On cutting blocking sets and their codes

Let PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} be the r-dimensional projective space over the finite field GF ⁡ ( q ) {\operatorname{GF}(q)} . A set 𝒳 {\mathcal{X}} of points of PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} is a cutting blocking set if for each hyperplane Π of PG ⁡ ( r , q ) {\operatorname{PG}(r,q)} the set Π ∩ 𝒳 {\Pi\cap\mathcal{X}} spans Π. Cutting blocking sets give rise to saturating sets and minimal linear codes, and those having size as small as possible are of particular interest. We observe that from a cutting blocking set obtained in [20], by using a set of pairwise disjoint lines, there arises a minimal linear code whose length grows linearly with respect to its dimension. We also provide two distinct constructions: a cutting blocking set of PG ⁡ ( 3 , q 3 ) {\operatorname{PG}(3,q^{3})} of size 3 ⁢ ( q + 1 ) ⁢ ( q 2 + 1 ) {3(q+1)(q^{2}+1)} as a union of three pairwise disjoint q-order subgeometries, and a cutting blocking set of PG ⁡ ( 5 , q ) {\operatorname{PG}(5,q)} of size 7 ⁢ ( q + 1 ) {7(q+1)} from seven lines of a Desarguesian line spread of PG ⁡ ( 5 , q ) {\operatorname{PG}(5,q)} . In both cases, the cutting blocking sets obtained are smaller than the known ones. As a byproduct, we further improve on the upper bound of the smallest size of certain saturating sets and on the minimum length of a minimal q-ary linear code having dimension 4 and 6.

On Factors of Independent Transversals in $k$-Partite Graphs

A $[k,n,1]$-graph is a $k$-partite graph with parts of order $n$ such that the bipartite graph induced by any pair of parts is a matching. An independent transversal in such a graph is an independent set that intersects each part in a single vertex. A factor of independent transversals is a set of $n$ pairwise-disjoint independent transversals. Let $f(k)$ be the smallest integer $n_0$ such that every $[k,n,1]$-graph has a factor of independent transversals assuming $n \geqslant n_0$. Several known conjectures imply that for $k \geqslant 2$, $f(k)=k$ if $k$ is even and $f(k)=k+1$ if $k$ is odd. While a simple greedy algorithm based on iterating Hall's Theorem shows that $f(k) \leqslant 2k-2$, no better bound is known and in fact, there are instances showing that the bound $2k-2$ is tight for the greedy algorithm. Here we significantly improve upon the greedy algorithm bound and prove that $f(k) \leqslant 1.78k$ for all $k$ sufficiently large, answering a question of MacKeigan.

Julia sets of Zorich maps

The Julia set of the exponential family $E_{\kappa }:z\mapsto \kappa e^z$ , $\kappa>0$ was shown to be the entire complex plane when $\kappa>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<\kappa \leq 1/e$ the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three-dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the whole of $\mathbb {R}^3$ , generalizing Misiurewicz’s result. Moreover, we show that the periodic points of the Zorich map are dense in $\mathbb {R}^3$ and that its escaping set is connected, generalizing a result of Rempe. We also generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics of the exponential.

On Weighted Pál Type (0,2)-interpolation on the Unit Circle

In this paper, we study the explicit representation of weighted Pál-type (0,2)-interpolation on two pairwise disjoint sets of nodes on the unit circle, which are obtained by projecting vertically the zeros of (1−x2)Pn(x) and Pn′′(x) respectively, where Pn(x) stands for nth Legendre polynomial.AMS Classification (2000): 41A05, 30E10.

Mostar index and edge Mostar index of polymers

Let $$G=(V,E)$$ G = ( V , E ) be a graph and $$e=uv\in E$$ e = u v ∈ E . Define $$n_u(e,G)$$ n u ( e , G ) be the number of vertices of G closer to u than to v. The number $$n_v(e,G)$$ n v ( e , G ) can be defined in an analogous way. The Mostar index of G is a new graph invariant defined as $$Mo(G)=\sum _{uv\in E(G)}|n_u(uv,G)-n_v(uv,G)|$$ M o ( G ) = ∑ u v ∈ E ( G ) | n u ( u v , G ) - n v ( u v , G ) | . The edge version of Mostar index is defined as $$Mo_e(G)=\sum _{e=uv\in E(G)} |m_u(e|G)-m_v(G|e)|$$ M o e ( G ) = ∑ e = u v ∈ E ( G ) | m u ( e | G ) - m v ( G | e ) | , where $$m_u(e|G)$$ m u ( e | G ) and $$m_v(e|G)$$ m v ( e | G ) are the number of edges of G lying closer to vertex u than to vertex v and the number of edges of G lying closer to vertex v than to vertex u, respectively. Let G be a connected graph constructed from pairwise disjoint connected graphs $$G_1,\ldots ,G_k$$ G 1 , … , G k by selecting a vertex of $$G_1$$ G 1 , a vertex of $$G_2$$ G 2 , and identifying these two vertices. Then continue in this manner inductively. We say that G is a polymer graph, obtained by point-attaching from monomer units $$G_1,\ldots ,G_k$$ G 1 , … , G k . In this paper, we consider some particular cases of these graphs that are of importance in chemistry and study their Mostar and edge Mostar indices.

On the VC-dimension of half-spaces with respect to convex sets

A family S of convex sets in the plane defines a hypergraph H = (S, E) as follows. Every subfamily S' of S defines a hyperedge of H if and only if there exists a halfspace h that fully contains S' , and no other set of S is fully contained in h. In this case, we say that h realizes S'. We say a set S is shattered, if all its subsets are realized. The VC-dimension of a hypergraph H is the size of the largest shattered set. We show that the VC-dimension for pairwise disjoint convex sets in the plane is bounded by 3, and this is tight. In contrast, we show the VC-dimension of convex sets in the plane (not necessarily disjoint) is unbounded. We provide a quadratic lower bound in the number of pairs of intersecting sets in a shattered family of convex sets in the plane. We also show that the VC-dimension is unbounded for pairwise disjoint convex sets in R^d , for d > 2. We focus on, possibly intersecting, segments in the plane and determine that the VC-dimension is always at most 5. And this is tight, as we construct a set of five segments that can be shattered. We give two exemplary applications. One for a geometric set cover problem and one for a range-query data structure problem, to motivate our findings.

Vanishing John–Nirenberg spaces

There still exist many unsolved problems on the study related to John–Nirenberg spaces. In this article, the authors introduce two new vanishing subspaces of the John–Nirenberg space JN p ⁢ ( ℝ n ) {\mathrm{JN}_{p}(\mathbb{R}^{n})} denoted, respectively, by VJN p ⁢ ( ℝ n ) {\mathrm{VJN}_{p}(\mathbb{R}^{n})} and CJN p ⁢ ( ℝ n ) {\mathrm{CJN}_{p}(\mathbb{R}^{n})} , and establish their equivalent characterizations which are counterparts of those characterizations for the classic spaces VMO ⁢ ( ℝ n ) {\mathrm{VMO}(\mathbb{R}^{n})} and CMO ⁢ ( ℝ n ) {\mathrm{CMO}(\mathbb{R}^{n})} obtained, respectively, by D. Sarason and A. Uchiyama. All these results shed some light on the mysterious space JN p ⁢ ( ℝ n ) {\mathrm{JN}_{p}(\mathbb{R}^{n})} . The approach strongly depends on the fine geometrical properties of dyadic cubes, which enable the authors to subtly classify any collection of interior pairwise disjoint cubes.

On complete decompositions of dihedral groups

Let G be a finite non-abelian group and B1, …, Bt be nonempty subsets of G for integer t ≥ 2. Suppose that B1, …, Bt are pairwise disjoint, then (B1, …, Bt) is called a complete decomposition of G of order t if the subset product Bi1 … Bit = {bi1 … bit | bij ∈ Bij, j = 1,2, …,t} coincides with G, where {Bi1 … Bit} = {B1, …, Bt} and the Bij are all distinct. Let D2n = ‹r,s| rn = s2 = 1, rs =srn-1› be the dihedral group of order 2n for integer n ≥3. In this paper, we shall give the constructions of the complete decompositions of D2n of order t, where 2 ≤ t ≤ n.

Disjoint Blocks in a MOL(6)

We prove that the maximal number of pairwise disjoint 4-blocks in a MOL(6) is 3. We recall various proofs for the non-existence of a MOL(6) and show: with the theorem the proofs can be simplified considerably.