This paper is the second part of the work begun in reference (BC I) and the terminology and notation will be carried over from this earlier part.

1·1. A group is called characteristically simple if it has no proper non-trivial subgroups which are left invariant by all of its automorphisms. One familiar class of characteristically simple groups consists of all direct powers of simple groups: this contains all finite characteristically simple groups, and, more generally, all characteristically simple groups having minimal normal subgroups. However not all characteristically simple groups lie in this class because, for instance, additive groups of fields are characteristically simple. Our object here is to construct finitely generated groups, and also groups satisfying the maximal condition for normal subgroups, which are characteristically simple but which are not direct powers of simple groups.

The Bruck-Slaby theorem on nilpotence of commutative Moufang loops may be combined with a theorem of Belousov to give a quick elementary proof of a stronger version of Fischer's Theorem on the solubility of right multiplication groups of finite distributive quasigroups.

1. Introduction. In this paper we consider Q-groups; that is, Q is a group and we consider groups N endowed with a Q-action, meaning a homomorphism of Q into the group of automorphisms of N. In (4) a lower central Q-series was defined for such a Q-group, N, generalizing the lower central series of a group, and results were obtained relating to the localization of such a series. Since the ideas in that paper were inspired by the homotopical localization theory of nilpotent spaces (see (6)), the main body of results in (4) was concerned with the case in which N is nilpotent, and perhaps also the group Q and the action of Q on N (in the sense that the lower central Q-series terminates after a finite number of steps with the trivial group {1}). We now adopt a broader view-point and only restrict ourselves to the nilpotent case when our results appear to require us to do so; thus the spirit of this paper is much more that of general group theory as presented in [(8), especially ch. VI]. Thus, while there is some overlap of results, the methods used are not the same and many results (for example, Theorem 3·1) are far more general than any obtained in (4). Moreover, the methods also appear to us to be more appropriate in that essential appeal was made in (4) to a sophisticated theorem of Norman Blackburn on nilpotent groups, whereas here we merely use homological methods, the construction of the semidirect product and, in section 4, some very classical facts of the commutator calculus.

Let k be a (smooth) knot in S3, → X the infinite cyclic covering of X = S3 – k corresponding to a generator of and let t generate the (infinite cyclic) group of covering transformations of . Various authors (1), (2), have shown that, for iterated torus knots, is of finite order (i.e. some power of t* is the identity).

It is known that the automorphism groups of various torsion-free associative algebras over two generators take a particularly simple form. It has been suggested (10), though never proved, that this fails over three or more generators. Here it is shown that this is indeed the case for the enveloping algebra of sl(2), a result which answers a question implicitly raised in (4). A weaker hypothesis is proposed for such automorphism groups and this is related to the structure of locally nilpotent derivations.

1. Ganea has shown (Theorem 2·2 of (2)) that h-cogroup structures on a CW space coincide, in the homotopy category, with coalgebra structures defined with respect to the cotriple determined by SΩ. The construction used by Ganea to prove this does not dualize. It seems possible moreover that the dual statement is false. In this note I show, using the results of (4), that, for suitable spaces, there is a surjection from the set of algebra structures defined with respect to the triple determined by ΩS to the set of h-group structures, and I construct a right inverse for this surjection.

The investigation of a topological, uniform or proximity space endowed with a preordering is by now well established. A topological space along with a preordering related to the topology in varying degrees has been studied, for example, in (3), (4), (5) and (6). ‘Uniform preordered spaces’ and ‘Proximity preordered spaces’ have been considered in (6) and (8) respectively. In each of these cases generalizations of varying proportions of the classical theories have been obtained.

The notion of basic topological binary relations is introduced, and corresponding to each such relation we define -regularity, -connectedness, and a separation axiom . The family of separation axioms so obtained is shown to include all the standard axioms T0, T1, …, T5, while familiar properties such as hyperconnectedness and ultraconnectedness are among the class of connectedness conditions. The regularity conditions include R0, R1, and regularity itself.

Arnol'd (1) remarks that the partition by Segre symbol of the endomorphisms of the complex vector space is a finite constructible stratification: in this paper we verify this statement, and show moreover that it is a Whitney stratification. Some of the arguments are by no means special to the case in hand; however, there seems little point in attempting a general formulation. It is a pleasure to acknowledge helpful conversations with Professor C. T. C. Wall and Dr E. J. N. Looijenga.

Let Xm and Yn be C1 manifolds embedded in , m < n < p, and let . In (4) C. T. C. Wall considered the following conditions:

(as) For any local C1 retraction at x, π: , x has a neighbourhood U such that is a submersion.

1. Introduction. Throughout this paper we shall take S to be a separately continuous, completely regular and Hausdorff, topological semigroup. We denote by Cb(S) the space of continuous and bounded complex-valued functions on S, and by βS the Stone-Čech compactification of S.

In (1) Akemann and Rosenfeld introduced a property for Banach algebras which they called (*). A Banach algebra satisfies (*) if every maximal one-sided ideal in is closed. They proved that certain classes of Banach algebra with satisfy (*), and they mentioned a conjecture that if is a Banach algebra with , then satisfies (*). In this paper we show that, if is a Banach algebra with a bounded right (left) approximate identity, then maximal left (right) ideals in are closed, and we give a counter-example to the above conjecture. We also give an independent proof that C*-algebras satisfy (*).

A condition of frequency domain type on the vector polynomial G(z) is obtained which is both necessary and sufficient for the Hurwitzian set of G(z) to include the special Hurwitzian set , where E(z) = col (1, z, …, zn). This result is extended to Hurwitzian sets arising from scalar delay-differential equations. Analogous conditions are also given for each of the inclusions where denotes the Schur set of G(z).

Under assumptions (1·1)–(1·3) given below, we prove discreteness of the positive singular spectrum of the forms realization of the operator – Δ + V. The limiting absorption principle is also proved.

The multivariate Gaussian process with the same variance/covariance structure as the multivariate kernel density estimator in Euclidean space of dimension d is considered. An exact result is obtained for the limit in probability of the maximum of the normalized process. In addition weak and strong bounds are placed on the asymptotic behaviour of the maximum of the process over a multidimensional interval which is allowed to increase as the sample size increases. All the bounds obtained on the process are

Only the uniform continuity of the underlying density is assumed; the conditions on the kernel are also mild.

1. Introduction. It has long been known that a connexion exists between reversible random walks and electrical networks ((2), (4) pp. 303–310, (6)). In section 2 we outline this connexion. More recently Kingman (5) has exhibited an electrical analogy for a flow model. In section 3 we show that this analogy follows from the results on random walks and we extend the analogy to cover the transient behaviour of Kingman's model. In so far as this model may be regarded as a naive description of the mechanism by which electrons move in a conductor it provides a physical explanation of the connexion between random walks and electrical networks. In section 4 we describe an invasion model which was inspired by the work of Clifford and Sudbury (1) on models to represent competition between species. Adapting their method we display the relationship of this model to random walks and hence obtain an electrical analogy for another class of stochastic processes.

Two different models of open, dissipative systems are considered. The stability and the asymptotic manifold of solutions are studied with respect to the variation of a model viscosity ν. Bifurcation theorems result from the analysis which shows the existence of a class of ‘prototurbulent’ motions. These are generalisations of the quasi-periodic asymptotic motions characteristic of Hopf's well-known mathematical model of turbulence. Given the statistics of the driving external field, it is shown how to calculate the correlation functions of the asymptotic motions (as t → ∞) for arbitrary values of the model viscosity. The systems treated are non-linear and except in special cases, cannot be solved in closed form. The relation to real systems is discussed. The effects of dimensionality on the qualitative nature of the motion is explored. It is found that for the models treated, perturbations of the external field do not affect the qualitative properties. Certain ‘mean’ quantities vary smoothly with the variation of the external field, but the details of the motion may be sensitive to the external perturbations. It is also found that even if the motion is strictly not quasiperiodic in the sense of Hopf and Landau, almost all asymptotic motions have a recurrent character under certain conditions. Some of the models also show the reverse of this behaviour, i.e. a non-dissipative, quasi-recurrent approach to ‘equilibrium’. This phenomenon can occur if the number of active degrees of freedom at a non-zero model viscosity is infinite.

This work amalgamates and solves certain problems arising in differential equation theory and in classical differential geometry.

It describes a novel technique for solving systems of non-linear partial differential equations of the form

where f(γ) and g(γ) are arbitrarily assigned functions. The circumstances are determined under which compatible solutions exist, not only when γ is real, but also when γ is complex, and all of the corresponding solutions are found. This is done by using a geometric technique that incorporates the equipotential surfaces of constant γ. In general, these surfaces are imaginary, and a fairly extensive treatment of such surfaces in (complexified) 3-dimensional Euclidean space is included. A close association is discovered between the set of equipotential surfaces and the class of surfaces of constant radii of principal normal curvature.

This is an exploration of the mechanical consequences of the symmetry inherent in a monatomic elastic crystal lattice. The investigation is based on a theorem by means of which the square of the right stretch tensor is uniquely decomposed. Generality is assured by the affine equivalence of all monatomic lattices. Response functions appropriate to perfect monatomic crystals are thereby constructed, and some simple deformation paths are considered.