A characterization of induced central extensions which gives an explicit relationship between induced central extensions and n-stem extensions is obtained. Using the characterization, necessary and sufficient conditions for a central extension of an abelian group by a nilpotent group of class n to be a Baer sum of an induced central extension and an extension of class n are obtained.

We announce results of Landesman-Lazer type for boundary-value problems for ordinary differential equations. Details will appear elsewhere.

I.N. Herstein has shown that an associative ring in which the nilpotent elements are “well-behaved”, and such that every element satisfies a certain polynomial identity, is commutative. This result is generalized here. Specifically, it is shown that an alternative ring R which satisfies the following three properties is commutative:

(i) for x ∈ R, there exists an integer n(x) and a polynomial px (t) with integer coefficients such that xn+1p(x) = xn;

(ii) for a fixed positive integer m, a a nilpotent and b an arbitrary element of R, a - am commutes with b - bm;

(iii) for the same m, a and b, (ab+b)m = (ba+b)m and (ab)m = ambm.

Examples are given to show that all three properties are essential, and it is shown that for associative rings certain modified versions of these properties are individually enough to assure that the commutator ideal of the ring is nil.

The following is a correct proof of the main theorem of [1]. It should be substituted for the published Section 3, which, as pointed out by Professor B.L.S. Prakasa Rao, contains an error in the equations following (15) on page 49.

It is shown that expanding self-maps of the interval with a finite number of turning points must have periodic points whose periods are not integral power of 2 and therefore are chaotic.

This paper considers a special class of stochastic optimal parameter selection problem subject to probability constraints on the state. The system dynamics are governed by a linear Ito stochastic differential equation with controllable parameters appearing nonlinearly in the dynamics. The problem seeks to optimise a cost functional which is quadratic in the state with weighting matrices being time invariant but depending nonlinearly on the parameters. Although the inclusion of the probability state constraints renders the problem insolvable by the conventional LQG theory, we show that the problem can in fact be transformed into an equivalent deterministic optimal parameter selection problem solvable by an existing software MISER. Numerical examples are presented to demonstrate the feasibility and efficiency of the proposed approach.

A unified theory of continuous and certain non-continuous functions, initiated in an earlier paper, is further elaborated. The proposed theory provides a common platform for dealing simultaneously with continuous functions and a host of non-continuous functions including lower (upper) semicontinuous functions, almost continuous functions, weakly continuous functions (encountered in functional analysis), c-continuous functions, δ-continuous functions, semiconnected functions, H-continuous functions s-continuous functions, ε-continuous functions of Klee and several other variants of continuity.

In previous work, Page and Butson [Algebra Universalis 3 (1973), 112–126] characterized all equationally complete classes (atoms) of m–semigroups (universal algebras with one m–ary associative operation), and hence m–groups, in the commutative case. The further characterization of the non-commutative m-group atoms was thought to hinge upon a conjecture by Page [PhD thesis, University of Miami, 1973] that a weaker form of commutativity held true. In this paper that conjecture is proved and then used to study the special case m = 4. Two additional infinite sets of atoms are thereby determined, although it is not proved that these examples exhaust the remaining atoms for m = 4.

Given a finitely generated multiplicative subgroup Us in a number field, we employ a simple argument from the geometry of numbers and an inequality on multiplicative dependence in number fields to obtain a minimal set of generators consisting of elements of relatively small height.

We prove that a Banach space X has the Radon-Nikodym property if, and only if, every weak*-lower semicontinuous convex continuous function f of X* is Gâteaux differentiable at some point of its domain with derivative in the predual space x.

We study the mod 2 homology of the moduli space of instantons associated with the prinicpal Sp (n) bundle over the four-sphere and the classifying space of the gauge group using the Serre spectral sequence and the homology operations.

In a recent paper, Cusack has given a criterion, in terms of the Fitting class “join” operation, for a normal Fitting class to be closed under the taking of Hall π-subgroups. Here we show that Cusack's result can be slightly modified so as to give a criterion for any Fitting class of finite soluble groups to be closed under taking Hall π-subgroups.

A new brief proof of the fact stated in the title is presented.

Andrews [‘Binary and semi-Fibonacci partitions’, J. Ramanujan Soc. Math. Math. Sci.7(1) (2019), 1–6] recently proved a new identity between the cardinalities of the set of semi-Fibonacci partitions and the set of partitions into powers of 2 with all parts appearing an odd number of times. We extend the identity to the set of semi-$m$-Fibonacci partitions of $n$ and the set of partitions of $n$ into powers of $m$ in which all parts appear with multiplicity not divisible by $m$. We also give a new characterisation of semi-$m$-Fibonacci partitions and some congruences satisfied by the associated number sequence.

The propogation of disturbance when a shear flow with a free surface, in a channel of infinite horizontal extent and finite depth, is disturbed by the application of time-oscillatory pressure, is studied. The initial value problem is solved by using transform techniques and the steady state solution is obtained therefrom in the limit t → ∞. The effect of the initial shear on the development of the wave system is investigated.

Using some techniques of perturbation theory for Banach space complexes, we obtain necessary and sufficient conditions for the stability of the topological index of an open linear relation under small (with respect to the gap topology) perturbations with linear relations.

Motivated by constructions of Witt rings in the algebraic theory of quadratic forms, the authors construct new classes of finite commutative rings and explore some of their properties. These rings are constructed as quotient rings of a special class of integral group rings for which the group is an elementary 2-group. The new constructions are compared to other rings in the literature.

Majorant functions for contractors can be defined in a natural war. Such a case is considered here in order to find iterative solutions of general equations in Banach spaces by means of contractors. A class of majorant functions is defined which contains in particular the linear majorant ones. Local and global existence and convergence theorems are proved.