A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
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48 results for “geometric comparison”
Minimal surfaces help compare geometric shapes.
problem Comparing different geometric shapes.
method Applications of minimal surfaces.
result New insights into geometric comparisons.
Rigidity phenomena on lower -weighted Ricci curvature bounds with -range for non-symmetric Laplacianmath.DG
Paper investigates rigidity phenomena for weighted Ricci curvature bounds with Laplacian comparison theorem.
problem Investigating rigidity phenomena for weighted Ricci curvature bounds.
method Derived comparison geometric estimates and generalized for non-symmetric Laplacian.
result Obtained rigidity results for Laplacian comparison theorem, diameter comparisons, and volume comparisons.
In this study, a pairwise comparison matrix is generalized to the case when coefficients create Lie group , non necessarily abelian. A necessary and sufficient criterion for pairwise comparisons matrices to be consistent is provided. Basic criteria for finding a nearest consistent pairwise comparisons matrix (extend…
New comparison theorem for submanifolds with geometric inequalities.
problem Geometric inequalities for submanifolds in ambient spaces.
method Explicit Jacobian determinant formula for normal exponential map.
result Establishes new comparison theorem related to Heintze-Karcher's.
We treat the vakonomic dynamics with general constraints within a new geometric framework which will be appropriate to study optimal control problems. We compare our formulation with Vershik-Gershkovich one in the case of linear constraints. We show how nonholonomic mechanics also admits a new geometrical description w…
Study embeds PC matrices into Grassmannian manifold for geometric interpretation.
problem Understanding algebraic consistency of pairwise comparisons matrices.
method Leverages Plücker coordinates and geometric interpretation of Grassmannian manifold.
result Algebraic consistency condition is equivalent to geometric consistency in .
Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.
problem Geometric and topological properties of Finsler metric measure manifolds with integral weighted Ricci curvature bounds.
method Establish Laplacian comparison theorem, volume comparison theorems, volume growth estimate, Gromov pre-compactness, local Dirichlet isoperimetric constant estimate.
result First Dirichlet eigenvalue estimate and gradient estimate for harmonic functions.
A novel method compares 3D point clouds using information geometry.
problem Comparing 3D point clouds in machine learning applications.
method Interprets point clouds as probability density functions on a statistical manifold, using GMM and Modified Symmetric KL divergence.
result Demonstrates effectiveness through various case studies.
We compare the covariant formulation of Quantum Mechanics on a curved spacetime fibred on absolute time with the standard Geometric Quantisation.
Transforms curves and surfaces for efficient geometric analysis.
problem Efficiently analyzing and comparing curves and surfaces.
method Square root velocity transformation for curves and intrinsic comparison for surfaces.
result Fundamental geometric properties of curves under the transformation.
Comparison theorems in centro-affine differential geometry
problem rigidity phenomena of comparison theorems
method study of centro-affine differential geometry
result examples of rigidity phenomena
Study eigenvalues on quaternion-Kähler manifolds with geometric bounds.
problem Estimating eigenvalues on quaternion-Kähler manifolds.
method Lower bounds derived from modulus of continuity estimates for heat equation solutions and Laplace comparison theorem.
result Established bounds for first nonzero eigenvalues in terms of dimension, diameter, and scalar curvature.
Study compares nonsmooth spaces with integrable Ricci bounds.
problem Comparing geometric and functional inequalities on nonsmooth spaces.
method Localization method and one-dimensional comparison estimates.
result Extension of comparison principles to nonsmooth settings.
The paper proves geometric comparisons on metric measure spaces with integral Bakry-Émery Ricci tensor bounds.
problem Geometric comparisons on metric measure spaces with specific tensor bounds.
method Integral radial Bakry-Émery Ricci tensor bounds and potential function/gradient bounds.
result Diameter and eigenvalue estimates on smooth metric measure spaces.
The paper explores dualities in differential equations and their applications in Riemannian geometry.
problem Developing comparison theorems for mixed type differential equations.
method Utilizing dualities in differential equations and inequalities, and applying them to Riemannian geometry.
result Proves Hessian and Laplacian comparison theorems under various curvature assumptions.
We develop the differential geometric and geometric analytic studies of Hamiltonian systems. Key ingredients are the curvature operator, the weighted Laplacian, and the associated Riccati equation. We prove the appropriate generalizations of Bochner--Weitzenböck formula and Laplacian comparison theorem, and study the h…
We observe that the comparison result of Barles-Biton-Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow.
We present a criterion for the stochastic completeness of a submanifold in terms of its distance to a hypersurface in the ambient space. This relies in a suitable version of the Hessian comparison theorem. In the sequel we apply a comparison principle with geometric barriers for establishing mean curvature estimates fo…
We study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition, and a suitable condition for the weighted mean curvature for the boundary, we ob…
Neural Point-Formscs.LG
Introduces neural point-forms for learning geometric features from noisy point clouds.
problem Learning geometric features from noisy point clouds with missing tangency information.
method Uses Laplacian-based techniques to build comparison matrices for point clouds, proving consistency under various assumptions.
result Neural point-forms provide a competitive and interpretable representation, especially beneficial for dense or manifold-like structures.
Study compares manifolds with boundary under weighted Ricci curvature bounds.
problem Understand geometric properties of manifolds with boundary under lower weighted Ricci curvature bounds.
method Use lower -weighted Ricci curvature bounds with -range to study comparison geometry.
result Conclude splitting theorems and comparison geometric results for inscribed radius, volume, and eigenvalues.
This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is two-fold: Give a concise and geometric review on the birth of weighted Ricci curvature and its applications; Explain recent results from a nonlinear an…
Geometrically transforms word embeddings into a common space for better comparison.
problem Comparing embeddings from different sources is challenging.
method Applies orthogonal rotations and Mahalanobis scaling to transform embeddings into a shared latent space.
result The method improves word similarity and analogy tasks.
Enhanced 3D shape analysis using information geometry.
problem Challenges in comparing 3D point clouds due to their unstructured nature and complex geometry.
method Information geometric framework for 3D point cloud shape analysis using Gaussian Mixture Models (GMMs) on a statistical manifold. Proposed MSKL divergence with upper and lower bounds.
result MSKL provides stable and monotonically varying values that directly reflect geometric variation, outperforming traditional distances and existing KL approximations.
The paper explores geometric relationships in manifolds with curvature constraints, proving new inequalities and rigidity results.
problem Understanding geometric features of manifolds with curvature constraints.
method Comparison theorems and spacetime harmonic functions.
result Partial resolution of Gromov's conjecture and new characterizations of geometries.
Study geodesic paths on flat surfaces, comparing length and singularity counts.
problem Comparing geometric length and singularity counts on geodesic paths.
method Apply counting limit laws to infinite graphs and then to flat surfaces.
result Statistical comparison of geometric length and singularity counts on geodesic paths.
The paper proves geometric rigidity using harmonic twisted spinors and scalar curvature comparison.
problem Proving geometric rigidity for closed Riemannian spin manifolds with specific properties.
method Using Gromov's exact-lift two-form method and harmonic spinors to analyze scalar curvature.
result The original metric is Einstein, and the universal cover is real hyperbolic in the positive-spectrum case.
Paper proves a sharp weighted Isoperimetric inequality for substatic manifolds.
problem Proving geometric results for substatic Riemannian manifolds.
method Comparison theory based on a newly discovered conformal connection.
result Sharp, weighted Isoperimetric inequality quantifying boundary minimization.
Counting ends on shrinkersmath.DG
The study counts ends on shrinkers using geometric covering methods.
problem Counting the number of ends on shrinkers.
method Geometric covering method to study the number of ends.
result Proves that the number of ends on any complete non-compact shrinker is at most polynomial growth with fixed degree.
Second part of Q-curvature research focusing on volume comparison.
problem Volume control and rigidity of Q-curvature.
method Volume comparison and local rigidity analysis.
result Volume comparison theorem for metrics close to strictly stable positive Einstein metrics.
Survey of three geometric frameworks for action-dependent field theories.
problem Understanding action-dependent field theories through geometric structures.
method Introduction and analysis of three geometric frameworks: k-contact, k-cocontact, and multicontact.
result Analysis of relationships among these geometric structures and comparison with other definitions.
Sharp inequality found for hypersurfaces in curved spaces.
problem Establishing geometric inequalities for hypersurfaces in curved spaces.
method Standard comparison methods in Riemannian Geometry.
result Sharp geometric inequality for closed hypersurfaces in manifolds with asymptotically nonnegative curvature.
We show that the computation of the Fredholm index of a fully elliptic pseudodifferential operator on an integrated Lie manifold can be reduced to the computation of the index of a Dirac operator, perturbed by a smoothing operator, canonically associated, via the so-called clutching map. To this end we adapt to our fra…
The paper studies volume and area comparisons in non-compact 3-manifolds with non-negative scalar curvature.
problem Volume and area comparisons in non-compact 3-manifolds with non-negative scalar curvature.
method Gradient integral estimates and level set analysis.
result Sharp volume and area comparisons derived from a gradient integral estimate.
Abstract cone operators prove scalar curvature comparisons on singular manifolds.
problem Proving scalar curvature inequalities on manifolds with cone singularities.
method Using index theory for twisted Dirac operators on Lipschitz bundles.
result Lipschitz rigidity for scalar curvature on odd-dimensional manifolds.
We prove Cheng's eigenvalue comparison theorems for geodesic balls within the cut locus under weaker geometric hypothesis, and we also show that there are certain geometric rigidity in case of equality of the eigenvalues. This rigidity becomes isometric rigidity under upper sectional curvature bounds or lower Ricci cur…
New method for curve comparison using iterated integrals and moving frames.
problem Comparing curves robustly to noise and transformations.
method Moving frame method paired with log-signature transform.
result Algorithmic construction of invariants for curve equivalence under rigid motions.
The paper examines torsional rigidity bounds under geometric flows.
problem Torsional rigidity behavior under geometric flows.
method Bounds on torsional rigidity derived under Ricci Flow and Inverse Mean Curvature Flow.
result Inequalities of comparison with the flat disk for torsional rigidity.
Online Streaming Feature Selection (OSFS) is a sequential learning problem where individual features across all samples are made available to algorithms in a streaming fashion. In this work, firstly, we assert that OSFS's main assumption of having data from all the samples available at runtime is unrealistic and introd…
This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps (minimal stretch maps) and geodesics for the `Lipschitz metric' are constructed. The extremal Lipschitz constant equals the maximum ratio of lengths of measured la…
Sharp gradient estimates for positive Ricci curvature manifolds.
problem Understanding geometric properties of manifolds with positive Ricci curvature.
method Proving sharp gradient estimates and monotonicity formulae.
result Sharp gradient estimates and monotonicity formulae for positive Ricci curvature manifolds.
We relate the Gromov norm on homology classes to the harmonic norm on the dual cohomology and obtain double sided bounds in terms of the volume and other geometric quantities of the underlying manifold. Along the way, we provide comparisons to other related norms and quantities as well.
Uniform eigenvalue bounds for Hodge Laplacian on manifolds with Ricci curvature and injectivity radius bounds.
problem Establishing uniform bounds for eigenvalues of the Hodge Laplacian on manifolds with specific geometric constraints.
method Using uniform bounds on Ricci curvature, injectivity radius, and diameter, we derive eigenvalue estimates for the Hodge Laplacian.
result Uniform upper bounds for eigenvalues of the Hodge Laplacian on differential forms on manifolds with given geometric constraints.
Geometric approach combines asset returns and investor views for better portfolio optimization.
problem Optimizing portfolios with investor-specific views.
method Generalized Wasserstein barycenter (GWB) to integrate statistical asset returns and investor views.
result The geometric approach offers more flexibility and rewards for correct investor views.
Study curvature flows on pinched Hadamard surfaces, proving convexity preservation and convergence.
problem Preserving convexity and convergence of curves under curvature flows on pinched Hadamard surfaces.
method Area- and length-preserving curvature flows, refined comparison arguments, delicate curvature estimates.
result Convexity is preserved and curves converge to a geodesic circle under certain conditions.
This paper compares two NUTS variants and analyzes their convergence and mixing times.
problem Theoretical comparison and convergence guarantees of NUTS variants.
method Deriving necessary and sufficient conditions for geometric ergodicity, and analyzing mixing times.
result NUTS-mul and NUTS-BPS have nearly identical qualitative behavior but differ quantitatively in convergence rates.
Interdisciplinary study linking potential theory and elliptic PDEs.
problem Understanding solutions to nonlinear elliptic PDEs.
method Combining geometric and potential theory approaches.
result Validity of comparison principle and existence/uniqueness of solutions.