Related papers: On uniqueness in Steiner problem
We investigate the existence and uniqueness of solutions for second-order semi-linear partial differential equations defined on a Riemannian manifold $M$. By combining differential geometry and analysis techniques, we establish the…
Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in $\mathbb{R}^d$. In a recent breakthrough, Le and Solomon (2019)…
In a recent paper, Chan, \L aba, and Pramanik investigated geometric configurations inside thin subsets of the Euclidean set possessing measures with Fourier decay properties. In this paper we ask which configurations can be found inside…
We give approximation schemes for Subset TSP and Steiner Tree on unit disk graphs, and more generally, on intersection graphs of similarly sized connected fat (not necessarily convex) polygons in the plane. As a first step towards this…
This paper addresses the question: how should N n-dimensional subspaces of m-dimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of N, n, m are…
We show that there is generically non-uniqueness for the anisotropic Calder\'on problem at fixed frequency when the Dirichlet and Neumann data are measured on disjoint sets of the boundary of a given domain. More precisely, we first show…
We introduce and study a novel problem of computing a shortest path tree with a minimum number of non-terminals. It can be viewed as an (unweighted) Steiner Shortest Path Tree (SSPT) that spans a given set of terminal vertices by shortest…
The present paper opens a new branch in the theory of variational problems with branching extremals, the investigation of one-dimensional minimal fillings of finite pseudo-metric spaces. On the one hand, this problem is a one-dimensional…
We give a short proof of the fact that there are no measurable subsets of Euclidean space (in dimension d > 2), which, no matter how translated and rotated, always contain exactly one integer lattice point. In dimension d=2 (the original…
We establish a lower bound for the number of sign changing solutions with precisely two nodal domains to the singularly perturbed nonlinear elliptic equation -{\epsilon}^{2}{\Delta}_{g}u+u=|u|^{p-2}u on an n-dimensional Riemannian manifold…
In 1953, Blumenthal showed that every semi-metric space that is isometrically embeddable in a Hilbert space has the n-point property; we have previously called such spaces supermetric spaces. Although this is a strictly stronger property…
We extend Yosida's 1941 version of Stone-Gelfand duality to metrically complete unital lattice-ordered groups that are no longer required to be real vector spaces. This calls for a generalised notion of compact Hausdorff space whose points…
We prove that the space of persistence diagrams on $n$ points (with the bottleneck or a Wasserstein distance) coarsely embeds into Hilbert space by showing it is of asymptotic dimension $2n$. Such an embedding enables utilisation of Hilbert…
A rectilinear Steiner tree for a set $P$ of points in $\mathbb{R}^2$ is a tree that connects the points in $P$ using horizontal and vertical line segments. The goal of Minimal Rectilinear Steiner Tree is to find a rectilinear Steiner tree…
This paper is concerned with the Hamiltonian elliptic system in dimension two\begin{equation*}\aligned \left\{ \begin{array}{lll} -\epsilon^2\Delta u+V(x)u=g(v)\ & \text{in}\quad \mathbb{R}^2,\\ -\epsilon^2\Delta v+V(x)v=f(u)\ &…
In this paper we show that on a complete Riemannian manifold of negative curvature and dimension $n>1$ every two points which realize a local maximum for the distance function are connected by at least $2n+1$ geometrically distinct geodesic…
A homemorphism between domains in $\mathbb R^n$, $n\ge 2$ is quasiconformal, with its intricate analytic and geometric consequences, if the (pointwise) linear dilatation -- a purely metric quantity -- is uniformly bounded. Gehring proved…
In this paper we study the Dirichlet problem for fully nonlinear second-order equations on a riemannian manifold. As in a previous paper we define equations via closed subsets of the 2-jet bundle. Basic existence and uniqueness theorems are…
We consider low connectivity variants of the Survivable Network with Minimum Number of Steiner Points (SN-MSP) problem: given a finite set $R$ of terminals in a metric space (M,d), a subset $B \subseteq R$ of "unstable" terminals, and…
If $H$ is a Hilbert space, the non-compact Stiefel manifold $St(n,H)$ consists of independent $n$-tuples in $H$. In this article, we contribute to the topological study of non-compact Stiefel manifolds, mainly by proving two results on the…