Related papers: The largest angle bisection procedure
We extend recent results by G. E. Andrews and G. Simay on the $m$th largest and $m$th smallest parts of a partition to the more general context of skew plane partitions. In order to do this, we introduce new objects called skew plane…
Given a family of sets on the plane, we say that the family is intersecting if for any two sets from the family their interiors intersect. In this paper, we study intersecting families of triangles with vertices in a given set of points. In…
An $N$-dimensional parallelepiped will be called a bar if and only if there are no more than $k$ different numbers among the lengths of its sides (the definition of bar depends on $k$). We prove that a parallelepiped can be dissected into…
The random greedy algorithm for constructing a large partial Steiner-Triple-System is defined as follows. We begin with a complete graph on $n$ vertices and proceed to remove the edges of triangles one at a time, where each triangle removed…
We construct isoperimetric regions from separating hypersurfaces in closed manifolds. This yields isoperimetric boundaries exhibiting a wide variety of topological types and singular sets.
In big data analysis, a simple task such as linear regression can become very challenging as the variable dimension $p$ grows. As a result, variable screening is inevitable in many scientific studies. In recent years, randomized algorithms…
The maximum graph bisection problem is a well known graph partition problem. The problem has been proven to be NP-hard. In the maximum graph bisection problem it is required that the set of vertices is divided into two partition with equal…
Many emerging computer applications require the processing of large numbers, larger than what a CPU can handle. In fact, the top of the line PCs can only manipulate numbers not longer than 32 bits or 64 bits. This is due to the size of the…
Starting from any given rational-sided, right triangle, for example the $(3,4,5)$-triangle with area $6$, we use Euclidean geometry to show that there are infinitely many other rational-sided, right triangles of the same area. We show…
If we label the vertices of a triangle with 1, 2 and 4, and the orthocentre with 7, then any of the four numbers 1, 2, 4, 7 is the nim-sum of the other three and is their orthocentre. Regard the triangle as an orthocentric quadrangle.…
The area of judicious partitioning considers the general family of partitioning problems in which one seeks to optimize several parameters simultaneously, and these problems have been widely studied in various combinatorial contexts. In…
We provide a remarkably simple algorithm to compute all (at most four) common tangents of two disjoint simple polygons. Given each polygon as a read-only array of its corners in cyclic order, the algorithm runs in linear time and constant…
The binomial interpolated transform of a sequence is a generalization of the well-known binomial transform. We examine a Pascal-like triangle, on which a binomial interpolated transform works between the left and right diagonals, focusing…
A construction to arbitrarily section a taxicab angle into an equal number of angles in (pure) taxicab geometry is presented.
A pair of non-adjacent edges is said to be separated in a circular ordering of vertices, if the endpoints of the two edges do not alternate in the ordering. The circular separation dimension of a graph $G$, denoted by $\pi^\circ(G)$, is the…
The problem of packing a set of circles into the smallest surrounding container is considered. This problem arises in different application areas such as automobile, textile, food, and chemical industries. The so-called circle packing…
A set of segments in the plane may form a Euclidean TSP tour or a matching, among others. Optimal TSP tours as well as minimum weight perfect matchings have no crossing segments, but several heuristics and approximation algorithms may…
Hierarchical Clustering is an unsupervised data analysis method which has been widely used for decades. Despite its popularity, it had an underdeveloped analytical foundation and to address this, Dasgupta recently introduced an optimization…
In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one…
A simple greedy refinement procedure for the generation of data-adapted triangulations is proposed and studied. Given a function of two variables, the algorithm produces a hierarchy of triangulations and piecewise polynomial approximations…