Spectral Geometry of Shapes

Spectral Geometry of Shapes
Author: Jing Hua,Zichun Zhong
Publsiher: Academic Press
Total Pages: 195
Release: 2020-01-15
ISBN 10: 0128138424
ISBN 13: 9780128138427
Language: EN, FR, DE, ES & NL

Spectral Geometry of Shapes Book Review:

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

Shape Optimization and Spectral Theory

Shape Optimization and Spectral Theory
Author: Antoine Henrot
Publsiher: De Gruyter Open
Total Pages: 474
Release: 2017-05-08
ISBN 10: 9783110550856
ISBN 13: 3110550857
Language: EN, FR, DE, ES & NL

Shape Optimization and Spectral Theory Book Review:

"Shape optimization and spectral theory" is a survey book aiming to give an overview of recent results in spectral geometry and its links with shape optimization. It covers most of the issues which are important for people working in PDE and differential geometry interested in sharp inequalities and qualitative behaviour for eigenvalues of the Laplacian with different kind of boundary conditions (Dirichlet, Robin and Steklov). This includes: existence of optimal shapes, their regularity, the case of special domains like triangles, isospectrality, quantitative form of the isoperimetric inequalities, optimal partitions, universal inequalities and numerical results. Much progress has been made in these extremum problems during the last ten years and this edited volume presents a valuable update to a wide community interested in these topics. List of contributors Antunes Pedro R.S., Ashbaugh Mark, Bonnaillie-Noel Virginie, Brasco Lorenzo, Bucur Dorin, Buttazzo Giuseppe, De Philippis Guido, Freitas Pedro, Girouard Alexandre, Helffer Bernard, Kennedy James, Lamboley Jimmy, Laugesen Richard S., Oudet Edouard, Pierre Michel, Polterovich Iosif, Siudeja Bartlomiej A., Velichkov Bozhidar

Processing Analyzing and Learning of Images Shapes and Forms Part 2

Processing  Analyzing and Learning of Images  Shapes  and Forms  Part 2
Author: Anonim
Publsiher: Elsevier
Total Pages: 706
Release: 2019-10-16
ISBN 10: 0444641416
ISBN 13: 9780444641410
Language: EN, FR, DE, ES & NL

Processing Analyzing and Learning of Images Shapes and Forms Part 2 Book Review:

Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2, Volume 20, surveys the contemporary developments relating to the analysis and learning of images, shapes and forms, covering mathematical models and quick computational techniques. Chapter cover Alternating Diffusion: A Geometric Approach for Sensor Fusion, Generating Structured TV-based Priors and Associated Primal-dual Methods, Graph-based Optimization Approaches for Machine Learning, Uncertainty Quantification and Networks, Extrinsic Shape Analysis from Boundary Representations, Efficient Numerical Methods for Gradient Flows and Phase-field Models, Recent Advances in Denoising of Manifold-Valued Images, Optimal Registration of Images, Surfaces and Shapes, and much more. Covers contemporary developments relating to the analysis and learning of images, shapes and forms Presents mathematical models and quick computational techniques relating to the topic Provides broad coverage, with sample chapters presenting content on Alternating Diffusion and Generating Structured TV-based Priors and Associated Primal-dual Methods

Partitioning a Many Dimensional Containment Space

Partitioning a Many Dimensional Containment Space
Author: Dr. Martin Concoyle
Publsiher: Trafford Publishing
Total Pages: 794
Release: 2014-01-16
ISBN 10: 1490723714
ISBN 13: 9781490723716
Language: EN, FR, DE, ES & NL

Partitioning a Many Dimensional Containment Space Book Review:

This book is an introduction to the simple math patterns used to describe fundamental, stable, spectral-orbital physical systems (represented as discrete hyperbolic shapes). The containment set has many dimensions, and these dimensions possess macroscopic geometric properties (which are discrete hyperbolic shapes). Thus, it is a description that transcends the idea of materialism (i.e., it is higher-dimensional), and it can also be used to model a life-form as a unified, high-dimension, geometric construct, which generates its own energy and which has a natural structure for memory, where this construct is made in relation to the main property of the description being the spectral properties of both material systems and of the metric-spaces that contain the material systems, where material is simply a lower dimension metric-space and where both material components and metric-spaces are in resonance with the containing space.

The Authority of Material Vs the Spirit

The Authority of Material Vs  the Spirit
Author: Douglas D Hunter
Publsiher: Trafford Publishing
Total Pages: 135
Release: 2006-12-22
ISBN 10: 1412240433
ISBN 13: 9781412240437
Language: EN, FR, DE, ES & NL

The Authority of Material Vs the Spirit Book Review:

A new mathematically-based structure for language allows for a new context with which one can make verifiable predictions about: material, life, mind, and the spiritual intent of (creative) existence.

Numerical Geometry of Non Rigid Shapes

Numerical Geometry of Non Rigid Shapes
Author: Alexander M. Bronstein,Michael M. Bronstein,Ron Kimmel
Publsiher: Springer Science & Business Media
Total Pages: 346
Release: 2008-09-18
ISBN 10: 0387733019
ISBN 13: 9780387733012
Language: EN, FR, DE, ES & NL

Numerical Geometry of Non Rigid Shapes Book Review:

Deformable objects are ubiquitous in the world surrounding us, on all levels from micro to macro. The need to study such shapes and model their behavior arises in a wide spectrum of applications, ranging from medicine to security. In recent years, non-rigid shapes have attracted growing interest, which has led to rapid development of the field, where state-of-the-art results from very different sciences - theoretical and numerical geometry, optimization, linear algebra, graph theory, machine learning and computer graphics, to mention several - are applied to find solutions. This book gives an overview of the current state of science in analysis and synthesis of non-rigid shapes. Everyday examples are used to explain concepts and to illustrate different techniques. The presentation unfolds systematically and numerous figures enrich the engaging exposition. Practice problems follow at the end of each chapter, with detailed solutions to selected problems in the appendix. A gallery of colored images enhances the text. This book will be of interest to graduate students, researchers and professionals in different fields of mathematics, computer science and engineering. It may be used for courses in computer vision, numerical geometry and geometric modeling and computer graphics or for self-study.

ECAI 2020

ECAI 2020
Author: G. De Giacomo,A. Catala,B. Dilkina
Publsiher: IOS Press
Total Pages: 3122
Release: 2020-09-11
ISBN 10: 164368101X
ISBN 13: 9781643681016
Language: EN, FR, DE, ES & NL

ECAI 2020 Book Review:

This book presents the proceedings of the 24th European Conference on Artificial Intelligence (ECAI 2020), held in Santiago de Compostela, Spain, from 29 August to 8 September 2020. The conference was postponed from June, and much of it conducted online due to the COVID-19 restrictions. The conference is one of the principal occasions for researchers and practitioners of AI to meet and discuss the latest trends and challenges in all fields of AI and to demonstrate innovative applications and uses of advanced AI technology. The book also includes the proceedings of the 10th Conference on Prestigious Applications of Artificial Intelligence (PAIS 2020) held at the same time. A record number of more than 1,700 submissions was received for ECAI 2020, of which 1,443 were reviewed. Of these, 361 full-papers and 36 highlight papers were accepted (an acceptance rate of 25% for full-papers and 45% for highlight papers). The book is divided into three sections: ECAI full papers; ECAI highlight papers; and PAIS papers. The topics of these papers cover all aspects of AI, including Agent-based and Multi-agent Systems; Computational Intelligence; Constraints and Satisfiability; Games and Virtual Environments; Heuristic Search; Human Aspects in AI; Information Retrieval and Filtering; Knowledge Representation and Reasoning; Machine Learning; Multidisciplinary Topics and Applications; Natural Language Processing; Planning and Scheduling; Robotics; Safe, Explainable, and Trustworthy AI; Semantic Technologies; Uncertainty in AI; and Vision. The book will be of interest to all those whose work involves the use of AI technology.

Medical Image Computing and Computer Assisted Intervention MICCAI 2016

Medical Image Computing and Computer Assisted Intervention   MICCAI 2016
Author: Sebastien Ourselin,Leo Joskowicz,Mert R. Sabuncu,Gozde Unal,William Wells
Publsiher: Springer
Total Pages: 641
Release: 2016-10-17
ISBN 10: 3319467263
ISBN 13: 9783319467269
Language: EN, FR, DE, ES & NL

Medical Image Computing and Computer Assisted Intervention MICCAI 2016 Book Review:

The three-volume set LNCS 9900, 9901, and 9902 constitutes the refereed proceedings of the 19th International Conference on Medical Image Computing and Computer-Assisted Intervention, MICCAI 2016, held in Athens, Greece, in October 2016. Based on rigorous peer reviews, the program committee carefully selected 228 revised regular papers from 756 submissions for presentation in three volumes. The papers have been organized in the following topical sections: Part I: brain analysis, brain analysis - connectivity; brain analysis - cortical morphology; Alzheimer disease; surgical guidance and tracking; computer aided interventions; ultrasound image analysis; cancer image analysis; Part II: machine learning and feature selection; deep learning in medical imaging; applications of machine learning; segmentation; cell image analysis; Part III: registration and deformation estimation; shape modeling; cardiac and vascular image analysis; image reconstruction; and MR image analysis.

The Mathematical Structure of Stable Physical Systems

The Mathematical Structure of Stable Physical Systems
Author: Dr. Martin Concoyle & G.P. Coatmundi
Publsiher: Trafford Publishing
Total Pages: 700
Release: 2014
ISBN 10: 1490723641
ISBN 13: 9781490723648
Language: EN, FR, DE, ES & NL

The Mathematical Structure of Stable Physical Systems Book Review:

This book is an introduction to the simple math patterns used to describe fundamental, stable spectral-orbital physical systems (represented as discrete hyperbolic shapes), the containment set has many-dimensions, and these dimensions possess macroscopic geometric properties (which are also discrete hyperbolic shapes). Thus, it is a description which transcends the idea of materialism (ie it is higher-dimensional), and it can also be used to model a life-form as a unified, high-dimension, geometric construct, which generates its own energy, and which has a natural structure for memory, where this construct is made in relation to the main property of the description being, in fact, the spectral properties of both material systems and of the metric-spaces which contain the material systems, where material is simply a lower dimension metric-space, and where both material-components and metric-spaces are in resonance with the containing space. Partial differential equations are defined on the many metric-spaces of this description, but their main function is to act on either the, usually, unimportant free-material components (to most often cause non-linear dynamics) or to perturb the orbits of the, quite often condensed, material trapped by (or within) the stable orbits of a very stable hyperbolic metric-space shape.

Perturbing Material Components on Stable Shapes

Perturbing Material Components on Stable Shapes
Author: Martin Concoyle Ph.D.
Publsiher: Trafford Publishing
Total Pages: 400
Release: 2014-01-16
ISBN 10: 1490723722
ISBN 13: 9781490723723
Language: EN, FR, DE, ES & NL

Perturbing Material Components on Stable Shapes Book Review:

This book is an introduction to the simple math patterns that can be used to describe fundamental, stable spectral-orbital physical systems (represented as discrete hyperbolic shapes, i.e., hyperbolic space-forms), the containment set has many dimensions, and these dimensions possess macroscopic geometric properties (where hyperbolic metric-space subspaces are modeled to be discrete hyperbolic shapes). Thus, it is a description that transcends the idea of materialism (i.e., it is higher-dimensional so that the higher dimensions are not small), and it is a math context can also be used to model a life-form as a unified, high-dimension, geometric construct that generates its own energy and which has a natural structure for memory where this construct is made in relation to the main property of the description being, in fact, the spectral properties of both (1) material systems and of (2) the metric-spaces, which contain the material systems where material is simply a lower dimension metric-space and where both material-components and metric-spaces are in resonance with (and define) the containing space.

Innovations for Shape Analysis

Innovations for Shape Analysis
Author: Michael Breuß,Alfred Bruckstein,Petros Maragos
Publsiher: Springer Science & Business Media
Total Pages: 496
Release: 2013-04-04
ISBN 10: 3642341411
ISBN 13: 9783642341410
Language: EN, FR, DE, ES & NL

Innovations for Shape Analysis Book Review:

The concept of 'shape' is at the heart of image processing and computer vision, yet researchers still have some way to go to replicate the human brain's ability to extrapolate meaning from the most basic of outlines. This volume reflects the advances of the last decade, which have also opened up tough new challenges in image processing. Today's applications require flexible models as well as efficient, mathematically justified algorithms that allow data processing within an acceptable timeframe. Examining important topics in continuous-scale and discrete modeling, as well as in modern algorithms, the book is the product of a key seminar focused on innovations in the field. It is a thorough introduction to the latest technology, especially given the tutorial style of a number of chapters. It also succeeds in identifying promising avenues for future research. The topics covered include mathematical morphology, skeletonization, statistical shape modeling, continuous-scale shape models such as partial differential equations and the theory of discrete shape descriptors. Some authors highlight new areas of enquiry such as partite skeletons, multi-component shapes, deformable shape models, and the use of distance fields. Combining the latest theoretical analysis with cutting-edge applications, this book will attract both academics and engineers.

Image Analysis and Recognition

Image Analysis and Recognition
Author: Mohamed Kamel,Aurelio Campilho
Publsiher: Springer
Total Pages: 809
Release: 2013-06-05
ISBN 10: 3642390943
ISBN 13: 9783642390944
Language: EN, FR, DE, ES & NL

Image Analysis and Recognition Book Review:

This book constitutes the thoroughly refereed proceedings of the 10th International Conference on Image Analysis and Recognition, ICIAR 2013, held in Póvoa do Varzim, Portugal, in June 2013, The 92 revised full papers presented were carefully reviewed and selected from 177 submissions. The papers are organized in topical sections on biometrics: behavioral; biometrics: physiological; classification and regression; object recognition; image processing and analysis: representations and models, compression, enhancement , feature detection and segmentation; 3D image analysis; tracking; medical imaging: image segmentation, image registration, image analysis, coronary image analysis, retinal image analysis, computer aided diagnosis, brain image analysis; cell image analysis; RGB-D camera applications; methods of moments; applications.

Transactions on Computational Science XXVI

Transactions on Computational Science XXVI
Author: Marina L. Gavrilova,C.J. Kenneth Tan,Andrés Iglesias,Mikio Shinya,Akemi Galvez,Alexei Sourin
Publsiher: Springer
Total Pages: 173
Release: 2016-01-22
ISBN 10: 3662492474
ISBN 13: 9783662492475
Language: EN, FR, DE, ES & NL

Transactions on Computational Science XXVI Book Review:

This, the 26th issue of the Transactions on Computational Science journal, is comprised of ten extended versions of selected papers from the International Conference on Cyberworlds 2014, held in Santander, Spain, in June 2014. The topics covered include areas of virtual reality, games, social networks, haptic modeling, cybersecurity, and applications in education and arts.

Advances in Geometric Modeling and Processing

Advances in Geometric Modeling and Processing
Author: Bernard Mourrain,Scott Schaefer,Guoliang Xu
Publsiher: Springer
Total Pages: 315
Release: 2010-06-09
ISBN 10: 3642134114
ISBN 13: 9783642134111
Language: EN, FR, DE, ES & NL

Advances in Geometric Modeling and Processing Book Review:

This volume contains the papers presented at 6th Conference on Geometric Modeling and Processing (GMP 2010) held in Castro Urdiales, Spain during June16–18,2010. GeometricModelingandProcessingisabiannualinternational conference series on geometric modeling, simulation and computing. Previously, GMPhasbeenheldinHongKong(2000),Saitama,Japan(2002),Beijing,China (2004), Pittsburgh, USA (2006) and Hangzhou, China (2008). GMP 2010 received a total of 30 submissions that were reviewed by three to four Program Committee members on average. While the number of subm- sions dropped signi?cantly from previous years, the quality did not and was still quite high overall. Based on the reviews received, the committee decided to - cept 20 papers for inclusion in the proceedings. Additionally, extended versions of selected papers were considered for a special issue of Computer-Aided - sign (CAD) and Computer-Aided Geometric Design (CAGD). The paper topics spanned a wide variety and include: – Solutions of transcendental equations – Volume parameterization – Smooth curves and surfaces – Isogeometric analysis – Implicit surfaces – Computational geometry Many people helped make this conference happen and we are grateful for their help. We would especially like to thank the Conference Chair, all of the authors who submitted papers, the ProgramCommittee members who reviewed the papers and all of the participants at the conference.

An Introduction to Laplacian Spectral Distances and Kernels

An Introduction to Laplacian Spectral Distances and Kernels
Author: Giuseppe Patanè
Publsiher: Morgan & Claypool Publishers
Total Pages: 139
Release: 2017-07-05
ISBN 10: 1681731401
ISBN 13: 9781681731407
Language: EN, FR, DE, ES & NL

An Introduction to Laplacian Spectral Distances and Kernels Book Review:

In geometry processing and shape analysis, several applications have been addressed through the properties of the Laplacian spectral kernels and distances, such as commute time, biharmonic, diffusion, and wave distances. Within this context, this book is intended to provide a common background on the definition and computation of the Laplacian spectral kernels and distances for geometry processing and shape analysis. To this end, we define a unified representation of the isotropic and anisotropic discrete Laplacian operator on surfaces and volumes; then, we introduce the associated differential equations, i.e., the harmonic equation, the Laplacian eigenproblem, and the heat equation. Filtering the Laplacian spectrum, we introduce the Laplacian spectral distances, which generalize the commute-time, biharmonic, diffusion, and wave distances, and their discretization in terms of the Laplacian spectrum. As main applications, we discuss the design of smooth functions and the Laplacian smoothing of noisy scalar functions. All the reviewed numerical schemes are discussed and compared in terms of robustness, approximation accuracy, and computational cost, thus supporting the reader in the selection of the most appropriate with respect to shape representation, computational resources, and target application.

Mathematics Of Shapes And Applications

Mathematics Of Shapes And Applications
Author: Sergey Kushnarev,Anqi Qiu,Laurent Younes
Publsiher: World Scientific
Total Pages: 220
Release: 2019-11-20
ISBN 10: 9811200149
ISBN 13: 9789811200144
Language: EN, FR, DE, ES & NL

Mathematics Of Shapes And Applications Book Review:

Understanding how a single shape can incur a complex range of transformations, while defining the same perceptually obvious figure, entails a rich and challenging collection of problems, at the interface between applied mathematics, statistics and computer science. The program on Mathematics of Shapes and Applications, was held at the Institute for Mathematical Sciences at the National University of Singapore in 2016. It provided discussions on theoretical developments and numerous applications in computer vision, object recognition and medical imaging.The analysis of shapes is an example of a mathematical problem directly connected with applications while offering deep open challenges to theoretical mathematicians. It has grown, over the past decades, into an interdisciplinary area in which researchers studying infinite-dimensional Riemannian manifolds (global analysis) interact with applied mathematicians, statisticians, computer scientists and biomedical engineers on a variety of problems involving shapes.The volume illustrates this wealth of subjects by providing new contributions on the metric structure of diffeomorphism groups and shape spaces, recent developments on deterministic and stochastic models of shape evolution, new computational methods manipulating shapes, and new statistical tools to analyze shape datasets. In addition to these contributions, applications of shape analysis to medical imaging and computational anatomy are discussed, leading, in particular, to improved understanding of the impact of cognitive diseases on the geometry of the brain.

Describing the Dynamics of Free Material Components in Higher Dimensions

Describing the Dynamics of  Free  Material Components in Higher Dimensions
Author: Dr. Martin Concoyle
Publsiher: Trafford Publishing
Total Pages: 828
Release: 2014
ISBN 10: 1490723706
ISBN 13: 9781490723709
Language: EN, FR, DE, ES & NL

Describing the Dynamics of Free Material Components in Higher Dimensions Book Review:

The issue which the new ideas of these new books really raise with our culture, is not about whether they are true, since these new ideas identify a valid context for physical description, and whereas the current context for math and physics (2014) cannot do that, ie they cannot describe the stable properties of a general many-(but-few)-body system. Whereas the new ideas about math and physics can be used to solve the most fundamental problems about the physical world, in regard to understanding physical stability, a problem which the current descriptive context of math and physics (2014) cannot solve. That is, "what now, in 2014, passes for math and physics knowledge are delusions."* Yet these delusions are the ideas expressed in our propaganda-education system about math and physics. Rather The real issue, which these new ideas present to our culture, is about our cultural relation to "what is beyond the material world." That is, it is about our cultural representation of religion, or the spirit. In particular, in relation to the "previous knowledge humans needed to possess" in order to make Gobekli-tepe, Puma Punku, Stonehenge, etc, ie simply to be able to lift and position such large stones, as well as the understanding which is needed to go beyond the context of the material world, and into the context of all the ancient mythologies in regard to the ancient religious stories, etc etc *The current paradigm (in 2014) describes a general state of indefi nable randomness in which there is always "a chaotic transitioning process" which exists as random elementary-particle collisions, and which, supposedly, is perpetually occurring. Thus, their description of the wide range of the generally stable states of the many-(but-few)-body systems..., into which this "forever chaotically transitioning" process supposedly settles but explicit descriptions of this process do not exist. Instead their answer is that "such stable, many-(but-few)-body systems are too complicated to describe."

Spectral Theory and Geometry

Spectral Theory and Geometry
Author: Scotland) Icms Instructional Conference (1998 Edinburgh,International Centre for Mathematical Sciences
Publsiher: Cambridge University Press
Total Pages: 328
Release: 1999-09-30
ISBN 10: 0521777496
ISBN 13: 9780521777490
Language: EN, FR, DE, ES & NL

Spectral Theory and Geometry Book Review:

Authoritative lectures from world experts on spectral theory and geometry.

On Perturbative Methods in Spectral Geometry

On Perturbative Methods in Spectral Geometry
Author: Mikhail Panine
Publsiher: Unknown
Total Pages: 122
Release: 2017
ISBN 10: 1928374650XXX
ISBN 13: OCLC:1036282641
Language: EN, FR, DE, ES & NL

On Perturbative Methods in Spectral Geometry Book Review:

The goal of spectral geometry is to establish how much information about the geometry of compact Riemannian manifolds is contained in the spectra of natural differential operators, especially Laplacians, defined on them. Ideally, one would like to be able to recover the Riemannian manifold, up to isometry, from the spectra of one or several such operators. This would be a very powerful result, as it would introduce an invariant way to describe the shape of Riemannian manifolds. The consequences of such a result would range from practical applications such as shape recognition to theoretical insights into quantum gravity. However, the most general form of such statements is known to be false. There are a number of known counterexamples, that is isospectral but not isometric manifolds. Indeed, there are even techniques to construct such counterexamples. Nonetheless, it is believed that almost all Riemannian manifolds can be identified by their spectra. In other words, the counterexamples are expected to be exceedingly rare special cases. This has been shown to be the case in some restricted classes of manifolds. The proof in the general case has remained elusive. The main goal of this thesis is to move towards such a proof by studying the structure of isospectral sets of metrics. The main tool we use for this purpose is perturbation theory, a method ubiquitous in physics, but strangely underused in spectral geometry. Consequently, a secondary goal of this work is to demonstrate the usefulness of perturbation theory to the study of spectral geometry. We begin by a numerical exploration of spectral geometry in a perturbative regime. Then, we show that sets of isospectral conformally equivalent metrics on boundaryless manifolds of dimension two contain no convex subsets. This is an entirely new type of result in spectral geometry. We argue that it could lead to a proof of the rarity of counterexamples in the program of identifying shapes by their spectra. The thesis also includes reviews of the fundamentals of the spectral theory of Laplace-type operators, of major results in spectral geometry and of perturbation theory.

Old and New Aspects in Spectral Geometry

Old and New Aspects in Spectral Geometry
Author: M.-E. Craioveanu,Mircea Puta,Themistocles RASSIAS
Publsiher: Springer Science & Business Media
Total Pages: 446
Release: 2013-03-14
ISBN 10: 940172475X
ISBN 13: 9789401724753
Language: EN, FR, DE, ES & NL

Old and New Aspects in Spectral Geometry Book Review:

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.