Using results from topological groups and topological dynamics for locally compact spaces, Aoki has shown that when a group automorphism of a locally compact totally disconnected group is ergodic under the Haar measure, the group is compact. We provide some remarks on Aoki's proof. Also we present a new proof of his result using the structure of locally compact totally disconnected groups established by Willis.

The Euler–Mascheroni constant $\gamma =0.5772\ldots \!$ is the $K={\mathbb Q}$ example of an Euler–Kronecker constant $\gamma _K$ of a number field $K.$ In this note, we consider the size of the $\gamma _q=\gamma _{K_q}$ for cyclotomic fields $K_q:={\mathbb Q}(\zeta _q).$ Assuming the Elliott–Halberstam Conjecture (EH), we prove uniformly in Q that

$$ \begin{align*} \frac{1}{Q}\sum_{Q<q\le 2Q} |{\gamma_q - \log q}| = o(\log Q).\end{align*} $$

In other words, under EH, the $\gamma _q /\!\log q$ in these ranges converge to the one point distribution at $1$ . This theorem refines and extends a previous result of Ford, Luca and Moree for prime $q.$ The proof of this result is a straightforward modification of earlier work of Fouvry under the assumption of EH.

We obtain a class of recurrence relations for the Bernoulli numbers that includes a recurrence formula proved recently by M. Kaneko. Analogous formulas are also derived for the Euler and Genocchi numbers.

We obtain exact formulas for the cumulative distribution function of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. From these formulas, we deduce exact formulas for the cumulative distribution function of the product of two correlated zero-mean normal random variables.

Starting from the well-known classical theorem of Archimedes about the volumes of spheres and circumscribing cylinders in three-dimensional Euclidean space, one considers circumscribing tubes of small geodesic spheres in general Riemannian manifolds and one derives new characterisations of two-point homogeneous spaces from it.

One form of Dirichlet's theorem on simultaneous diophantine approximation asserts that if α1, α2, …, αn are any real numbers and m ≥ 2 is an integer, then there exist integers q, p1, p2, …, pn such that 1 ≤ q < m and |qαi.-pi| ≤ m–1/n holds for 1 < i < n. The paper considers the problem of the extent to which this theorem can be improved by replacing m–1/n by a smaller number. A general solution to this problem is given. It is also shown that a recent result of Kurt Mahler [Bull. Austral. Math. Soc. 14 (1976), 463–465] amounts to a solution of the case n = 1 of the above problem. A related conjecture of Mahler is proved.

In this note we present a simple, short proof of the sum formula for subdifferentials of convex functions.

We show that trigonometric Lagrange interpolating approximation with arbitrary real distinct nodes in Lp space for 1 ≤ p < ∞, as that with equally spaced nodes in Lp space for 1 < p < ∞ in an earlier paper by T.F. Xie and S.P. Zhou, may also be arbitrarily “bad”. This paper is a continuation of this earlier work by Xie and Zhou, but uses a different method.

Suppose that H is a closed subgroup of a locally compact group G. We show that a unitary representation U of H is the restriction of a unitary representation of G if and only if a dual representation Û of a crossed product C*(G) ⋊ (G/H) is regular in an appropriate sense. We then discuss the problem of deciding whether a given representation is regular; we believe that this problem will prove to be an interesting test question in non-Abelian duality for crossed products of C*-algebras.

Let ${\mathcal{A}}$ be a unital torsion-free algebra over a unital commutative ring ${\mathcal{R}}$. To characterise Lie $n$-higher derivations on ${\mathcal{A}}$, we give an identity which enables us to transfer problems related to Lie $n$-higher derivations into the same problems concerning Lie $n$-derivations. We prove that: (1) if every Lie $n$-derivation on ${\mathcal{A}}$ is standard, then so is every Lie $n$-higher derivation on ${\mathcal{A}}$; (2) if every linear mapping Lie $n$-derivable at several points is a Lie $n$-derivation, then so is every sequence $\{d_{m}\}$ of linear mappings Lie $n$-higher derivable at these points; (3) if every linear mapping Lie $n$-derivable at several points is a sum of a derivation and a linear mapping vanishing on all $(n-1)$th commutators of these points, then every sequence $\{d_{m}\}$ of linear mappings Lie $n$-higher derivable at these points is a sum of a higher derivation and a sequence of linear mappings vanishing on all $(n-1)$th commutators of these points. We also give several applications of these results.

In 1996 Poland and Rhemtulla proved that the number v (G) of conjugacy classes of non-normal subgroups of a non-Hamiltonian nilpotent group G is at least c − 1, where c is the nilpotency class of G. In this paper we consider the map that associates to every conjugacy class of subgroups of a finite p-group the conjugacy class of the normaliser of any of its representatives. In spite of the fact that this map need not be injective, we prove that, for p odd, the number of conjugacy classes of normalisers in a finite p-group is at least c (taking into account the normaliser of the normal subgroups). In the case of p-groups of maximal class we can find a better lower bound that depends also on the prime p.

An axiomatic approach to relativity theory is proposed which is based entirely on a single reflexive relation called signalling. This generalises and simplifies earlier schemes of Kronheimer and Penrose and of Carter which are based on causality relations. Many of the features of space-times, such as photon paths and topology, have their counterparts in general signal spaces.

We show that a right duo ring R is strongly regular if and only if for each ideal I of R, the coset product of I in the factor ring R/I is the same as their set product.

We establish upper bounds for the distance to finite-dimensional subspaces in inner product spaces and improve some generalisations of Bessel's inequality obtained by Boas, Bellman and Bombieri. Refinements of the Hadamard inequality for Gram determinants are also given.

The cyclotomic identity

where and μ is the classical Möbius function, is shown to be a consequence of a natural isomorphism of species.

Diophantine problems involving recurrence sequences have a long history. We consider the equation $B_{m}B_{m+d}\cdots B_{m+(k-1)d}=y^{\ell }$ in positive integers $m,d,k,y$ with $\gcd (m,d)=1$ and $k\geq 2$, where $\ell \geq 2$ is a fixed integer and $B=(B_{n})_{n=1}^{\infty }$ is an elliptic divisibility sequence, an important class of nonlinear recurrences. We prove that the equation admits only finitely many solutions. In fact, we present an algorithm to find all possible solutions, provided that the set of $\ell$th powers in $B$ is given. We illustrate our method by an example.

Let p > 1 real. We Doob-Meyer-decompose Lp(ℙ)-valued positive submartingales such that the martingale and predictable parts are also in Lp(ℙ). We give two variants of such a decomposition. The first one handles also not necessarily right continuous submartingales, since its proof is as discrete in its nature as Doob's archaically decomposition. The second decomposition acts in Lp (ℝ × Ω ℬ ⊗ ℱ, μ ⊗ ℙ) for some finite measure μ on ℝ.

D.L. Berman has proved several divergence theorems about “extended” Hermite-Fejér interpolation on the Chebyshev nodes of the first kind. These are surprising in light of the classical convergence theorem of L. Fejér concerning ordinary Hermite-Fejér interpolation on these nodes. However there is one case which has been neglected so far: the case of quasi-Hermite-Fejér interpolation on these nodes. In this paper it is proved that quasi-Hermite-Fejér interpolation polynomials on the Chebyshev nodes converge uniformly to the continuous function being interpolated. In addition, an estimate for the rate of convergence is established.

Let u(x1,…,xn) = x11 … x1m be a word in the alphabet x1, …,xn such that x1i ≠ x1i for all i = 1,…, m − 1. If (H1, …, Hn) is an n-tuple of subgroups of a group G then denote by u(H1, …, Hn) the set {u(h1,…,hn) | hi ∈ Hi}. If σ ∈ Sn then denote by uσ(H1,…,Hn) the set u(Hσ(1),…,Hσ(n)). We study groups G with the property that for each n-tuple (H1,…,Hn) of subgroups of G, there is some σ ∈ Sn σ ≠ 1 such that u(H1,…,Hn) = uσ(H1,…,Hn). If G is a finitely generated soluble group then G has this property for some word u if and only if G is nilpotent-by-finite. In the paper we also look at some specific words u and study the properties of the associated groups.