arXiv research

Study examines maximal domains of radial harmonic functions across different curvature types.

The study characterizes harmonic spaces and their radial eigen-functions and vector fields.

Sharp inequalities for radial functions on hyperbolic spaces without boundary conditions.

Study proves radial symmetry in convex cones using subharmonic functions.

New topologies for star-shaped sets without boundedness.

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 ${\mathbb R}^3$. In particular, radially symmetric entire Willmore graphs in ${\mathbb R}^3$ 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 …

This paper derives 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.

Gradient-based training and pruning for radial basis function networks in materials physics.

A core operation in reinforcement learning (RL) is finding an action that is optimal with respect to a learned value function. This operation is often challenging when the learned value function takes continuous actions as input. We introduce deep radial-basis value functions (RBVFs): value functions learned using a de…

We prove that if $M$ is a complete hypersurface in $\mathbb{R}^{n+1}$ which is graph of a real radial function, then the spectrum of the Laplace operator on M is the interval $[0,\infty)$.

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 $S^2$. 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…

New RBF networks can approximate any continuous function.

We prove the existence and uniqueness of radial graphs over a given domain of $\mathbb{S}^{n}$ having boundary on the sphere $\mathbb{S}^{n}$ 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.

The study examines exceptional sets for radial limits of superharmonic functions on curved manifolds.

OKSVM optimizes RBF kernel hyperparameter for SVMs, improving classification performance.

Local classification of surfaces and hypersurfaces with radial 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…

We determine the optimal structure of couplings for the \emph{Martingale transport problem} between radially symmetric initial and terminal laws $μ, ν$ on $\R^d$ and show the uniqueness of optimizer. Here optimality means that such solutions will minimize the functional $\E |X-Y|^p$ where $0<p \leq 1$, and the dimensio…

Method learns radial basis function distributions from samples.

Bayesian approach approximates probability functions of Gaussian mixtures.

Study differential operators on non-compact harmonic manifolds, finding conditions for radial fundamental solutions and dense heat-semigroups.

s-RBFN integrates multiple hypotheses for efficient and diverse prediction.

The paper generalizes radial curvature bounds on 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 $ω=\frac{i}{2}\partial\bar\partialΦ$. We prove that if $g$ is $g_{eucl}$-balanced of height 3 (where $g_{eucl}$ is the standard Euclidean metric on ${\complex}=…

While Bayesian neural networks have many appealing characteristics, current priors do not easily allow users to specify basic properties such as expected lengthscale or amplitude variance. In this work, we introduce Poisson Process Radial Basis Function Networks, a novel prior that is able to encode amplitude stationar…

Study dynamics of $L^p$-multipliers on harmonic manifolds with exponential volume growth.

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…

It is well known that the Euclidean Sobolev inequality holds on any Cartan-Hadamard manifold of dimension $n\ge 3$, i.e. any complete, simply connected Riemannian manifold with nonpositive sectional curvature. As a byproduct of the Cartan-Hadamard conjecture, a longstanding problem in the mathematical literature sett…

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…

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.

No radial balanced metrics found on Kepler manifold unit ball with mild boundary conditions.

This paper develops efficient surrogate models for optimization of complex dynamical systems.

In this paper we have generalized the notion of $λ$-radial contraction in complete Riemannian manifold and developed the concept of $p^λ$-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…

Study complete gradient Ricci solitons with zero radial Weyl curvature.

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…

The paper explores properties of functions on Teichmüller space, proving theorems about limits and non-ergodicity.

On a flat plane, convexity of a set is preserved by both radial expansion and contraction of the set about any point inside it. Using the Poincaré disk model of hyperbolic geometry, we prove that radial expansion of a hyperbolic convex set about a point inside it always preserves hyperbolic convexity. Using stereograph…

The paper studies biharmonic functions and bi-eigenfunctions on spheres and model spaces.