Related papers: Orthogonal systems in vector spaces over finite fi…
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…
Let V be an n-dimensional vector space and let On be the orthogonal group. Motivated by a question of B. Szegedy (B. Szegedy, Edge coloring models and reflection positivity, Journal of the American Mathematical Society Volume 20, Number 4,…
It was recently shown by Aka, Einsiedler and Shapira that if d>2, the set of primitive vectors on large spheres when projected to the d-1-dimensional sphere coupled with the shape of the lattice in their orthogonal complement equidistribute…
We show that the hyperplane conjecture holds for the classes of $k$-intersection bodies with arbitrary measures in place of volume.
In this note, we give a new necessary condition for the existence of non-trivial partitions of a finite vector space. Precisely, we prove that, if V is a finite vector space over a field of order q, then the number of the subspaces of…
We partition in classes the set of matroids of fixed dimension on a fixed vertex set. In each class we identify two special matroids, respectively with minimal and maximal h-vector in that class. Such extremal matroids also satisfy a…
Let D be a division ring with centre F. Let T(D) be the vector space over F generated by all multiplicative commutators in D. In [1], authors have conjectured that every division ring is generated as a vector space over its centre by all of…
Suppose that for some unit vectors $b_1,\ldots b_n$ in $\mathbb C^d$ we have that for any $j\neq k$ $b_j$ is either orthogonal to $b_k$ or $|\langle b_j,b_k\rangle|^2 = 1/d$ (i.e. $b_j$ and $b_k$ are unbiased). We prove that if $n=d(d+1)$,…
An orthogonal n-frame is an ordered set of n pairwise orthogonal vectors. The set of all orthogonal n-frames in a d-dimensional quadratic vector space is an algebraic variety V(d,n). In this paper, we investigate the variety V(d,n) as well…
An arc is a set of vectors of the $k$-dimensional vector space over the finite field with $q$ elements ${\mathbb F}_q$, in which every subset of size $k$ is a basis of the space, i.e. every $k$-subset is a set of linearly independent…
With every family of finitely many subsets of a finite-dimensional vector space over the Galois-field with two elements we associate a cyclic transversal polytope. It turns out that those polytopes generalize several well-known polytopes…
We introduce the concept of bi-conformal transformation, as a generalization of conformal ones, by allowing two orthogonal parts of a manifold with metric $\G$ to be scaled by different conformal factors. In particular, we study their…
Given a $k$-point configuration $x\in (\mathbb{R}^d)^k$, we consider the $\binom{k}{d}$-vector of volumes determined by choosing any $d$ points of $x$. We prove that a compact set $E\subset \R^d$ determines a positive measure of such volume…
We study skew-orthogonal polynomials with respect to the weight function $\exp[-2V(x)]$, with $V(x)=\sum_{K=1}^{2d}(u_{K}/{K})x^{K}$, $u_{2d} > 0$, $d > 0$. A finite subsequence of such skew-orthogonal polynomials arising in the study of…
We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard…
The notion of a Heffter array, which received much attention in the last decade, is equivalent to a pair of orthogonal Heffter systems. In this paper we study the existence problem of a set of $r$ mutually orthogonal Heffter systems for any…
Vector spaces over finite fields and Anzahl formulas of subspaces were studied by Wan (Geometry of Classical Groups over Finite Fields, Science Press, 2002). As a generalization, we study vector spaces and singular linear spaces over…
Suppose that each proper subset of a set $S$ of points in a vector space is contained in the union of planes of specified dimensions, but $S$ itself is not contained in any such union. How large can $|S|$ be? We prove a general upper bound…
Motivated by a model in quantum computation we study orthogonal sets of integral vectors of the same norm that can be extended with new vectors keeping the norm and the orthogonality. Our approach involves some arithmetic properties of the…
Let $K$ be a differential field over $\C$ with derivation $D$, $G$ a finite linear automorphism group over $K$ which preserves $D$, and $K^G$ the fixed point subfield of $K$ under the action of $G$. We show that every finite-dimensional…