Related papers: Bounds of sub-triangles

We give a complete investigation of Morley's trisector theorem. If the intersections of the half lines starting from the adjacent vertices of a triangle form an equilateral triangle for an arbitrary triangle, then the half lines are the…

We present the error analysis of Lagrange interpolation on triangles. A new \textit{a priori} error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on…

A triangulation of a surface is \emph{irreducible} if there is no edge whose contraction produces another triangulation of the surface. We prove that every irreducible triangulation of a surface with Euler genus $g\geq1$ has at most $13g-4$…

A tripartite-circle drawing of a tripartite graph is a drawing in the plane, where each part of a vertex partition is placed on one of three disjoint circles, and the edges do not cross the circles. The tripartite-circle crossing number of…

Given a handle decomposition of a 4-manifold with boundary, and an open book decomposition of the boundary, we show how to produce a trisection diagram of a trisection of the 4-manifold inducing the given open book. We do this by making the…

We give upper and lower bounds on the spectral radius of a graph in terms of the number of walks. We generalize a number of known results.

We prove matching upper and lower bounds for the average of the 6-torsion of class groups of quadratic fields. Furthermore, we count the number of integer solutions on an affine quartic threefold.

We survey some results on toric topology.

We apply polynomial techniques (linear programming) to obtain lower and upper bounds on the covering radius of spherical designs as function of their dimension, strength, and cardinality. In terms of inner products we improve the lower…

A pseudo-triangle is a simple polygon with three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as data structures in computational geometry, as…

We study integrals over the triangle with vertices (1,0), (0,1), (1,1) that give linear combinations of multiple zeta values.

We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of…

Using ideas from the geometry of compression, we improve on the current upper and lower bounds of the Heilbronn triangle problem. In particular, let $\Delta(s)$ denote the minimal area of the triangle induced by $s$ points on a unit disk.…

We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration…

We show there is an upper bound on the diameter of a closed, hyperbolic 3-manifold in terms of the length of any presentation of its fundamental group.

We establish the upper bound in the multiplicity conjecture of Herzog, Huneke and Srinivasan for the codimension three almost complete intersections. We also give some partial results in the case where I is the aci linked to a complete…

We address the problem of topology of quadrature domains, namely we give upper bounds on the connectivity of the domain in terms of the number of nodes and their multiplicities in the quadrature identity.

A common representation of a three dimensional object in computer applications, such as graphics and design, is in the form of a triangular mesh. In many instances, individual or groups of triangles in such representation need to satisfy…

We first introduce a configuration of arbitrary isogonal conjugates related to a known property concerning the spiral center of two pairs of isogonal conjugates. We then consider a special case where two conics are tangent at exactly two…

The complete sets of irreducible triangulations are known for the orientable surfaces with genus of 0, 1, or 2 and for the nonorientable surfaces with genus of 1, 2, 3, or 4. By examining these sets we determine some of the properties of…