Related papers: On particular examples of planar integral point se…
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…
Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of $n$ points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$ outside of the segment delimited by $x$…
We determine those maps between affine or projective spaces that are linear in the abstract sense of transforming collinear points into collinear points and whose restriction to any line is constant or injective. Our results are extensions…
We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in…
A simplicial set is non-singular if the representing map of each non-degenerate simplex is degreewise injective. The simplicial mapping set $X^K$ has $n$-simplices given by the simplicial maps $\Delta[n] \times K \to X$. We prove that $X^K$…
We determine the set of catenary degrees, the set of distances, and the unions of sets of lengths of the monoid of nonzero ideals and of the monoid of invertible ideals of orders in quadratic number fields.
We study the quadratic integral points-that is, (S-)integral points defined over any extension of degree two of the base field-on a curve defined in P_3 by a system of two Pell equations. Such points belong to three families explicitly…
Neutrosophic set is a part of neutrosophy which studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. Neutrosophic set is a powerful general formal framework that has been…
According to [1] an $n$-dimensional $\mathcal{N}$--set is a compact subset $A$ of $\mathbb{R}^n$ such that for every $x$ in $\mathbb{R}^n$ there is $y$ in $A$ with $y-x$ in $\mathbb{Z}^n$. We prove that every two dimensional…
Certain types of bilinearly defined sets in $\mathbb{R}^n$ exhibit a higher degree of linearity than what is apparent by inspection.
We establish asymptotic formulas for the number of integral points of bounded height on partial equivariant compactifications of vector groups.
An ordinary circle of a set $P$ of $n$ points in the plane is defined as a circle that contains exactly three points of $P$. We show that if $P$ is not contained in a line or a circle, then $P$ spans at least $\frac{1}{4}n^2 - O(n)$…
We consider the notion of multiple gap as a finite set of ideals that cannot be separated. We study the different types of such objects that can be found in the Boolean algebra of subsets of the natural numbers modulo finite sets.
We define the notion of extrinsic symplectic symmetric spaces and exhibit some of their properties. We construct large families of examples and show how they fit in the perspective of a complete classification of these manifolds. We also…
We show that there are five types of planar curves such that arrangements of its translates are combinatorially equivalent to an arrangement of lines. These curves can be used to define norms giving constructions with many unit distances…
An ideal on a set $X$ is a collection of subsets of $X$ closed under the operations of taking finite unions and subsets of its elements. Ideals are a very useful notion in topology and set theory and have been studied for a long time. We…
Theories in physics can provide a kind of map of the physical system under investigation, showing all of the possible types of behavior which may occur. Certain points on the map are of greater significance than others, because they…
An arc in $\Z^2_n$ is defined to be a set of points no three of which are collinear. We describe some properties of arcs and determine the maximum size of arcs for some small $n$.
A \emph{complete geometric graph} consists of a set $P$ of $n$ points in the plane, in general position, and all segments (edges) connecting them. It is a well known question of Bose, Hurtado, Rivera-Campo, and Wood, whether there exists a…
This article describes some aspects of Cauchy integrals and related geometry of sets and measures in Euclidean spaces, etc.