of the plenary speakers Peter Sarnak and Victor Guillemin, address the flurry of recent progress in such areas as quantum unique ergodicity, isospectrality, semiclassical measures, the geometry of nodal lines of eigenfunctions, methods of numerical computation, and spectra of quantum graphs. This volume also contains mini-courses on spectral theory for hyperbolic surfaces, semiclassical analysis, and orbifold spectral geometry that prepared the participants, especially graduate students and young researchers, for conference lectures.
Author : Peter B. Gilkey
Publisher : CRC Press
ISBN 13 : 0203490460
Total Pages : 312 pages
Book Rating : 4.2/5 (34 download)
Download or read book Asymptotic Formulae in Spectral Geometry written by Peter B. Gilkey and published by CRC Press. This book was released on 2003-12-17 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 computi
Author : Michael Levitin
Publisher : American Mathematical Society
ISBN 13 : 1470475251
Total Pages : 346 pages
Book Rating : 4.4/5 (74 download)
Download or read book Topics in Spectral Geometry written by Michael Levitin and published by American Mathematical Society. This book was released on 2023-11-30 with total page 346 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Pavel Kurasov
Publisher : Springer Nature
ISBN 13 : 3662678721
Total Pages : 644 pages
Book Rating : 4.6/5 (626 download)
Download or read book Spectral Geometry of Graphs written by Pavel Kurasov and published by Springer Nature. This book was released on 2023-12-09 with total page 644 pages. Available in PDF, EPUB and Kindle. Book excerpt: This open access book gives a systematic introduction into the spectral theory of differential operators on metric graphs. Main focus is on the fundamental relations between the spectrum and the geometry of the underlying graph. The book has two central themes: the trace formula and inverse problems. The trace formula is relating the spectrum to the set of periodic orbits and is comparable to the celebrated Selberg and Chazarain-Duistermaat-Guillemin-Melrose trace formulas. Unexpectedly this formula allows one to construct non-trivial crystalline measures and Fourier quasicrystals solving one of the long-standing problems in Fourier analysis. The remarkable story of this mathematical odyssey is presented in the first part of the book. To solve the inverse problem for Schrödinger operators on metric graphs the magnetic boundary control method is introduced. Spectral data depending on the magnetic flux allow one to solve the inverse problem in full generality, this means to reconstruct not only the potential on a given graph, but also the underlying graph itself and the vertex conditions. The book provides an excellent example of recent studies where the interplay between different fields like operator theory, algebraic geometry and number theory, leads to unexpected and sound mathematical results. The book is thought as a graduate course book where every chapter is suitable for a separate lecture and includes problems for home studies. Numerous illuminating examples make it easier to understand new concepts and develop the necessary intuition for further studies.
Author : Krzysztof P. Wojciechowski
Publisher : American Mathematical Soc.
ISBN 13 : 082183536X
Total Pages : 338 pages
Book Rating : 4.8/5 (218 download)
Download or read book Spectral Geometry of Manifolds with Boundary and Decomposition of Manifolds written by Krzysztof P. Wojciechowski and published by American Mathematical Soc.. This book was released on 2005 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Michael Ruzhansky
Publisher : CRC Press
ISBN 13 : 0429780575
Total Pages : 366 pages
Book Rating : 4.4/5 (297 download)
Download or read book Spectral Geometry of Partial Differential Operators written by Michael Ruzhansky and published by CRC Press. This book was released on 2020-02-07 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 : Peter Buser
Publisher : Springer Science & Business Media
ISBN 13 : 0817649921
Total Pages : 456 pages
Book Rating : 4.8/5 (176 download)
Download or read book Geometry and Spectra of Compact Riemann Surfaces written by Peter Buser and published by Springer Science & Business Media. This book was released on 2010-10-29 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is a self-contained introduction to the geometry of Riemann Surfaces of constant curvature –1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces. Research workers and graduate students interested in compact Riemann surfaces will find here a number of useful tools and insights to apply to their investigations.
