Related papers: On directions determined by subsets of vector spac…
In this paper we study some generalized versions of a recent result due to Covert, Koh, and Pi (2015). More precisely, we prove that if a subset $\mathcal{E}$ in a regular variety satisfies $|\mathcal{E}|\gg…
The set of primitive vectors on large spheres in the euclidean space of dimension d>2 equidistribute when projected on the unit sphere. We consider here a refinement of this problem concerning the direction of the vector together with the…
It is shown that certain transformations on quiver-dimension vector pairs induce isomorphisms on the corresponding moduli spaces of quiver representations and map a stable dimension vector to a stable dimension vector. This result combined…
We generalize to sets with cardinality more than $p$ a theorem of R\'edei and Sz\H{o}nyi on the number of directions determined by a subset $U$ of the finite plane $\mathbb F_p^2$. A $U$-rich line is a line that meets $U$ in at least…
We prove a new upper bound on the number of $r$-rich lines (lines with at least $r$ points) in a `truly' $d$-dimensional configuration of points $v_1,\ldots,v_n \in \mathbb{C}^d$. More formally, we show that, if the number of $r$-rich lines…
We study the Erd\"os/Falconer distance problem in vector spaces over finite fields. Let ${\Bbb F}_q$ be a finite field with $q$ elements and take $E \subset {\Bbb F}^d_q$, $d \ge 2$. We develop a Fourier analytic machinery, analogous to…
We consider arrangements of n hyperplanes of codimension one in a real projective space of dimension d. Let us denote by F the maximal possible number f of connected components of the complement in the projective space of dimension d to the…
We prove that a subset of $\mathbb{F}_q^n$ that contains a hyperplane in any direction has size at least $q^{n}-O(q^2)$.
We show that the order dimension of the weak order on a Coxeter group of type A, B or D is equal to the rank of the Coxeter group, and give bounds on the order dimensions for the other finite types. This result arises from a unified…
In this paper we study the kaleidoscopic pseudo-randomness of finite Euclidean graphs using probabilistic methods. Roughly speaking, we show that sufficiently large subsets of d-dimensional vector spaces over finite fields contain every…
First, we study constructible subsets of $\A^n_k$ which contain a line in any direction. We classify the smallest such subsets in $\A^3$ of the type $R\cup\{g\neq 0\},$ where $g\in k[x_1,...,x_n]$ is irreducible of degree $d$, and $R\subset…
We characterize pairs (Q,d) consisting of a quiver Q and a dimension vector d, such that over a given algebraically closed field k there are infinitely many representations of Q of dimension vector d. We also present an application of this…
We present a family of nonconforming vector finite elements of arbitrary order for problems posed on the space (curl) intersected with H(div) on a bidimensional domain. This result was first stated as a conjecture by Brenner and Sung. In…
Fix a point in a finite-dimensional complex vector space and consider the sequence of iterates of this point under the composition of a unitary map with the orthogonal projection on the hyperplane orthogonal to the starting point. We prove…
Given $n+1$ unit vectors in $\mathbf{R}^n$ or $\mathbf{C}^n,$ consider the absolute values of the determinants of the vectors taken $n$ at a time. By taking a geometric perspective, we show that the minimum of these determinants is…
The paper is devoted to developing subdifferential theory for set-valued mappings taking values in ordered infinite-dimensional spaces. This study is motivated by applications to problems of vector and set optimization with various…
It is well known that a positive proportion of all points in a $d$-dimensional lattice is visible from the origin, and that these visible lattice points have constant density in $\mathbb{R}^d$. In the present paper we prove an analogous…
We study the structure of two-sided vector spaces over a perfect field $K$. In particular, we give a complete characterization of isomorphism classes of simple two-sided vector spaces which are left finite-dimensional. Using this…
Every set of natural numbers determines a generating function convergent for $q \in (-1,1)$ whose behavior as $q \rightarrow 1^-$ determines a germ. These germs admit a natural partial ordering that can be used to compare sets of natural…
Given $E \subset \mathbb{R}^d$, $d \ge 2$, define ${\mathcal D}(E) \equiv {(x-y)/|x-y|: x,y \in E} \subset S^{d-1},$ the set of directions determined by $E$. We prove that if the Hausdorff dimension of $E$ is greater than $d-1$, then…