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Spectral Geometry of Shapes

Spectral Geometry of Shapes
A Book

by Jing Hua,Zichun Zhong

  • Publisher : Academic Press
  • Release : 2020-01-15
  • Pages : 195
  • ISBN : 0128138424
  • Language : En, Es, Fr & De
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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 Analysis Using Spectral Geometry

Shape Analysis Using Spectral Geometry
A Book

by Jiaxi Hu

  • Publisher : Unknown Publisher
  • Release : 2015
  • Pages : 95
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Finally we prove the shape spectrum is a continuous function to a scale function on the conformal factor of the manifold. The derivatives of the eigenvalues are analytically expressed with those of the scale function. The property applies to both continuous domain and discrete triangle meshes. On the triangle meshes, a spectrum alignment algorithm is developed. Given two closed triangle meshes, the eigenvalues can be aligned from one to the other and the eigenfunction distributions are aligned as well. This extends the shape spectra across non-isometric deformations, supporting a registration-free analysis of general motion data.

Shape Optimization and Spectral Theory

Shape Optimization and Spectral Theory
A Book

by Antoine Henrot

  • Publisher : De Gruyter Open
  • Release : 2017-05-08
  • Pages : 474
  • ISBN : 9783110550856
  • Language : En, Es, Fr & De
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"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

Old and New Aspects in Spectral Geometry

Old and New Aspects in Spectral Geometry
A Book

by M.-E. Craioveanu,Mircea Puta,Themistocles RASSIAS

  • Publisher : Springer Science & Business Media
  • Release : 2013-03-14
  • Pages : 446
  • ISBN : 940172475X
  • Language : En, Es, Fr & De
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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.

The Application of Spectral Geometry to 3D Molecular Shape Comparison

The Application of Spectral Geometry to 3D Molecular Shape Comparison
A Book

by Matthew Seddon

  • Publisher : Unknown Publisher
  • Release : 2017
  • Pages : 329
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Dirac Operators and Spectral Geometry

Dirac Operators and Spectral Geometry
A Book

by Giampiero Esposito

  • Publisher : Cambridge University Press
  • Release : 1998-08-20
  • Pages : 209
  • ISBN : 9780521648622
  • Language : En, Es, Fr & De
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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.

Steklov Geometry Processing

Steklov Geometry Processing
An Extrinsic Approach to Spectral Shape Analysis

by Wang Yu (S.M.)

  • Publisher : Unknown Publisher
  • Release : 2018
  • Pages : 80
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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We propose using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator, cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our operators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.

Spectrum Geometry

Spectrum Geometry
A Book

by Spectrum

  • Publisher : Carson-Dellosa Publishing
  • Release : 2015-02-15
  • Pages : 128
  • ISBN : 1483816621
  • Language : En, Es, Fr & De
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With the help of Spectrum Geometry(R) for grades 6 to 8, children develop problem-solving math skills they can build on. This standards-based workbook focuses on middle school geometry concepts like points, lines, rays, angles, triangles, polygons, circles, perimeter, area, and more. --Middle school is known for its challengesÑlet Spectrum(R) ease some stress. Developed by education experts, the Spectrum Middle School Math series strengthens the important home-to-school connection and prepares children for math success. Filled with easy instructions and rigorous practice, Spectrum Geometry helps children soar in a standards-based classroom!

The Changing Shape of Geometry

The Changing Shape of Geometry
Celebrating a Century of Geometry and Geometry Teaching

by Mathematical Association,Mathematical Association of America

  • Publisher : Cambridge University Press
  • Release : 2003-01-09
  • Pages : 541
  • ISBN : 9780521531627
  • Language : En, Es, Fr & De
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Celebrating a century of geometry and geometry teaching, this book will give the reader an enjoyable insight into all things geometrical. There are a wealth of popular articles including sections on Pythagoras, the golden ratio and recreational geometry. Historical items, drawn principally from the Mathematical Gazette, are authored by mathematicians such as G. H. Hardy, Rouse Ball, Thomas Heath and Bertrand Russell as well as some more recent expositors. Thirty 'Desert Island Theorems' from distinguished mathematicians and educationalists give light to some surprising and beautiful results. Contributors include H. S. M. Coxeter, Michael Atiyah, Tom Apostol, Solomon Golomb, Keith Devlin, Nobel Laureate Leon Lederman, Carlo Séquin, Simon Singh, Christopher Zeeman and Pulitzer Prizewinner Douglas Hofstadter. The book also features the wonderful Eyeball Theorems of Peruvian geometer and web designer, Antonio Gutierrez. For anyone with an interest in mathematics and mathematics education this book will be an enjoyable and rewarding read.

Spectral Geometric Methods for Deformable 3D Shape Retrieval

Spectral Geometric Methods for Deformable 3D Shape Retrieval
A Book

by Chunyuan Li

  • Publisher : Unknown Publisher
  • Release : 2013
  • Pages : 329
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Mathematical Analysis of Evolution, Information, and Complexity

Mathematical Analysis of Evolution, Information, and Complexity
A Book

by Wolfgang Arendt,Wolfgang P. Schleich

  • Publisher : John Wiley & Sons
  • Release : 2009-07-10
  • Pages : 502
  • ISBN : 3527628037
  • Language : En, Es, Fr & De
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Mathematical Analysis of Evolution, Information, and Complexity deals with the analysis of evolution, information and complexity. The time evolution of systems or processes is a central question in science, this text covers a broad range of problems including diffusion processes, neuronal networks, quantum theory and cosmology. Bringing together a wide collection of research in mathematics, information theory, physics and other scientific and technical areas, this new title offers elementary and thus easily accessible introductions to the various fields of research addressed in the book.

The Geometry of Walker Manifolds

The Geometry of Walker Manifolds
A Book

by Miguel Brozos-Vázquez

  • Publisher : Morgan & Claypool Publishers
  • Release : 2009
  • Pages : 159
  • ISBN : 1598298194
  • Language : En, Es, Fr & De
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Basic algebraic notions -- Introduction -- A historical perspective in the algebraic context -- Algebraic preliminaries -- Jordan normal form -- Indefinite geometry -- Algebraic curvature tensors -- Hermitian and para-Hermitian geometry -- The Jacobi and skew symmetric curvature operators -- Sectional, Ricci, scalar, and Weyl curvature -- Curvature decompositions -- Self-duality and anti-self-duality conditions -- Spectral geometry of the curvature operator -- Osserman and conformally Osserman models -- Osserman curvature models in signature (2, 2) -- Ivanov-Petrova curvature models -- Osserman Ivanov-Petrova curvature models -- Commuting curvature models -- Basic geometrical notions -- Introduction -- History -- Basic manifold theory -- The tangent bundle, lie bracket, and lie groups -- The cotangent bundle and symplectic geometry -- Connections, curvature, geodesics, and holonomy -- Pseudo-Riemannian geometry -- The Levi-Civita connection -- Associated natural operators -- Weyl scalar invariants -- Null distributions -- Pseudo-Riemannian holonomy -- Other geometric structures -- Pseudo-Hermitian and para-Hermitian structures -- Hyper-para-Hermitian structures -- Geometric realizations -- Homogeneous spaces, and curvature homogeneity -- Technical results in differential equations -- Walker structures -- Introduction -- Historical development -- Walker coordinates -- Examples of Walker manifolds -- Hypersurfaces with nilpotent shape operators -- Locally conformally flat metrics with nilpotent Ricci operator -- Degenerate pseudo-Riemannian homogeneous structures -- Para-Kaehler geometry -- Two-step nilpotent lie groups with degenerate center -- Conformally symmetric pseudo-Riemannian metrics -- Riemannian extensions -- The affine category -- Twisted Riemannian extensions defined by flat connections -- Modified Riemannian extensions defined by flat connections -- Nilpotent Walker manifolds -- Osserman Riemannian extensions -- Ivanov-Petrova Riemannian extensions -- Three-dimensional Lorentzian Walker manifolds -- Introduction -- History -- Three dimensional Walker geometry -- Adapted coordinates -- The Jordan normal form of the Ricci operator -- Christoffel symbols, curvature, and the Ricci tensor -- Locally symmetric Walker manifolds -- Einstein-like manifolds -- The spectral geometry of the curvature tensor -- Curvature commutativity properties -- Local geometry of Walker manifolds with -- Foliated Walker manifolds -- Contact Walker manifolds -- Strict Walker manifolds -- Three dimensional homogeneous Lorentzian manifolds -- Three dimensional lie groups and lie algebras -- Curvature homogeneous Lorentzian manifolds -- Diagonalizable Ricci operator -- Type II Ricci operator -- Four-dimensional Walker manifolds -- Introduction -- History -- Four-dimensional Walker manifolds -- Almost para-Hermitian geometry -- Isotropic almost para-Hermitian structures -- Characteristic classes -- Self-dual Walker manifolds -- The spectral geometry of the curvature tensor -- Introduction -- History -- Four-dimensional Osserman metrics -- Osserman metrics with diagonalizable Jacobi operator -- Osserman Walker type II metrics -- Osserman and Ivanov-Petrova metrics -- Riemannian extensions of affine surfaces -- Affine surfaces with skew symmetric Ricci tensor -- Affine surfaces with symmetric and degenerate Ricci tensor -- Riemannian extensions with commuting curvature operators -- Other examples with commuting curvature operators -- Hermitian geometry -- Introduction -- History -- Almost Hermitian geometry of Walker manifolds -- The proper almost Hermitian structure of a Walker manifold -- Proper almost hyper-para-Hermitian structures -- Hermitian Walker manifolds of dimension four -- Proper Hermitian Walker structures -- Locally conformally Kaehler structures -- Almost Kaehler Walker four-dimensional manifolds -- Special Walker manifolds -- Introduction -- History -- Curvature commuting conditions -- Curvature homogeneous strict Walker manifolds -- Bibliography.

On Perturbative Methods in Spectral Geometry

On Perturbative Methods in Spectral Geometry
A Book

by Mikhail Panine

  • Publisher : Unknown Publisher
  • Release : 2017
  • Pages : 122
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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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.

Geometry, Grade 6

Geometry, Grade 6
A Book

by Anonim

  • Publisher : Carson-Dellosa Publishing
  • Release : 2013-12-02
  • Pages : 96
  • ISBN : 1483809285
  • Language : En, Es, Fr & De
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New to the Spectrum(R) series, Geometry, is a skill-specific math resource designed to completely support and challenge sixth graders in geometry. This 96-page book goes into greater depth about geometry and provides a wide range of examples, practice problems, and assessments to measure progress. The best–selling Spectrum(R) series now provides students with focused practice based on the essential skills they need to master for Common Core success. With explicit skill instruction, step-by-step examples, and ample practice, as well as assessment tools for progress monitoring, students are provided everything they need to master specific math skills. Skill–specific Spectrum(R) books are the perfect supplement for home or school.

Old and New Aspects in Spectral Geometry

Old and New Aspects in Spectral Geometry
A Book

by M. -E. Craioveanu,Mircea Puta,Themistocles Rassias

  • Publisher : Unknown Publisher
  • Release : 2014-01-15
  • Pages : 460
  • ISBN : 9789401724760
  • Language : En, Es, Fr & De
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Old and New Aspects in Spectral Geometry

Old and New Aspects in Spectral Geometry
A Book

by M.-E. Craioveanu,Mircea Puta,Themistocles M. Rassias

  • Publisher : Springer Science & Business Media
  • Release : 2001-10-31
  • Pages : 446
  • ISBN : 9781402000522
  • Language : En, Es, Fr & De
GET BOOK

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.

Perturbing Material-Components on Stable Shapes

Perturbing Material-Components on Stable Shapes
A Book

by Martin Concoyle Ph.D.

  • Publisher : Trafford Publishing
  • Release : 2014
  • Pages : 400
  • ISBN : 1490723692
  • Language : En, Es, Fr & De
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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.

Geometric Approaches for 3D Shape Denoising and Retrieval

Geometric Approaches for 3D Shape Denoising and Retrieval
A Book

by Anis Kacem

  • Publisher : Unknown Publisher
  • Release : 2013
  • Pages : 329
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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Geometric Approximation Algorithms

Geometric Approximation Algorithms
A Book

by Sariel Har-Peled

  • Publisher : American Mathematical Soc.
  • Release : 2011
  • Pages : 362
  • ISBN : 0821849115
  • Language : En, Es, Fr & De
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Exact algorithms for dealing with geometric objects are complicated, hard to implement in practice, and slow. Over the last 20 years a theory of geometric approximation algorithms has emerged. These algorithms tend to be simple, fast, and more robust than their exact counterparts. This book is the first to cover geometric approximation algorithms in detail. In addition, more traditional computational geometry techniques that are widely used in developing such algorithms, like sampling, linear programming, etc., are also surveyed. Other topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings. The topics covered are relatively independent and are supplemented by exercises. Close to 200 color figures are included in the text to illustrate proofs and ideas.

Shapes and Shadows

Shapes and Shadows
A Book

by Madison Julius Cawein

  • Publisher : Good Press
  • Release : 2019-12-04
  • Pages : 140
  • ISBN : 9876543210XXX
  • Language : En, Es, Fr & De
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"Shapes and Shadows" by Madison Julius Cawein. Published by Good Press. Good Press publishes a wide range of titles that encompasses every genre. From well-known classics & literary fiction and non-fiction to forgotten−or yet undiscovered gems−of world literature, we issue the books that need to be read. Each Good Press edition has been meticulously edited and formatted to boost readability for all e-readers and devices. Our goal is to produce eBooks that are user-friendly and accessible to everyone in a high-quality digital format.