The free field equations for particles with spin are invariant under a group SL(2, c) whose transformations correspond to changes of representation of the twocomponent spinor algebra. The generalization of the equations which extends this invariance to a guage invariance in the Yang–Mills sense necessitates the introduction of auxiliary fields (which are also necessary to maintain Lorentz covariance). These fields can be interpreted as the potentials of a spin-2 field, just as the auxiliary fields for the charge gauge group are the potentials of a spin-l field (electromagnetism); this spin-2 field is then self-interacting. The Bargmann–Wigner formulation of the linear spin-2 field, when modified by the proposed self-interaction, yields a non-linear theory of a spin-2 field which is shown to be identical with Einstein's gravitational theory. With this interpretation the auxiliary fields take on an extra role of Yang–Mills field for the general coordinate transformation group – that is, they are the components of the affine connexion.

We consider the steady co-current flow of two immiscible viscous liquids in a horizontal pipe, the fluid interface being ripple-free and plane. It is shown that if the cross-section of the duct is symmetric with respect to the interface, the velocity distribution may be expressed in terms of two separate pipe-flow solutions. One corresponds to the flow of a single fluid occupying the whole of the pipe, and the second to a similar flow in a pipe whose cross-section coincides with that of the region occupied by one fluid in the two-phase motion. Flow rates and other quantities of engineering interest are evaluated, and several particular flows are discussed.

1. Introduction. Let X be a complex Banach space and let (X) denote the Banach algebra of all bounded linear operators on X. In this paper we study subalgebras of (X) that contain non-zero compact operators and that are Banach algebras with respect to some norm dominating the operator norm. Our main aim is to give conditions for the existence of minimal one-sided ideals in . Barnes ((l) Theorem 2·2) has shown that if every element of a semi-simple Banach algebra has countable spectrum then the algebra has minimal one-sided ideals. If, in particular, is a uniformly closed sub-algebra of the compact operators on X, then every element of has countable spectrum by the Riesz–Schauder theory and so has minimal one-sided ideals. We extend this latter result by showing that if is a semi-simple uniformly closed subalgebra of (X) that contains some non-zero compact operator, then has minimal one-sided ideals. The proof depends heavily on the fact (see e.g. Bonsall (3)) that the spectral projection at a non-zero eigenvalue of a compact operator T belongs to the least closed subalgebra of (X) that contains T. The uniform norm is quite critical for Bonsall's result, but we are able to give a mild generalization which leads to conditions for the existence of minimal one-sided ideals in subalgebras of (X, Y,〈,〉) where (X, Y,〈,〉) are Banach spaces in normed duality.

1. Introduction. Let {Xk: k ≥ 1} be a sequence of independent, but not necessarily identically distributed, random variables. Suppose that the random variables Xn are uniformly bounded by a random variable X in the sense that

(1) P(|Xn| ≥x) ≤ P(|X| ≥ x)for all x > 0. Write qn(x) = P(|Xn| ≥ x) and q(x) = P(|X| ≥ x).

The classical theory of electromagnetism with both electric and magnetic charges is discussed. Following Gamblin(l) we proceed from an action principle and investigate the resulting particle equations. A consistent theory in which electrically and magnetically charged particles interact amongst themselves and with each other, without any constraints, is shown to be possible, in contrast to the conclusions of Gamblin. The introduction of constraints is considered.

1. Introduction. Let P be a collection of functions mapping some set into the complex plane C and let δ be a subset of C. A function F: δ → C is said to operate on P provided the composition F o φ belongs to P whenever φ ∈ P and the range of φ is contained in δ. The basic problem is to characterize the functions which operate.

1. Introduction. The relationship between sufficient statistics and the exponential family was first investigated by Pitman (8) and Koopman (7). The treatments assumed a density function f(x; θ) which was at least differentiable with respect to both arguments.

It is shown that the formula for the gravitational red shift predicted by the theory of general relativity can also be derived by classical quantum mechanics combined with relativistic arguments. The agreement between the two derivations is a consequence of the separability of the time-dependent wave function, and of the first-order time differential in the wave equation.

The investigation of a recently devised model in which the permittivity profile takes the form of an asymmetrical Epstein layer is adapted for use in a gyrational medium. Reflexion and transmission coefficients for normally incident electromagnetic waves are obtained and resonance properties are discussed.

A number of special results exist for summability methods B which, include Riesz summability (R,λ,k)—for example, when B is generalized Abel summability (A,λ,ρ) [Kuttner(5)], or Riemann summability (,λ,μ) [Russell(14)], or Riemann-Cesàro summability (,λ,p,α) [Rangachari(12)], or generalized Cesàro summability (C,λ,k) [Meir (9); Borwein and Russell (l)]. The question of necessary and sufficient conditions to be satisfied by an arbitrary method B in order that B ⊇ (R,λ,k) has received an answer only for limited values of λ and k—for example, by Lorentz [(6), Theorem 10] for k = 1; the restrictions on λ in this case were removed by Maddox [(8), Theorem 1]. Thus (apart from the well-known case k = 0) the case k = 1 is the only one for which a complete solution exists, though application of a theorem of Russell [(13), Theorem 1A] yields one form of a result for 0 < k ≤ 1. Maddox's results, however, suggest an alternative form capable of generalization to all k ≥ 0, and in this paper we obtain a complete solution for 0 < k ≤ 1 in that form, without restriction on λ. We first recall the following definitions.

1. Definitions and notations. Let be a given infinite series with the sequence of its partial sums {sn}. Let {pn} be a sequence of constants, real or complex, and let us write

A geodesic in space-time is complete if it can be extended to infinite values of its affine parameter: it is shown that all strongly causal spaces are conformal to space-times in which all null geodesies are complete, and that a wide class of space-times are conformal to ones in which almost no null geodesies are complete.

1. A theorem of Montel (14), states that if f(z) is analytic and bounded in the half-strip

and if there exists an x0 in (a, b) such that

as y → ∞, then

as y → ∞ l. u. on (a, b)(locally uniformly on (a, b), i.e. uniformly on every compact subset of (a, b)). Bohr (3), has proved a version of this result applicable to functions analytic, but not necessarily bounded, in S(a, b) and Hardy, Ingham and Pólya (10), have considered whether or not f(z) can be replaced by |f(z)| in (1·1) and (1·2). They show that this is not so but prove that if f(z) is analytic and bounded in S(a, b) and if

as y → ∞ for three distinct values x1, x2 and z3 in (a, b) then there exist constants A and B such that

as y → ∞, l.u. on (a, b). Cartwright (Theorem 5, (6)) has proved that if f(z) is analytic and satisfies |f(z)| < 1 in, S(a, b) and if for some x0 in (a, b)

asy → ∞, then

as y → ∞ co, l.u. on (a, b) and Bowen (5), has shown that if (1·4) is replaced by the weaker condition

as n → ∞ for some suitable sequence of points Zn in S(a, b) then (1·5) is still valid. In sections 2–5 of this paper we shall consider whether or not these results are valid |f(z)| is replaced by a subharmonic function.

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

If u is subharmonic in some open subset of Euclidean space, RN (N ≥ 1), which contains the ball |x| ≤ ρ and if S(r) and B(r) denote the mean values of u taken over the sphere |x| = r and the ball |x| ≤ r respectively (0 ≤ r ≤ ρ) then, ((2), p. 196),

The following work arises from the study of a rotating fluid contained in a rectangular vessel whose depth depends on one coordinate only.

Keller and Rubinow(l) have considered the force on a spinning sphere which is moving through an incompressible viscous fluid by employing the method of matched asymptotic expansions to describe the asymmetric flow. Childress(2) has investigated the motion of a sphere moving through a rotating fluid and calculated a correction to the drag coefficient. Brenner(3) has also obtained some general results for the drag and couple on an obstacle which is moving through the fluid. The present paper is concerned with a similar problem, namely the axially symmetric flow past a rotating sphere due to a uniform stream of infinity. It is shown that leading terms for the flow consist of a linear superposition of a primary Stokes flow past a non-rotating sphere together with an antisymmetric secondary flow in the azimuthal plane induced by the spinning sphere. For a3n2 > 6Uv, where n is the angular velocity of the sphere, U the speed of the uniform stream, and a the radius of the sphere, there is in the azimuthal plane a region of reversed flow attached to the rear portion of the sphere. The structure of the vortex is described and is shown to be confined to the rear portion of the sphere. A similar phenomenon occurs for a sphere rotating about an axis oblique to the direction of the uniform stream but the analysis will be given in a separate paper.

Let p = 1 + 5n be a rational prime congruent to 1 (mod 5). Let ζ = e2πi/p and let g be a primitive root mod p. Let the non-zero residues g, g2, …, gp-1 (mod) p be divided into five classes , ℬ, , , ℰ, where gν ∈ , ℬ, , , ℰ according as ν ≡ 0, 1, 2, 3, 4 (mod 5). Let

be the 5-nomial periods. Then it is well known (see (3)) that they are the roots of a monic polynomial with integral coefficients. Our object is to determine these coefficients in terms of the quantities A, B, C, D, E, Y, Z considered in a previous paper (2), p. 65. A large number of relations connecting these quantities have been obtained in the above-mentioned paper and we shall use these relations to simplify the coefficients and get them in a reasonably compact and symmetrical form.