1995, DMV Seminar, ISBN 3764352426, Volume Bd. 25, v, 112

Book

IEEE Transactions on Pattern Analysis and Machine Intelligence, ISSN 0162-8828, 07/2011, Volume 33, Issue 7, pp. 1415 - 1428

This paper introduces a square-root velocity (SRV) representation for analyzing shapes of curves in euclidean spaces under an elastic metric...

path straightening method | Additives | Shape | elastic geodesics | square-root representations | Extraterrestrial measurements | Orbits | shape models | Elastic curves | Fisher-Rao metric | Manifolds | Analytical models | parallel transport | Riemannian shape analysis | elastic metric | FACE RECOGNITION | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Space and time | Measurement | Speed | Technology application | Usage | Geodesic domes | Innovations | Riemann integral | Mathematical optimization | Studies | Straightening | Deformation | Face recognition | Intelligence | Mathematics | Representations | Pattern analysis | Optimization

path straightening method | Additives | Shape | elastic geodesics | square-root representations | Extraterrestrial measurements | Orbits | shape models | Elastic curves | Fisher-Rao metric | Manifolds | Analytical models | parallel transport | Riemannian shape analysis | elastic metric | FACE RECOGNITION | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Space and time | Measurement | Speed | Technology application | Usage | Geodesic domes | Innovations | Riemann integral | Mathematical optimization | Studies | Straightening | Deformation | Face recognition | Intelligence | Mathematics | Representations | Pattern analysis | Optimization

Journal Article

SIAM journal on optimization, ISSN 1095-7189, 2014, Volume 24, Issue 3, pp. 1542 - 1566

The geometric median as well as the Frechet mean of points in a Hadamard space are important in both theory and applications...

Mean | Computational phylogenetics | Tree space | Median | The law of large numbers | Hadamard space | Proximal point algorithm | Diffusion tensor imaging | NONLINEAR MARKOV OPERATORS | CONVEX FUNCTIONALS | MATHEMATICS, APPLIED | HARMONIC MAPS | METRIC-SPACES | computational phylogenetics | GEODESIC DISTANCES | the law of large numbers | median | tree space | PRODUCTS | mean | diffusion tensor imaging | CONVERGENCE | PROJECTIONS | proximal point algorithm | MONOTONE-OPERATORS | Trees | Splitting | Algorithms | Computation | Mathematical analysis | Tools | Mathematical models | Optimization

Mean | Computational phylogenetics | Tree space | Median | The law of large numbers | Hadamard space | Proximal point algorithm | Diffusion tensor imaging | NONLINEAR MARKOV OPERATORS | CONVEX FUNCTIONALS | MATHEMATICS, APPLIED | HARMONIC MAPS | METRIC-SPACES | computational phylogenetics | GEODESIC DISTANCES | the law of large numbers | median | tree space | PRODUCTS | mean | diffusion tensor imaging | CONVERGENCE | PROJECTIONS | proximal point algorithm | MONOTONE-OPERATORS | Trees | Splitting | Algorithms | Computation | Mathematical analysis | Tools | Mathematical models | Optimization

Journal Article

Advances in Mathematics, ISSN 0001-8708, 07/2018, Volume 332, pp. 287 - 310

Botelho, Jamison, and Molnár [1], and Gehér and Šemrl [4] have recently described the general form of surjective isometries of Grassmann spaces of all projections of a fixed finite rank on a Hilbert space H...

Subspace | Projection | Isometry | Geodesic structure | Grassmann space | Gap metric | MATHEMATICS | SET | N-DIMENSIONAL SUBSPACES | TRANSFORMATIONS

Subspace | Projection | Isometry | Geodesic structure | Grassmann space | Gap metric | MATHEMATICS | SET | N-DIMENSIONAL SUBSPACES | TRANSFORMATIONS

Journal Article

SIAM Journal on Discrete Mathematics, ISSN 0895-4801, 2017, Volume 31, Issue 3, pp. 2015 - 2038

We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all ultrametrics. The CAT(0...

Polytope | Phylogenetic tree | CAT space | Geodesic triangle | Ultrametric | Tropical convexity | Billera-Holmes-Vogtman metric | MATHEMATICS, APPLIED | phylogenetic tree | APPROXIMATION | COMPLEXES | ALGORITHM | tropical convexity | SEMIMODULES | PHYLOGENETIC TREES | ultrametric | geodesic triangle | polytope | GEOMETRY

Polytope | Phylogenetic tree | CAT space | Geodesic triangle | Ultrametric | Tropical convexity | Billera-Holmes-Vogtman metric | MATHEMATICS, APPLIED | phylogenetic tree | APPROXIMATION | COMPLEXES | ALGORITHM | tropical convexity | SEMIMODULES | PHYLOGENETIC TREES | ultrametric | geodesic triangle | polytope | GEOMETRY

Journal Article

Potential Analysis, ISSN 0926-2601, 2/2012, Volume 36, Issue 2, pp. 317 - 337

... ≤ 2 in proper geodesic metric spaces. By means of a general Hamilton–Jacobi semigroup we prove that these are equivalent, and moreover equivalent to the hypercontractivity of the Hamilton–Jacobi semigroup...

Geodesic metric space | Hamilton–Jacobi semigroup | Probability Theory and Stochastic Processes | Mathematics | Secondary 36C05 | Geometry | Primary 70H20 | 49L99 | 47D06 | Potential Theory | Functional Analysis | Poincaré inequalities | Logarithmic–Sobolev inequalites | Talagrand inequalites | Metric-measure space | Hamilton-Jacobi semigroup | Logarithmic-Sobolev inequalites | METRIC-MEASURE-SPACES | TRANSPORTATION COST | HOPF-LAX FORMULA | BRASCAMP | MATHEMATICS | MAPS | Poincare inequalities | GEOMETRY

Geodesic metric space | Hamilton–Jacobi semigroup | Probability Theory and Stochastic Processes | Mathematics | Secondary 36C05 | Geometry | Primary 70H20 | 49L99 | 47D06 | Potential Theory | Functional Analysis | Poincaré inequalities | Logarithmic–Sobolev inequalites | Talagrand inequalites | Metric-measure space | Hamilton-Jacobi semigroup | Logarithmic-Sobolev inequalites | METRIC-MEASURE-SPACES | TRANSPORTATION COST | HOPF-LAX FORMULA | BRASCAMP | MATHEMATICS | MAPS | Poincare inequalities | GEOMETRY

Journal Article

Pattern Recognition, ISSN 0031-3203, 12/2016, Volume 60, pp. 802 - 812

The sample mean is one of the most fundamental concepts in statistics. Properties of the sample mean that are well-defined in Euclidean spaces become unclear in graph spaces...

Majorize–minimize algorithm | Geometric midpoint | Graph edit distance | Fréchet mean | Consistent estimator | Graph matching | Majorize-minimize algorithm | MANIFOLDS | Frechet mean | COMPUTATION | ANALYSIS GEODESIC PCA | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Analysis | Algorithms

Majorize–minimize algorithm | Geometric midpoint | Graph edit distance | Fréchet mean | Consistent estimator | Graph matching | Majorize-minimize algorithm | MANIFOLDS | Frechet mean | COMPUTATION | ANALYSIS GEODESIC PCA | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | Analysis | Algorithms

Journal Article

Journal of the Royal Statistical Society: Series B (Statistical Methodology), ISSN 1369-7412, 03/2017, Volume 79, Issue 2, pp. 463 - 482

...‐valued response in a Riemannian symmetric space (RSS) and its association with multiple covariates of interest, such as age or gender, in Euclidean space. Such RSS...

Group action | Generalized method of moment | Geodesic | Regression | Link function | Lie group | Riemannian symmetric space | STATISTICS & PROBABILITY | LEAST-SQUARES | EXTRINSIC SAMPLE MEANS | SMOOTHING SPLINES | CORPUS-CALLOSUM | MANIFOLDS | Machine vision | Models | Information management | Studies | Data analysis | Statistical methods | Euclidean space | Medical imaging | Statistics | Euclidean geometry | Parameters | Mathematical analysis | Mathematical models | Statistical tests | Estimates | Symmetry | RS space

Group action | Generalized method of moment | Geodesic | Regression | Link function | Lie group | Riemannian symmetric space | STATISTICS & PROBABILITY | LEAST-SQUARES | EXTRINSIC SAMPLE MEANS | SMOOTHING SPLINES | CORPUS-CALLOSUM | MANIFOLDS | Machine vision | Models | Information management | Studies | Data analysis | Statistical methods | Euclidean space | Medical imaging | Statistics | Euclidean geometry | Parameters | Mathematical analysis | Mathematical models | Statistical tests | Estimates | Symmetry | RS space

Journal Article

ACM TRANSACTIONS ON GRAPHICS, ISSN 0730-0301, 07/2007, Volume 26, Issue 3

.... Shapes - triangular meshes or more generally straight line graphs in Euclidean space - are treated as points in a shape space...

Riemannian geometry | GEODESIC PATHS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | geodesic | parallel transport | PARAMETERIZATION | isometric | shape exploration | deformation | shape space

Riemannian geometry | GEODESIC PATHS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | geodesic | parallel transport | PARAMETERIZATION | isometric | shape exploration | deformation | shape space

Journal Article

Communications in mathematical physics, ISSN 1432-0916, 2018, Volume 364, Issue 2, pp. 441 - 504

We show that a shift space on a finite alphabet with a non-uniform specification property can be modeled by a strongly positive recurrent countable-state Markov shift to which every equilibrium state lifts...

Quantum Physics | Mathematical Physics | Classical and Quantum Gravitation, Relativity Theory | Theoretical, Mathematical and Computational Physics | Complex Systems | Physics | THERMODYNAMIC FORMALISM | UNIQUE EQUILIBRIUM STATES | GEODESIC-FLOWS | MAPS | INTRINSIC ERGODICITY | SYSTEMS | PIECEWISE MONOTONIC TRANSFORMATIONS | BETA-SHIFTS | PHYSICS, MATHEMATICAL | SYMBOLIC DYNAMICS | ENTROPY

Quantum Physics | Mathematical Physics | Classical and Quantum Gravitation, Relativity Theory | Theoretical, Mathematical and Computational Physics | Complex Systems | Physics | THERMODYNAMIC FORMALISM | UNIQUE EQUILIBRIUM STATES | GEODESIC-FLOWS | MAPS | INTRINSIC ERGODICITY | SYSTEMS | PIECEWISE MONOTONIC TRANSFORMATIONS | BETA-SHIFTS | PHYSICS, MATHEMATICAL | SYMBOLIC DYNAMICS | ENTROPY

Journal Article

ACM Transactions on Graphics (TOG), ISSN 0730-0301, 07/2007, Volume 26, Issue 3, pp. 64 - es

.... Shapes -- triangular meshes or more generally straight line graphs in Euclidean space -- are treated as points in a shape space...

Riemannian geometry | geodesic | parallel transport | shape space | isometric deformation | shape exploration | Geodesic | Shape space | Shape exploration | Isometric deformation | Parallel transport

Riemannian geometry | geodesic | parallel transport | shape space | isometric deformation | shape exploration | Geodesic | Shape space | Shape exploration | Isometric deformation | Parallel transport

Journal Article

Ergodic theory and dynamical systems, ISSN 0143-3857, 07/2020, Volume 40, Issue 7, pp. 1991 - 2016

We generalize the higher rank rigidity theorem to a class of Finsler spaces, i.e. Berwald spaces. More precisely, we prove that a complete connected Berwald...

MATHEMATICS | MATHEMATICS, APPLIED | Berwald spaces | non-positive flag curvature | CURVATURE | higher rank rigidity | MANIFOLDS | geodesic flow | Finsler spaces | Minkowski space | Rigidity

MATHEMATICS | MATHEMATICS, APPLIED | Berwald spaces | non-positive flag curvature | CURVATURE | higher rank rigidity | MANIFOLDS | geodesic flow | Finsler spaces | Minkowski space | Rigidity

Journal Article

Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 03/2017, Volume 447, Issue 1, pp. 154 - 166

The angular geometry of asymptotic Teichmüller spaces is studied. Although it has been proved that the angles between any two geodesic rays from the base point of AT(X...

Geodesic | Teichmüller space | Asymptotic Teichmüller space | Angle | MATHEMATICS | MATHEMATICS, APPLIED | Teichmuller space | Asymptotic Teichmuller space

Geodesic | Teichmüller space | Asymptotic Teichmüller space | Angle | MATHEMATICS | MATHEMATICS, APPLIED | Teichmuller space | Asymptotic Teichmuller space

Journal Article

Annals of global analysis and geometry, ISSN 1572-9060, 2019, Volume 56, Issue 3, pp. 597 - 611

We prove that for continuous Lorentz–Finsler spaces timelike completeness implies inextendibility...

Geometry | Mathematical Physics | Analysis | Geodesics | Global Analysis and Analysis on Manifolds | Mathematics | Differential Geometry | Geodesics in weak regularity | Extension of spacetimes | MATHEMATICS | Continuity (mathematics) | Geodesy | Physics - General Relativity and Quantum Cosmology

Geometry | Mathematical Physics | Analysis | Geodesics | Global Analysis and Analysis on Manifolds | Mathematics | Differential Geometry | Geodesics in weak regularity | Extension of spacetimes | MATHEMATICS | Continuity (mathematics) | Geodesy | Physics - General Relativity and Quantum Cosmology

Journal Article

Journal of mathematical imaging and vision, ISSN 1573-7683, 2014, Volume 50, Issue 1-2, pp. 60 - 97

This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics...

Riemannian geometry | Shape space | Mathematical Methods in Physics | Manifolds of mappings | Signal, Image and Speech Processing | Surface matching | Computer Science | Image Processing and Computer Vision | Applications of Mathematics | Diffeomorphism group | Landmark space | RIEMANNIAN METRICS | HUNTER-SAXTON EQUATION | MATHEMATICS, APPLIED | DISTANCE | ONE-DIMENSIONAL MODEL | SHALLOW-WATER EQUATION | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | COMPUTER SCIENCE, SOFTWARE ENGINEERING | SOBOLEV METRICS | MOTION | HYPERSURFACES | GEODESIC-FLOW | MANIFOLD

Riemannian geometry | Shape space | Mathematical Methods in Physics | Manifolds of mappings | Signal, Image and Speech Processing | Surface matching | Computer Science | Image Processing and Computer Vision | Applications of Mathematics | Diffeomorphism group | Landmark space | RIEMANNIAN METRICS | HUNTER-SAXTON EQUATION | MATHEMATICS, APPLIED | DISTANCE | ONE-DIMENSIONAL MODEL | SHALLOW-WATER EQUATION | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | COMPUTER SCIENCE, SOFTWARE ENGINEERING | SOBOLEV METRICS | MOTION | HYPERSURFACES | GEODESIC-FLOW | MANIFOLD

Journal Article

Studia Logica, ISSN 0039-3215, 06/2017, Volume 105, Issue 3, pp. 611 - 624

.... In this paper we adapt this to the class of CAT)-spaces X for and establish a new metatheorem that explains specific bound extractions that recently have been achieved...

CAT(κ) -spaces | Effective bounds | Proof mining | MATHEMATICS | GEODESIC SPACES | PHILOSOPHY | FUNCTIONAL-ANALYSIS | CAT(kappa)-spaces | LOGICAL METATHEOREMS | LOGIC | Mineral industry | Numerical analysis | Mining industry | Analysis

CAT(κ) -spaces | Effective bounds | Proof mining | MATHEMATICS | GEODESIC SPACES | PHILOSOPHY | FUNCTIONAL-ANALYSIS | CAT(kappa)-spaces | LOGICAL METATHEOREMS | LOGIC | Mineral industry | Numerical analysis | Mining industry | Analysis

Journal Article

SIAM journal on imaging sciences, ISSN 1936-4954, 2012, Volume 5, Issue 1, pp. 244 - 310

.... Then shape space is either the manifold of submanifolds of R-n of type M or the orbifold of immersions from M to Rn modulo the group of diffeomorphisms of M...

Shape space | Geodesic equation | Numerical experiments | Surface matching | Curvature | Almost local Riemannian metrics | RIEMANNIAN METRICS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | surface matching | MATHEMATICS, APPLIED | almost local Riemannian metrics | geodesic equation | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | shape space | numerical experiments | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | curvature | Mathematics - Differential Geometry

Shape space | Geodesic equation | Numerical experiments | Surface matching | Curvature | Almost local Riemannian metrics | RIEMANNIAN METRICS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | surface matching | MATHEMATICS, APPLIED | almost local Riemannian metrics | geodesic equation | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | shape space | numerical experiments | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | curvature | Mathematics - Differential Geometry

Journal Article

Annals of global analysis and geometry, ISSN 1572-9060, 2018, Volume 54, Issue 2, pp. 173 - 185

...) -spaces with Boeckx invariant $$I \le -1$$ I≤-1 . In particular, we prove that such submanifolds must be, up to local isometries, among the examples that we explicitly...

(\kappa , \mu )$$ ( κ , μ ) -space | 53C40 | Mathematical Physics | Totally umbilical | Mathematics | 53C15 | 53C25 | Geometry | Legendrian submanifold | Totally geodesic | Analysis | Global Analysis and Analysis on Manifolds | Differential Geometry | (κ, μ) -space | MATHEMATICS | (kappa, mu)-space | Resveratrol | Manifolds (mathematics) | Mathematics - Differential Geometry

(\kappa , \mu )$$ ( κ , μ ) -space | 53C40 | Mathematical Physics | Totally umbilical | Mathematics | 53C15 | 53C25 | Geometry | Legendrian submanifold | Totally geodesic | Analysis | Global Analysis and Analysis on Manifolds | Differential Geometry | (κ, μ) -space | MATHEMATICS | (kappa, mu)-space | Resveratrol | Manifolds (mathematics) | Mathematics - Differential Geometry

Journal Article

International Journal of Mathematics, ISSN 0129-167X, 02/2016, Volume 27, Issue 2, p. 1650002

We define a Larotonda space as a quotient space = / ℬ of the unitary groups of C ∗ -algebras 1 ∈ ℬ...

geodesic | Finsler metric | unitary group of a C-algebra | homogeneous space | MATHEMATICS

geodesic | Finsler metric | unitary group of a C-algebra | homogeneous space | MATHEMATICS

Journal Article

Transactions of the American Mathematical Society, ISSN 0002-9947, 05/2014, Volume 366, Issue 5, pp. 2699 - 2718

ää The space of geodesics L^{\pm }(\mathbb{S}^{n+1}_{p,1}) \mathbb{S}^{n+1}_{p,1} (\mathbb{J},\mathbb{G}) L^{\pm }(\mathbb{S}^{n+1}_{p,1}) (\mathbb{J}',\mathbb...

Geometry | Tensors | Mathematical theorems | Hypersurfaces | Geodesy | Scalars | Mathematical vectors | Mathematical congruence | Lagrangian function | Curvature | MATHEMATICS | ORIENTED GEODESICS | LINES | SURFACES | GEOMETRY

Geometry | Tensors | Mathematical theorems | Hypersurfaces | Geodesy | Scalars | Mathematical vectors | Mathematical congruence | Lagrangian function | Curvature | MATHEMATICS | ORIENTED GEODESICS | LINES | SURFACES | GEOMETRY

Journal Article