Let S be both a topological space and a semigroup. For s ∈ S, define the maps λs and ρs of S to S by

We shall say that S is a right-topological (resp. left-topological) semigroup if ρs (resp. λs) is continuous for each s in S. We denote by Λ(S) the set

this is a subsemigroup of S.

Let $P$ be the self-similar probability corresponding to an iterated function system of contracting similitudes $S_1,\ldots,S_N$ with probabilities $p_1,\ldots,p_N$ and contraction constants $s_1,\ldots, s_N$. Let $D_r$ be the quantization dimension of $P$ of order $r\in(0,+\infty)$, and let $P_r$ be the self-similar probability corresponding to $S_1,\ldots,S_N$ with probabilities $(p_1s_1^r)^{D_r/(r+D_r)},\ldots,(p_Ns_N^r)^{D_r/(r+D_r)}$. It is shown that the quanitzation coefficient of $P$ of order $r$ exists and that the empirical measures on an arbitrary sequence of asymptotically optimal quantizing sets (of order $r$) weakly converges to $P_r$ provided $\log(p_1s_1^r),\ldots,\log(p_Ns_N^r)$ is not arithmetic and $S_1,\ldots,S_N$ satisfies the open set condition.

The magnetic field observable outside a body of conducting fluid in which field is imbedded may be considerably altered by convection currents in the fluid. One possible explanation of the geomagnetic secular variation foci is that localized convection cells in the earth's core disturb the main field present. An analytic solution for such a process is readily obtained by assuming the form and dimensions for such a cell, and shows that the magnitude of the secular variation cannot easily be explained on these lines without the presence of a subsurface toroidal magnetic field of some hundreds of gauss which is ‘convected through’ the surface of the core.

In papers (7, 8), Vizing studied graphs which are critical with respect to edge colourings. In particular, he obtained bounds on the number of edges and on the circuit length of such graphs in terms of its maximum valency. The object of this paper is to improve these bounds by obtaining others which also take into account the order of the graph.

In solid mechanics an outstanding lacuna is a general method for solving problems of plane strain for materials other than classically elastic, or visco-elastic, or rigid–plastic. Even with restriction to incrementally linear response, and to infinitesimal deformations superimposed on uniform pre-strains, there is no body of theory with the scope and power of the complex variable approach in isotropic elasticity. This lack has been highlighted recently by the growing interest in bifurcation phenomena in finitely deformed materials. So far, all analytical solutions to plane boundary-value problems with bifurcations (1,2,3,4) have relied on a separation-of-variables technique, which succeeds only for certain types of data.

In a previous paper (3), to which this is a sequel, a central limit theorem was presented for the homogeneous additive processes defined on a finite Markov chain, a class of processes treated extensively by Miller (4). A typical homogeneous process {R(t), X(t)} takes its values in the space

and is described by a vector distribution F(x, t) with components

and an increment matrix distribution B(x) governing the transitions. The present paper treats the motion of the process in the same space when its homogeneity is modified by the presence of a set of boundary states in x. Such bounded processes have many applications to the theory of queues, dams, and inventories. Indeed, this paper and its predecessor were motivated initially by a desire to discuss queuing systems with many servers and many service phases. We will treat both absorbing boundaries and associated passage time densities, and reflecting boundaries. For the latter our main objective is an asymptotic discussion of the tails of the ergodic distribution.

We introduce Γ-homology, the natural homology theory for E∞-algebras, and a cyclic version of it. Γ-homology specializes to a new homology theory for discrete commutative rings, very different in general from André–Quillen homology. We prove its general properties, including at base change and transitivity theorems. We give an explicit bicomplex for the Γ-homology of a discrete algebra, and elucidate connections with stable homotopy theory.

Tricot [27] provided apparently dual representations of the Hausdorff and packing dimensions of any analytic subset of Euclidean d-space in terms of, respectively, the lower and upper pointwise dimension maps of the finite Borel measures on ℝd. In this paper we show that Tricot's two representations, while similar in appearance, are in fact not duals of each other, but rather the duals of two other ‘missing’ representations. The key to obtaining these missing representations lies in extended Frostman and antiFrostman lemmas, both of which we develop in this paper. This leads to the formulation of two distinct characterizations of dim (A) and Dim (A), one which we call the weak duality principle and the other the strong duality principle. In particular, the strong duality principle is concerned with the existence, for each analytic set A, of measures on A that are (almost) of the same exact dimension (Hausdorff or packing) as A. The connection with Rényi (or information) dimension and a variational principle of Cutler and Olsen[12] is also established.

It is shown that a spacetime with collisionless matter evolving from data on a compact Cauchy surface with hyperbolic symmetry can be globally covered by compact hypersurfaces on which the mean curvature is constant and by compact hypersurfaces on which the area radius is constant. Results for the related cases of spherical and plane symmetry are reviewed and extended. The prospects of using the global time coordinates obtained in this way to investigate the global geometry of the spacetimes concerned are discussed.

An important unsolved problem in the theory of non-linear oscillations is to establish the boundedness or unboundedness of the general solution of

where dots denote differentiation with respect to t. When p(t) is periodic, we may seek periodic solutions. This search is interesting for its own sake, and of course leads us to special bounded solutions. In three previous papers (2, 3, 4) I have exhibited the equation

as tractable: on the assumption that e(t) is even and periodic, it was shown that the equation has an infinity of periodic solutions.

Let A be the generator of an analytic C0-semigroup on a Banach space X. We associate a closed operator ${\cal A}_{1}$ with A defined on Rad(X) and show that when X is a UMD-space, the Cauchy problem associated with A has maximal regularity if and only if the operator ${\cal A}_{1}{\rm g}$ generates an analytic C0-semigroup on Rad(X). This allows us to exploit known results on analytic C0-semigroups to study maximal regularity. Our results show that ${\cal R}$-boundedness is a local property for semigroups: an analytic C0-semigroup T of negative type is ${\cal R}$-bounded if and only if it is ${\cal R}$-bounded at z = 0. As applications, we give a perturbation result for positive semigroups. Finally, we show the following: when X is a UMD-space, T is an analytic C0-semigroup of negative type, then for every $f\in L^{p}(\RR_{+}; X)$, the mild solution of the corresponding inhomogeneous Cauchy problem with initial value 0 belongs to $W^{\theta,p}(\RR_{+}; X)$ for every $0<\theta < 1$.

The paper is concerned with linear second-order differential equations in one dimension. The arguments are developed for these equations in general and the examples given are drawn from quantum mechanics, where the accuracies required are in general higher than in classical mechanics and in engineering. An examination is made of the convergence of the eigenvalue Λ(h) of the corresponding finite difference equations towards the eigenvalue λ of the differential equation itself and it is shown that

where h is the size of the interval of the grid covering the range of the independent variable; the constant ν is usually a negative number and consequently Λ(h) may well be a lower bound to λ. This convergence property is used in the numerical calculation of λ by a simple extrapolation technique to a high degree of accuracy. Three examples are given of bounded problems in quantum mechanics. The corresponding eigenfunction can be calculated by a refinement of the familiar relaxation technique by using differences higher than the second, and an example is given.

For a classical knot K in the 3-shere, the unknotting number u(K) is defined to be the smallest number of crossing changes required to obtain the unknot, the minimum taken over all the regular projections. This dependence on projection makes the unknotting number a difficult knot invariant. While some algebriac methods exist to give a lower bound for u(K) the unknotting number for approximately one-sixth of the 84 knots with nine or fewer crossings remains unddetermined, see [9] or [7]. For an upper bound, one is usually forced to intellgently experiment knotting various projections of the knot under study. Usual practice is to work with a minimal crossingprojection; indeed, it has long been a ‘folk’ conjecture that the unknotting number is realized in such a projection ([5], p. 21). This note shows by example that this conjecture is false. This remarkable knot is rational, i.e. a 2-bridge knot, and hence alternating. Thus there is also the surprising result that the unknotting number of an alternating kniot is not necessarily realized in a minimal alternating projecting.

Using the results of the preceding paper of this series (by M. Bradburn), the equation of state and the elastic constants for a monatomic face-centred cubic lattice are calculated. Central forces with a potential of the form −ar−m+br−n are assumed to act between the particles. Numerical results are obtained for five sets of values (m, n) and represented in tables and diagrams. The general features of these are discussed and compared with previous computations.

I take this opportunity of thanking Prof. Born who suggested this problem to me, for his interest in my work and his advice on many occasions. I am also indebted to Dr R. Fürth for his numerous helpful suggestions.

Given an ‘extraordinary cohomology theory’, that is, a cohomology theory satisfying all the axioms of Eilenberg and Steenrod (7) except the dimension axiom, it is well known that there exists a spectral sequence relating the ordinary cohomology of a space with the extraordinary theory (see, for example, (3) in the case of K*(X)). Obviously, it would be useful to know the differentials in this spectral sequence, and it is the purpose of this paper to identify them in terms of cohomology operations defined by certain k-invariants. We shall make use of E. H. Brown's recent theorem (5) on the representability of extraordinary cohomology theories, to construct a second spectral sequence, in which the differentials are readily identifiable, which we shall prove is isomorphic to the usual one.

In most theories for the construction of finite groups with given properties a major difficulty is the ‘isomorphism problem’, which consists of specifying how one representative of each class of isomorphic groups may be selected from the totality of groups constructed by the process laid down. To do this we need a practical criterion for the isomorphism of two constructed groups. The main object of the present paper is to establish such a criterion in a particular case, which in spite of its simplicity is important because it gives a method for the construction of A-groups (i.e. soluble groups whose Sylow subgroups are all Abelian). All soluble groups of cube-free order are included in this class of groups, and to exemplify the application of the criterion we summarize that part of our unpublished dissertation (7) which deals in detail with the groups of order 22.32.52.

The theoretical problem of the probable time of recurrence of a given state of a physical system which changes according to a specified random or stochastic process is in general not a very tractable one, but many situations are covered by the more particular methods and formulae given in this paper. Several of these are believed new, but others are also included for reference. We shall usually limit our attention to homogeneous processes, i.e. the transition probabilities do not depend explicitly on the time. Even so, a generally useful solution of the complete distributional problem of recurrence times would appear rather unlikely; for example, it would have to include as a special case the recurrence theory for trigonometrical or power series, problems of some complexity which have separately engaged the attention of mathematicians (see, for example, Kac (7)). Thus a manageable simplification of the complete distributional problem emerges only in particular but important classes of physical problems classifiable theoretically as Markoff processes or, somewhat more generally, as ‘renewal’ processes (cf. Feller (4)). In other cases, if we are content with mean values (which do not always exist), and are dealing with stationary processes with complete ergodic properties, such as a statistical mechanical system in equilibrium, the simple but powerful arguments originally used by Smoluchowski (11) are available. We first formulate the probability problem in symbolic terms, being more concerned in this paper with general structural relations than with special points of rigour, or specific applications.

An analytical study is made of the boundary-layer approximation for the steady laminar flow of a viscous incompressible fluid past a paraboloid of revolution. The functional form of the vorticity distribution is found and shown to involve exponential decay with distance from the surface of the body, and inequalities are established concerning the displacement thickness.

The exact results are compared with various linearizations. The Oseen linearization (12) turns out to be incorrect except in the limit of vanishing Reynolds number; this result is connected with the infinite drag of any paraboloid of non-zero thickness. An alternate linearization, related to a method of Burgers (3) and allowing for the displacement effect of the body and of its boundary layer, is shown to give the correct functional form for these quantities. Numerical results are included which bear out these conclusions.

It is shown that the space-time region inside an axially symmetric, infinite, rotating, cylindrical mass distribution is necessarily Minkowskian.

Let Bn denote the open unit ball in Cn. We write V to denote Lebesgue volume measure on Bn normalized so that V(Bn)=1. Fix −1<γ<∞ and let Vγ denote the measure given by dVγ(z)=cγ (1−[mid ]z[mid ]2)γdV(z), for z∈Bn, where cγ=Γ(n+γ+1)/ (n!Γ(γ+1)); then Vγ(Bn)=1. The weighted Bergman space A2,γ(Bn) is the space of all analytic functions in L2(Bn, dVγ). This is a closed linear subspace of L2(Bn, dVγ). Let Pγ denote the orthogonal projection of L2(Bn, dVγ) onto A2,γ(Bn). For a function f∈L∞(Bn) the Toeplitz operator Tf is defined on A2,γ(Bn) by Tfh=Pγ(fh), for h∈A2,γ(Bn). It is clear that Tf is bounded on A2,γ(Bn) with ∥Tf∥[les ]∥f∥∞. In this paper we will consider the question for which f∈L∞(Bn) the operator Tf is compact on A2,γ(Bn). Although a complete answer has been given by the author and D. Zheng (see the next section), the condition for compactness is somewhat unnatural. In this article we will give a more natural description for compactness of Toeplitz operators with sufficiently nice symbols. We will describe compactness in terms of behaviour of the so-called Berezin transform of the symbol, which has been useful in characterizing compactness of Toeplitz operators with positive symbols (see [5, 9]). Before we can define this Berezin transform we need to introduce more notation.