Related papers: Bounds of sub-triangles

In this article we prove some theorems related to the triplets of triangles, homological two by two. These theorems will be used later to build triplets of triangles two by two tri-homological.

We establish a bound for the length of vectors involved in a unimodular triangulation of simplicial cones. The bound is exponential in the square of the logarithm of the multiplicity, and improves previous bounds significantly. The proof is…

In this paper we study underlying graphs corresponding to a set of halving lines. We establish many properties of such graphs. In addition, we tighten the upper bound for the number of halving lines.

A lower bound is obtained for the greatest possible number of colors in an interval colourings of some regular graphs.

In this note, we give answers to three questions from the paper [A. Das, Triameter of graphs, Discuss. Math. Graph Theory, 41 (2021), 601--616]. Namely, we obtain a tight lower bound for the triameter of trees in terms of order and number…

We provide analytical lower and upper bounds for entanglement of formation for bipartite systems, which give a direct relation between the bounds of entanglement of formation and concurrence, and improve the previous results. Detailed…

We provide a new formulation and proof of the triangle altitudes theorem in hyperbolic plane geometry, together with an easily computed discriminant to distinguish between different basic configurations of the altitudes of such a triangle.

We give three new proofs of the triangle inequality in Euclidean Geometry. There seems to be only one known proof at the moment. It is due to properties of triangles, but our proofs are due to circles or ellipses. We aim to prove the…

We introduce series-triangular graph embeddings and show how to partition point sets with them. This result is then used to improve the upper bound on the number of Steiner points needed to obtain compatible triangulations of point sets.…

We discuss some basic properties of the graded center of a triangulated category and compute examples arising in representation theory of finite dimensional algebras.

We define essential and strongly essential triangulations of 3-manifolds, and give four constructions using different tools (Heegaard splittings, hierarchies of Haken 3-manifolds, Epstein-Penner decompositions, and cut loci of Riemannian…

There exist tilings of the plane with pairwise noncongruent triangles of equal area and bounded perimeter. Analogously, there exist tilings with triangles of equal perimeter, the areas of which are bounded from below by a positive constant.…

We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.

We describe a new algorithm to compute the geometric intersection number between two curves, given as edge vectors on an ideal triangulation. Most importantly, this algorithm runs in polynomial time in the bit-size of the two edge vectors.…

We prove, under suitable conditions, a lower bound on the number of pinned distances determined by small subsets of two-dimensional vector spaces over fields. For finite subsets of the Euclidean plane we prove an upper bound for their…

The paper is devoted to finding the colorings of the edges of the 1-skeleton of triangulations of the 2-sphere in three colors so that for each face all three of its sides have different colors. First, by the method of adding one vertex…

Se enuncia los principales teoremas empleados en la resoluci'on de tri'angulos oblicu'angulos. Con ellos, se ilustra c'omo resolver los cinco casos de resoluci'on que se presentan, incluyendo algunos caso at'ipicos (cuando se conoce el…

In this talk, some aspects of duality symmetries are presented.

We provide upper and lower bounds on the least-perimeter way to enclose and separate n regions of equal area in the plane. Along the way, inside the hexagonal honeycomb, we provide minimizers for each n .

We find explicit formulas for the Hilbert series of residual intersections of a scheme in terms of the Hilbert series of its conormal modules. In a previous paper we proved that such formulas should exist. We give applications to the…