1. A locus Vn, of dimension n, in [2n + 1], for example a curve in [3] or a surface in [5], has ∞ 2n chords, of which a finite number pass through a general point of the space.

Some topological identifications are introduced into the DS-spaces of Robinson and Trautman, and the symmetry and asymptotic symmetry groups of the resulting spaces are calculated. The latter have structures similar to the GBM group, each being the semi-direct product of a group of supertranslations by a group of classical type.

Larmor has shown that if the upper atmosphere contains electrons (charge ε, mass m, density ν) and if collisions between these electrons and molecules—and also the forces between the electrons themselves—are negligible, then electric waves are propagated as if the dielectric constant of the medium were reduced by , from which it appears that, so long as the approximations are valid, the velocity of propagation of the waves can be increased indefinitely by increasing either the electron density or the wave-length λ. Several later authors have attempted to take account of the collisions between electrons and molecules, assuming free paths or velocities according to Maxwell's laws for a uniform gas, and it appears that the above law holds only for short waves; but it is doubtful how far the properties of a uniform gas can be assumed when periodic forces are acting. In the first part of this paper an alternative method of solution is given by means of Boltzmann's integral equation for a non-uniform gas, the analysis being similar to that used by Lorentz in discussing the motion of free electrons in a metal. Only the case when ν is small is considered, i.e. the interactions of electrons with one another and with positive ions are neglected. How far it is possible to increase the velocity of propagation by increasing ν is a more difficult question, but it seems possible that the forces between the electrons and ions may impose a limit just as collisions with neutral molecules limit the effect of increasing the wave-length.

Let F be the finite field of q = pn elements and let F0 be its prime subfield; thus, card F0 = p. For polynomials f ∈ F[x] and non-principal additive characters η of F A. Weil (1) proved the estimate

For knots in S3 we obtain new criteria for periodicity. We show that the Casson–Gordon invariants of a periodic knot are preserved under the periodic action lifted to the cyclic covers. As an application, we consider a family of knots with the Seifert form of a period 3 knot, and using Casson–Gordon invariants show that knots in this family do not have period 3. We also obtain periodicity criteria in terms of the homology groups of cyclic branched covers of S3.

Consider a system of diagonal equations \begin{equation}\sum_{j=1}^sa_{ij}x_j^k=0\quad (1\le i\le r),\end{equation} satisfying the property that the (fixed) integral coefficient matrix $(a_{ij})$ contains no singular $r\times r$ submatrix. A recent paper of the authors [3] establishes that whenever $k\ge 3$ and $s>(3r+1)2^{k-2}$, then the expected asymptotic formula holds for the number $N(P)$ of integral solutions ${\bf x}$ of ($1{\cdot}1$) with $|x_i|\le P$$(1\le i\le s)$.

Throughout this paper q = 2h with h ≥ 4, and PG(r, q) stands for the r-dimensional projective space over the finite field GF(q) with q elements.

Let D be the open unit disc in the complex plane C and let dA be the normalized area measure on D. The Bergman space is the space of analytic functions f in D such that

An extension is proposed to the theory of Verwey and Overbeek on the force between two colloidal particles in a dilute sol. The effect of the Coulomb interaction of the ions in the dispersion medium is considered. It is shown that this contributes an additional repulsion term to the mutual free energy of the two particles. This correction term behaves as an inverse power and is the predominant effect at large separations, although it is a second-order effect at small separations. For parallel plates, this power is the fourth, and for two spheres, the sixth. A tentative calculation shows that, at least for thin plates and low electrolyte concentrations, our new repulsion will outweigh the attractive London-van der Waals force considered by Verwey and Overbeek, and more recently by Casimir and Polder, when the plates are far apart.

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.

We obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato–Tate density. Examples of such sequences come from coefficients of several L-functions of elliptic curves and modular forms. In particular, we show that |τ(n)| ⩽ n11/2(logn)−1/2+o(1) for a set of n of asymptotic density 1, where τ(n) is the Ramanujan τ function while the standard argument yields log 2 instead of −1/2 in the power of the logarithm. Another consequence of our result is that in the number of representations of n by a binary quadratic form one has slightly more than square-root cancellations for almost all integers n.

In addition, we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato–Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally and might be within reach unconditionally using the currently established potential automorphy.

For a bounded open set U ⊂ ℂ, we denote by H∞(U) the collection of all bounded analytic functions on U. We let X denote bdy (U), the boundary of U, Y denote the polynomial hull of U (the complement of the unbounded component of ℂ / X), and U* denote mt (Y), the interior of Y. We denote the sup norm of a function f: A → ℂ by ∥f∥A:

We denote the space of all analytic polynomials by ℂ[z], and we denote the open unit disc by D and the unit circle by S1.

Three kinds of proper actions of increasing strength are defined. We prove that the three definitions specialize to the definitions by Bourbaki, by Palais and by Baum, Connes and Higson in their respective settings. The third of these, which thus turns out to be the strongest, originally only concerns actions of second countable locally compact groups on metrizable spaces. In this situation, it is shown to coincide with the other two definitions if the total space locally has the Lindelöf property and the orbit space is regular.

Let Xnj, 1 ≤ j ≤ kn, be independent, asymptotically negligible random variables for each n ≥ 1. In certain cases there exists a duality between the behaviour of ΣjXnj and . We extend one of the known forms of this duality, and show that, under mild conditions on the truncated moments of the Xnj, the convergence of to 1 in the mean of order p (p ≥ 1) is equivalent to the convergence of ΣjXnj to the standard normal law, together with the convergence of its 2pth absolute moment to that of a standard normal variable. A similar result holds in the case of convergence to a Poisson law.

This sequence of papers is concerned with a group of theorems on Fourier transforms, and with the corresponding results for Fourier series. The transform theorems deal with the equations

when two of the functions concerned are monotonic; and the series theorems, with the equations

when two of the functions or sequences of coefficients are monotonic. A general survey of the problem was given in two earlier papers, which we shall call M.F. (I) and M.F. (II), and the results obtained were tabulated in M.F. (II). There is thus no need to restate the position in detail here. We shall merely recall as briefly as possible what has been proved already, so as to show the scope of the present paper.

We shall be concerned with the Laguerre polynomials, defined by Some time ago Szegö (8) showed that for (a, b, c = 0, 1, …). It is easily deduced from this (cf. (5)) that for all A λ ≥ 3.

1. Introduction. The open subsets of a topological space X form a complete Brouwerian, i.e. distributive and pseudo-complemented lattice L(X). Many topological properties of X can be formulated as properties of L(X), although, in general, X is not determined by L(X). Obvious examples are quasi-compactness and connectivity: X is quasi-compact if any set of elements of L(X) whose join is I has a finite subset whose join is I; X is connected if no element of L(X) has a complement. These properties which we shall call ‘paratopological’, can be defined for lattices that are not of the form L(X), and many topological theorems can be proved in this more general context. The purpose of this paper is to develop sections of general topology as part of the theory of complete Brouwerian lattices (CBL).

In a recent paper of the same title (2), Conway considered the problem of dividing a square of side N, into squares of integral side. The case when we demand that all these squares are of unequal size has been considered in some detail by various authors ((l),(4),(6),(7)). Conway considered the case with the condition relaxed. Dudeney ((3), Problem 173) has considered the same problem for the special case N = 13; hence the title of this paper.

In this paper we will study properties of twisted Alexander polynomials of knots corresponding to metabelian representations. In particular we answer a question of Wada about the twisted Alexander polynomial associated to the tensor product of two representations, and we settle several conjectures of Hirasawa and Murasugi.

For an odd prime number p and a finite set S of prime numbers congruent to 1 modulo p, we consider the Galois group of the maximal pro-p-extension unramified outside S over the ${\mathbb Z}_p$-extension of the rational number field. In this paper, we classify all S such that the Galois group is a metacyclic pro-p group.