Lower and upper bounds on the size of a covering of subspaces in the Grassmann graph Gq(n, r) by subspaces from the Grassmann graph Gq(n, k), k ≥ r , are discussed. The problem is of interest from four points of view: coding theory, combinatorial designs, q-analogs, and projective geometry. In particular we examine coverings based on lifted maximum rank distance codes, combined with spreads and...

OBJECTIVE A collaborative project between Hull CHCP and Smith and Nephew wound management, which sought to enhance practice efficiency, specifically in relation to the optimisation of dressing change frequency. The project involved the use of a new foam dressing (Allevyn Life, Smith & Nephew, Hull) designed to manage wound exudate more effectively, be more acceptable to patients by masking stai...

The chiral hypermaps with at most four hyperfaces were classified in [A. Breda d’Azevedo and R. Nedela,Chiral hypermaps with few hyperfaces, Math. Slovaca, 53 (2003), n.2, 107–128]. It arises from this classification that all chiral hypermaps are “canonical metacyclic”, that is, the one-step rotation about a hypervertex, or about a hyperedge, or about a hyperface, generates a normal subgroup in...

We prove that octants are cover-decomposable into multiple coverings, i.e., for any k there is an m(k) such that any m(k)-fold covering of any subset of the space with a nite number of translates of a given octant can be decomposed into k coverings. As a corollary, we obtain that any m(k)-fold covering of any subset of the plane with a nite number of homothetic copies of a given triangle can be...

We discuss the existence of vertex disjoint path coverings with prescribed ends for the n-dimensional hypercube with or without deleted vertices. Depending on the type of the set of deleted vertices and desired properties of the path coverings we establish the minimal integer m such that for every n ≥ m such path coverings exist. Using some of these results, for k ≤ 4, we prove Locke’s conjectu...

We develop an algebraic representation for (1, 1)-knots using the mapping class group of the twice punctured torus MCG2(T ). We prove that every (1, 1)-knot in a lens space L(p, q) can be represented by the composition of an element of a certain rank two free subgroup of MCG2(T ) with a standard element only depending on the ambient space. As a notable examples, we obtain a representation of th...

A generalised Paley map is a Cayley map for the additive group of a finite field F , with a subgroup S = −S of the multiplicative group as generating set, cyclically ordered by powers of a generator of S. We characterise these as the orientably regular maps with orientation-preserving automorphism group acting primitively and faithfully on the vertices; allowing a non-faithful primitive action ...

Given a curve C of genus 2 defined over a field k of characteristic different from 2, with Jacobian variety J , we show that the two-coverings corresponding to elements of a large subgroup of H ` Gal(k/k), J [2](k) ́ (containing the Selmer group when k is a global field) can be embedded as intersection of 72 quadrics in P k , just as the Jacobian J itself. Moreover, we actually give explicit equ...

Let M = B/Γ be a smooth ball quotient of finite volume with first betti number b1(M) and let E(M) ≥ 0 be the number of cusps (i.e., topological ends) of M . We study the growth rates that are possible in towers of finite-sheeted coverings of M . In particular, b1 and E have little to do with one another, in contrast with the well-understood cases of hyperbolic 2and 3-manifolds. We also discuss ...

It is a well-known result of the covering groups that a subgroup G of the fundamental group at the identity of a semilocally simply connected topological group determines a covering morphism of topological groups with characteristic group G . In this paper we generalize this result to a large class of algebraic objects called topological groups with operations, including topological groups. We ...