We denote by ω the space of all complex sequences with the topology given by the semi-norms

where δn(x) = xn. An FK-space, E, is a subspace of ω on which there exists a complete metrizable locally convex topology τ, such that the inclusion (E, τ) ⊂ ω is continuous; if τ is given by a single norm then E is a BK-space.

Let ℘ be a Weierstrass elliptic function satisfying the differential equation

and let ζ(z) be the associated Weierstrass ζ-function satisfying (z) = −℘(z). Corresponding to a pair of fundamental periods ω1, ω2 of ℘(z), there is a pair of quasi-periods η1, η2 of ζ(z) defined by

and we have ηi = 2ζ(ωi/2) for i = 1, 2.

Let (λm), (µn) (m, n = 0, 1, 2,…) satisfy

respectively. Let vp (p = 0, 1, 2, …) be the sequence (λm+µn) arranged in ascending order, equal sums λm+µn being considered as giving just one vp Then for given formal series Σam, Σbn the formal series C = Σ cp where

is called the general Dirichlet product of Σamand Σbn (see Hardy (2), p. 239). When λn = µn = n we have the Cauchy product. In the case λn = logm, µn = logn (m, n = 1, 2,…) we have vp =log p(p = 1, 2, …)and it is natural to call C the ordinary Dirichlet product.

1. Let G be a compact Abelian group with character group X. Let be an increasing sequence of finite symmetric subsets of X, and consider a symmetric subset P of . For any Hermitian complex-valued function u on P, we write snu for the real-valued trigonometric polynomial . Edwards, Hewitt and Ross(4) investigated the following property for a non-void measurable subset W of G satisfying W ⊂ (int W)−:

The validity of this implication was shown to be independent of the choice of . Accordingly, if (*) holds, P is called an FZ(W)-set. If P is an FZ(W)-set for all W, then P is termed a full FZ-set or full Fatou-Zygmund set. In this paper, we characterize the full FZ-sets as FZ(G)-sets satisfying a certain algebraic condition. In particular, we show that if G is connected, then a symmetric subset of X is an FZ(G)-set if and only if it is a full FZ-set. Some of the techniques are adaptations of those of Mme Déchamps-Gondim(1), (2). The class of full FZ-sets is not always closed under the operation of finite unions; this contrasts with the situation for Sidon sets and for FZ(G)-sets.

The union of 2 disjoint sets of interpolation need not be of interpolation even if every finite subset is of interpolation with bounded constant.

Mills(7), Gordon(5), Birnbaum(1), and the author(6) have studied the ratio of the area of the standardized normal curve from x to ∞ and the ordinate at x. Des Raj (4) established the monotonic character of, and obtained lower and upper bounds for this ratio for the standardized type III population. This ratio, as shown by Cohen (2) and Des Raj (3), has to be calculated for several values of x when solving approximately the equations involved in the problem of estimating the para-meters of type III populations from truncated samples. When the areas and ordinates are small, either this ratio cannot be obtained from existing tables prepared by Salvosa(8) or that very few significant digits are available for its calculation. The object of this note is to obtain an upper bound which could satisfactorily locate this ratio over the range x > −1.

Recently Wang (5) has given some results concerning characterizations of Poisson, normal, gamma, binomial and negative binomial distributions. We shall show here that these results are incorrect and also attempt to rectify them.

In this paper we are concerned with solutions of the three-dimensional Helmholtz equation which are of class C2 (i.e. regular) in the exterior of a bounded domain D. In cylindrical polar coordinates (r, z, φ) such solutions satisfy the equation

in which we have dimensionalized the radial coordinate r so that the wave number is normalized to unity. If we further assume that u satisfies the Sommerfeld radiation condition

then u may be regarded as being generated by volume sources, surface sources, or point singularities, all of which are contained in D.

1. Introduction. In a recent paper (2) Daykin has made the Conjecture. If t is real, then

where

He has proved (1, 2) that (i) Φ(t) is a convex function of t, (ii) Φ(t)≥ n if t ≥ 0 or 2, and (iii) Φ(t) takes values arbitrarily close to ½n if n is even, x2 = x4 = … = xn = 1, and xl = x3 = … = xn-1 and t have suitable small positive values. It is known (3) that (1) is true if t = 1.

Consider a real-valued random walk

which is defined on a Markov chain {Xn: n ≥ 0} with countable state space I. We assume that a matrix Q(.) = (qij(.)) is given such that

If the series Σαn is convergent, it follows that Σαnzn is uniformly convergent in a Stolz angle at z = 1 ((7), p. 229). It was shown in (3) that for the Ces methods (C, α) with α > – 1, and for a certain class of general Nörlund means (containing (C, α) for 0 < α ≤ 1), summability of Σαn implies uniform summability in a Stolz angle at z = 1. In section 2 we prove this theorem for a class of methods which satisfy a mean value property ((6), p. 31), and for a wider class of Nörlund methods which includes (C, α) for all α > 0. An analog is also proved for absolute summability.

The phase space of a finite dimensional classical Hamiltonian system is a C∞ differentiable manifold M which carries a C∞ differential 2-form ω which is closed, dω = 0, and non-singular in the sense that there is a bijective map α→ Xα from covariant vector fields to contravariant vector fields satisfying the identity

Such a pair (M, μ) is called a symplectic manifold.

This paper is concerned with the steady plane flow of a viscous fluid in symmetrical channels with slowly curving walls. The product of local channel half-width and local wall curvature is bounded by a small parameter ∈. We review the essentials of the formal approximation, in powers of ∈, proposed in (4) and (5); resolve a question, left open there, regarding the existence of the approximate series for the stream function for any value of the Reynolds number and to arbitrary order in ∈ and prove that, under certain restrictions on the Reynolds number and the divergence angle of the channel walls, this formal series is in fact a strict asymptotic expansion (for ∈ → 0) of an exact solution of the Navier–Stokes equations. As a result, the traditional picture of laminar separation, due to Prandtl, emerges as part of the steady flow field predicted by an exact solution that is known explicitly to arbitrary asymptotic order.

In (6) Plancherel considered linear, bounded transforms which satisfy

where the linear, bounded transform is defined by

Among other things he proved: if W1, W2 and W3 are linear, bounded transforms satisfying (1), then

An analysis is made of the response of a laminar boundary layer to a perturbation, either in the mainstream, or of the boundary conditions at the wall. The disturbance propagates with the mainstream velocity, and the manner in which it decays at large distances downstream is determined by eigensolutions of the boundary-layer equations. The elucidation of the structure of these eigensolutions requires division of the boundary layer into three regions. Comparison of the asymptotic formula obtained for the displacement thickness is made with the numerical results of Ackerberg and Phillips (1).

An initial-boundary-value problem for the equation

is considered for x, t ≥ 0. This system is a model for long water waves of small but finite amplitude, generated in a uniform open channel by a wavemaker at one end. It is shown that, in contrast to an alternative, more familiar model using the Korteweg–deVries equation, the solution of (a) has good mathematical properties: in particular, the problem is well set in Hadamard's classical sense that solutions corresponding to given initial data exist, are unique, and depend continuously on the specified data.

The assumption that disturbances are confined to the neighbourhood of the wall in the problem of Part I is invalidated by an exact solution of the resulting equation which shows that disturbances rapidly travel out of the region in which the equation is applicable. The asymptotic methods of Part I are also applied to this equation, which was studied by Lam and Rott (1), and an assessment is made of the role of the eigensolutions found by these authors.

One of the topics covered by most textbooks on partial differential equations is the classification problem for second-order equations in the plane. In a typical treatment, it is shown that under some smoothness conditions on the coefficients, hyperbolic and parabolic equations can be reduced locally to normal form (uniformized) by an elementary procedure. For elliptic equations, the procedure fails unless an extraneous hypothesis (analyticity of the coefficients) is introduced. It is then pointed out that a different and much deeper method (essentially the general uniformization theorem) is effective for the elliptic case and even yields a global result. It is striking, but not surprising in view of recent developments on generalized solutions, that the alternate (global) procedure for elliptic equations requires much less smoothness of the coefficients than is needed for a sensible local result in the other cases.