Let R be a semiprime ring in which for each x in R there exists a positive integer n = n(x) < 1 such that (xy)n = xnyn for all y in R. Then R is commutative.

A simple formula for the inverse of a block matrix with non-zero blocks in the principal diagonal and the first sub-diagonal only is proved. The matrix had arisen in an investigation of a difference equation.

We prove the reciprocity law for the twisted second moments of Dirichlet $L$-functions over rational function fields, corresponding to two irreducible polynomials. This formula is the analogue of the formulas for Dirichlet $L$-functions over $\mathbb{Q}$ obtained by Conrey [‘The mean-square of Dirichlet $L$-functions’, arXiv:0708.2699 [math.NT] (2007)] and Young [‘The reciprocity law for the twisted second moment of Dirichlet $L$-functions’, Forum Math. 23(6) (2011), 1323–1337].

By applying the theory of exponential dichotomies and contraction mapping, we establish some existence and uniqueness results for weighted pseudo almost periodic solutions of some differential equations with piecewise constant arguments. For this purpose, we also describe some basic properties of weighted pseudo almost periodic sequences.

The classification of the finite simple groups has had far-reaching consequences for many branches of algebra. This paper is a discussion of several problems about primitive permutation groups which have been solved using the simple group classification.

The concept of superspace is fundamental for some recent physical theories, notably supersymmetry, and a mathematical feedback for it is provided by superanalysis and supergeometry. We survey the state of affairs in superanalysis, shifting our attention from supermanifold theory to “plain” superspaces. The two principal existing approaches to superspaces are sketched and links between them discussed. We examine a problem by Manin of representing even geometry (analysis) as a collective effect in infinite-dimensional purely odd geometry (analysis), by applying the technique of nonstandard (infinitesimal) analysis.

For any integer $m\neq 0$, we prove that $f(x)=x^{9}+9mx^{6}+192m^{3}$ is irreducible over $\mathbb{Q}$ and that the Galois group of $f(x)$ over $\mathbb{Q}$ is the dihedral group of order 18. Moreover, we show that for infinitely many values of $m$, the splitting fields for $f(x)$ are distinct.

All subsets of a field that are intersections of almost-orders of the field are characterised, and all ring topologies on a field which are not finer than any nontrivial locally bounded ring are characterised.

A generalised topology is a collection of subsets of a given nonempty set containing the empty set and arbitrary unions of the elements in the collection. By using the concept of hereditary classes, a generalised topology can be extended to a new one, called a generalised topology via a hereditary class. We study continuity on generalised topological spaces via hereditary classes in various situations.

Torsion topological groups with the additional property that the intersection of open sets is open are considered and their topological structure is determined:

1) the topology is uniquely determined by a normal subgroup;

2) each group is uniquely an extension of an indiscrete group by a discrete group;

3) the topology may be changed within limits without changing the dual group.

In this paper we describe the irreducible decomposition of the Magnus representation of the Torelli group.

If G is a finite p-group of order pn, P. Hall determined the number of conjugacy classes of G, r(G), modulo (p2 − 1)(p − 1). Namely, he proved the existence of a constant k ≥ 0 such that r(G) = n(p2 − 1) + pe + k(p2 − 1)(p − 1). In this paper, we denote by the group of the upper unitriangular matrices over , the finite field with q = pt elements, and we determine the number of classes of modulo (q − 1)5.

In this paper we determine the mod-2 cohomology rings and the 2-primary part of the integral cohomology rings of finite groups with semi-dihedral Sylow 2-subgroups. The method used here is algebraic and can be considered as elementary.

We give bounds for the canonical height of rational and integral points on cubic twists of the Fermat elliptic curve. As a corollary we prove that there is no integral arithmetic progression on certain curves in this family.

A class of problems in forced convection give rise to quadratic eigenvalue problems. In this paper it is shown that the eigenvalues are necessarily real. The extension of this result to a wider class of problems is also discussed.

Let R be the field of real numbers. For α in R, let ‖α‖ be the distance of α from the nearest integer. The following conjecture of Kurt Mahler [Bull. Austral. Math. Soc. 14 (1976), 463–465] is proved.

Let m, n be two positive integers n ≥ 2m. Let S be a finite or infinite set of positive integers with the following properties:

(Q1) S contains the integers m, m+1, …, n−m;

(Q2) every element of S satisfies

Then

Let λ1, λ2, λ3 be non-zero reals, not all of the same sign and such that at least one ratio λi/λj is irrational. Then it is proved that for any given integer k ≥ 1 and real η, the inequaltiy

is solvable for every ε > 0. More general and sharper results are also proved.

We introduce some methods of constructing stable graphs and characterize a few classes of stable graphs. We also give a counter example to disprove Holton's conjecture.

The definition of division d.g. near-rings given by the author in 1976 has been found to be too restrictive. In this paper we generalise the definition of division d.g. near-rings and extend the results obtained earlier for division d.g. near-rings to the new wider class of near-rings.