Let S be the sum-of-digits function in base 2, which returns the number of 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t, define the asymptotic density

$${c_t} = \mathop {\lim }\limits_{N \to \infty } {1 \over N}|\{ 0 \le n < N:s(n + t) \ge s(n)\} |.$$

Equations have been given by Moullin to determine the current in a receiving antenna in wireless telegraphy and they have been discussed and extended by Colebrooke, who in particular considered the reception of a plane polarised wave with its electric vector vertical by an antenna which is partly vertical and partly horizontal. His result shows that the intensity of the received signal is independent of the orientation of the receiver. In practice, however, the incident wave front is not vertical, its tilt depending on the wave length and on the resistance of the earth, as shown by Zenneck, and the purpose of this paper is to investigate the effect of the orientation of a bent antenna in the reception of this type of wave. An approximate theory of the Beverage antenna is obtained by the same method.

The solution by Powell (6) of the diffraction of a complete expansion wave by a corner is extended to an incomplete wave. It is shown that the pressure is discontinuous across one of the boundaries and becomes logarithmically singular at a point on it. Graphs of the pressure distribution on the fixed boundary are given.

In this paper we consider Dehn surgery and essential annuli whose two boundary components are in distinct components of the boundary of a 3-manifold.

Let Nl be an orientable 3-manifold with boundary, Kl a knot in Nl, and N2 the 3-manifold obtained by performing γ-Dehn surgery Kl. In detail, let Vl be a regular neighbourhood Kl, X = Nl − int Vl the exterior of Kl, T the toral component ∂Vl of ∂X, and γ a slope on T. Then we obtain the 3-manifold N2 by attaching a solid torus V2 to X so that γ bounds a disc in V2. Let K2 be the core of V2. Let π be the slope of a meridian loop of Kl, and Δ the distance between the slopes π and γ, i.e. the minimal number of intersection points of the two slopes on T. Suppose for i = 1 and 2 that Ni contains a proper annulus Ai such that the two components of ∂Ai are essential loops on distinct incompressible components of ∂Ni. Then note that Ai is essential, i.e. incompressible and ∂-incompressible in Ni.

In [12] we elaborate the vague principle that the behaviour at infinity of the decreasing sequence of singular numbers sn(K) of a Hilbert–Schmidt kernel K is at least as good as that of the sequence {n−1/qω(n−1;K)}, where ωp is an Lp-modulus of continuity of K and q = p/(p − 1), where 1 ≤ p ≤ 2. Despite the author's effort to justify his study of refinements of the half-century old theorem of Smithies [13], that theorem remains the central result of the subject (viz. that for 0 < a ≤ 1, K∈Lip(a, p) implies that sn(K) = O(n−α−1/q)). For example, Cochran's omnibus theorems [5, 6] that delimit the Schatten classes to which a kernel belongs are based on the blending of ‘smoothness’ conditions and emphasize the pivotal role of the principal corollary of Smithies' theorem (viz. {sn}∈lr if r−1 < α + q−1). Cochran later offered in [7] a very simple derivation of the corollary from a Fourier series theorem of Konyushkov (see [2], vol. II, p. 197), whose proof was, however, at least as intricate as Smithies' demonstration.

We prove the following theorem:

Theorem A. Let G be a finite simple group of order divisible by 3. Suppose

(i) G contains no element of order 6;

(ii) G has cyclic Sylow 3-subgroups;

(iii) G does not involve A4;

(iv) Every 3′-simple section of G is a Suzuki group.

Then G ≅ SL(2, 2n) for some odd n ≥ 3.

We characterize those countable rooted trees with non-trivial components whose full automorphism group has uncountable strong cofinality, and those whose full automorphism group contains an open subgroup with ample generics.

Let T be an operator on a complex Banach space such that sup {‖Tn‖: n ≥ 0} < ∞; such operators are said to be power-bounded. For such an operator one defines the peripheral spectrum to be the set

where σ(T) denotes the spectrum of T, and T is the unit circle in ℂ. Esterle proved in [3] that limnn→∞ ‖Tn−Tn+1‖ = 0 provided that ‖T‖ ≤ 1 and σ{T} = {1}. Later Katznelson and Tzafriri [6] proved the same result for power-bounded operators such that σΓ(T) = {1}, and in fact they proved a more general result. More precisely, let be an absolutely convergent Taylor series, i.e. , and assume that f is of spectral synthesis for σΓ(T) (in particular f|σΓ(T) ≡ 0). Then limn→∞ ‖f(T) Tn‖ = 0. Katznelson and Tzafriri prove an analogous result, in which T is replaced by a commuting k-tuple (T1, T2, …, Tk) of contractions. In this case they consider functions of the form f(T1, T2, …, Tk), where f is either an absolutely convergent Taylor series if k is odd, or an element of a smaller algebra if k is even. The result of Katznelson and Tzafriri was further extended in the case of a single contraction T by Esterle, Strouse and Zouakia[4], who consider functions f(T) where f is allowed to be in a more general function algebra. For instance, if T acts on a Hilbert space, they allow f to be an element of the disk algebra.

1. The relation

in which we suppose each of a, b, c, a2 − c2 to be different from zero, leads, from a value θ, to two values of φ, which we may denote by θ1 and θ−1. Each of these, put in place of θ, leads, beside the value θ of φ, to another value of φ, say, respectively, θ2 and θ−2. If θ2 or θ−2 be put for θ, the same relation leads to two values of φ, say, θ1 and θ3 or θ−1 and θ−3, respectively. And so on. It may happen that θn is the same as θ, in which case also θ−n is the same as θ; this we may express by saying that the relation is closed, or that there is closure, after n links. It is the object of the present note to express in reduced form, in terms of a, b, c, the condition that this may be so. Evidently, if n = pq, the condition of closure after n links is satisfied when the condition for closure after p links (or also after q links) is satisfied. But there is a condition for closure after n links which is not satisfied for any less number of links; this is the condition which we call the reduced or proper condition for closure after n links, and it is this which we seek to express.

1. Introduction. This note is a sequel to (1), the notation of which will be used without further explanation. In (2), certain deductions from the theorems proved in (1) were made in the case where K(x, t) is derived from a function K(x), a.c. in [0, 1], by the equation

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.

Let O denote the origin of ℝn (where n ≥ 2), let B(X, r) denote the open ball in ℝn of centre X and radius r, and define φn: (0, + ∞) → ℝ by φn(t) = t2-n if n ≥ 3 and φ2(t) = log (1/t). A sequence (Xk) in B(O, 1) is said to converge regularly to O if Xk → O and there is a constant k > 1 such that φn(|Xk+1|) < Kφn(|Xk|) for each k ∈ ℕ. If n ≥ 3, this is clearly equivalent to saying that there is a constant k′ ε (0, 1) such that |Xk+1| > k′|Xk| for each k (cf. [18], p. 149).

The problem considered is that of the diffraction of sound pulses by a plane interface separating two stratified media in each of which the refractive index increases with distance from the interface. Two types of diffraction phenomena occur, glancing ray diffraction and total reflection. The solution is obtained as a series of ‘pulse propagation modes’, using a double transform technique. The main results are some formulae for the pulse field near a diffracted wavefront.

The equations of motion in a surge-tank fitted to a hydraulic pipe-line are non-linear. Various methods of solving them were described by Durand: (1) simplification of the equations and the use of empirical coefficients, (2) simplification by taking the friction forces as proportional to the first power of the velocity, (3) a step-by-step method of solution. As a revolt against the serious expense of time involved in (3) (which was later developed by Jakobsen and by Cole), he was led to devise other methods including the use of models, which he dealt with subsequently in more detail. Nevertheless, there remains a need for a quick approximate method of calculating the maximum surges which may be used with confidence at least in the early stages of design. A promising line of attack was suggested by the success which has attended the approximate analysis of mechanical vibrating systems having non-linear characteristics. This method has been set out by Den Hartog, who dismisses step-by-step solutions as ‘too laborious’.

§ 1. The general quaternary cubic

can be expressed in various interesting canonical forms involving suitably chosen linear forms Xi, Yi, Ai. Thus, referred to the pentahedron. X1X2X3X4X5 of Sylvester the cubic becomes the sum of five cubes

with its Hessian in the form

where the coefficients ai may if necessary be taken as equal to unity and the five linear forms Xi each contain four independent parameters, making a total of twenty parameters which is the number of coefficients aijk in the given cubic C3

If a receiving aerial is placed in the field of a wave produced by a distant transmitter, currents are induced in the receiving aerial and these dissipate energy in the ohmic resistances of the aerial. This energy may be thought of as being absorbed directly from the incident wave, but this viewpoint gives no insight into the mechanism by which the absorption takes place. In this paper the following alternative way of regarding the phenomenon is considered. The currents induced in the aerial cause it to re-radiate a secondary wave. The electric and magnetic fields of this secondary wave combine with those of the unabsorbed incident wave to give resultant fields which have, in general, a direction different from those of the incident wave. These resultant fields give rise to energy flows which are different from those in the incident wave, both in magnitude and direction. In this way a flow of energy into the aerial is produced, by the superposition of the incident and the re-radiated fields. It is important to notice that if this viewpoint is adopted we must add the secondary re-radiated wave to the unabsorbed main wave, the addition, in itself, producing the requisite absorption.

Let $N$ be a prime number, and $X$ a curve that is intermediate to the cover $X_1(N)\rightarrow X_0(N)$. We study the automorphism group of $X$, and prove that in most cases it is generated by the Galois group of the cover $X\rightarrow X_0(N)$, and any lift of the Atkin–Lehner involution to $X$.

The quantal quartic oscillator, characterized by two real and two complex conjugate transition points (simple zeros), is studied by means of the phase-integral method developed by Fröman and Fröman, and various quantization conditions are obtained. The main results, obtained in §3·3 and §4, are summarized below.

A correction to the generalized Bohr–Sommerfeld quantization condition, due to the complex conjugate transition points, is obtained from estimates of the. F-matrix for a path passing above the complex conjugate transition points. This quantization condition is closely related to the quantization condition for a double-oscillator. The correction is shown to be rigorously valid when the distance between the real transition points is large compared to the distance between the complex conjugate ones, which may lie at an unspecified distance from each other.

At sufficiently large distances from the cluster of four transition points the solutions of the Schrödinger equation can be characterized by the Stokes constants between anti-Stokes lines emerging from the cluster in six directions towards infinity. There are initially six non-trivial such Stokes constants in the problem under consideration, but, from the results in previous papers by N. Fröman and the present authors, these Stokes constants are linked by a number of algebraic relations, so that one or two Stokes constants suffice to describe the solutions far away from the cluster of transition points. The quantization condition is expressed in terms of these Stokes constants. Approximate values of the relevant Stokes constants are given in the three limiting cases when all four transition points coalesce (and the first-order phase-integral approximation is used), when the distance between the real transition points is sufficiently large (extreme double-oscillator), and in the Bohr-Sommerfeld limit.