Related papers: Nonlocal Ordered Mean Curvature with Integrable Ke…
We propose a notion of integral Menger curvature for compact, $m$-dimensional sets in $n$-dimensional Euclidean space and prove that finiteness of this quantity implies that the set is $C^{1,\alpha}$ embedded manifold with the H{\"o}lder…
We initiate the study of nonsmooth optimization problems under bounded local subgradient variation, which postulates bounded difference between (sub)gradients in small local regions around points, in either average or maximum sense. The…
Machine learning models can represent climate processes that are nonlocal in horizontal space, height, and time, often by combining information across these dimensions in highly nonlinear ways. While this can improve predictive skill, it…
In this paper we construct the action describing dynamics of the particle moving in curved spacetime, with a non-trivial momentum space geometry. Curved momentum space is the core feature of theories where relative locality effects are…
This paper studies random lozenge tilings of general non-convex polygonal regions. We show that the pairwise interaction of the non-convexities leads asymptotically to new kernels and thus to new statistics for the tiling fluctuations. The…
We consider the class of measurable functions defined in all of $\mathbb{R}^n$ that give rise to a nonlocal minimal graph over a ball of $\mathbb{R}^n$. We establish that the gradient of any such function is bounded in the interior of the…
On a compact Riemannian manifold with boundary, we study the set of conformal metrics of negative constant scalar curvature in the interior and positive constant mean curvature on the boundary. Working in the case of positive Yamabe…
We address the study of quantum metrology enhanced by indefinite causal order, demonstrating a quadratic advantage in the estimation of the product of two average displacements in a continuous variable system. We prove that no setup where…
A sequential quadratic programming (SQP) algorithm is designed for nonsmooth optimization problems with upper-C^2 objective functions. Upper-C^2 functions are locally equivalent to difference-of-convex (DC) functions with smooth convex…
The invertibility of integral linear operators is a major problem of both theoretical and practical importance. In this paper we investigate the relation between an operator invertibility and the rank of its integral kernel to develop a…
We study the problem of moving a vertex in an unstructured mesh of triangular, quadrilateral, or tetrahedral elements to optimize the shapes of adjacent elements. We show that many such problems can be solved in linear time using…
When thermal energies are weak, two dimensional lamellar structures confined on a curved substrate display complex patterns arising from the competition between layer bending and compression in the presence of geometric constraints. We…
Families of conformal field theories are naturally endowed with a Riemannian geometry which is locally encoded by correlation functions of exactly marginal operators. We show that the curvature of such conformal manifolds can be computed…
The purpose of this article is to determine explicitly the complete surfaces with parallel mean curvature vector, both in the complex projective plane and the complex hyperbolic plane. The main results are as follows: When the curvature of…
We give a relationship that yields an effective geometric way of evaluating mean curvature of surfaces. The approach is reminiscent of the Gauss's contour based evaluation of intrinsic curvature. The presented formula may have a number of…
Computational analysis with the finite element method requires geometrically accurate meshes. It is well known that high-order meshes can accurately capture curved surfaces with fewer degrees of freedom in comparison to low-order meshes.…
We derive intrinsic curvature and radius estimates for compact disks embedded in $\mathbb{R}^3$ with nonzero constant mean curvature and apply these estimates to study the global geometry of complete surfaces embedded in $\mathbb{R}^3$ with…
In 2014, Gromov vaguely conjectured that a sequence of manifolds with nonnegative scalar curvature should have a subsequence which converges in some weak sense to a limit space with some generalized notion of nonnegative scalar curvature.…
We study the geometry of non-minimal surfaces of supercritical constant mean curvature invariant under screw motions in the homogeneous 3-manifolds $\mathbb{E}(\kappa,\tau)$ including the space-forms of non-negative curvature. We give a…
The medial axis $M_X$ of a closed set $X\subset \mathbb{R}^n$ is the set of points from the ambient space that admit more than one closest point in $X$. We study the problem of reaching the singularities, i.e. of characterising the points…