Related papers: A note on a problem invoving a square in a curvili…
Traditional clustering identifies groups of objects that share certain qualities. Tangles do the converse: they identify groups of qualities that often occur together. They can thereby identify and discover 'types': of behaviour, views,…
There is an elegant relation found by Fabricius-Bjerre [Math. Scand 40 (1977) 20--24] among the double tangent lines, crossings, inflections points, and cusps of a singular curve in the plane. We give a new generalization to singular curves…
Given $n$ continuous open curves in the plane, we say that a pair is touching if they have only one interior point in common and at this point the first curve does not get from one side of the second curve to its other side. Otherwise, if…
We construct an associative ring which is a deformation of the quantum cohomology ring of the projective plane. Just as the quantum cohomology encodes the incidence characteristic numbers of rational plane curves, the contact cohomology…
The correspondence between 2-parameter families of oriented lines in ${\Bbb{R}}^3$ and surfaces in $T{\Bbb{P}}^1$ is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences…
We search for triangular numbers that are multiples of other triangular numbers. It is found that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers that are triangular numbers and…
Our goal is to present, in what we believe is the most efficient way possible, a construction of four mutually tangent circles.
It is shown that there exists a dihedral acute triangulation of the three-dimensional cube. The method of constructing the acute triangulation is described, and symmetries of the triangulation are discussed.
We study the relations between the triviality of the tangent bundle $TM$ and the generalized tangent bundle $\mathbb{T}M = TM\oplus T^*M$ of a manifold. We show that the generalized tangent bundle of a paralellizable manifold is trivial. We…
We investigate when the tangent bundle of a projective manifold has a non-trivial first order (or positive-dimensional) deformation. This leads to a new conjectural characterization of the complex projective space.
We characterise the quartic (i.e. 4-regular) multigraphs with the property that every edge lies in a triangle. The main result is that such graphs are either squares of cycles, line multigraphs of cubic multigraphs, or are obtained from…
Counting Euclidean triangulations with vertices in a finite set $\C$ of the convex hull $\conv(\C)$ of $\C$ is difficult in general, both algorithmically and theoretically. The aim of this paper is to describe nearly convex polygons, a…
Let M be a smooth strictly convex closed surface in space and denote by H the set of points x in the exterior of M such that all the tangent segments from x to M have equal lengths. In this note we prove that if H is either a closed surface…
A fast algorithm for counting intersections of two normal curves on a triangulated surface is proposed. It yields a convenient way for treating mapping class groups of punctured surfaces by presenting mapping classes by matrices, and the…
In the paper we prove generalization of Schl\"omilch's and Zetel's theorems about concurrent lines in a triangle. This generalization is obtained as a corollary of sharp geometric inequality about the ratio of triangular areas which is…
This note provides formula for determinant and inverse of r-circulant matrices with general sequences of third order. In other words, the study combines many papers in the literature.
The point-line geometry known as a \textit{partial quadrangle} (introduced by Cameron in 1975) has the property that for every point/line non-incident pair $(P,\ell)$, there is at most one line through $P$ concurrent with $\ell$. So in…
We consider two incidence problems for integral curves of vector fields. The first is an analogue of the Euclidean joints problem, in which lines are replaced by integral curves of smooth vector fields taken from some finite-dimensional…
In Descartes' five circle problem integer curvatures (inverse radii) are considered. The positive integer curvature triple [c_1, c_2, c_3] (dimensionless), with non-decreasing entries for three given mutually touching circles, leading to…
Curves in ${\mathbb R}^n$ for which the ratios between two consecutive curvatures are constant are characterized by the fact that their tangent indicatrix is a geodesic in a flat torus. For $n= 3,4$, spherical curves of this kind are also…