Related papers: New inductive constructions of complete caps in $P…
The aim of this paper is to give not only an explicit upper bound of the total Q-curvature but also an induced isoperimetric deficit formula for the complete conformal metrics on $\mathbb R^n$, $n\ge 3$ with scalar curvature being…
For nonautonomous, nonuniformly elliptic integrals with so-called $(p,q)$-growth conditions, we show a general interpolation property allowing to get basic higher integrability results for H\"older continuous minimizers under improved…
We describe a new construction of a subset of P^4 with no four points on a plane over any finite field of order q in which 3 is not a square. This set has size 2q + 1, is maximal with respect to inclusion, and is the largest known such set.
Let $\mathrm{PG}(k-1,q)$ be the $(k-1)$-dimensional projective space over the finite field $\mathbb{F}_q$. An arc in $\mathrm{PG}(k-1,q)$ is a set of points with the property that any $k$ of them span the entire space. The notion of…
The {\em Fibonacci cube} of dimension $n$, denoted as $\Gamma\_n$, is the subgraph of the $n$-cube $Q\_n$ induced by vertices with no consecutive 1's. In an article of 2016 Ashrafi and his co-authors proved the non-existence of perfect…
We study the computational complexity of approximating the 2->q norm of linear operators (defined as ||A||_{2->q} = sup_v ||Av||_q/||v||_2), as well as connections between this question and issues arising in quantum information theory and…
Minimal 1-saturating sets in the projective plane $PG(2,q)$ are considered. They correspond to covering codes which can be applied to many branches of combinatorics and information theory, as data compression, compression with distortion,…
In 2016, Ellenberg and Gijswijt established a new upper bound on the size of subsets of $\mathbb{F}^n_q$ with no three-term arithmetic progression. This problem has received much mathematical attention, particularly in the case $q = 3$,…
Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of narrow-sense BCH codes $\mathcal{C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^m$ and established that the codewords of the minimum (or the second minimum) weight in these…
We show that the problem of covering a set of points in the plane with a minimum number of guillotine cuts is NP-complete. To that end, first we present a new NP-completeness proof for the problem of covering points with disjoint line…
We present simple, geometric constructions for small regular graphs of girth 7 from the incidence graphs of some generalized quadrangles. We obtain infinite families of (q-1)-regular, q-regular and (q + 1)-regular graphs of girth 7, for q a…
We provide new families of minimal codes in any characteristic. Also, an inductive construction of minimal codes is presented.
Conjunctive query (CQ) evaluation is NP-complete, but becomes tractable for fragments of bounded hypertreewidth. Approximating a hard CQ by a query from such a fragment can thus allow for an efficient approximate evaluation. While…
Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph $\cG_q(n,r)$ by subspaces from the Grassmann graph $\cG_q(n,k)$, $k \geq r$, are discussed. The problem is of interest from four points of view: coding…
In this paper we are interested in the {\it{Cage Problem}} that consists in constructing regular graphs of given girth $g$ and minimum order. We focus on girth $g=5$, where cages are known only for degrees $k \le 7$. We construct regular…
In this paper, we prove several Poincar\'e inequalities of fractional type on conformally flat manifolds with finite total Q-curvature. This shows a new aspect of the $Q$-curvature on noncompact complete manifolds.
Using an Euclidean approach, we prove a new upper bound for the number of closed points of degree 2 on a smooth absolutely irreducible projective algebraic curve defined over the finite field $\mathbb F\_q$.This bound enables us to provide…
A family of sets $F$ is said to satisfy the $(p,q)$ property if among any $p$ sets in $F$, some $q$ have a non-empty intersection. Hadwiger and Debrunner (1957) conjectured that for any $p \geq q \geq d+1$ there exists $c=c_d(p,q)$, such…
In this thesis we construct 3-parameter families $G(p,q,r)$ of embedded arcs with fixed boundary in a 4-manifold. We then analyze these elements of $\pi_3\mathsf{Emb}_\partial(I,M)$ using embedding calculus by studying the induced map from…
Let $n$ be a positive integer. Denote by $\mathrm{PG}(n,q)$ the $n$-dimensional projective space over the finite field $\mathbb{F}_q$ of order $q$. A blocking set in $\mathrm{PG}(n,q)$ is a set of points that has non-empty intersection with…