Related papers: Projective Cross-Ratio on Hypercomplex Numbers
The paper deals with a particular type of a projective ring plane defined over the ring of double numbers over Galois fields, R\_{\otimes}(q) \equiv GF(q) \otimes GF(q) \cong GF(q)[x]/(x(x-1)). The plane is endowed with (q^2 + q + 1)^2…
We study the Hopf monoid of convex geometries, which contains partial orders as a Hopf submonoid, and investigate the combinatorial invariants arising from canonical characters. Each invariant consists of a pair: a polynomial and a more…
We discuss representations of the projective line over a ring $R$ with 1 in a projective space over some (not necessarily commutative) field $K$. Such a representation is based upon a $(K,R)$-bimodule $U$. The points of the projective line…
We prove a $p$-adic version of the Integral Geometry Formula for averaging the intersection of two $p$-adic projective algebraic sets. We apply this result to give bounds on the number of points in the modulo $p^m$ reduction of a projective…
The Hilbert metric on convex subsets of $\mathbb R^n$ has proven a rich notion and has been extensively studied. We propose here a generalization of this metric to subset of complex projective spaces and give examples of applications to…
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…
Graphs are fundamental tools for modeling pairwise interactions in complex systems. However, many real-world systems involve multi-way interactions that cannot be fully captured by standard graphs. Hypergraphs, which generalize graphs by…
In this work, we study a mathematically rigorous metric of a graph visualization quality under conditions that relate to visualizing a bipartite graph. Namely we study rectilinear crossing number in a special arrangement of the complete…
General structure of the multivariate plain and q-hypergeometric terms and univariate elliptic hypergeometric terms is described. Some explicit examples of the totally elliptic hypergeometric terms leading to multidimensional integrals on…
We propose to consider ensembles of cycles (quadrics), which are interconnected through conformal-invariant geometric relations (e.g. "to be orthogonal", "to be tangent", etc.), as new objects in an extended Moebius--Lie geometry. It was…
Projective splitting is a family of methods for solving inclusions involving sums of maximal monotone operators. First introduced by Eckstein and Svaiter in 2008, these methods have enjoyed significant innovation in recent years, becoming…
This is a follow-up paper of arXiv:1805.00115, where rational curves in surfaces that satisfy general positioned point and cross-ratio conditions were enumerated. A suitable correspondence theorem provided in arXiv:1509.07453 allowed us to…
We consider the set of pairs of orthogonal vectors in Hilbert space, which is also called the cross because it is the union of the horizontal and vertical axes in the Euclidean plane when the underlying space is the real line. Crosses,…
A generalization of von Staudt's theorem that every permutation of the projective line that preserves harmonic quadruples is a projective semilinear map is given. It is then concluded that any proper supergroup of permutations of the…
In this paper, we introduce a generalized concept of vertex transitivity in graphs called generalized vertex transitivity. We put forward a new invariant called transitivity number of a graph. The value of this invariant in different…
On any convex domain in $\mathbb{R}^n$ we can define the Hilbert metric. A projective transformation is an example of an isometry of the Hilbert metric. In this thesis we will prove that the group of projective transformations on a convex…
Geophysical inversion should ideally produce geologically realistic subsurface models that explain the available data. Multiple-point statistics is a geostatistical approach to construct subsurface models that are consistent with…
We show that the visual angle metric and the triangular ratio metric are comparable in convex domains. We also find the extremal points for the visual angle metric in the half space and in the ball by use of a construction based on…
This article emphasizes an extension of the study of metric and par- tition dimension to hypergraphs. We give a sharp lower bounds for the metric and partition dimension of hypergraphs in general and give exact values under specified…
We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We…