Boundedness and compactness of little Hankel operators from H∞ to the Bloch space and the little Bloch space are characterised.

Let $F$ be an integral linear recurrence, $G$ an integer-valued polynomial splitting over the rationals and $h$ a positive integer. Also, let ${\mathcal{A}}_{F,G,h}$ be the set of all natural numbers $n$ such that $\gcd (F(n),G(n))=h$. We prove that ${\mathcal{A}}_{F,G,h}$ has a natural density. Moreover, assuming that $F$ is nondegenerate and $G$ has no fixed divisors, we show that the density of ${\mathcal{A}}_{F,G,1}$ is 0 if and only if ${\mathcal{A}}_{F,G,1}$ is finite.

We consider the symmetric inverse monoid ℐn of an n-element chain and its inverse submonoids ℐn, ℐn, ℐn and ℘ℐn of all order-preserving, order-preserving or order-reversing, orientation-preserving and orientation-preserving or orientation-reversing transformations, respectively, and give descriptions of their Abelian kernels relative to decidable pseudovarieties of Abelian groups.

Let Γ be a G-symmetric graph admitting a nontrivial G-invariant partition ℬ of block size υ. For blocks B, C of ℬ adjacent in the quotient graph Γℬ, let k be the number of vertices in B adjacent to at least one vertex in C. In this paper we classify all possibilities for (Γ Γℬ, G) in the case where k = υ − 1 ≥ 2 and ℬ(α) = ℬ(β) for adjacent vertices α β of Γ where for a vertex of Γ, say γ ∈ B, ℬ(γ) denotes the set of blocks C such that γ is the only vertex in B not adjacent to any vertex in C.

Let $G$ be a group, $p$ be a prime and $P\in \text{Syl}_{p}(G)$. We say that a $p$-Brauer character $\unicode[STIX]{x1D711}$ is monolithic if $G/\ker \unicode[STIX]{x1D711}$ is a monolith. We prove that $P$ is normal in $G$ if and only if $p\nmid \unicode[STIX]{x1D711}(1)$ for each monolithic Brauer character $\unicode[STIX]{x1D711}\in \text{IBr}(G)$. When $G$ is $p$-solvable, we also prove that $P$ is normal in $G$ and $G/P$ is nilpotent if and only if $\unicode[STIX]{x1D711}(1)^{2}$ divides $|G:\ker \unicode[STIX]{x1D711}|$ for all monolithic irreducible $p$-Brauer characters $\unicode[STIX]{x1D711}$ of $G$.

Let W be an n−dimensional vector space over a field F. It is shown that the expected dimension of a vector subspace of W is n/2. If F is infinite, the expected dimension of a sum of a pair of subspaces of W is (n + 1)/2 if n > 1; and 3/4 if n = 1. If F is finite, with q elements, the expected dimension of a sum of subspaces of W depends on q and n. For fixed n, the limiting value of this expectation as q → ∞ is n if n is even; and n − 1/4 if n is odd. Moreover, if F is finite and n > 1, the expected dimension of a sum of three (not necessarily distinct) subspaces of W has limit n as q → ∞.

Groups having exactly one normaliser are well known. They are the Dedekind groups. All finite groups having exactly two normalisers were classified by Pérez-Ramos [‘Groups with two normalizers’, Arch. Math.50 (1988), 199–203], and Camp-Mora [‘Locally finite groups with two normalizers’, Comm. Algebra28 (2000), 5475–5480] generalised that result to locally finite groups. Then Tota [‘Groups with a finite number of normalizer subgroups’, Comm. Algebra32 (2004), 4667–4674] investigated properties (such as solubility) of arbitrary groups with two, three and four normalisers. In this paper we prove that every finite group with at most 20 normalisers is soluble. Also we characterise all nonabelian simple (not necessarily finite) groups with at most 57 normalisers.

Evidence is presented which suggests that the following assertions about a variety ⊻ of groups may be equivalent:

(a) ⊻ is locally finite,

(b) all ⊻-groups satisfying the maximal condition for normal subgroups are finite, and

(c) all ⊻-groups satisfying the minimal condition for normal subgroups are finite.

We show that the space of smooth maps from a compact smooth manifold into another smooth manifold can be endowed with the structure of a smooth manifold if we use the Γ-differentiation of Yamamuro. We then generalise the Smale Density Theorem to mappings between these manifolds.

Using the concept of equi-integrability we introduce a measure of weak noncompactness in the Lebesgue space L1(0, l). We show that this measure is equal to the classical De Blasi measure of weak noncompactness.

The object of this paper is to prove a conjecture given recently by Nunokawa that if f(z) ∈ Α(p) satisfies Re{1 + zf″(z)/f′(z)} < p + 1/2 in the unit disk U, then f(z) is p-valently starlike in U.

In this note, Conjecture Cl of Toby Lewis (1976), concerning a reciprocal pair of characteristic variables, is shown to be false.

In this paper we shall consider orthodox bands of commutative groups, together with a ring of endomorphisms. We shall generalize the concept of a left module by introducing orthodox bands of left modules; we shall also deal with linear mappings, the transpose of a linear mapping and with the dual of an orthodox band of left modules.

An automorphism of a graph product of groups is conjugating if it sends each factor to a conjugate of a factor (possibly different). In this article, we determine precisely when the group of conjugating automorphisms of a graph product satisfies Kazhdan’s property (T) and when it satisfies some vastness properties including SQ-universality.

It is remarked that, if A is a complete boolean algebra and δ is an A-valued equivalence relation on a non-empty set I, then the set of δ-extensional functions from I to A can be regarded as a complete boolean algebra extension of A and a characterization is given of the complete extensions which arise in this way.

We show that a C1-flow on a compact Riemannian manifold is structurally stable and topologically stable if and only if it satisfies Axiom A and the strong transversality condition. This improves Smale's conjecture for flows.

For $k\geq 2$ and $r\geq 1$, we prove that the number of odd $k$-perfect numbers with $r$ distinct prime factors is at most $4^{r^2}(k-1)^{2r^2+3}$.