Let Bn denote the unit ball in ℂn, n≥1. Given an α>0, let ℱα(n) denote the class of functions defined for z∈Bn by integrating the kernel (1−〈z,w〉)−α against a complex Borel measure dμ(w), w∈Bn. The family ℱ0(n) corresponds to the logarithmic kernel log (1/(1−〈z,w〉)). Various properties of the spaces ℱα(n), α≥0, are obtained. In particular, pointwise multiplies for ℱα(n) are investigated.

In this paper, it is shown that there exists a connected topological group which is not homeomorphic to any $\omega $-narrow topological group, and also that there exists a zero-dimensional topological group $G$ with neutral element $e$ such that the subspace $X = G\setminus \{e\}$ is not homeomorphic to any topological group. These two results give negative answers to two open problems in Arhangel’skii and Tkachenko [Topological Groups and Related Structures (Atlantis Press, Amsterdam, 2008)]. We show that if a compact topological group is a $K$-space, then it is metrisable. This result gives an affirmative answer to a question posed by Malykhin and Tironi [‘Weakly Fréchet–Urysohn and Pytkeev spaces’, Topology Appl. 104 (2000), 181–190] in the category of topological groups. We also prove that a regular $K$-space $X$ is a weakly Fréchet–Urysohn space if and only if $X$has countable tightness.

The aim of this paper is to prove uniqueness theorems for the Darboux problem of neutral type in the space L∞ and L1.

Let F be a system of polynomial equations in one or more variables with integer coefficients. We show that there exists a univariate polynomial $D \in \mathbb {Z}[x]$ such that F is solvable modulo p if and only if the equation $D(x) \equiv 0 \pmod {p}$ has a solution.

We propose generating functions, $\textrm {RGF}_p(x)$, for the quotients of numerical semigroups which are related to the Sylvester denumerant. Using MacMahon’s partition analysis, we can obtain $\textrm {RGF}_p(x)$ by extracting the constant term of a rational function. We use $\textrm {RGF}_p(x)$ to give a system of generators for the quotient of the numerical semigroup $\langle a_1,a_2,a_3\rangle $ by p for a small positive integer p, and we characterise the generators of ${\langle A\rangle }/{p}$ for a general numerical semigroup A and any positive integer p.

In this note we extend a result of Drewnowski concerning copies of C0 in the Banach space of all countably additive vector measures and study some properties of complemented copies of C0 in several Banach spaces of vector measures.

The notion of approximations relative to a functor is introduced and several characterisations of relative (dual) Maschke functors are given by using them. As an application, the injective objects in the category of comodules over a coring are described.

We prove that the sequence {log ζ(n)}n≥2 is not holonomic, that is, does not satisfy a finite recurrence relation with polynomial coefficients. A similar result holds for L-functions. We then prove a result concerning the number of distinct prime factors of the sequence of numerators of even indexed Bernoulli numbers.

In this paper, we obtain a new result for overtwisted contact $(+1/n)$-surgery. We also give a counterexample to a conjecture by James Conway on overtwistedness of manifolds obtained by contact surgery.

Dedicated to Professor George Szekeres

Short amicable sets were introduced recently and have many applications. The construction of short amicable sets has lead to the construction of many orthogonal designs, weighing matrices and Hadamard matrices. In this paper we give some constructions for short amicable sets as well as some multiplication theorems. We also present a table of the short amicable sets known to exist and we construct some infinite families of short amicable sets and orthogonal designs.

An area-diameter relation and an area-width relation for plane lattice-point-free-convex bodies is proved. This implies relations on covering sets with respect to general lattices.

In this paper, the amenability and approximate amenability of weighted ℓp-direct sums of Banach algebras with unit, where 1≤p<∞, are completely characterized. Applications to compact groups and hypergroups are given.

For a finite presentation of a group, or more generally, a two-complex, we define a function analogous to the Dehn function that we call the annular Dehn function. This function measures the combinatorial area of maps of annuli into the complex as a function of the lengths of the boundary curves. A finitely presented group has solvable conjugacy problem if and only if its annular Dehn function is recursive.

As with standard Dehn functions, the annular Dehn function may change with change of presentation. We prove that the type of function obtained is preserved by change of presentation. Further we obtain upper bounds for the annular Dehn functions of free products and, more generally, amalgamations or HNN extensions over finite subgroups.

The local connectivity, νk(G), of a graph G is the minimum of the connectivities of neighbourhoods of the vertices of G. G is minimally locally n-connected if νk(G) = n and for every edge x of G, νk(G−x) = n − 1. A necessary and sufficient condition for a locally connected graph to be minimally locally 1-connected is given, and it is shown that for n ≥ 7, is minimally locally 1-connected.

For a pythagorean field F with semiordering Q and associated preordering T, it is shown that the Witt ring WT (F) is isomorphic to the Witt ring W (K) whre K is a closure of F with respect to Q. For an arbitrary preordering T, it is shown how the covering number of T relates to the construction of WT (F).

In this paper we consider some notions of amenability such as ideal amenability, n-ideal amenability and approximate n-ideal amenability. The first two were introduced and studied by Gordji, Yazdanpanah and Memarbashi. We investigate some properties of certain Banach algebras in each of these classes. Results are also given for Segal algebras on locally compact groups.