arXiv research

Study on flat singularities of area-minimizing currents in codimension one.

Extends potential theory to Carnot groups, estimating Hausdorff dimension.

We consider the two body problem with central interaction on two point homogeneous spaces from point of view of the invariant differential operators theory. The representation of the two particle Hamiltonian in terms of the radial differential operator and invariant operators on the symmetry group is found. The connect…

We show that any homogeneous polynomial solution of |\nabla F(x)|^2=m^2|x|^(2m-2), m>1, is either a radially symmetric polynomial F(x)=\pm |x|^m (for even m's) or it is a composition of a Chebychev polynomial and a Cartan-Münzner polynomial.

A Riemannian manifold is called harmonic if its volume density function expressed in polar coordinates centered at any point is radial. Flat and rank-one symmetric spaces are harmonic. The converse (the Lichnerowicz Conjecture) is true for manifolds of nonnegative scalar curvature and for some other classes of manifold…

We prove that the quasi-homogenous symbols on the projective space $\mathbb{P}^n(\mathbb{C})$ yield commutative algebras of Toeplitz operators on all weighted Bergman spaces, thus extending to this compact case known results for the unit ball $\mathbb{B}^n$. These algebras are Banach but not $C^*$. We prove the existen…

We prove the existence of commutative $C^*$-algebras of Toeplitz operators on every weighted Bergman space over the complex projective space $\mathbb{P}^n(\mathbb{C})$. The symbols that define our algebras are those that depend only on the radial part of the homogeneous coordinates. The algebras presented have an assoc…

Study bi-Lipschitz equivalence of mixed polynomials under specific conditions.

Proposes new deviation measures using Minkowski gauges.

Following the spirit of a previous work of ours, we investigate the group of those General Coordinate Transformations (GCTs) which preserve manifest spatial homogeneity. In contrast to the case of Bianchi Type Models we, here, permit an isometry group of motions $G_{4}=SO(3)\otimes T_{r}$, where $T_{r}$ is the translat…

New research shows logistic regression can achieve optimal error rate for agnostic learning of halfspaces.

We show that up to automorphisms of $\mathbb P^2_{\mathbb C}$ there are $14$ homogeneous convex foliations of degree $5$ on $\mathbb P^2_{\mathbb C}.$ We establish some properties of the Fermat foliation $\mathcal F_{0}^{d}$ of degree $d\geq2$ and of the Hilbert modular foliation $\mathcal{F}_H^{5}$ of degree $5.$ As a…

The paper generalizes radial curvature bounds on manifolds.

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

Convexity properties are preserved under radial transformations in hyperbolic and spherical geometries.

Study geometry and topology of manifolds with radial curvature bounds.

We consider a mean curvature flow in a cone, that is, a hypersurface in a cone which moves toward the opening with normal velocity equaling to the mean curvature, and the contact angle between the hypersurface and the cone boundary being $\varepsilon$-periodic in its position. First, by constructing a family of self-si…

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

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

The two-body problem with a central interaction on simply connected constant curvature spaces of an arbitrary dimension is considered. The explicit expression for the quantum two-body Hamiltonian via a radial differential operator and generators of the isometry group is found. We construct a self-adjoint extension of t…

Study complete gradient Ricci solitons with zero radial Weyl curvature.

We consider smooth radial solutions to the Hamiltonian stationary equation which are defined away from the origin. We show that in dimension two all radial solutions on unbounded domains must be special Lagrangian. In contrast, for all higher dimensions there exist non-special Lagrangian radial solutions over unbounded…

New topologies for star-shaped sets without boundedness.

New algorithm radVI improves variational inference by optimizing radial profiles.

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 …

Study shows nonextendibility of warped spacelike singularities in specific spacetimes.

Proves conditions for radial Kaehler metrics to be Kaehler-Einstein.

Real projective structures on $n$-orbifolds are useful in understanding the space of representations of discrete groups into $\mathrm{SL}(n+1, \mathbb{R})$ or $\mathrm{PGL}(n+1, \mathbb{R})$. A recent work shows that many hyperbolic manifolds deform to manifolds with such structures not projectively equivalent to the o…

Study radial processes in sub-Riemannian Brownian motions, proving stochastic completeness and eigenvalue estimates.

We propose Radial Bayesian Neural Networks (BNNs): a variational approximate posterior for BNNs which scales well to large models while maintaining a distribution over weight-space with full support. Other scalable Bayesian deep learning methods, like MC dropout or deep ensembles, have discrete support-they assign zero…

The study computes indicial roots and metric convergence orders for Ricci-flat conifolds.

Study on gradient ρ-Einstein solitons with radially nonnegative Bach tensor.

Radial graphs with constant mean curvature found in Euclidean space.

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.

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 …

We classify the radially symmetric connections in vector bundles over round spheres by proving that they are all parallel.

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C^{1+ε}, showing the opt…

We give a proof of the existence of radial (smooth) parallel sections of vector bundles endowed with a linear connection.

Study on manifolds with density using modified Hessians for curvature comparison.

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

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.

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…

This paper derives radial fields on manifolds of symmetric positive definite matrices.

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…

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…