1. It is pointed out that the signal strength of a wireless wave at a distant point depends on:

(a) the electrical constants of the ground,

(b) the curvature of the earth,

(c) the existence of an “atmospheric” ray coming downwards from the Heaviside layer.

2. We can eliminate the effect of (b) and (c), and so obtain direct evidence about the electrical constants of the ground, by making measurements near the transmitter.

3. Measurements on wave lengths of 300 m. and upwards give information about the resistivity of the ground, and on shorter wave lengths (15 m.) give the dielectric constant of the ground.

4. Attenuation measurements have been made over short distances for wave lengths of 1600 m. and 360 m.

5. The results are compared with those calculated from Sommerfeld's theory and show close correlation for distances beyond 10 wave lengths, but show deviations from the theory for shorter distances.

They give as values for the resistivity of the ground

p = 1·8 × 1013 E.M.U.

for the Daventry signals, and

p = 0·6 × 1013 E.M.U.

for the London signals.

Suppose that we are given a random sample of size n chosen according to the uniform distribution on the unit interval. Let Zn(x) = Zn(x, ω) be the length of the unique left-closed and right-open sample spacing that contains x. The purpose of this paper is to examine the almost sure, and exceptional, growth rates of the process {Zn}. The typical maximum growth rate and the growth rate of the maximum can be of quite different orders of magnitude as is shown by the following two results.

Theorem 2. With probability one we have

for almost all x.

Theorem 3. With probability one we have

Let X0 be an infinite-type hyperbolic surface (whose boundary components, if any, are closed geodesics) which has an upper bounded pants decomposition. The length spectrum Teichmüller space Tls(X0) consists of all surfaces X homeomorphic to X0 such that the ratios of the corresponding simple closed geodesics are uniformly bounded from below and from above. Alessandrini, Liu, Papadopoulos and Su [1] described the Fenchel–Nielsen coordinates for Tls(X0) and using these coordinates they proved that Tls(X0) is path connected. We use the Fenchel–Nielsen coordinates for Tls(X0) to induce a locally bi-Lipschitz homeomorphism between l∞ and Tls(X0) (which extends analogous results by Fletcher [9] and by Allessandrini, Liu, Papadopoulos, Su and Sun [2] for the unreduced and the reduced Tqc(X0)). Consequently, Tls(X0) is contractible. We also characterize the closure in the length spectrum metric of the quasiconformal Teichmüller space Tqc(X0) in Tls(X0).

In this paper, a method is given for solving certain types of integral equations. It involves a technique of building up the solution of the given integral equation from that of a much simpler auxiliary equation and a consideration of this auxiliary equation forms a large section of the work. The method of solution is applied to a particular problem in semi-circular electron focusing in metals.

In this paper, we shall prove a result which identifies the differentials in the Adams spectral sequence (see (1,2)) with certain cohomology operations of higher kinds, in the sense of (4). This theorem will be stated precisely at the end of section 2, after a summary of the necessary information about the Adams spectral sequence and higher-order cohomology operations; the proof will follow in section 3. Finally, in section 4, we shall consider, by way of example, the Adams spectral sequence for the stable homotopy groups of spheres: we show how our theorem gives a proof of Liulevicius's result that , where the elements hn (n ≥ 0) are base elements of

corresponding to the elements Sq2n in A, the mod 2 Steenrod algebra.

We classify the Lagrangian orientable surfaces in complex space forms with the property that the ellipse of curvature is always a circle. As a consequence, we obtain new characterizations of the Clifford torus in the complex projective plane and of the Whitney spheres in the complex projective, complex Euclidean and complex hyperbolic planes.

We investigate the density of square-free values of polynomials with large coefficients over the rational function field 𝔽q[t]. Some interesting questions answered as special cases of our results include the density of square-free polynomials in short intervals, and an asymptotic for the number of representations of a large polynomial N as a sum of a k-th power of a small polynomial and a square-free polynomial.

The purpose of this paper is to compare some classical positions of convex bodies. We provide exact quantitative results which show that the minimal surface area position and the minimal mean width position are not necessarily M-positions. We also construct examples of unconditional convex bodies of minimal surface area that exhibit the worst possible behavior with respect to their mean width or their minimal hyperplane projection.

1. In this paper we consider the following problems, for continuous linear transformations in the space ω, whose points are arbitrary sequences of complex numbers, with the usual topology:

(a) the establishment of a spectral representation theorem; and

(b) the general description of the spectrum.

For arithmetical semigroups modelled on the positive integers, there is an ‘abstract prime number theorem’ (see, for example, [1]). In order to study enumeration problems in the several arithmetical categories whose prototype instead is the ring of polynomials in an indeterminate over a finite field of order q, Knopfmacher[2, 3] introduced the following modification. An additive arithmetical semigroup G is a free commutative semigroup with an identity, generated by a countable set of ‘primes’ P and admitting an integer-valued degree mapping ∂ with the properties

(i) ∂(l) = 0,∂(p) > 0 for p∈P;

(ii) ∂(ab) = ∂(a) + ∂(b) for all a, b in G;

(iii) the number of elements in G of degree n is finite. (This number will be denoted by G(n).)

1·1. The aim of the present paper is to prove the following

Theorem. Let Vi and Mn be closed smooth manifolds and i < n. Let f: Vi → Mn be a generic smooth map such that all of its singular points are of the type Σ;1, 0. (See [2].) Then there exists a non-zero integer N such that the bordism class N·[f]εΩi(Mn) contains an immersion.

Introduction. When the distribution of a statistical function could not be determined exactly, a suitable form for an approximate formula has sometimes been found by the use of information about the order of smoothness of the distribution and about any singularities it might have (see, for example, Geary ((4))). The object of the present paper is to develop a general theory that will provide information of this kind. For the singularity in the distribution of a function at an end of its range Hotelling ((8)) obtained formulae for cases where the original distribution is uniform over a curved space. However, some special sampling distributions are known to have singularities in the interior of the range, for example, the distribution of √ b1 (see (4)) and those of various serial correlation coefficients (see (6)). In view of this I shall deal below with orders of smoothness and with singularities over the whole range. For simplicity I shall assume that the original distribution has a fairly high order of smoothness: attention will therefore be concentrated on the singularities (if any) that arise from the form of the statistical function.

A classical theorem of Jordan and Schur states that if G is a finite group of s × s matrices over a field K whose characteristic does not divide |G|, then G has an abelian subgroup of index bounded by a function of s. There are several direct and elegant proofs of this, leading to explicit bounds (4), (18).

The success of approximate methods in the calculation of transmission coefficients of waves penetrating potential barriers rests, not so much on any satisfactory approximation theory, but on a comparison with a particular barrier, namely the parabolic barrier. The paucity of exact analytical solutions of barrier problems has prevented an explicit detailed analysis of the errors involved, but two recently devised barriers with analytical solutions are investigated here, in order to compare the exact values of the transmission coefficients (both as to their mathematical structure and numerically) with their approximate forms. An estimate is also made of the effect of neglecting poles existing in the complex propagation plane.

Solutions of the bi-harmonic equation valid in the region bounded externally by parallel lines and internally by a circle midway between the lines have been given by one of the Authors in a recent paper [2]. These solutions were adapted to the requirements of certain problems in the theory of elasticity, but modified solutions satisfying the boundary conditions characteristic of viscous fluid motion are easily derived. These modified solutions will here be given and will be used to find the stream function corresponding to the slow rotation of a cylinder placed symmetrically between parallel walls.

The purpose of this article is to derive a set of ‘easily verifiable’ sufficient conditions for the asymptotic stability of the trivial solution of a class of linear neutral integro-delay-differential equations of the form

where ẋ denotes the right derivative, and then generalize the result to a vector system. Asymptotic stability of the trivial solution of the form (1·1) and several of its variants has been considered by many authors (see the discussion at the end). Also, there exists a well developed fundamental theory for neutral delay-differential equations (e.g. existence, uniqueness, continuous dependence etc. of solutions; see for instance the survey article by Akhmerov[1] and the references in it); however there exists no ‘easily verifiable’ sufficient condition for the asymptotic stability of the trivial solution of (1·1). By the term ‘easily verifiable’ we mean a verification which is as easy as in the case of Routh-Hurwitz criterion or the diagonal dominance condition of a matrix. A number of results which are valid for linear autonomous ordinary and delay-differential equations cannot be generalized (or extended) to neutral equations. It has been shown by Gromova and Zverkin [8] that a linear neutral differential equation can have unbounded solutions even though the related characteristic equation has only purely imaginary roots (see also Gromova[7], Datko[3], Snow [16], Brumley [2]); such behaviour is not possible in the case of ordinary or (non-neutral) delay-differential equations. It is known (theorem 6·1 of Henry [10]) that if the characteristic equation associated with a only with negative real parts and if all the roots are uniformly bounded away from the imaginary axis, then asymptotic stability of the trivial solution can be asserted.

Let G be a finite group and n a natural number. Set Hn(G) = ≺g ∈ G|gn ≠ 1≻. The aim of this paper is to prove the following result.

In this paper we conduct a systematic, computerized search for groups generated by small, but highly symmetric, sets of involutions. Many classical groups are readily obtained in this way, as are a number of sporadic simple groups. The techniques of symmetric generation developed elsewhere are described afresh, and the results are presented in a convenient tabular form.

In this paper we study small cancellation quotients of free products. In particular we calculate their cohomology and, via a theorem of Serre, classify their finite subgroups. The results obtained are the natural analogues of the corresponding results for small cancellation quotients of free groups obtained in Huebschmann [6] (but see also Collins-Huebschmann[3]). They also generalize results of McCool[9] on elements of finite order in quotients of free products and similar conclusions have been obtained by Howie [5] for one-relator quotients of free products of locally indicable groups.

There comes a time in the development of a cohomology theory when a discussion of cohomology operations becomes necessary. In the case of complex K-theory, the subject of the present paper, such operations have of course already been investigated by Adams (see (2)), so that any further discussion might appear superfluous. Powerful as Adams's results are, however, the situation still leaves something to be desired: it is not known just what other operations can be defined in K-theory, and it is an inconvenience from the standpoint of stable homotopy theory that Adams's operations are not themselves stable.