Related papers: Exending pseudo-arcs in odd characteristic
Due to the applications in network coding, subspace codes and designs have received many attentions. Suppose that $k\mid n$ and $V(n,q)$ is an $n$-dimensional space over the finite field $\mathbb{F}_{q}$. A $k$-spread is a…
This note offers a probabilistic proof of Girard's angle excess formula for the area of a spherical triangle, based on the observation that an unbounded 3-dimensional convex cone, with single vertex at the origin, has only three kinds of…
Pach showed that every $d+1$ sets of points $Q_1,\dotsc,Q_{d+1} \subset \mathbb{R}^d$ contain linearly-sized subsets $P_i\subset Q_i$ such that all the transversal simplices that they span intersect. We show, by means of an example, that a…
For an affine, toric Q-Gorenstein variety Y (given by a lattice polytope Q) the vector space T^1 of infinitesimal deformations is related to the complexified vector spaces of rational Minkowski summands of faces of Q. Moreover, assuming Y…
This article looks at subconics of order $q$ of $PG(2,q^2)$ and characterizes them in the Bruck-Bose representation in $PG(4,q)$. In common with other objects in the Bruck-Bose representation, the characterisation uses the transversals of…
The theorem of three circles in real algebraic geometry guarantees the termination and correctness of an algorithm of isolating real roots of a univariate polynomial. The main idea of its proof is to consider polynomials whose roots belong…
Pseudo-cones are a class of unbounded closed convex sets, not containing the origin. They admit a kind of polarity, called copolarity. With this, they can be considered as a counterpart to convex bodies containing the origin in the…
Let $K$ be a set of $q^2+2q+1$ points in $PG(4,q)$. We show that if every 3-space meets $K$ in either one, two or three lines, a line and a non-degenerate conic, or a twisted cubic, then $K$ is a ruled cubic surface. Moreover, $K$…
We consider the only one known class of non-K\"ahler irreducible holomorphic symplectic manifolds, described in the works of D. Guan and the first author. Any such manifold $Q$ of dimension $2n-2$ is obtained as a finite degree $n^2$ cover…
In this paper we study combinatorial invariants of the equivalence classes of pencils of cubics on $\mathrm{PG}(1,q)$, for $q$ odd and $q$ not divisible by 3. These equivalence classes are considered as orbits of lines in…
It is proved that for every $d\ge 2$ such that $d-1$ divides $q-1$, where $q$ is a power of 2, there exists a Denniston maximal arc $A$ of degree $d$ in $\PG(2,q)$, being invariant under a cyclic linear group that fixes one point of $A$ and…
The ample hierarchy of geometries of stables theories is strict. We generalise the construction of the free pseudospace to higher dimensions and show that the n-dimensional free pseudospace is \omega-stable n-ample yet not (n+1)-ample. In…
Halfspaces or linear threshold functions are widely studied in complexity theory, learning theory and algorithm design. In this work we study the natural problem of constructing pseudorandom generators (PRGs) for halfspaces over the sphere,…
Based on a closed formula for a star product of Wick type on $\CP^n$, which has been discovered in an earlier article of the authors, we explicitly construct a subalgebra of the formal star-algebra (with coefficients contained in the…
We characterize the symmetry algebra of the generic superintegrable system on a pseudo-sphere corresponding to the homogeneous space $SO(p,q+1)/SO(p,q)$ where $p+q={\cal N}$, ${\cal N}\in\mathbb N$. We show that this algebra is independent…
A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of k-level vertices in an arrangement of n hyperplanes in d dimensions (vertices with exactly k of the hyperplanes passing below…
Let F_q be a finite field of order q with characteristic p. An arc is an ordered family of at least k vectors in (F_q)^k in which every subfamily of size k is a basis of (F_q)^k. The MDS conjecture, which was posed by Segre in 1955, states…
An $(r, s)$-${\textit set}$ in ${\rm PG}(n, q)$ is a set of points, say $\mathcal X$, such that each $s$-dimensional projective subspace contains at most $r$ points of $\mathcal X$. We investigate $(n, n-2)$-sets and $(n-2, n-3)$-sets in…
We construct the quantized enveloping algebra of any simple Lie algebra of type ADE as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group…
In \cite{ThasHVM}, a characterization of the finite quadric Veronesean $\mathcal{V}_{n}^{2^{n}}$ by means of properties of the set of its tangent spaces is proved. These tangent spaces form a {\em regular generalised dual arc}. We prove an…