The paper generalizes radial curvature bounds on manifolds.
problem Bounding radial curvature on Riemannian manifolds.
method Introducing a new class of manifolds with radial curvature constraints.
result New insights into curvature properties of manifolds.
Proof of smooth radial sections in vector bundles.
problem Existence of smooth radial sections in vector bundles.
method Proof of existence using linear connection.
result Existence of smooth radial sections in vector bundles.
All radially symmetric connections on round spheres are parallel.
problem Classifying radially symmetric connections in vector bundles over spheres.
method Proving all such connections are parallel.
result Radially symmetric connections on round spheres are parallel.
Convexity properties are preserved under radial transformations in hyperbolic and spherical geometries.
problem Preserving convexity in hyperbolic and spherical geometries under radial transformations.
method Used Poincaré disk model for hyperbolic geometry and stereographic projection for spherical geometry to prove preservation of convexity under radial expansion and contraction.
result Radial expansion and contraction preserve hyperbolic and spherical convexity, respectively.
Study examines maximal domains of radial harmonic functions across different curvature types.
problem Understanding maximal domains of radial harmonic functions in various curvature settings.
method Analysis of harmonic spaces with positive, zero, and negative curvature.
result Characterization of maximal domains for radial harmonic functions in different curvature contexts.
Study geometry and topology of manifolds with radial curvature bounds.
problem Understanding manifolds with specific curvature constraints.
method Systematic investigation of manifolds with integral radial curvature bounds.
result Obtained many interesting and important conclusions about manifold geometry and topology.
Study on radial solutions in higher dimensions, finding special cases.
problem Analyzing radial solutions to Hamiltonian stationary equations in various dimensions.
method Examined smooth radial solutions defined away from the origin, focusing on dimensions two and higher.
result In higher dimensions, non-special Lagrangian radial solutions exist near the origin, with continuity conditions.
The study characterizes harmonic spaces and their radial eigen-functions and vector fields.
problem Characterizing harmonic spaces and their geometric properties.
method Examining radial eigen-spaces of Laplacians and using duality.
result Results extend to spaces harmonic with respect to a single point.
No radial balanced metrics found on Kepler manifold unit ball with mild boundary conditions.
problem Finding radial balanced metrics on the unit ball of the Kepler manifold.
method Analyzing boundary behavior and weights of metrics.
result Explicit weights for radial metrics satisfying balanced condition identified.
Study complete gradient Ricci solitons with zero radial Weyl curvature.
problem Characterize complete gradient Ricci solitons with specific curvature properties.
method Classify complete gradient Ricci solitons with zero radial Weyl curvature for dimensions n≥4. result Completely classified complete gradient Ricci solitons with zero radial Weyl curvature.
Radial BNNs offer a scalable, continuous weight distribution for Bayesian deep learning.
problem Discrete support in Bayesian deep learning methods like MC dropout.
method Radial BNNs with full support over weight-space.
result Radial BNNs outperform discrete-support methods in real-world applications.
New topologies for star-shaped sets without boundedness.
problem Defining convergence for unbounded star-shaped sets.
method Introducing radial distance functionals and new topologies.
result New topologies τWr and τAWr for star-shaped sets. New algorithm radVI improves variational inference by optimizing radial profiles.
problem Gaussian approximations often fail to capture the radial profile of complex distributions.
method Optimizes over radial profiles in variational inference, providing theoretical guarantees.
result Theoretical convergence guarantees for radVI, improving over existing VI methods.
New capillary surface found without radial limits at corner.
problem Existence of capillary surfaces with no radial limits at corners.
method Investigated a convex corner domain with bounded contact angle.
result Found a capillary surface with no radial limits at (0,0) with bounded contact angle. The aim of this article is to establish a Toponogov type triangle comparison theorem for Finsler manifolds, in the manner of radial curvature geometry. We consider the situation that the radial flag curvature is bounded below by the radial curvature function of a non-compact surface of revolution, the edge opposite to …
Real projective structures on n-orbifolds are useful in understanding the space of representations of discrete groups into SL(n+1,R) or PGL(n+1,R). A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the o…
Proves conditions for radial Kaehler metrics to be Kaehler-Einstein.
problem Characterizing radial Kaehler metrics as Kaehler-Einstein.
method Analyzes conditions for radial Kaehler metrics to be Kaehler-Einstein.
result Conditions for radial Kaehler metrics to be Kaehler-Einstein are provided.
Study radial processes in sub-Riemannian Brownian motions, proving stochastic completeness and eigenvalue estimates.
problem Analyzing sub-Riemannian Brownian motions and their radial processes.
method Application of Itô's formula and sub-Laplacian comparison theorems to prove stochastic completeness and eigenvalue estimates.
result Proved Cheng's type estimates for Dirichlet eigenvalues of sub-Riemannian metric balls.
Study on gradient ρ-Einstein solitons with radially nonnegative Bach tensor.
problem Characterizing gradient ρ-Einstein solitons with specific tensor properties.
method Analyzing the properties of Bach tensor and using local warping to classify solitons.
result Gradient ρ-Einstein solitons with radially nonnegative Bach tensor are locally warped products of an interval and an Einstein manifold.
Radial graphs with constant mean curvature found in Euclidean space.
problem Existence of hypersurfaces with constant mean curvature.
method Radial graphs over domains of the unit sphere, Dirichlet problem.
result Existence of hypersurfaces with positive constant mean curvature.
Let us fix two different radial eigenfunctions of a hyperbolic Laplacian and assume that both of them have the same value at the origin. Both eigenvalues can be complex numbers. The main goal of this paper is to estimate the lower bound for the interval (0,T], where these two eigenfunctions must assume different values…
Study proves radial symmetry in convex cones using subharmonic functions.
problem Proving radial symmetry in convex cones with boundary conditions.
method Using maximum principle and integral identities for subharmonic functions.
result Proves radial symmetry and Serrin-type results for partially overdetermined problems.
We generalize the maximal diameter sphere theorem due to Toponogov by means of the radial curvature. As a corollary to our main theorem, we prove that for a complete connected Riemannian n-manifold M having radial sectional curvature at a point bounded from below by the radial curvature function of an ellipsoid of …
Classifies radial scalar flat metrics with a constant third coefficient.
problem Classifying radial scalar flat metrics with a specific coefficient.
method Analyzing the TYZ expansion and Simanca's scalar flat metric.
result Characterization of Simanca's scalar flat metric as a byproduct.
Sharp inequalities for radial functions on hyperbolic spaces without boundary conditions.
problem Establishing inequalities for radial functions on hyperbolic spaces without zero boundary conditions.
method Novel approach considering both bounded and unbounded domains, focusing on weighted Sobolev and Adams-Trudinger-Moser embeddings.
result Theorems 1.2, 1.3, and 1.4 for weighted Sobolev embedding theorems, and Theorems 1.5 and 1.6 for Adams-Trudinger-Moser type embedding theorems.
Study on manifolds with density using modified Hessians for curvature comparison.
problem Developing comparison geometry on manifolds with density.
method Modified Hessian approach based on weighted sectional curvature framework.
result Derivation of Hessian comparison and shape operator comparison theorems.
It is showed that on a plane with a radial density the Four Vertex Theorem holds for the class of all simple closed curves if and only if the density is constant. But for the class of simple closed curves that are invariant under a rotation about the origin, the Four Vertex Theorem holds for every radial density.
The paper generalizes λ-radial contraction and introduces pλ-convex sets in Riemannian manifolds.
problem Generalizing λ-radial contraction and defining pλ-convex sets in Riemannian manifolds. method Developed the concept of pλ-convex function and provided counterexamples and relations between geodesic convex sets and pλ-convex sets. result Under certain conditions, geodesic convex sets and pλ-convex sets are equivalent. We show that a smooth radially symmetric solution u to the graphic Willmore surface equation is either a constant or the defining function of a half sphere in R3. In particular, radially symmetric entire Willmore graphs in R3 must be flat. When u is a smooth radial solution over a puncture…
This paper derives radial fields on manifolds of symmetric positive definite matrices.
problem Lack of an expression for radial fields on manifolds of symmetric positive definite matrices.
method Derives an expression for radial fields on manifolds of symmetric positive definite matrices.
result Derives an expression for radial fields on manifolds of symmetric positive definite matrices.
Study examines conditions for entire bounded or large solutions to elliptic equations under radiality.
problem Conditions for existence of entire bounded or large solutions to elliptic equations.
method Analyzes Laplace operator and Laplace Beltrami operator on harmonic NA groups and Euclidean spaces, with focus on radiality of the equation. result Necessary and sufficient conditions for existence of entire bounded or large solutions are provided.
This paper introduces the dual Orlicz-Brunn-Minkowski theory for star sets. A radial Orlicz addition of two or more star sets is proposed and a corresponding dual Orlicz-Brunn-Minkowski inequality is established. Based on a radial Orlicz linear combination of two star sets, a formula for the dual Orlicz mixed volume is…
In this paper we are concerned with the problem of finding hypersurfaces of constant curvature and prescribed boundary in the Euclidean space, using the theory of fully nonlinear elliptic equations. We prove that if the given data admits a suitable radial graph as a subsolution, then we can find a radial graph with con…
We establish a classification of cubic minimal cones in case of the so-called radial eigencubics. Our principal result states that any radial eigencubic is either a member of the infinite family of eigencubics of Clifford type, or belongs to one of 18 exceptional families. We prove that at least 12 of the 18 families a…
New non-trivial Kaehler-Ricci solitons found in infinite dimensional complex space forms.
problem Non-trivial Kaehler-Ricci solitons in infinite dimensional complex space forms.
method Exhibited families of non trivial radial Kaehler-Ricci solitons in infinite dimensional complex space forms.
result Non-trivial Kaehler-Ricci solitons exist in infinite dimensional complex space forms, contradicting previous finite dimensional results.
Proves resurgence properties for Habiro elements from radial limits of theta series.
problem Proving resurgence properties for Habiro elements.
method Using strange identities and Borel transform of formal power series.
result Proves Costin and Garoufalidis conjecture for two families of torus knots.
Paper proves MS convergence for radially symmetric kernels with large bandwidths.
problem Proving convergence of mean shift algorithm with radially symmetric kernels.
method Analyzes convergence of mean shift algorithm with radially symmetric, positive definite kernels.
result Guaranteed convergence for sufficiently large bandwidth in any dimension.
Study proves radial symmetry of solutions to certain nonlinear equations in space forms.
problem Proving radial symmetry of solutions to nonlinear equations in space forms.
method Establishing Rellich-Pohožaev type identities for Hessian quotient and k-Hessian equations.
result Radial symmetry of solutions for Hessian quotient and k-Hessian equations in space forms.
The study provides criteria for solving the Dirichlet problem at infinity on Cartan-Hadamard surfaces.
problem Solvability of the Dirichlet problem at infinity on Cartan-Hadamard surfaces.
method General criterion and upper radial curvature bounds.
result Solvability of the Dirichlet problem at infinity implies a natural continuous surjection of the Martin boundary onto the geometric boundary.
New isoperimetric inequalities in the plane with radial weights identified.
problem Finding isoperimetric shapes in the plane with radial power weights.
method Analyzing the punctured plane with specific volume and perimeter densities.
result Centred balls are uniquely isoperimetric under certain conditions.
Reformulates RBF networks for graph-based data.
problem Applying RBF networks to graph data.
method Reformulate RBF networks for adjacency matrices, derive gradient updates.
result Guaranteed same responses as vector-based RBF networks.
We describe min-max formulas for the principal eigenvalue of a V-drift Laplacian defined by a vector field V on a geodesic ball of a Riemannian manifold N. Then we derive comparison results for the principal eigenvalue with the one of a spherically symmetric model space endowed with a radial vector field, under p…
In this work we study surfaces in radial conformally flat spaces. We characterize surfaces of rotation with constant Gaussian and Extrinsic curvature in these radial 3-spaces. We prove that all the spheres in the conformal 3-space have constant Gaussian curvature K=1 if, and only if, the conformal factor is special. …
We prove that if M is a complete hypersurface in Rn+1 which is graph of a real radial function, then the spectrum of the Laplace operator on M is the interval [0,∞).
Under the quadratic-decay-conditions of the radial curvatures of an end, we shall derive growth estimates of solutions to the eigenvalue equation and show the absence of eigenvalues.
The study of radial prescribing scalar curvature of X.Xu and P.C.Yang [2] in 1993 showed a nonexistence result on S2. Later in 1995, W.Chen and C.Li [2] generalized the nonexistence result to higher dimensions. G.Bianchi and E.Egnell [1] suggested that there may exist some non-negative smooth radial function which c…
Two RBF methods solve complex financial derivatives pricing problems.
problem Pricing derivatives in models with multiple stochastic factors.
method Radial Basis Function Partition of Unity and Radial Basis Function generated Finite Differences methods.
result Both methods achieve high accuracy and are efficient for solving multi-dimensional PDEs.
The paper establishes a new sphere theorem for certain types of manifolds.
problem Finding conditions under which compact manifolds are spheres.
method Developed a generalized sphere theorem for manifolds with radial Ricci curvature.
result Established conditions for compact manifolds to be topologically spheres.