Related papers: Minimum weight disk triangulations and fillings

Let $\|n\|$ stand for the integer complexity of the number $n$, i.e. for the least number of $1$'s needed to write $n$ using arbitrary many additions, multiplications, and parentheses. The two-sided inequality $3\log_3 n\leq\|n\|\leq…

We prove the following proposition. Under linear interpolation on a Euclidean $n$-dimensional ball $B$, an interpolation projector whose nodes coincide with the vertices of a regular simplex inscribed into the boundary sphere has the…

We introduce and investigate numerically a minimal class of dynamical triangulations of two-dimensional gravity on the sphere in which only vertices of order five, six or seven are permitted. We show firstly that this restriction of the…

We analyze the touring regions problem: find a ($1+\epsilon$)-approximate Euclidean shortest path in $d$-dimensional space that starts at a given starting point, ends at a given ending point, and visits given regions $R_1, R_2, R_3, \dots,…

Let P be a closed triangulated manifold, dim P=n. We consider the group of simplicial 1-chains C_1(P) and the homology group H_1(P). We also use some nonnegative weighting function L on C_1(P). For any homological class from H_1(P) method…

We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $3$-manifold $\mathcal{N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $2$-sided surface. We prove a…

Let $\Poly$ be a simple polygon with $n$ vertices. The \emph{dual graph} $\triang^*$ of a triangulation~$\triang$ of~$\Poly$ is the graph whose vertices correspond to the bounded faces of $\triang$ and whose edges connect those faces…

Let $K$ be a closed polygonal curve in $\RR^3$ consisting of $n$ line segments. Assume that $K$ is unknotted, so that it is the boundary of an embedded disk in $\RR^3$. This paper considers the question: How many triangles are needed to…

The notion of a layered triangulation of a lens space was defined by Jaco and Rubinstein in earlier work, and, unless the lens space is L(3,1), a layered triangulation with the minimal number of tetrahedra was shown to be unique and termed…

In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a…

We show that the minimum number of orientations of the edges of the n-vertex complete graph having the property that every triangle is made cyclic in at least one of them is $\lceil\log_2(n-1)\rceil$. More generally, we also determine the…

Thurston's circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin-Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a…

In a recent paper A. Fraser and R. Schoen have proved the existence of free boundary minimal surfaces $\Sigma\_n$ in $B^3$ which have genus $0$ and $n$ boundary components, for all $ n \geq 3$. For large $n$, we give an independent…

Linear complementary dual codes (or codes with complementary duals) are codes whose intersections with their dual codes are trivial. We study the largest minimum weight $d(n,k)$ among all binary linear complementary dual $[n,k]$ codes. We…

We show that if a disc triangulation has all internal vertex degrees at least 6, then the full triangulation may be determined from the pairwise graph distance between boundary vertices. A similar result holds for quadrangulations with all…

In this paper, we consider a facility location problem to find a minimum-sum coverage of n points by disks centered at a fixed line. The cost of a disk with radius r has a form of a non-decreasing function f(r) = r^a for any a >= 1. The…

We obtain bounds for the minimum and maximum mass/radius ratio of a stable, charged, spherically symmetric compact object in a $D$-dimensional space-time in the framework of general relativity, and in the presence of dark energy. The total…

We show that every minimum area isosceles triangle containing a given triangle $T$ shares a side and an angle with $T$. This proves a conjecture of Nandakumar motivated by a computational problem. We use our result to deduce that for every…

Consider the problem of fnding the smallest area convex $k$-gon containing $n\in\mathbb{N}$ congruent disks without an overlap. By using Wegner inequality in sphere packing theory we give a lower bound for the area of such polygons. For…

We prove the following results on flag triangulations of 2- and 3-manifolds. In dimension 2, we prove that the vertex-minimal flag triangulations of $\mathbb{R} P^2$ and $\mathbb{S}^1\times \mathbb{S}^1$ have 11 and 12 vertices,…