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.
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.
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 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.
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. Radial-basis-function networks are traditionally defined for sets of vector-based observations. In this short paper, we reformulate such networks so that they can be applied to adjacency-matrix representations of weighted, directed graphs that represent the relationships between object pairs. We re-state the sum-of-squ…
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…
It is well known that quasi-isometric embeddings of Gromov hyperbolic spaces induce topological embeddings of their Gromov boundaries. A more general question is to detect classes of functions between Gromov hyperbolic spaces that induce continuous maps between their Gromov boundaries. In this paper we introduce the cl…
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 …
Nonexistence of radial optimal functions on certain Cartan-Hadamard manifolds.
problem Proving nonexistence of radial optimal functions for the Sobolev inequality on Cartan-Hadamard manifolds.
method Ad hoc arguments not relying on the Cartan-Hadamard conjecture.
result If the optimal constant in the Sobolev inequality is achieved by a radial function, then the manifold must be isometric to Euclidean space.
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.
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 …
Study biharmonic functions on vector bundles with spherical symmetry.
problem Investigate biharmonic functions on vector bundles with spherically symmetric metrics.
method Analyze vertical lifts and radial functions of functions on vector bundle manifolds.
result Construct an infinite two-parameter family of proper biharmonic functions.
Gradient-based training and pruning for radial basis function networks in materials physics.
problem Interpretable and robust machine learning for materials physics problems.
method Gradient-based training and pruning of radial basis function networks with closed-form optimization criteria.
result Pruned models provide compact and interpretable versions of larger models, offering insights into atom-level migration processes.
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,∞).
Novel Bayesian prior for neural networks encodes amplitude and lengthscale.
problem Lack of user-friendly priors for specifying basic properties in Bayesian neural networks.
method Introduced Poisson Process Radial Basis Function Networks (PP-RBFN) as a novel prior.
result PP-RBFN allows decoupled specification of amplitude and lengthscale, and estimated function is consistent.
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…
We propose two localized Radial Basis Function (RBF) methods, the Radial Basis Function Partition of Unity method (RBF-PUM) and the Radial Basis Function generated Finite Differences method (RBF-FD), for solving financial derivative pricing problems arising from market models with multiple stochastic factors. We demons…
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…
Deep RBVFs improve continuous control in RL.
problem Challenges in finding optimal actions for continuous actions in RL.
method Introduced deep radial-basis value functions (RBVFs) for continuous control.
result RBF-DQN significantly outperforms value-function-only baselines and is competitive with actor-critic algorithms.
New RBF networks can approximate any continuous function.
problem Approximating any continuous function on a compact subset.
method Replacing smoothing factors with shifts in RBF networks and proving approximation under certain conditions.
result RBF networks can approximate any continuous function on any compact subset.
We prove the existence and uniqueness of radial graphs over a given domain of Sn having boundary on the sphere Sn and whose mean curvature at every point equals a prescribed positive function satisfying suitable barrier-type and monotonicity conditions.
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.
The study examines exceptional sets for radial limits of superharmonic functions on curved manifolds.
problem Understanding exceptional sets for radial limits of superharmonic functions on curved manifolds.
method Analysis of radial geodesic rays, Poisson integrals, Green potentials, and Riesz decomposition.
result Sharp bounds on Hausdorff dimensions of exceptional sets for superharmonic functions.
OKSVM optimizes RBF kernel hyperparameter for SVMs, improving classification performance.
problem Intrinsic dependence of RBF kernel hyperparameter on SVM performance.
method Gradient descent method for automatic hyperparameter learning and SVM weights adjustment.
result OKSVM outperforms classical SVM regardless of initial RBF hyperparameter values.
Local classification of surfaces and hypersurfaces with radial mean curvature.
problem Classifying surfaces and hypersurfaces with specific curvature properties.
method Local classification and recursive construction method.
result Local classification of hypersurfaces with vanishing second mean curvature.
Global optimization problems whose objective function is expensive to evaluate can be solved effectively by recursively fitting a surrogate function to function samples and minimizing an acquisition function to generate new samples. The acquisition step trades off between seeking for a new optimization vector where the…
Method learns radial basis function distributions from samples.
problem Learning radial basis function distributions from training samples.
method Projected particle Langevin optimization method with distributionally robust optimization.
result Empirical measure of Langevin particles converges to a reflected Itô diffusion-drift process.
We determine the optimal structure of couplings for the \emph{Martingale transport problem} between radially symmetric initial and terminal laws μ,ν on Rd and show the uniqueness of optimizer. Here optimality means that such solutions will minimize the functional $\E |X-Y|^p$ where 0<p≤1, and the dimensio…
Bayesian approach approximates probability functions of Gaussian mixtures.
problem Approximating probability functions of non-spherical Gaussian mixtures.
method Bayesian decomposition, spherical radial decomposition, random sampling.
result Established differentiability and integral representation of gradient for probability functions.
Study differential operators on non-compact harmonic manifolds, finding conditions for radial fundamental solutions and dense heat-semigroups.
problem Conditions for differential operators on non-compact harmonic manifolds to have specific properties.
method Analyzing the algebra of differential operators, their commutation properties, and using geometric averages.
result Algebra of differential operators on non-compact harmonic manifolds has specific properties related to radial fundamental solutions and dense heat-semigroups.
s-RBFN integrates multiple hypotheses for efficient and diverse prediction.
problem Integrating multiple hypotheses into learning models for regression.
method Structured Radial Basis Function Network (s-RBFN) using Voronoi tessellations and least-squares training.
result s-RBFN achieves superior generalization and efficiency compared to other models.
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.
Let Φ be a strictly plurisubharmonic and radial function on the unit disk ${\cal D}\subset {\complex}$ and let g be the \K metric associated to the \K form ω=2i∂∂ˉΦ. We prove that if g is geucl-balanced of height 3 (where geucl is the standard Euclidean metric on ${\complex}=…
Study dynamics of Lp-multipliers on harmonic manifolds with exponential volume growth.
problem Characterize the behavior of Lp-multipliers on harmonic manifolds of purely exponential volume growth. method Analyzing the dynamics of Lp-multipliers on non-compact harmonic manifolds, using Fourier transformation and properties of radial functions. result Show that Lp-multipliers acting nicely on smooth functions with compact support for p≤2 cannot be chaotic. 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. …
Radial Basis Functions Neural Networks (RBFNNs) are tools widely used in regression problems. One of their principal drawbacks is that the formulation corresponding to the training with the supervision of both the centers and the weights is a highly non-convex optimization problem, which leads to some fundamentally dif…
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.
Based on the tree architecture, the objective of this paper is to design deep neural networks with two or more hidden layers (called deep nets) for realization of radial functions so as to enable rotational invariance for near-optimal function approximation in an arbitrarily high dimensional Euclidian space. It is show…
The paper simplifies electricity market curves with less parameters.
problem Modeling electricity prices and demands efficiently.
method Mesh-free interpolation using radial basis functions.
result The method reduces parameters needed to represent curves.
We study conformally flat surfaces with prescribed Gaussian curvature, described by solutions u of the PDE: Δu(x)+K(x)exp(2u(x))=0, with K(x) the Gauss curvature function at $x\in\RR^2$. We assume that the integral curvature is finite. For radially symmetric K we introduce the notion of a least integrally curv…
Study on optimal ReLU networks with weight decay for interpolation.
problem Interpolating data with radially symmetric distributions using shallow ReLU networks.
method Weight decay regularization in infinite neuron, infinite data limit; analysis of growth rates.
result Existence and growth rates of unique radially symmetric minimizers with weight decay.
This paper develops efficient surrogate models for optimization of complex dynamical systems.
problem Computational expense in solving complex dynamical systems through numerical simulation.
method Combination of proper orthogonal decomposition and radial basis functions for constructing low-dimensional surrogate models.
result Surrogate models reduce computational time for optimization problems while maintaining accuracy.
In this paper we have generalized the notion of λ-radial contraction in complete Riemannian manifold and developed the concept of pλ-convex function. We have also given a counter example proving the fact that in general λ-radial contraction of a geodesic is not necessarily a geodesic. We have also deduced some r…
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.
The aim of this chapter is to show how option prices in jump-diffusion models can be computed using meshless methods based on Radial Basis Function (RBF) interpolation. The RBF technique is demonstrated by solving the partial integro-differential equation (PIDE) in one-dimension for the American put and the European va…
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.
The paper explores properties of functions on Teichmüller space, proving theorems about limits and non-ergodicity.
problem Properties of bounded pluriharmonic and holomorphic functions on Teichmüller space.
method Analyzes the boundary behavior of functions and proves theorems about limits and non-ergodicity.
result Proves the existence of radial limits for bounded pluriharmonic functions and non-constant bounded holomorphic functions.