The two-dimensional configuration is considered of a fixed, semi-infinite, vertical barrier extending downwards from a fluid surface and having, at some depth, a gap of arbitrary width. A train of surface waves, incident on the barrier, is partly transmitted and partly reflected. The velocity potential of the resulting fluid motion is determined by a reduction procedure and also by an integral equation formulation. It is shown that the two methods lead to the same Riemann–Hilbert problem. Transmission and reflexion coefficients are calculated for several values of the ratio gap width/mean gap depth.

A relationship between the Hamiltonian of a system and its distribution function in phase space is sought which will guarantee that the average energy is the weighted mean of the Hamiltonian over phase space. This relationship is shown to imply the existence of a wave function satisfying the Schrödinger equation, and dictates the possible forms of time-dependence of the distribution function.

It is well-known (see [2]) that the finite symmetric group Sn has rank 2. Specifically, it is known that the cyclic permutations

generate Sn,. It easily follows (and has been observed by Vorob'ev [9]) that the full transformation semigroup on n (< ∞) symbols has rank 3, being generated by the two generators of Sn, together with an arbitrarily chosen element of defect 1. (See Clifford and Preston [1], example 1.1.10.) The rank of Singn, the semigroup of all singular self-maps of {1, …, n}, is harder to determine: in Section 2 it is shown to be ½n(n − 1) (for n ≽ 3). The semigroup Singn it is known to be generated by idempotents [4] and so it is possible to define the idempotent rank of Singn as the cardinality of the smallest possible set P of idempotents for which <F> = Singn. This is of course potentially greater than the rank, but in fact the two numbers turn out to be equal.

In a recent note I attempted to obtain the postulation formula for the Grassmannian of k-spaces in [n] by the consideration of forms of a certain type in k + 1 sets of r + 1 homogeneous variables, which I called k-connexes. My attempt was not entirely successful; I obtained a formula for k-connexes which suggested what the required postulation formula should be, but was unable to prove it. D. E. Littlewood has now written a paper to show that my problem is intimately connected with the theory of invariant matrices, and has thereby established the truth of the postulation formula which I had conjectured. Littlewood's proof requires a considerable knowledge of the theory of invariant matrices, and this paper results from an attempt to re-write his proof in a form which is intelligible to a student not having this specialized knowledge. Prof. H. W. Turnbull has pointed out to me the importance of the so-called k-connexes in the theory of forms, particularly in connexion with the Gordan-Capelli series, and for this reason I am taking the k-connexes as the principal topic of this paper, leaving the deduction of certain postulation formulae which are the more immediate concern of a geometer to the end.

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.

Let Fp be the field of a prime order p and F*p be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the boundfor any subset A ⊂ Fp with 1 < |A| < p12/23. Then we apply our estimate to obtain explicit bounds for some exponential sums in Fp. We show that for any subsets X, Y, Z ⊂ F*p and any complex numbers αx, βy, γz with |αx| ≤ 1, |βy| ≤ 1, |γz| ≤ 1, the following bound holds:We apply this bound further to show that if H is a subgroup of F*p with |H| > p1/4, thenFinally we show that if g is a generator of F*p then for any M < p the number of solutions of the equationis less than . This implies that if p1/2 < M < p, then

Let f and h be transcendental entire functions and let g be a continuous and open map of the complex plane into itself with g∘f=h∘g. We show that if f satisfies a certain condition, which holds, in particular, if f has no wandering domains, then g−1(J(h))=J(f). Here J(·) denotes the Julia set of a function. We conclude that if f has no wandering domains, then h has no wandering domains. Further, we show that for given transcendental entire functions f and h, there are only countably many entire functions g such that g∘f=h∘g.

The title problem is solved by a semi-inverse method involving plane wave expansions. Expressions are given for stress singularities near the crack edge and for the radiation patterns produced by the crack. The case of an elliptical crack in an isotropic medium expanding under shear is examined in detail. Finally, some generalizations are discussed. Solutions are outlined for when the centre of the crack is allowed to move and for a self-similar phase transformation problem, in which a phase change takes place in an expanding ellipsoidal region.

Wright (14), Feller (2) and others have proposed various stochastic models of genetics to study the fluctuations of gene frequency under the influence of mutation and selection. One of their simplest models has the following structure. There are a fixed number N of gametes each of which may be of two types a or A. The process X(n), n = 0, 1, 2, … which is assumed to have stationary transitions, is said to be in state j when there are j gametes of type a, and N − j of type A. Let γ1 denote the probability that immediately after formation an a gamete mutates into an A gamete and let γ2 denote the probability of an A gamete mutating into an a gamete. Each of the gametes of the next generation is independently formed by making a random selection from the gametes of the present generation. The probability of a particular gamete in the next generation being of type a is then and of type A where j represents the state of the process. Roughly speaking, the chance of a mating resulting in a gamete of a prescribed kind for the next generation is proportional to the fraction of these gametes present in this generation allowing for mutation effects.

Let G be an extension of the normal subgroup K by its quotient group H. Suppose we are given free resolutions for H, K (see below for definition). We shall show how to construct from them, by a ‘twisted tensor product’, a free resolution for G, which may also be used to give the spectral sequence of the extension.

This paper obtains an exact description of the statistical mechanics of a one-dimensional system of charged particles moving in a uniform neutralizing background of positive charge. These results are compared with the behaviour of a one-dimensional system of equal numbers of positive and negative charged particles. Although the thermodynamics of the two systems do differ, the discrepancy is small enough to indicate that the assumption of a uniform background of positive charge, common in statistical mechanical treatments of a plasma in equilibrium, may provide a good approximation to a real system of discrete charges.

A number of important Markoff processes, with a continuous time parameter, can be represented approximately by a discrete process, interesting in its own right, of the following type. A class of individuals gives rise seasonally (in January say) to a number of new individuals (children), the probabilities of an individual having 0, 1, 2, … children being p0, p1, p2, …. These probabilities are the same for all individuals and are independent. The individuals formed each January are regarded as a new generation, and only this generation is capable of reproducing in the next January. Let

so that F(x) is the probability generating function (p.g.f.) of the number of children of an individual. Clearly the series for F(x) is absolutely convergent when |x| < |1.

Let S be a set and let {X1, …, Xn} = be a family of distinct subsets of S. The intersection graph Ω() of has vertex set {X1, …, Xn} and XiXj (i ≠ j) is an edge of Ω() if and only if Xi ∩ Xi ≠ Ø (c.f. (6)). It is easily seen, (7), that every graph is an intersection graph, in other words every graph can be represented by subsets ofa set. Moreover, it was proved by Erdös, Goodman and Pósa (5) that if a graph has n ≥ 4 vertices then one can find a representing set with at most [n2/4] elements. This assertion is an immediate consequence of the result, (5), that the edges of a graph with n ≥ 1 vertices can be covered with at most [n2/4] edge disjoint triangles and edges. We say that a graph G is covered with the subgraphs G1, …, Gk, if every edge of G is in at least one Gi. One of the aims of this note is to prove an extension of this result, pro-posed by Erdös (4).

When a probability distribution is specified by more than one parameter, the statistical information in a sample on an unknown α will usually depend on the values of the other unknowns. For large samples where the joint distribution of the estimates tends to normality, and the efficiency of the estimation to a maximum, little theoretical difficulty exists. In a discussion(1) of the more general theory of small samples, I have stressed the importance of properties of sufficiency.

The function τ (n), defined by the equation

is considered by Ramanujan* in his memoir “On certain arithmetical functions”

The associated Diriċhlet series

converges when σ = > σ0, for a sufficiently large positive σ0.

The wreath product is a useful method for constructing new soluble groups from given ones (cf. P. Hall (3)). Now although the wreath product of one soluble group by another is (obviously) always soluble, the corresponding result is no longer true for nilpotent groups. It is the object of § 3 of this note to determine precisely when the wreath product W of a non-trivial nilpotent group A by a non-trivial nilpotent group B is nilpotent; in fact I prove that W is nilpotent if and only if both A and B are (nilpotent) p–groups with A of finite exponent and B finite.

Statement of results. Let g(x) be a given real finite function of a real variable x. Let F(g) denote the class of all real Lebesgue-measurable and finite (though possibly unbounded) functions f(x) which satisfy where η may depend upon f Let F* denote the class of all real f(x) such that exists and satisfies − ∞ ≤l<+∞.

The two theorems in the title give conditions on Banach lattices E and F under which a positive operator from E into F, dominated by another positive operator with some property, must also have that property. The Dodds-Fremlin theorem says that this is true for compactness provided both E′ and F have order continuous norms, whilst the Kalton–Saab theorem establishes such a result for Dunford–Pettis operators provided F has an order continuous norm. These results were originally provided, in their full generality, in [3] and [5], respectively, whilst very readable proofs may be found in chapter 5 of [2] or §3·7 of [6].