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Modern Approaches to Discrete Curvature 1st ed. 2017 [Minkštas viršelis]

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  • Formatas: Paperback / softback, 353 pages, aukštis x plotis: 235x155 mm, weight: 5737 g, 35 Illustrations, color; 45 Illustrations, black and white; XXVI, 353 p. 80 illus., 35 illus. in color., 1 Paperback / softback
  • Serija: Lecture Notes in Mathematics 2184
  • Išleidimo metai: 05-Oct-2017
  • Leidėjas: Springer International Publishing AG
  • ISBN-10: 3319580019
  • ISBN-13: 9783319580012
Kitos knygos pagal šią temą:
Modern Approaches to Discrete Curvature 1st ed. 2017Didesnis paveikslėlis
  • Formatas: Paperback / softback, 353 pages, aukštis x plotis: 235x155 mm, weight: 5737 g, 35 Illustrations, color; 45 Illustrations, black and white; XXVI, 353 p. 80 illus., 35 illus. in color., 1 Paperback / softback
  • Serija: Lecture Notes in Mathematics 2184
  • Išleidimo metai: 05-Oct-2017
  • Leidėjas: Springer International Publishing AG
  • ISBN-10: 3319580019
  • ISBN-13: 9783319580012
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783319580012.html
Raktažodžiai:

This book provides a valuable glimpse into discrete curvature, a rich new field of research which blends discrete mathematics, differential geometry, probability and computer graphics. It includes a vast collection of ideas and tools which will offer something new to all interested readers. Discrete geometry has arisen as much as a theoretical development as in response to unforeseen challenges coming from applications. Discrete and continuous geometries have turned out to be intimately connected. Discrete curvature is the key concept connecting them through many bridges in numerous fields: metric spaces, Riemannian and Euclidean geometries, geometric measure theory, topology, partial differential equations, calculus of variations, gradient flows, asymptotic analysis, probability, harmonic analysis, graph theory, etc. In spite of its crucial importance both in theoretical mathematics and in applications, up to now, almost no books have provided a coherent outlook on this emerging field.

1 The Geometric Meaning of Curvature: Local and Nonlocal Aspects of Ricci Curvature
1(62)
Frank Bauer
Bobo Hua
Jurgen Jost
Shiping Liu
Guofang Wang
1.1 The Origins of the Concept of Curvature
2(1)
1.2 A Primer on Riemannian Geometry: Not Indispensable
3(7)
1.2.1 Tangent Vectors and Riemannian Metrics
3(2)
1.2.2 Differentials, Gradients, and the Laplace-Beltrami Operator
5(3)
1.2.3 Lengths and Distances
8(1)
1.2.4 Volumes
9(1)
1.3 Curvature of Riemannian Manifolds
10(10)
1.3.1 Covariant Derivatives
10(1)
1.3.2 The Curvature Tensor; Sectional and Ricci Curvature
11(2)
1.3.3 The Geometric Meaning of Sectional Curvature
13(1)
1.3.4 The Geometric Meaning of Ricci Curvature
14(5)
1.3.5 Harmonic Functions
19(1)
1.4 A Nonlocal Approach to Geometry
20(4)
1.5 Generalized Ricci Curvature
24(7)
1.5.1 Forman's Ricci Curvature
25(2)
1.5.2 Ollivier's Ricci Curvature
27(2)
1.5.3 Curvature Dimension Inequality
29(2)
1.6 Ricci Curvature and the Geometry of Graphs
31(32)
1.6.1 Basic Notions from Graph Theory
31(4)
1.6.2 Ricci Curvature and Clustering
35(13)
1.6.3 Curvature Dimension Inequality and Eigenvalue Ratios
48(3)
1.6.4 Exponential Curvature Dimension Inequality on Graphs
51(1)
1.6.5 Li-Yau Gradient Estimate on Graphs and Its Applications
52(2)
1.6.6 Applications to Network Analysis
54(1)
1.6.7 Other Curvature Notions for Graphs
55(4)
References
59(4)
2 Metric Curvatures Revisited: A Brief Overview
63(52)
Emil Saucan
2.1 Introduction
63(1)
2.2 Metric Curvature for Curves
64(6)
2.2.1 Menger Curvature
65(1)
2.2.2 Haantjes Curvature
66(4)
2.3 Metric Curvature for Surfaces: Wald Curvature
70(10)
2.3.1 Wald Curvature
70(10)
2.4 Wald Curvature Under Gromov--Hausdorff Convergence
80(5)
2.4.1 Intrinsic Properties and Gromov--Hausdorff Convergence
80(3)
2.4.2 Wald Curvature and Gromov--Hausdorff Convergence
83(2)
2.5 Wald and Alexandrov Curvatures Comparison
85(3)
2.5.1 Alexandrov Curvature
85(2)
2.5.2 Alexandrov Curvature vs. Wald Curvature
87(1)
2.6 A Metric Approach to Ricci Curvature
88(13)
2.6.1 Metric Ricci Flow for PL Surfaces
89(8)
2.6.2 PL Ricci for Cell Complexes
97(4)
2.7 Metric Curvatures for Metric Measure Spaces
101(14)
2.7.1 The Basic Idea: The Snowflaking Operator
102(9)
References
111(4)
3 Distances Between Datasets
115(18)
Facundo Memoli
3.1 Introduction
115(1)
3.1.1 Notation and Background Concepts
115(1)
3.2 The Gromov--Hausdorff Distance
116(7)
3.2.1 An Example
117(1)
3.2.2 A Simplification
118(1)
3.2.3 Another Expression
119(1)
3.2.4 The Case of Subsets of Euclidean Space
120(1)
3.2.5 Another Expression and Consequences
121(2)
3.3 The Modified Gromov-Hausdorff and Curvature Sets
123(4)
3.3.1 Curvature Sets
124(1)
3.3.2 Comparing Curvature Sets?
125(1)
3.3.3 Asking for More
126(1)
3.4 A Metric on Mw
127(4)
3.4.1 Pre-compactness
128(1)
3.4.2 Completeness
128(1)
3.4.3 Other Properties: Geodesics and Alexandrov Curvature
129(1)
3.4.4 The Metric dGW,p in Applications
130(1)
3.5 Discussion and Outlook
131(2)
References
131(2)
4 Inference of Curvature Using Tubular Neighborhoods
133(26)
Frederic Chazal
David Cohen-Steiner
Andre Lieutier
Quentin Merigot
Boris Thibert
4.1 Introduction
134(1)
4.2 Distance Function and Sets with Positive Reach
134(5)
4.2.1 Gradient of the Distance and Sets with Positive Reach
135(2)
4.2.2 Generalized Gradient and Sets with Positive μ-Reach
137(2)
4.3 Boundary Measures and Federer's Curvature Measures
139(11)
4.3.1 Boundary Measures
139(1)
4.3.2 Stability of Boundary Measures
140(3)
4.3.3 Tube Formulas and Federer's Curvature Measures
143(2)
4.3.4 Stability of Federer's Curvature Measures
145(2)
4.3.5 Computation of Boundary Measures and Visualization
147(3)
4.4 Voronoi Covariance Measures and Local Minkowski Tensors
150(2)
4.4.1 Covariance Matrices of Voronoi Cells
150(1)
4.4.2 Voronoi Covariance Measure
151(1)
4.5 Stability of Anisotropic Curvature Measures
152(7)
4.5.1 Anisotropic Curvature Measures of Sets with Positive Reach
152(1)
4.5.2 Stability of the Curvature Measures of the Offsets
153(1)
4.5.3 Computation of the Curvature Measures of 3D Point Clouds
154(1)
4.5.4 Sketch of Proof of Theorem 4.9
155(2)
References
157(2)
5 Entropic Ricci Curvature for Discrete Spaces
159(16)
Jan Maas
5.1 Ricci Curvature Lower Bounds for Geodesic Measure Spaces
159(2)
5.1.1 Discrete Spaces
161(1)
5.2 The Heat Flow as Gradient Flow of the Entropy
161(2)
5.2.1 The Benamou-Brenier Formula
162(1)
5.3 A Gradient Flow Structure for Reversible Markov Chains
163(4)
5.3.1 Discrete Transport Metrics
164(2)
5.3.2 A Riemannian Structure on the Space of Probability Measures
166(1)
5.3.3 The Discrete JKO-Theorem
166(1)
5.4 Discrete Entropic Ricci Curvature
167(8)
5.4.1 Discrete Spaces with Lower Ricci Curvature Bounds
169(1)
5.4.2 Examples
170(2)
References
172(3)
6 Geometric and Spectral Consequences of Curvature Bounds on Tessellations
175(36)
Matthias Keller
6.1 Introduction
175(7)
6.1.1 Graphs
177(1)
6.1.2 Tessellations of Surfaces
177(1)
6.1.3 Curvature
178(3)
6.1.4 Duality
181(1)
6.2 Geometry
182(12)
6.2.1 Gauss--Bonnet Theorem
182(1)
6.2.2 Approximating Flat and Infinite Curvature
183(1)
6.2.3 Finiteness
184(1)
6.2.4 Absence of Cut Locus
185(1)
6.2.5 Volume Growth
186(4)
6.2.6 Isoperimetric Inequalities
190(4)
6.3 Spectral Theory
194(9)
6.3.1 The Combinatorial Laplacian
194(2)
6.3.2 Bottom of the Spectrum
196(2)
6.3.3 Discrete Spectrum, Eigenvalue Asymptotics and Decay of Eigenfunctions
198(2)
6.3.4 Unique Continuation of Eigenfunctions
200(2)
6.3.5 lp Spectrum
202(1)
6.4 Extensions to More General Graphs
203(8)
6.4.1 Curvature on Planar Graphs
203(2)
6.4.2 Sectional Curvature for Polygonal Complexes
205(1)
References
206(5)
7 The Geometric Spectrum of a Graph and Associated Curvatures
211(48)
Paul Baird
7.1 Introduction
211(3)
7.2 The Geometric Spectrum
214(3)
7.3 Invariant Stars and the Lifting Problem
217(10)
7.3.1 Configured Stars
218(2)
7.3.2 Invariant Stars
220(4)
7.3.3 The Lifting Problem
224(3)
7.4 Edge Length and the Gauss Map
227(9)
7.4.1 Graph Orientation
227(2)
7.4.2 Edge Curvature and Edge Length
229(5)
7.4.3 Path Metric Space Structure and Curvature in the Sense of Alexandrov
234(2)
7.5 Gaussian Curvature
236(8)
7.5.1 The Theorem of Descartes
236(1)
7.5.2 Vertex Curvature
237(7)
7.6 Other Curvatures
244(15)
References
257(2)
8 Discrete Minimal Surfaces of Koebe Type
259(34)
Alexander I. Bobenko
Ulrike Bucking
Stefan Sechelmann
8.1 Introduction
259(1)
8.2 Discrete Minimal Surfaces
260(9)
8.2.1 Discrete Curvatures
263(1)
8.2.2 Characterization of Discrete Minimal Surfaces
264(5)
8.3 Construction of Koebe Polyhedra and Discrete Minimal Surfaces
269(6)
8.3.1 Construction of Koebe Polyhedra and Spherical Circle Patterns
270(2)
8.3.2 Construction of S-Isothermic Discrete Minimal Surfaces with Special Boundary Conditions
272(3)
8.4 Examples of Discrete Minimal Surfaces
275(11)
8.4.1 Gergonne's Surface
275(2)
8.4.2 Schwarz's CLP Surface
277(1)
8.4.3 Schwarz's D Surface
278(1)
8.4.4 Neovius's Surface
279(1)
8.4.5 Schwarz's H Surface
280(1)
8.4.6 Schoen's I-6 Surface and Generalizations
281(1)
8.4.7 Polygonal Boundary Frames
282(4)
8.5 Weierstrass Representation and Convergence of Discrete Minimal Surfaces
286(7)
8.5.1 Discrete Weierstrass Representation
287(1)
8.5.2 Convergence of Discrete Minimal Surfaces
287(3)
References
290(3)
9 Robust and Convergent Curvature and Normal Estimators with Digital Integral Invariants
293(56)
Jacques-Olivier Lachaud
David Coeurjolly
Jeremy Levallois
9.1 Curvature Estimation on Discrete Data
294(3)
9.2 Background: Multigrid Convergence and Integral Invariants
297(6)
9.2.1 Digital Space and Digitizations Operators
297(3)
9.2.2 Multigrid Convergence of Global and Local Geometric Estimators
300(1)
9.2.3 Integral Invariants in the Continuous Setting
301(2)
9.3 Digital Moments
303(11)
9.3.1 Moments and Digital Moments
303(1)
9.3.2 General Results for Volume Estimation Errors
304(2)
9.3.3 Volume Approximation Within a Ball of Radius R
306(5)
9.3.4 Errors on Volume and Moment Estimation Within a Ball of Radius R
311(2)
9.3.5 Conclusion
313(1)
9.4 Multigrid Convergence of Mean Curvature in 2D and 3D
314(6)
9.4.1 Definition of Mean Curvature Estimators
314(1)
9.4.2 Convergence at Points of X (Weak Formulation)
315(2)
9.4.3 Multigrid Convergence for Smooth Enough Shapes
317(3)
9.5 Multigrid Convergence of Curvature Tensor in 3D
320(11)
9.5.1 Digital Covariance Matrix and Digital Principal Curvature Estimators
320(1)
9.5.2 Some Properties of Covariance Matrix and Moments
321(1)
9.5.3 Multigrid Convergence of Digital Covariance Matrix
322(4)
9.5.4 Useful Results of Matrix Perturbation Theory
326(1)
9.5.5 Multigrid Convergence of Integral Principal Curvature Estimators
327(4)
9.6 Parameter-Free Digital Curvature Estimation
331(3)
9.6.1 Asymptotic Laws of Straight Segments Along the Boundary of Digitized Shapes and Scale Determination
331(1)
9.6.2 Parameter-Free Digital Curvature Estimators
332(1)
9.6.3 Local Parameter-Free Digital Curvature Estimator and 3D Case
333(1)
9.7 Experimental Evaluation
334(7)
9.7.1 Implementation Details
334(2)
9.7.2 Multigrid Convergence Analysis
336(2)
9.7.3 Parameter-Free Estimators
338(3)
9.8 Discussion, Applications and Further Works
341(8)
9.8.1 Robustness to Noise
341(2)
9.8.2 Feature Detection with Multiscale Approach
343(1)
9.8.3 Current Limitations and Possible Research Directions
344(1)
References
345(4)
Index 349