Related papers: A Generalized Cross Ratio
The crossing number of a graph $G$ is the least number of crossings over all possible drawings of $G$. We present a structural characterization of graphs with crossing number one.
This is an introductory note concerning the distribution vectors in a unitary representation of a Lie group. We discuss the definition of matrix coefficients associated with a pair of distributions and how one can compute them. Most of the…
Computing the crossing number of a graph is one of the most classical problems in computational geometry. Both it and numerous variations of the problem have been studied, and overcoming their frequent computational difficulty is an active…
This article represents a major step in the unification of the theory of algebraic, topological and singular transition matrices by introducing a definition which is a generalization that encompasses all of the previous three. When this…
A plane drawing of a graph is {\em cylindrical} if there exist two concentric circles that contain all the vertices of the graph, and no edge intersects (other than at its endpoints) any of these circles. The {\em cylindrical crossing…
The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position…
This paper proposes that common measures for network transitivity, based on the enumeration of transitive triples, do not reflect the theoretical statements about transitivity they aim to describe. These statements are often formulated as…
The crossing number is the smallest number of pairwise edge-crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant…
The crossing number of a graph is the least number of crossings of edges among all drawings of the graph in the plane. In this article, we prove that the crossing number of the generalized Petersen graph P(10, 3) is equal to 6.
Inspired by the increasingly popular research on extending partial graph drawings, we propose a new perspective on the traditional and arguably most important geometric graph parameter, the crossing number. Specifically, we define the…
The critical behaviour of a system of two coupled scalar fields in three dimensions is studied within the formalism of the effective average action. The fixed points of the system are identified and the crossover between them is described…
In this article we study an exact analogue of the cross-ratio for the algebra of quaternions H and use it to derive several interesting properties of quaternionic fractional linear transformations. In particular, we show that there exists a…
A linear map $\Phi$ between matrix spaces is called cross-positive if it is positive on orthogonal pairs $(U,V)$ of positive semidefinite matrices in the sense that $\langle U,V\rangle:=\text{Tr}(UV)=0$ implies $\langle…
Geometric crossover is a representation-independent definition of crossover based on the distance of the search space interpreted as a metric space. It generalizes the traditional crossover for binary strings and other important…
The crossing number of a graph is the minimum number of edge crossings that a graph can have when drawn in the plane. Determining this number, known as the Crossing Number problem, is a celebrated problem in combinatorial optimization. It…
We generalize the natural cross ratio on the ideal boundary of a rank one symmetric spaces, or even $\mathrm{CAT}(-1)$ space, to higher rank symmetric spaces and (non-locally compact) Euclidean buildings - we obtain vector valued cross…
Many systems near criticality can be described by Hamiltonians involving several relevant couplings and possessing many nontrivial fixed points. A simple and physically appealing characterization of the crossover lines and surfaces…
A multifractal analysis is performed on a three-dimensional grayscale image associated with a complex system. First, a procedure for generating 3D synthetic images (2D image stacks) of a complex structure exhibiting multifractal behaviour…
We use three different methods to count the number of lines in the plane whose intersection with a fixed general quintic has fixed cross-ratios. We compare and contrast these methods, shedding light on some classical ideas which underly…
Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). We introduce a special kind of simple drawings that we call generalized…