Related papers: On Euclidean Steiner $(1+\epsilon)$-Spanners
We consider the $(1+\epsilon)$-approximate nearest neighbor search problem: given a set $X$ of $n$ points in a $d$-dimensional space, build a data structure that, given any query point $y$, finds a point $x \in X$ whose distance to $y$ is…
In this paper, by invoking the appropriate decomposition of pressure to exploit the energy hidden in pressure, we present some new $\varepsilon$-regularity criteria for suitable weak solutions of the 3D Navier-Stokes equations at one scale:…
Let ${\cal P}$ be a set of $n$ points embedded in the plane, and let ${\cal C}$ be the complete Euclidean graph whose point-set is ${\cal P}$. Each edge in ${\cal C}$ between two points $p, q$ is realized as the line segment $[pq]$, and is…
We consider a 2D smectics model \begin{equation*} E_{\epsilon }\left( u\right) =\frac{1}{2}\int_\Omega \frac{1}{\varepsilon }\left( u_{z}-\frac{1% }{2}u_{x}^{2}\right) ^{2}+\varepsilon \left( u_{xx}\right) ^{2}dx\,dz. \end{equation*} For…
We construct new stationary weak solutions of the 3D Euler equation with compact support. The solutions, which are piecewise smooth and discontinuous across a surface, are axisymmetric with swirl. The range of solutions we find is different…
We consider the problem of computing an approximate weighted shortest path in a weighted subdivision, with weights assigned from the set $\{0, 1, \infty\}$. We present a data structure $B$, which stores a set of convex, non-overlapping…
Erd\H{o}s' unit distance problem and Erd\H{o}s' distinct distances problem are among the most classical and well-known open problems in discrete mathematics. They ask for the maximum number of unit distances, or the minimum number of…
We obtain sharp upper bounds for the first two nonzero Steklov eigenvalues among bounded domains in Euclidean spaces of dimension $d \geq 7$ under a natural normalization involving volume and boundary measure. These bounds are derived from…
We reinvestigate the stability properties of ultracompact spinning boson stars with a stable light ring using fully nonlinear 3+1 and 2+1 numerical relativity simulations and two different formulations of the Einstein equations. We find no…
We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the…
In this paper we devise an optimal construction of fault-tolerant spanners for doubling metrics. Specifically, for any $n$-point doubling metric, any $\eps > 0$, and any integer $0 \le k \le n-2$, our construction provides a…
The following result was proved by Barany in 1982: For every d >= 1 there exists c_d > 0 such that for every n-point set S in R^d there is a point p in R^d contained in at least c_d n^{d+1} - O(n^d) of the simplices spanned by S. We…
$\newcommand{\Re}{\mathbb{R}}$We study the minWSPD problem of computing the minimum-size well-separated pairs decomposition of a set of points, and show constant approximation algorithms in low-dimensional Euclidean space and doubling…
In this article, we construct a Steiner system with the parameters $S(3,6,42)$, settling one of the smallest open parameter sets of Steiner $3$-designs. Furthermore, we establish the existence of rotational Steiner quadruple systems on $46$…
$\newcommand{\eps}{\varepsilon}\newcommand{\tldO}{\widetilde{O}}$Consider the problem of constructing weak $\eps$-nets where the stabbing elements are lines or $k$-flats instead of points. We study this problem in the simplest setting where…
In this paper, we study the stability and minimizing properties of higher codimensional surfaces in Euclidean space associated with the $f$-weighted area-functional $$\mathcal{E}_f(M)=\int_M f(x)\; d \mathcal{H}_k$$ with the density…
According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an $\eps$-net of size $O\left(\frac{1}{\eps}\log\frac1{\eps}\right)$. Using probabilistic techniques, Pach and Woeginger (1990)…
In this paper we discuss the Steiner property for minimal clusters in the plane with an anisotropic double density. This means that we consider the classical isoperimetric problem for clusters, but volume and perimeter are defined by using…
In this article, we study the 2D incompressible steady Navier-Stokes equation in a channel $(-L,0)\times(-1,1)$ with the no-slip boundary condition on $\{Y = \pm 1\}$, and consider the inviscid limit $\varepsilon \to 0$. In the special case…
We study spherically symmetric solutions to the Einstein-Euler equations which model an idealized relativistic neutron star surrounded by vacuum. These are barotropic fluids with a free boundary, governed by an equation of state which sets…