We present a boundary version of a theorem about solenoidal unit vector fields with minimum energy on a spherical domain of an odd dimensional Euclidean sphere.
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48 results for “spherical domains”
We present in this paper a \boundary version" for theorems about minimality of volume and energy functionals on a spherical domain of threedimensional Euclidean sphere.
We consider the lower order eigenvalues of poly-Laplacian with any order on spherical domains. We obtain universal inequalities for them and show that our results are optimal.
Paper explores neural network approximations on sphere domains.
problem Approximating functionals on sphere domains using neural networks.
method Encoder-decoder framework with spherical harmonics for infinite-dimensional domain.
result Approximation rates of neural networks with different encoder structures.
The paper examines the stability of Minkowski inequality for nearly spherical domains.
problem Stability of Minkowski inequality for nearly spherical domains.
method Analyzes stability inequalities for perturbations of a ball and axially symmetric perturbations.
result Established stability inequalities for curvature integrals of nearly spherical domains.
Developed a diffusion model on spherical data, addressing geometric and stochastic challenges.
problem Diffusion models on spherical data face unique geometric and stochastic issues.
method Extended spectral diffusion to spherical harmonics, introducing modified stochastic differential equations.
result Introduced a geometry-dependent inductive bias in spectral diffusion models.
In this paper, we study the first two eigenvalues of the buckling problem on spherical domains. We obtain an estimate on the second eigenvalue in terms of the first eigenvalue, which improves one recent result obtained by Wang-Xia in [7].
We apply a spherical CR Dehn surgery theorem in order to obtain infinitely many Dehn surgeries of the Whitehead link complement that carry spherical CR structures. We consider as starting point the spherical CR uniformization of the Whitehead link complement constructed by Parker and Will, using a Ford domain in the co…
Rigidity theorem for spherical sectors in Riemannian manifolds.
problem Rigidity of spherical sectors in Riemannian manifolds under overdetermined conditions.
method Analyzing solutions to the inhomogeneous Helmholtz equation with constant Dirichlet and Neumann boundary conditions.
result Spherical sectors are the only solutions under given conditions.
Paper introduces S3W distance for spherical probability distributions.
problem Comparing spherical probability distributions efficiently and accurately.
method S3W distance using stereographic projection and generalized Radon transform.
result Extensive theoretical analysis and evaluation of S3W performance.
New rigidity found for 3D warped product domains.
problem Finding rigidity conditions for warped product domains.
method Developed scalar curvature rigidity for a general class of domains.
result Identified domains satisfying a boundary condition analogous to logarithmic concavity.
The present paper considers two infinite families of cone-manifolds endowed with spherical metric. The singular strata is either the torus knot or the torus link . Domains of existence for a spherical metric are found in terms of cone angles and volume formulæ are presented.
We carry out the harmonic analysis on four Platonic spherical three-manifolds with different topologies. Starting out from the homotopies (Everitt 2004), we convert them into deck operations, acting on the simply connected three-sphere as the cover, and obtain the corresponding variety of deck groups. For each topology…
Develops spherical density-equalizing maps for closed surfaces.
problem Lack of methods for genus-0 closed surfaces.
method Conformal parameterization onto unit sphere, density equalization, quasi-conformal theory, harmonic energy, landmark constraints.
result Landmark-aligned spherical density-equalizing maps balancing different distortion measures.
Transformers adapted to spherical geometry using space-filling curves.
problem Generalizing transformers to geometric domains like spheres.
method Attention heads following a space-filling curve.
result Introduction of the Spiroformer on a 2-sphere.
Quantifies nearly spherical subsets in complex ball geometry.
problem Isoperimetric inequality for nearly spherical domains in Bergman ball.
method Proves a quantitative isoperimetric inequality for nearly spherical subsets of Bergman ball.
result First result on isoperimetric phenomenon in Bergman ball.
In this paper, we define a certain "proportional volume property" for an unit vector field on a spherical domain in S3. We prove that the volume of these vector fields has an absolute minimum and this value is equal to the volume of the Hopf vector field. Some examples of such vector fields are given. We also study the…
Study stable capillary hypersurfaces with planar boundaries in half-spaces and domains.
problem Characterize stable capillary hypersurfaces with planar boundaries in bounded domains.
method Analyzes hypersurfaces in half-spaces and domains bounded by hyperplanes, proving conditions for stability and shape.
result Stable hypersurfaces in certain domains are spherical caps or pieces of spheres.
Enhances Gaussian processes with spherical features for better scalability and flexibility.
problem Lack of representation learning in Gaussian processes compared to deep neural networks.
method Introduces spherical inter-domain features to improve GP approximation and scalability.
result The method alleviates limitations and improves scalability compared to alternative strategies.
Proposes variational Wasserstein barycenters for geometric clustering.
problem Geometric clustering problems, especially K-means and co-clustering.
method Solves for Monge maps using variational principle, explores connections to K-means and co-clustering.
result Demonstrates feasibility and use of variational Wasserstein barycenters in clustering.
In this paper we study the eigenvalues of buckling problem on domains in a unit sphere. By introducing a new parameter and using Cauchy inequality, we optimize the inequality obtained by Wang and Xia in [12].
In this new version, we give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in is Kähler-Einstein if and only if the domain is biholomorphic to the ball. We establish versions of various …
New flow for capillary surfaces converges to spherical caps.
problem Optimizing capillary surfaces in space forms.
method Constrained mean curvature flow.
result Flow converges to spherical caps globally.
Study sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces.
problem Finding sharp upper bounds for eigenvalues of magnetic Laplacian.
method Isoperimetric inequalities and bounds in terms of Gaussian curvature.
result Maximal first eigenvalue for geodesic disk on simply connected surfaces.
We describe a simple fundamental domain for the holonomy group of the boundary unipotent spherical CR uniformization of the figure eight knot complement, and deduce that small deformations of that holonomy group (such that the boundary holonomy remains parabolic) also give a uniformization of the figure eight knot comp…
A new GP model uses spherical harmonics for faster inference.
problem Efficiently fitting large datasets with Gaussian processes.
method Sparse Gaussian processes with spherical harmonic features.
result Significant speed-up in inference for large datasets.
Small bubbles sliding on a boundary maintain half-spherical shape.
problem Preserving the shape of small bubbles sliding on a boundary.
method Area-preserving Willmore flow, asymptotic analysis, convergence proof.
result The flow keeps a half-spherical shape for all times.
In this paper, we revisit the analyses of Antonie Stern (1925) and Hans Lewy (1977) devoted to the construction of spherical harmonics with two or three nodal domains. Our method yields sharp quantitative results and a better understanding of the occurrence of bifurcations in the families of nodal sets.This paper is a …
In this paper, we establish universal inequalities for eigenvalues of the clamped plate problem on compact submanifolds of Euclidean spaces, of spheres and of real, complex and quaternionic projective spaces. We also prove similar results for the biharmonic operator on domains of Riemannian manifolds admitting spherica…
New geometric approach for analyzing compositional data like gut microbiomes.
problem Analyzing non-negative compositional data with relative values only.
method Reinterpret compositional data as quotient topology of a sphere, using spherical harmonics and reflection group actions.
result Construction of Reproducing Kernel Hilbert Space (RKHS) for compositional data.
Study concavity of solutions to elliptic equations under conformal deformations.
problem Establish concavity estimates for the principle eigenfunction of weighted Schrödinger operators.
method Analyzing the Dirichlet problem for the weighted Schrödinger operator \[-Δu + Vu = λρu\] with conformal connections.
result Partial resolution of Nguyen's conjecture on fundamental gap of horoconvex domains and power convexity estimate for solutions in spherical geometry.
Research proves limits on harmonic map orders into Euclidean buildings.
problem Limits on the possible orders of harmonic maps from surfaces to Euclidean buildings.
method Direct analysis of homogeneous maps and related spherical billiards problem.
result The order of harmonic maps is of the form where divides .
We develop a solution theory for a generalized electro-magneto static Maxwell system in an exterior domain with anisotropic coefficients converging at infinity with a certain rate towards the identity. Our main goal is to treat right hand side data from some polynomially weighted Sobolev spaces and obtain solutions whi…
Paper proves stability of quermassintegral inequalities using inverse curvature flow.
problem Stability of quermassintegral inequalities for nearly spherical sets.
method Inverse curvature flow with special rescaling to study quermassintegral inequalities.
result Decreasing rate of k-th quermassintegral is faster than Fraenkel asymmetry for nearly spherical sets.
Statistical models with constrained probability distributions are abundant in machine learning. Some examples include regression models with norm constraints (e.g., Lasso), probit, many copula models, and latent Dirichlet allocation (LDA). Bayesian inference involving probability distributions confined to constrained d…
Study of immersions with Willmore energy leading to spherical and catenoid bubbles.
problem Classifying immersions with specific energy properties.
method Analyzing sequences of weak immersions with diverging conformal classes, applying Möbius transformations, and strong -limits.
result Obtaining spherical and catenoid bubbles as limits of immersions.
Needlets have been recognized as state-of-the-art tools to tackle spherical data, due to their excellent localization properties in both spacial and frequency domains. This paper considers developing kernel methods associated with the needlet kernel for nonparametric regression problems whose predictor variables are de…
An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1). We show that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius in the visual…
In this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities, whose equality cases are saturated by domains with spherical symmetry. In particular, for a regular bounded domain , , we prove that if the mean curvature…
We compute the Szego kernel of the unit circle bundle of a negative line bundle dual to a regular quantum line bundle over a compact Kaehler manifold. As a corollary we provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds (disk bundles over homogeneous Hodge manifolds) for w…
Developed an ellipsoidal density-equalizing map for genus-0 closed surfaces.
problem Large geometric distortion when using spherical domain for genus-0 closed surfaces.
method Developed a novel method for ellipsoidal density-equalizing maps and combined with quasi-conformal maps.
result Significantly improved surface remeshing performance for genus-0 closed surfaces.
In this paper, we study a mean curvature type flow with capillary boundary in the unit ball. Our flow preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases an energy functional . We show that it has the longtime existence and subconverges to spherical caps. As an applic…
Spherical Knot Mosaicsmath.GT
Paper introduces spherical knot mosaics for knot and link invariants.
problem Representing knots on a sphere with tiles.
method Tiling a 2-sphere with 11 knot mosaic tiles to define new invariants.
result New knot invariants derived from spherical mosaic tiling.
Paper explores curvature flows on spheres to prove inequalities.
problem Prove inequalities for convex domains on spheres.
method Designs locally constrained curvature flows to preserve quermassintegrals.
result Flow convergence to a round sphere would settle inequalities.
We formulate certain inequalities for the geometric quantities characterizing causal diamonds in curved and Minkowski spacetimes. These inequalities involve the red-shift factor which, as we show explicitly in the spherically symmetric case, is monotonic in the radial direction and it takes its maximal value at the cen…
Study geodesics on spherical polyhedra, estimating their number.
problem Counting simple closed geodesics on spherical polyhedra.
method Examined regular spherical octahedra, cubes, and tetrahedra.
result Estimated the number of simple closed geodesics on spherical polyhedra.
We define a complex connection on a real hypersurface of $\C^{n+1}$ which is naturally inherited from the ambient space. Using a system of Codazzi-type equations, we classify connected real hypersurfaces in $\C^{n+1}$, , which are Levi umbilical and have non zero constant Levi curvature. It turns out that such …
In this paper, we are interested in the construction of quasiconformal mappings between domains of the Heisenberg group H that minimise a mean distortion functional. We propose to construct such mappings by considering a corresponding problem between domains of Poincaré half-plane . The first map we construc…