Related papers: $M_{0,5}$: Towards the Chabauty-Kim method in high…

Let $X$ be a normed space of a finite dimension at least two, and $C\subsetneq X$ a closed convex set with nonempty interior. We are interested in extending Lipschitz quasiconvex functions on $C$ to quasiconvex functions on $X$. We show…

In this paper we prove the finiteness of the set of S-integral points of a punctured rational elliptic curve without complex multiplication using the Chabauty-Kim method. This extends previous results of Kim in the complex multiplication…

We consider a variational problem for submanifolds Q $\subset$ M with nonempty boundary $\partial$Q = K. We propose the definition that the boundary K of any critical point Q have constant mean curvature, which seems to be a new perspective…

Surface reconstruction from a set of scattered points, or a point cloud, has many applications ranging from computer graphics to remote sensing. We present a new method for this task that produces an implicit surface (zero-level set)…

Generalizing the well-known relations on characteristic functions on a plane to the case of a one-dimensional regular surface (curve) with compact support, we establish implicit equations for these functions. Introducing an approximation,…

We consider 3D versions of the Zernike polynomials that are commonly used in 2D in optics and lithography. We generalize the 3D Zernike polynomials to functions that vanish to a prescribed degree $\alpha\geq0$ at the rim of their supporting…

This note records two results which were inexplicably omitted from our paper on Pin structures on low dimensional manifolds, [KT]. Kirby chose not to be listed as a coauthor. A Pin^- structure on a surface F induces a quadratic enhancement…

This paper generalizes recent advances on quadratic manifold (QM) dimensionality reduction by developing kernel methods-based nonlinear-augmentation dimensionality reduction. QMs, and more generally feature map-based nonlinear corrections,…

The advent of high resolution imaging has made data on surface shape widespread. Methods for the analysis of shape based on landmarks are well established but high resolution data require a functional approach. The starting point is a…

A high-symmetry crystal surface may undergo a kinetic instability during the growth, such that its late stage evolution resembles a phase separation process. This parallel is rigorous in one dimension, if the conserved surface current is…

With the development of low order scaling methods for performing Kohn-Sham Density Functional Theory, it is now possible to perform fully quantum mechanical calculations of systems containing tens of thousands of atoms. However, with an…

This article deals with the existence of hypersurfaces minimizing general shape functionals under certain geometric constraints. We consider as admissible shapes orientable hypersurfaces satisfying a so-called reach condition, also known as…

We study the optimization of (strongly) quasar-convex functions, a class that arises naturally in many machine learning and data science applications due to its favorable properties. The fundamental properties of this class are first…

The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems:…

A class of quasi-distribution evaluation criteria based on piecewise Bezier curves is proposed to address the issue of the inability to objectively evaluate finite element models. During the optimization design of mechanical parts, finite…

A piecewise Pad\'e-Chebyshev type (PiPCT) approximation method is proposed to minimize the Gibbs phenomenon in approximating piecewise smooth functions. A theorem on $L^1$-error estimate is proved for sufficiently smooth functions using a…

For the minimal graph with strict convex level sets, we find an auxiliary function to study the Gaussian curvature of the level sets. We prove that this curvature function is a concave function with respect to the height of the minimal…

Let $G\curvearrowright M$ be an isometric action of a Lie Group on a complete orientable Riemannian manifold. We disintegrate absolutely continuous measures with respect to the volume measure of $M$ along the principal orbits of…

This article presents a new and efficient alternative to well established algorithms for molecular geometry optimization. The new approach exploits the approximate decoupling of molecular energetics in a curvilinear internal coordinate…

Consider an $s$-dimensional function being evaluated at $n$ points of a low discrepancy sequence (LDS), where the objective is to approximate the one-dimensional functions that result from integrating out $(s-1)$ variables. Here, the…