The paper is concerned with an extension of the inequality 1 - u(2nt) ≤ 4n[1-u(t)] for u(t) the real part of a characteristic function. The main result is that the inequality in fact holds for all positive integer n and not only powers of 2. Certain consequences are deduced and a brief discussion is given of the circumstances under which equality holds.

Let f(x) ε Lipα, 0 < α < 1, in the range (-π, π), and periodic with period 2π, outside this range. Also let

.

We define the norm as

and let the degree of approximation be given by

where Tn (x) is some n–th trigonometric polynomial.

Let n, s, t be integers with s > t > 1 and n > (t+2)2s–t–1. We prove that if n subsets of a set S with s elements have intersection I and union J then some t of them have intersection J and union J. The result is best possible.

For a noncompact riemannian manifold R, let MP (R) be the P-algebra, and R*P the P-compactification, with the assumption that ∫RPdV = ∞. If s is the P-singular point of the P-harmonic boundary ΔP, and Δ is the harmonic boundary of Royden's compactification R*, we construct a continuous mapping π: R* → R*P such that π(Δ) = Δp or π(Δ) = ΔP – s.

The least upper bound for the nilpotent lengths of soluble linear groups of degree n is calculated. For each n it is , where r(n) = [log3 (2n–1)/4] and [x] is the integral part of x.

A ring is a left Q-ring if all of its left ideals are quasi-injective. For an integer m ≤ 2, a sfield D, and a null D-algebra V whose left and right D-dimensions are both equal to one, let H(m, D, V) be the ring of all m x m matrices whose only non-zero entries are arbitrary elements of D along the diagonal and arbitrary elements of V at the places (2, 1), …, (m, m-l) and (l, m). We show that the only indecomposable non-local left Q-rings are the simple artinian rings and the rings H(m, D, V). An arbitrary left Q-ring is the direct sum of a finite number of indecomposable non-local left Q-rings and a Q-ring whose idempotents are all central.

A tournament T is called symmetric if its automorphism group is transitive on the points and arcs of T. The main result of this paper is that if T is a finite symmetric tournament then T is isomorphic to one of the quadratic residue tournaments formed on the points of a finite field GF(pn), pn ≡ 3 (4), by the following rule: If a, b ∈ GF(pn) then there is an are directed from a to b exactly when b – a is a non-zero quadratic residue in GF(pn).

We prove that a translation plane π of odd order is a generalized Hall plane if and only if π is derived from a translation plane of semi-translation class 1–3a. Also, a derivable translation plane of even order and class 1–3a derives a generalized Hall plane. We also show that the generalized Hall planes of Kirkpatrick form a subclass of the class of planes derived from the Dickson semifield planes.

A weighted average of the partial sums of a series provides a quick and moderately powerful sum for any series in which the ratio of successive terms varies slowly along the series and this ratio is not close to +1. Some properties of the sum are examined.

If {Ln} is a sequence defined by Ln = min {Ln-a +Ln-b, Ln-c +Ln-d}, with a, b, c, d positive integers, then one can ask if necessarily Ln = Ln-b + Ln-b, for all sufficiently large n.

The answer is yes if a and b are relatively prime, Ln > 0 initially, and λ < μ, where λ-a + λ-b = 1, μ-c + μ-d = 1. The answer is no if instead a and b have greatest common divisor k ≥2, with c ≡ 0 (mod k), d ≢ 0 (mod k).

This work is concerned with the forced convection of heat in a circular tube. The fluid flow is assumed to be laminar Poiseuille flow, and the physical parameters; viscosity, density, conductivity; are assumed to be independent of temperature changes. Viscous dissipation terms are also ignored, and there are no heat sources in the fluid. The problem is treated for the case of a step change in the wall temperature, and the eigenvalues have been obtained as an expansion in powers of the Péclet number for the smaller values, and in an asymptotic form for the larger values. The temperature distribution in the fluid in the neighbourhood of the temperature jump has been calculated for two values of the Péclet number, as have the Nusselt numbers.

In this paper, the problem of heat transfer to laminar Poiseuille flow in a circular tube is discussed for the case of an insulated tube with a ring source of heat on the boundary. The solution is developed analytically for low values of the Péclet number, and formulae for calculating the eigenvalues and coefficients have been obtained. The temperature distributions in the neighbourhood of the source have been calculated for two values of the Péclet number. The extension to the case of arbitrary wall flux has also been discussed.

Let X be a locally convex linear topological space. A point z in an ultralimit enlargement of X is pre-near-standard if and only it is finite and for every equicontinuous subset S′ of the dual space X′, a point z′ belongs to *S′ ∩ μσ(X′, X) (0) only if z′ (z) is infinitesimal.

Let be a hermitian matrix in partitioned form. Let the eigenvalues of A, B, C be α1 ≥ … ≥ αa, β1 ≥ … ≥ βb, γ1 ≥ … ≥ γn, respectively. In this paper four classes of inequalities are proved comparing the αi. and βj with the γk. The simplest of these is: if the subscripts is, js satisfy 1 ≤ i1 < … < im ≤ α, 1 ≤ j1 < … < jm ≤ b.

It is shown that Schauder decompositions exist in non-separable weakly compactly generated spaces and in certain non-separable conjugate spaces. Some results are obtained concerning shrinking and boundedly complete Schauder decompositions in non-separable spaces.

In this paper the diffraction of a planetary wave by a semi-infinite plate of arbitrary inclination is investigated using a β-plane approximation. The Wiener-Hopf technique is used to obtain an integral representation of the solution and an asymptotic description of the diffracted wave is obtained by the method of steepest descent.