The two-dimensional flow in a jet, falling under gravity from a slot in a horizontal plane, is studied. The fluid is considered to be incompressible and inviscid; the flow is taken to be irrotational; and the reciprocal ε of the Froude number is considered to be small. By taking the complex potential as the independent variable we overcome the difficulty that the boundary geometry is not known in advance. The method of matched asymptotic expansions is applied. The first two terms of an inner asymptotic expansion and the first three of an outer one are found: the inner expansion is valid above and near the slot, but is inappropriate far downstream, while the outer expansion is valid far downstream, but fails to satisfy the conditions upstream. The two expansions are matched and ‘composite’ approximations, covering the whole flow field, are derived.

The classical divergence theorem for an $n$-dimensional domain $A$ and a smooth vector field $F$ in $n$-space \[\int_{\partial A} F \cdot n = \int_A div F\] requires that a normal vector field $n(p)$ be defined a.e. $p \in \partial A$. In this paper we give a new proof and extension of this theorem by replacing $n$ with a limit $\star \partial A$ of 1-dimensional polyhedral chains taken with respect to a norm. The operator $\star$ is a geometric dual to the Hodge star operator and is defined on a large class of $k$-dimensional domains of integration $A$ in $n$-space called chainlets. Chainlets include a broad range of domains, from smooth manifolds to soap bubbles and fractals. We prove as our main result the Star theorem \[ \displaystyle\int_{\star A} \omega = (-1)^{k(n-k)}\int_A \star \omega \] where $\omega$ is a $k$-form and $A$ is an $(n-k)$-chainlet. When combined with the general Stokes' theorem ([H1, H2]) \[\int_{\partial A} \omega = \int_A d \omega\] this result yields optimal and concise forms of Gauss' divergence theorem \[\int_{\star \partial A}\omega = (-1)^{(k)(n-k)} \int_A d\star \omega\] and Green's curl theorem \[\int_{\partial A} \omega = \int_{\star A} \star d\omega.\]

The aim of this article is to discuss a paradox of importance in the quantum theory of measurement. The paradox was first propounded by Einstein, Podolsky and Rosen in 1935(1), and was discussed subsequently by Bohr(2), Furry(3), Schrödinger(4, 5), as well as in texts (6, 7). We begin by formulating the paradox.

It is clear from § 1 that we shall have quite a large number of theorems to consider, as we may distribute the conditions of monotony and existence in a good many different ways. It is therefore convenient to collect together the various results for series and for transforms in tabular form, for reference and comparison; this is done at the end of this section.

The transform table is very simple. The theorems contained in it have been discussed at length in M.F.(I); they may be summarized by saying that (1) holds in all the cases considered, except perhaps in the consine case of section [B] where the truth is not known. The series results are apparently more complicated as well as more numerous, since there are extra conditions to be imposed in [A, 1], [B, 1], [C′, 1] and [C, 2]. However, these are certainly satisfied if we suppose f and g to be positive whenever they are given to be monotonic (a condition which automatically holds in the transform theorems, since in these the monotonic functions tend to zero at infinity). If we confine our attention to this case, the series results are what we should expect from the transform ones, except that the difficulty in section [B] for transforms is not reproduced for series in [B, 1] or [B′, 1]; it is reproduced in [B, 2].

This paper is in two parts. In Part I we are concerned with one or more linear series on an algebraic curve; we consider a set of points on the curve which are contained with assigned multiplicities in a set of each of the linear series and, by persistent use of Severi's equivalence relation for the united points of an algebraic correspondence with valency, we derive formulae for the number of such sets of points when the constants involved are such as to make this number finite. All this is essentially a generalization of the formula for the number of points in the Jacobian set of a linear series of freedom 1, and the main result is Theorem 3.

Posner (1) proved the following result:

Theorem. Let R be a prime ring and d a derivation of R such that, for all a∈R, ad(a)–d(a) a∈Z (centre of R). Then, if d is not a zero derivation, R is commutative.

The purpose of this note is to give a short proof of this theorem. We state the following known result ((1), Lemma 1).

In (1) and (2), Gould has discussed some generalized Vandermonde-type convolution identities, for C coefficients satisfying

where

and

In a note in the April 1942 issue of these Proceedings, Hodge gives a formula for the number of independent terms in what he calls a k-connex, but while conjecturing its general validity he proves it only for very restricted cases. Its truth is here demonstrated in the general case.

The application of the operational method of Oliver Heaviside to the solution of linear differential equations has been fully described in a recent Cambridge Tract by Dr H. Jeffreys.

The group Aut Γ of automorphisms of Rado's universal graph Γ (otherwise known as the ‘random’ graph: see [1]) and the corresponding groups Aut Γc for C a set of ‘colours’ with 2 ≤ |C| ≤ ℵ0, were studied in [4]. It was shown that Aut Γc is a simple group, and the possible cycle types of its members were classified. A natural extension of Aut Γc to a highly transitive permutation group on the same set is obtained by considering the ‘almost automorphisms’ of Γ. It is the purpose of the present paper to answer similar questions about the resulting group AAut Γc. Namely we shall classify its normal subgroups and the cycle types of its members. The main result on normal subgroups is summed up in Corollary 2·9, which says that the non-trivial normal subgroups of AAut Γc form a lattice isomorphic to the lattice of subgroups of the free Abelian group of rank n where n = |C| – 1, and for cycle types it will be shown that those occurring in AAut Γc are precisely the same as in Aut Γc except for those which are the product of finitely many cycles.

An algebra of Colombeau generalized ultradistributions in which the corresponding space of ultradistributions is embedded via regularizations and in which the multiplication of ultradifferentiable functions of an appropriate class is the ordinary multiplication is constructed. This enables the development of the local and micro-local analysis in the framework of a new scale of Colombeau algebras. Sufficient conditions for a generalized ultradistribution being an ultradifferentiable function are given.

We study the free group of rank two from the point of view of Stallings core graphs. The first half of the paper examines primitive elements in this group, giving new and self-contained proofs for various known results about them. In particular, this includes the classification of bases of this group. The second half of the paper is devoted to constructing a counterexample to a conjecture by Miasnikov, Ventura and Weil, which seeks to characterize algebraic extensions in free groups in terms of Stallings graphs.

1. The theoretical study of the transport phenomena in metals requires the solution of a certain integral equation, which is formed by equating the rate of change of the distribution function of the conduction electrons due to the applied fields and temperature gradients to the rate of change due to the mechanism responsible for the scattering. This integral equation has so far been solved only in the simplest case where the electrons are assumed to be quasi-free, the energy being proportional to the square of the wave vector, and this assumption will be made throughout the present paper. The scattering of the electrons is due to two causes: the thermal vibration of the crystal lattice and the presence of impurities or strains. The scattering due to the impurities can always be described in terms of a free path l (or more conveniently a time of relaxation τ, were l = τν, ν being the average velocity of the conduction electrons), whereas for the scattering due to the thermal vibration this is possible only at high temperatures such that (Θ/T)2 can be neglected, where Θ is the Debye temperature (1). At very low temperatures the scattering due to the thermal vibration can be neglected compared with that due to the impurities. When a time of relaxation exists, the integral equation for the distribution function reduces to an ordinary equation, and its solution is then a comparatively simple matter. In the general case the problem is much more difficult, and no rigorous and generally valid solution has so far been obtained.

In a recent paper (2) Rankin has proved that for all positive integral n where As stated by Rankin, his proof is not simple although it is based on elementary ideas. Also, as noted by him, no other proof is available for any inequality of the form with λ a positive constant independent of n. It is the main purpose of this note to give a simple proof for the following result which implies the truth of (3) with λ = ¼, though not the truth of (1).

Let S be a normal projective algebraic surface over C with at worst quotient singularities. S is a quasi-log del Pezzo surface if the anti-canonical divisor — Ks is nef (= numerically effective) and big, i.e. — Ks. C ≥ 0 for all curves C on S and (−Ks)2 > 0. Further, if — Ks is ample we say S is a log del Pezzo surface.

This paper deals with wave propagation in inhomogeneous compressible electron plasmas. The waves are described by Maxwell's equations coupled to the linearized single-fluid equations of hydrodynamics. Coupled wave equations are derived which describe the propagation of coupled electromagnetic and electron acoustic waves. Results are obtained for generally inhomogeneous plasmas and are specialized to planar and cylindrically stratified media. Particular attention is given to the planar case and several approximate techniques for treating the equations are discussed. The fields in a region of coupling are investigated.

It is shown that slight departures from independence and equal weight of the sample observations introduce a bias into Fisher's z, but that the distributions of the variances “within and between groups”, and of Fisher's z would appear to maintain their form to the first order, although the extent to which the approximations hold in practice has not been considered.

It is a classical result (5) that the algebraic surfaces which admit a continuous group of birational self-transformations, or automorphisms, and which are neither rational nor scrollar, belong to one or other of two families: (i) the elliptic surfaces; these possess an elliptic group of ∞1 automorphisms; (ii) the hyperelliptic surfaces of rank 1 (which, for brevity, we shall call hyperelliptic); these possess a completely transitive permutable group of ∞2 automorphisms. The chief properties of the two classes of surface, in so far as they are required in the present work, are described in § 1.

We show that every locally compact Polish group is isomorphic to the isometry group of a proper separable metric space. This answers a question of Gao and Kechris. We also analyze the natural action of the isometry group of a separable ultrametric space on the space. This leads us to a structure theorem representing an arbitrary separable ultrametric space as a bundle with an ultrametric base and with ultrahomogeneous fibers which are invariant under the action of the isometry group.