arXiv research

New Harnack inequality for curve shortening flow without convexity.

Distribution grid is the medium and low voltage part of a large power system. Structurally, the majority of distribution networks operate radially, such that energized lines form a collection of trees, i.e. forest, with a substation being at the root of any tree. The operational topology/forest may change from time to …

New Fourier features improve high-precision approximation in large-scale problems.

$σ_k$-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In an earlier work YanYan Li proved that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $σ_k$-Yamabe equation is asymptotically radially symmetric. In this work we prove that an admis…

The Blum medial axis rigidity is studied in terms of cross ratios and differential geometry.

Given a complete isometric immersion $φ: P^m \longrightarrow N^n$ in an ambient Riemannian manifold $N^n$ with a pole and with radial sectional curvatures bounded from above by the corresponding radial sectional curvatures of a radially symmetric space $M^n_w$, we determine a set of conditions on the extrinsic curvatur…

The paper proves geometric and spectral alignment for deep neural networks.

Stable capillary surfaces in weighted balls are disks.

Study minimal graphs on non-negative Ricci curvature manifolds.

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.

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.

Study complete gradient Ricci solitons with zero radial Weyl curvature.

The increasing penetration of distributed energy resources poses numerous reliability issues to the urban distribution grid. The topology estimation is a critical step to ensure the robustness of distribution grid operation. However, the bus connectivity and grid topology estimation are usually hard in distribution gri…

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 …

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…

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 …

The topology of a power grid affects its dynamic operation and settlement in the electricity market. Real-time topology identification can enable faster control action following an emergency scenario like failure of a line. This article discusses a graphical model framework for topology estimation in bulk power grids (…

New multi-spiral approach improves packaging of thick membranes.

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

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.

Let $(M, g)$ be a Kaehler manifold whose associated Kaehler form $ω$ is integral and let $(L, h)\rightarrow (M, ω)$ be a quantization hermitian line bundle. In this paper we study those Kaehler manifolds $(M, g)$ admitting a finite TYCZ expansion. We show that if the TYCZ expansion is finite then $T_{mg}$ is indeed a p…

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…

In this paper, we systematically investigate the geometry and topology of manifolds with integral radial curvature bounds, and obtain many interesting and important conclusions.

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…

New non-trivial Kaehler-Ricci solitons found in infinite dimensional complex space forms.

Proves resurgence properties for Habiro elements from radial limits of theta series.

Paper proves MS convergence for radially symmetric kernels with large bandwidths.

Study proves radial symmetry of solutions to certain nonlinear equations in space forms.