In this paper, we apply Stein's method for distributional approximations to prove a quantitative form of the Erdös–Kac Theorem. We obtain our best bound on the rate of convergence, on the order of log log log n (log log n)−1/2, by making an intermediate Poisson approximation; we believe that this approach is simpler and more probabilistic than others, and we also obtain an explicit numerical value for the constant implicit in the bound. Different ways of applying Stein's method to prove the Erdös–Kac Theorem are discussed, including a Normal approximation argument via exchangeable pairs, where the suitability of a Poisson approximation naturally suggests itself.

In this note we determine the greatest lower bound of the number of independent edges in a graph in terms of the number of vertices, the minimal degree and the maximal degree. More generally we examine the greatest lower bound of the number of independent edges in a K-connected (or λ-edge-connected) graph with given number of vertices and given minimal and maximal degrees. These results extend those of Erdös and Pósa (3) and Weinstein (7). Our proof makes heavy use of Berge's slight extension of Tutte's characterization of graphs with 1-factors.

The protective unitary representations of the mapping class groups of surfaces corresponding to the Jones–Witten topological quantum field theory for SU(2) are expressed as representations in algebras of skeins in the surface. The skein-theoretic construction of the representations uses neither Kirby's surgery theorem nor a presentation of the group. Using these representations and the Reidemeister–Singer classification of Heegaard splittings gives a proof of the existence of the moduli of the Witten invariants of 3-manifolds.

The notion of fibre product in a category is quite basic and has been studied by many authors. Also in ring theory, it is known that the fibre product is useful in the construction of examples. (See for example [3], [4] and references of [1].) Unfortunately, most such examples are non-noetherian and so are unsatisfactory from the viewpoint of commutative algebra.

In order to determine the mathematical form of the distribution of gene frequencies in a population, R. A. Fisher (1, 2) applied two different but complementary methods. First, differential equations were set up for certain cases and solved. An extension of these results arose from the expression by Wright (10,11,12) of the conditions as an integral equation. But strictly, neither integral nor differential equations can represent the conditions exactly, since integration is substituted for summation, and differentials for minimum steps (1/2n) in gene frequency. The second method employed by Fisher was exact, even though the solutions involved new functions. The properties of generating functions were utilized to yield functional equations, and the special case of rare mutations was dealt with in detail. In the present paper this powerful method is extended to the general case where mutation rates are not necessarily small.

In this paper we consider topological properties of ordinary as well as graph directed iterated function systems. First we give the basic definitions.

Apart from the study of the integral

where {X(u)} is a stationary Gaussian process with autocorrelation function ρ(t), by Kac and Siegert(1), most stochastic functionals of the general type

have been considered for {X(u)} either additive or Markovian (see, for example, (2), (3)), and in the Markovian case only for diffusion-type processes (Darling and Siegert (4)). More general approaches exist (e.g. (5), (6)), but seem less concerned with the investigation of specific problems. Some preliminary remarks here are therefore aimed at examining the structure of integrals of type (2), or such further extensions of the formal Riemann sum type

that would be expected to have well-behaved distributional properties for {X(t)} Markovian, and associated equations for studying these properties. As an example, the sum

say, is considered (i) for the normal linear Markovian process, (ii) for simple birth-and-death and emigration—immigration processes.

In this paper we investigate the local form of the tangent developable of a curve in 3-space, and obtain results relating this to the order of vanishing of the torsion. In particular,

Theorem 1. If at the point γ(t) on the curve γ, the torsion vanishes to order k (0 ≼ k ≼ 4), then the germ of the tangent developable at that point has one curve of self intersection if k is odd, and none if k is even.

We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called a locally cubical bicategory, this being a bicategory enriched in the monoidal 2-category of pseudo double categories. Finally, we show that every sufficiently well-behaved locally cubical bicategory gives rise to a tricategory, and thereby deduce the existence of a tricategory of tricategories.

A tempered distribution on ℝ whose Fourier transform is supported in an interval [−Ω,Ω], where Ω>0, can be characterized by the behaviour of its successive derivatives. On the other hand, a tempered distribution on ℝ whose Fourier transform vanishes in an interval (−Ω,Ω), where Ω>0, can be characterized by the behaviour of a particular sequence of successive antiderivatives. Similar considerations apply to general convolution operators acting on J′(ℝn) and yield characterizations for tempered distributions having their Fourier transforms supported in sets of the form or , where and Ω>0.

Explicit formulae are given for calculating the matrix elements of the irreducible representations of the three-dimensional pure rotation group by the direct method. In addition explicit formulae are derived to calculate the representations of the finite elements of any group when the eigenvalues of the matrix representing the corresponding infinitesimal elements are given.

1. When crystals of a metal are highly perfect, their elastic limit is low, plastic flow taking place when the applied shear stress is very small. The elastic limit reaches higher values as the perfect structure is progressively broken up by cold-work. A steady state is finally reached where further distortion of the metal does not increase the elastic limit. The stress at which a metal passes beyond the elastic limit and begins to yield is ill-defined, as in general the crystal begins to ‘creep’ appreciably near this point and the time element enters into the definition of the stress-strain curve. Nevertheless, there is a fairly well-defined stress beyond which a metal, which has been brought into a steady state by being saturated with cold-work, ceases to behave elastically and undergoes permanent deformation. An attempt is made in this paper to derive an expression for the ultimate elastic limit or yield stress of a cold-worked metal, in terms of its structure and elastic constants.

We describe the integral equivariant cohomology ring of a weighted projective space in terms of piecewise polynomials, and thence by generators and relations. We deduce that the ring is a perfect invariant, and prove a Chern class formula for weighted projective bundles.

The equation to be considered is of the type

where p (x) is continuous for all real values of x, even, and periodic. It is no restriction to suppose that the period is π, and this assumption will be made, so that the equation is virtually Hill's equation.

The protons emitted from boron, when it is bombarded by α-particles from radium C′ and polonium, have been examined. All have been ascribed to the disintegration of the B10 isotope following the capture of an α-particle. Four energy changes with values 3·1 m.v., 0·4 m.v., − 0·1 m.v., and − 1·0 m.v. are necessary to explain the range distribution observed. Experiments with α-particles from polonium also show that penetration through the top of the potential barrier stops at an energy of 3·6 m.v. and that below this there is a resonance level at about 2·9 m.v.

It is shown that the theorem stated in Born's paper, and proved for the case of a linear lattice of N equal particles under certain restrictions concerning the forces between the particles, that macroscopic stability (stability for long waves) implies microscopic stability, may be extended to three dimensions for the particular case of a face-centred cubic lattice, where the effects of all neighbours, other than the first twelve neighbours, are neglected.

I take this opportunity of expressing my sincere thanks to Prof. Born for much valuable advice.

In the early sixties Effros[9] and Prosser[14] studied, in independent work, the duality of the faces of the positive cones in a von Neumann algebra and its predual space. In an implicit way, this work was generalized to certain ordered Banach spaces in papers of Alfsen and Shultz [3] in the seventies, the duality being given in terms of faces of the base of the cone in a base norm space and the faces of the positive cone of the dual space. The present paper is concerned with the facial structure of the unit balls in an ordered Banach space and its dual as well as the duality that reigns between these structures. Specifically, the main results concern the sets of norm-exposed and norm-semi-exposed faces of the unit ball V1 in a GL-space or complete base norm space V and the sets of weak*-exposed and weak*-semi-exposed faces of the unit ball in its dual space V* which forms a unital GM-space or a complete order unit space.

Let (Gi | i ∈ I) be a family of groups, let F be a free group, and let the free product of F and all the Gi.

Let denote the set of all finitely generated subgroups H of G which have the property that, for each g ∈ G and each i ∈ I, By the Kurosh Subgroup Theorem, every element of is a free group. For each free group H, the reduced rank of H, denoted r(H), is defined as To avoid the vacuous case, we make the additional assumption that contains a non-cyclic group, and we defineWe are interested in precise bounds for . In the special case where I is empty, Hanna Neumann proved that ∈ [1,2], and conjectured that = 1; fifty years later, this interval has not been reduced.

With the understanding that ∞/(∞ − 2) is 1, we define

Generalizing Hanna Neumann's theorem we prove that , and, moreover, whenever G has 2-torsion. Since is finite, is closed under finite intersections. Generalizing Hanna Neumann's conjecture, we conjecture that whenever G does not have 2-torsion.

The effect on the boundary-layer equations of a weak shock wave of strength ∈ has been investigated, and it is shown that if R is the Reynolds number of the boundary layer, separation occurs when ∈ = o(R−i). The boundary-layer assumptions are then investigated and shown to be consistent. It is inferred that separation will occur if a shock wave meets a boundary and the above condition is satisfied.

The requirements of Kolmogoroff's theory of the equilibrium spectrum are satisfied only at very high Reynolds numbers, higher than any at which experiments have yet been done. In particular, when the theory holds, the rate of decay of the mean-square vorticity ω must be negligible compared with either its rate of increase due to the stretching of the vortex filaments or the rate of dissipation due to viscosity.

An extended version of Kolmogoroff's hypothesis may be proposed, in which the statistical properties of the turbulence in a range of wave-numbers (range of eddy sizes) depend not only on the rate of dissipation ∈ per unit volume and the viscosity ν, but also on the time rate of change d∈/dt of ∈. The result is to introduce a dependence on the Reynolds number R of the turbulence into quantities and constants which, on Kolmogoroff's original hypothesis, were independent of R. The Reynolds number R is defined from the decay law; u2t, with an origin of time suitably chosen, is a function of t, finite when t = 0, and R is defined as (u2t)t=0/ν. Lin's decay law follows logically from the extended hypothesis, according to which the rates of change of ω−1, due to the causes mentioned above, are constant during the decay; Lin's decay law would also follow if this less general part only of the extended hypothesis be assumed. The same decay law is also obtained if the similarity spectrum of Heisenberg is taken to apply not to the whole of the energy-bearing eddies, but only to the energy-dissipating eddies. But it is suggested that further generalization of the theory of the similarity spectrum, and of the decay law, is necessary; that the similarity spectrum is probably only asymptotically correct for a range of large wave-numbers, the range depending on the initial conditions and decreasing as the decay proceeds; that the general decay law is u2t = μRd(t), where d(t) is an integral function of t, such that d(0) = 1, and with an asymptotic value for large t to give correctly the law of decay in the final period; and that d(t), and the number of constants needed to specify it approximately, depend on the initial conditions. An experiment is suggested to test the dependence of the law of decay on the initial conditions. It is also suggested that the recently observed constancy of u2t in the initial period in the turbulence behind a single grid is only approximate, and this approximate constancy still needs explanation. Remarks are also included on the range of application of the equilibrium spectrum, for which some formulae are given when there is a definite cut-off in the spectrum.