Related papers: Classification of spreads of Tits quadrangles of o…
This is the first in a series of articles devoted to providing a foundation for a theory of flocks of arbitrary cones in PG(3,q). The desire to have such a theory stems from a need to better understand the very significant and applicable…
In this article we consider a set C of points in PG(4,q), q even, satisfying certain combinatorial properties with respect to the planes of PG(4,q). We show that there is a regular spread in the hyperplane at infinity, such that in the…
We present a new explicit family of polynomials orthogonal on the unit circle with a dense point spectrum. This family is expressed in terms of q-hypergeometric function of type ${_2}\phi_1$. The orthogonality measure is the wrapped…
A spread of a Hermitian unital in PG(2,q^2) is a set of q^2+q+1 pairwise disjoint blocks that partition the points of the unital. In this paper, we discuss the results of an exhaustive computer search for spreads of Hermitian unitals of…
We give an account of well known calculations of the RO(Q)-graded coefficient rings of some of the most basic Q-equivariant cohomology theories, where Q is a group of order 2. One purpose is to advertise the effectiveness of the Tate…
The edge-to-edge tilings of the sphere by congruent quadrilaterals of Type $a^2bc$ are classified as $3$ classes: a sequence of two-parameter families of $2$-layer earth map tilings with $2n$ $(n\ge3)$ tiles, a one-parameter family of…
We completely classify edge-to-edge tilings of the sphere by congruent quadrilaterals. As part of the classification, we also present a modern version of the classification of edge-to-edge tilings of the sphere by congruent triangles.…
Square-triangle-rhombus ($\mathcal{STR}$) tilings are encountered in various self-organized multi-component systems. They exhibit a rich structural diversity, encompassing both periodic tilings and long-range ordered quasicrystals,…
Ovoids in $\PG(3, \gf(q))$ have been an interesting topic in coding theory, combinatorics, and finite geometry for a long time. So far only two families of ovoids are known. The first is the elliptic quadratics and the second is the Tits…
Ovoids of the Klein quadric $Q^+(5,q)$ of $\mathrm{PG}(5,q)$ have been studied in the last 40 year, also because of their connection with spreads of $\mathrm{PG}(3,q)$ and hence translation planes. Beside the classical example given by a…
We show that a dense subset of a sufficiently large group multiplication table contains either a large part of the addition table of the integers modulo some $k$, or the entire multiplication table of a certain large abelian group, as a…
We classify affine rank three Tits arrangements whose roots are contained in the locus of a homogeneous cubic polynomial. We find that there exist irreducible affine Tits arrangements which are not locally spherical.
We study a family of bivariate orthogonal polynomials associated to the deltoid curve. These polynomials arise when classifying bivariate diffusion operators that have discrete spectral decomposition given by orthogonal polynomials with…
We present a description of maximal partial ovoids of size $q^2-1$ of the parabolic quadric $\q(4,q)$ as sharply transitive subsets of $\SL(2,q)$ and show their connection with spread sets. This representation leads to an elegant explicit…
In this article, the partial plane spreads in $PG(6,2)$ of maximum possible size $17$ and of size $16$ are classified. Based on this result, we obtain the classification of the following closely related combinatorial objects: Vector space…
We show that, if the family \cal{O} of orbits of all vector fields on a subcartesian space P is locally finite and each orbit in \cal{O} is locally closed, then \cal{O} defines a smooth Whitney A stratification of P. We also show that the…
We classify the special families of dihedral folding tilings of the sphere derived from the M\"obius triangle $(2,3,4)$. Our study emerges from the study of isometric foldings in the Riemann sphere and meets at the juncture of the triangle…
We study the Oort groups for a prime p, i.e. finite groups G such that every G-Galois branched cover of smooth curves over an algebraically closed field of characteristic p lifts to a G-cover of curves in characteristic 0. We prove that all…
Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This last one classifies the case of $a^3b$-quadrilaterals with some irrational angle: there are a sequence of…
S. Rottey and G. Van de Voorde characterized regular pseudo-ovals of PG(3n - 1, q), q = 2^h, h >1 and n prime. Here an alternative proof is given and slightly stronger results are obtained.