In this paper we demonstrate how the geometrically motivated algorithm to determine whether a two generator real Mobius group acting on the Poincare plane is or is not discrete can be interpreted as a non-Euclidean Euclidean algorithm. That is, the algorithm can be viewed as an application of the Euclidean division alg…
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48 results for “Euclidean”
4-manifolds with non-positive curvature are essentially Euclidean.
problem Understanding the structure of 4-manifolds with specific curvature properties.
method Proving homeomorphism to Euclidean space using globally non-positive curvature.
result CAT(0) 4-manifolds are homeomorphic to Euclidean space.
The study shows how strictly convex domains in Euclidean spaces are rigid.
problem Understanding the rigidity of strictly convex domains in Euclidean spaces.
method Proved a rigidity theorem for smooth strictly convex domains in Euclidean spaces.
result Smooth strictly convex domains in Euclidean spaces are rigid.
Prove Gromov's Euclidean endpoint rigidity conjecture for positive mass theorem.
problem Prove Gromov's Euclidean endpoint rigidity conjecture for positive mass theorem.
method Prove Gromov's Euclidean endpoint rigidity conjecture for positive mass theorem.
result Prove Gromov's Euclidean endpoint rigidity conjecture for positive mass theorem.
Researchers describe a specific type of submanifolds in Euclidean space.
problem Understanding inhomogeneous almost symmetric submanifolds.
method Completely describing submanifolds as unions of parallel symmetric submanifolds.
result Described inhomogeneous properly embedded almost symmetric submanifolds as unions of symmetric submanifolds.
Study classifies graphs in Euclidean and non-Euclidean spaces with specific curvature conditions.
problem Classifying graphs with prescribed curvature in various spaces.
method Proves rigidity and classification results for graphs in Riemannian manifolds, focusing on and .
result Provides general splitting theorems for graphs in these settings.
We calculate Euclidean distance degrees for common manifold optimization types.
problem Optimizing on manifold structures.
method Closed-form expressions for stationary points of Euclidean distance function.
result Closed-form expressions for all stationary points on manifold optimization.
Flat subsets in Euclidean buildings are contained within apartments.
problem Understanding the structure of flat subsets in Euclidean buildings.
method Proving containment within apartments.
result Convex flat subsets are contained in apartments.
The paper classifies conformal solitons in pseudo-Euclidean spaces.
problem Classifying conformal solitons in pseudo-Euclidean spaces.
method Classification through pseudo-Riemannian hypersurfaces and position vector fields.
result Complete classification of conformal solitons in pseudo-Euclidean spaces.
For a submanifold M in a Euclidean space, the tangential component x^T of the position vector field x of M is the most natural vector field tangent to the Euclidean submanifold, called the canonical vector field of M. In this article, first we prove that the canonical vector field of every Euclidean submanifold is alwa…
Conditions for Riemannian manifolds to be Euclidean spheres or spaces.
problem Characterizing Riemannian manifolds with specific vector fields.
method Analyzing conformal Killing vector fields and Ricci solitons.
result Conditions for nontrivial closed affine conformal Killing vector fields.
For all 0<t \leq 1, we define a locally Euclidean metric ρ_t on R^3. These metrics are invariant under Euclidean isometries and, if t increases to 1, converges to the Euclidean metric d_E. This research is motivated by expanding universe.
Euclidean volumes of hyperbolic knots are algebraic numbers.
problem Understanding the algebraic nature of Euclidean volumes in hyperbolic knots.
method Deforming hyperbolic structures into Euclidean structures and analyzing the normalised Euclidean volumes.
result Normalised Euclidean volumes of hyperbolic knots are always algebraic numbers.
The paper discusses algorithms for reconstructing curves with given Euclidean or affine curvatures.
problem Reconstructing planar curves with specified Euclidean or affine curvatures.
method The paper presents algorithms for curve reconstruction under the special Euclidean and equi-affine groups.
result The reconstructed curves are close to the original curves in terms of the specified curvatures.
Study on hyperspheres in 4-spaces as special Riemannian manifolds.
problem Characterizing hyperspheres in Euclidean and Minkowski 4-spaces as specific Riemannian manifolds.
method Constructing and studying hyperspheres in 4-dimensional spaces (Euclidean and pseudo-Euclidean) as almost paracontact almost paracomplex Riemannian manifolds.
result Characterization and geometric properties of these manifolds.
Study infinite Euclidean distance discriminants of algebraic varieties.
problem Understanding the structure of data points with infinitely many critical points in Euclidean distance correspondence.
method Developed computer code to compute discriminants and proved properties of fibers.
result Infinite Euclidean distance discriminants contain all data points with infinitely many critical points for the nearest-point problem.
The study explores properties and mutations in oriented matroids, proving new results on Euclidean and non-Euclidean structures.
problem Investigating the Euclidean and non-Euclidean properties of oriented matroids.
method Analyzing the minimum number of mutations, using lexicographic extensions, and mutation-flips to prove properties.
result For rank 4 uniform oriented matroids, the minimum number of mutations adjacent to an element is at most 3.
Log-Euclidean Lie Groupsmath.DG
Develops log-Euclidean Lie groups for SPD and correlation matrices.
problem Unifies various log-Euclidean constructions for SPD and correlation matrices.
method Theory and explicit isometries linking different log-Euclidean metrics.
result Explicit log-Euclidean metrics on SPD and correlation matrices.
Study uses crochet to visualize non-Euclidean geometry.
problem Understanding non-Euclidean surfaces through physical models.
method Parametrization of crochet models to represent Lobachevskian surface.
result Crochet models reflect non-Euclidean geometry characteristics.
The study characterizes canal hypersurfaces in Euclidean spaces and their curvature properties.
problem Characterizing canal hypersurfaces in Euclidean spaces.
method Analyzing canal hypersurfaces in Euclidean n-space, focusing on E4, computing curvature properties, and proving specific cases.
result Flat canal hypersurfaces in Euclidean 4-space are only circular hypercylinders or circular hypercones, and minimal canal hypersurfaces are only generalized catenoids.
Euclidean systems and real PK arrangements linked via geometry.
problem Establishing a connection between Euclidean systems and real PK arrangements.
method Proving a correspondence between Euclidean -systems and real PK arrangements, and showing homeomorphism of moduli spaces.
result Moduli space of Euclidean -systems is homeomorphic to a polytope's interior, and hyperplane arrangements are simplicial.
Paper defines minimal hypersurfaces in Euclidean and Riemannian spaces.
problem Characterizing minimal hypersurfaces in different spaces.
method Analyzes conditions for hypersurfaces to be minimal or stable.
result Minimal and stable hypersurfaces are hyperplanes in Euclidean spaces and totally geodesic submanifolds in Riemannian manifolds.
The study characterizes round spheres in Euclidean space based on r-mean curvature conditions.
problem Characterizing round spheres in Euclidean space under specific curvature conditions.
method Characterization based on r-mean curvature conditions.
result Characterizes round spheres in Euclidean space under suitable r-mean curvature conditions.
Paper uses non-Euclidean analysis to classify brain structure variations.
problem Classifying joint variations in multi-object brain structures.
method Combines non-Euclidean statistics and non-parametric integrative analysis.
result Effective, robust, and interpretable joint structure found.
The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold . The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component x^T of the position vector field is the most natural vector field tangent to the …
We prove that any asymptotically Euclidean metric on with no conjugate points must be isometric to the Euclidean metric.
Examines medial axis in pseudo-Euclidean spaces.
problem No specific problem stated; focuses on new context.
method Follows Birbrair and Denkowski's approach.
result Feasibility of medial axis in pseudo-Euclidean spaces checked.
The paper studies special surfaces in pseudo-Euclidean space.
problem Characterizing and classifying -isothermic surfaces in pseudo-Euclidean 3-space.
method Analyzing the pseudo-Calapso equation and providing explicit coordinates for Dupin surfaces.
result Explicit solutions to the pseudo-Calapso equation are provided.
Tensor approach simplifies Euclidean space descriptions.
problem Simplifying tensor descriptions of Euclidean spaces.
method Emphasizes geometric vectors in tensor description.
result Proved integral identities with vector integrands.
We complete a minor gap in Gromoll and Walschap classification of metric fibrations from the Euclidean space, thus completing the classification of Riemannian foliations on Euclidean spaces.
The paper defines a metric on Euclidean triangles and polygons, proving properties and completeness.
problem Defining and analyzing a metric space for Euclidean triangles and polygons.
method Introducing and proving properties of a metric on marked Euclidean triangles, extending to polygons and triangulated surfaces.
result The metric is Finsler and complete, providing formulas for its infinitesimal structure.
New approach uses isotropic geometry to solve Euclidean problems.
problem Solving systems of constraints in Euclidean geometry.
method Start with analogous problems in isotropic geometry to initialize optimization algorithms.
result Solutions in isotropic geometry provide insight and initialize Euclidean problem solutions.
Study on hypersurfaces in pseudo-Euclidean space with constant curvature or rotational properties.
problem Characterizing hypersurfaces in pseudo-Euclidean space.
method Defined and studied warped product hypersurfaces with constant sectional curvature or rotational properties.
result Hypersurfaces in pseudo-Euclidean space either have constant curvature or are contained in rotational hypersurfaces.
Local and global classifications of Einstein submanifolds in Euclidean space.
problem Classifying Einstein submanifolds in Euclidean space.
method Local and global parametric classifications with emphasis on intrinsic assumptions.
result Local and global classifications of Einstein submanifolds of codimension two.
Piecewise flat approximations for curvature in Euclidean and non-Euclidean spaces.
problem Approximating local extrinsic curvature on discrete manifolds.
method Constructing discrete curvature forms on piecewise flat manifolds, using weighted sums of hinge angles.
result Converges to smooth curvature values as mesh refinement occurs, favorably comparing with other discrete approaches.
Optimal Euclidean structure minimizes energy in weighted toroidal graphs.
problem Finding the optimal Euclidean structure for weighted toroidal graphs.
method Minimizing Dirichlet energy over all possible Euclidean structures and realizations within a fixed homotopy class.
result The optimal Euclidean structure induces a weighted Delaunay decomposition.
Constructs hyperspheres with prescribed mean curvature in Euclidean space.
problem Creating hyperspheres with a specific curvature in Euclidean space.
method Constructs families of smooth functions to fill Euclidean space with hyperspheres of prescribed mean curvature.
result Euclidean space can be filled with hyperspheres of prescribed mean curvature.
Optimal Liouville theorem for half-Euclidean space equations.
problem Optimal Liouville-type theorems for conformally invariant equations.
method Established optimal Liouville-type theorems for conformally invariant second-order elliptic equations.
result Proved an optimal Liouville-type theorem for equations in the half-Euclidean space.
Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.
problem Proving theorems for minimal and constant mean curvature graphs in Euclidean and Lorentz-Minkowski spaces.
method Explains several proofs and provides mean curvature estimates for graphs in Euclidean and Lorentz-Minkowski spaces.
result Bernstein-type theorems for constant mean curvature graphs in Euclidean 3-space and space-like graphs in Lorentz-Minkowski 3-space.
Study on biharmonic hypersurfaces with specific recurrent operators in Euclidean space.
problem Characterizing biharmonic hypersurfaces with recurrent operators.
method Analysis of various recurrent operators and their impact on biharmonic hypersurfaces.
result Some well-known recurrent operators play a significant role in making biharmonic hypersurfaces minimal.
This paper proposes a spectral clustering algorithm for hyperbolic spaces, improving efficiency over Euclidean methods.
problem Inefficient clustering in Euclidean spaces for complex data structures.
method Developed a spectral clustering algorithm using hyperbolic similarity matrices.
result The algorithm converges at least as fast as Euclidean spectral clustering and performs better on complex datasets.
I consider compact metric spaces which admit intrinsic isometries to Euclidean d-space. The main result roughly states that the class of these spaces coincides with class of inverse limits of Euclidean d-polyhedra.
We study the Ricci flow for initial metrics which are C^0 small perturbations of the Euclidean metric on R^n. In the case that this metric is asymptotically Euclidean, we show that a Ricci harmonic map heat flow exists for all times, and converges uniformly to the Euclidean metric as time approaches infinity. In provin…
Given a closed orientable Euclidean cone 3-manifold C with cone angles less than or equal to pi, and which is not almost product, we describe the space of constant curvature cone structures on C with cone angles less than pi. We establish a regeneration result for such Euclidean cone manifolds into spherical or hyperbo…
Minimal biharmonic hypersurfaces in Euclidean spaces are ideal.
problem Proving biharmonic ideal hypersurfaces are minimal.
method Analyzing δ(r)-ideal biharmonic hypersurfaces in Euclidean spaces.
result Every δ(r)-ideal biharmonic hypersurface in Euclidean space is minimal.
Study shows no new Euclidean factors can appear in the limit of CAT(0) spaces.
problem Stability of Euclidean factors in CAT(0) spaces under convergence.
method GH-convergence of CAT(0) spaces with uniformly cocompact discrete groups of isometries.
result Dimension of the maximal Euclidean factor is the same for large .
New methods create full discretized isothermic tori in Euclidean spaces.
problem Creating full discretized isothermic tori in Euclidean spaces.
method Using Darboux transformations and periodic curvature line systems.
result Discrete and semi-discrete k-dimensional isothermic tori in n-dimensional Euclidean space.
Study on sets with positive reach in Euclidean and Riemannian spaces.
problem Understanding sets with positive reach in various spaces.
method Structural results on subsets of positive reach.
result New insights into sets with positive reach in Euclidean and Riemannian spaces.