This paper deals with some details of the application of the Lorentz-Dirac equations of motion of an electron to two simple cases, (a) with no incident field, and (b) with an incident pulse of radiation. In case (a), the field-energy distribution in the self-accelerating motion of the electron when the electron has built up a velocity close to the velocity of light is considered. Numerical and graphical methods are used to form a picture showing how the field energy tends to be concentrated in a disk for this state of motion. A discontinuity of the advanced field variables of the self-accelerating electron is also examined. In case (b), the general solution of the equations of motion for the velocity of the electron is worked out accurately. In the physical solution, this brings out the effects of radiation damping when the pulse has great intensity. Some aspects of the non-physical solutions are pointed out.

In this paper, we investigate cycle integrals of weakly holomorphic modular forms. We show that these integrals coincide with the cycle integrals of classical cusp forms. We use these results to define a Shintani lift from integral weight weakly holomorphic modular forms to half-integral weight holomorphic modular forms.

Let M be a closed, connected, orientable 3-manifold. In Row [10], Jaco and Myers [3] and Myers [7], it was pointed out that the topological type of M is closely related to the knot theory in M. Therefore it is an interesting problem to find knots in M with nice properties. Alexander proved M contains a fibred link (see [9]). Myers proved, in [7], M contains a hyperbolic knot, and, in [8], every link in M is concordant to a hyperbolic link. In this paper we consider the fibred version of his results.

A. J. Miller has proposed a stochastic model for the study of highway traffic in one direction along a two-lane road in which the desired speed of a vehicle is sampled from a continuous probability distribution and queues of vehicles are Poisson distributed on the highway. Miller has derived the equilibrium expression for the relation between the spacial densities of all vehicles and the spacial densities of those which are freely travelling. This relationship takes the form of a non-linear integral equation which is left unsolved. The present model differs from the original only in the assumption of a discrete distribution of desired speeds. Although the derivations of the equilibrium equations are based upon different arguments from those of Miller, these relations transform to the corresponding integral equations in the limit of a continuous speed distribution. The main body of the paper is devoted to a detailed investigation of the case of two speeds and asymptotic results are obtained for three speeds.

Given positive integers k and n, let [Xfr ] be the class of all groups G such that γk(G) is locally nilpotent and [x1, x2, …, xk]n = 1 for any x1, x2, …, xk ∈ G. It is shown that [Xfr ] is a variety.

Let G be a finite solvable group and A a group of automorphisms of G such that (|A|, |G|) = 1. We denote by h(G) the Fitting height of G and by l(A) the length of the longest chain of subgroups of A. Then, under some additional hypotheses, it is known from [5] that h(G) ≤ 2l(A) + h(CG(A)) and from [8] that, when CG(A) = 1, h(G) ≤ l(A), both results being best possible (see [6, 7]). The present paper attempts to explain the difference in the coefficient of l(A) in the two inequalities, from 2 to 1.

In this paper we consider sound pulses generated in the fluid outside an impermeable sphere by axisymmetrical time-dependent deformations of the sphere. Following an account of some analytic techniques applicable to this class of problems, a detailed study is made of the case in which the movement of the spherical boundary is confined to a cap of angle α (0 < α < π). In this case a rather complicated pattern of wavefronts develops consisting of a direct (spherical) wavefront, diffracted wave-fronts and a toroidal wavefront having as its centre circle the edge of the cap. Approximations to the velocity potential near the various wavefronts are obtained and some numerical results are given.

It can be very difficult to determine whether or not certain rings are really full matrix rings. For example, let p be an odd prime, let H be the ring of quaternions over the integers localized at p, and set

Then T is not presented as a full matrix ring, but there is a subring W of H such that T ≅ M2(W). On the other hand, if we take H to be the ring of quaternions over the integers and form T as above, then it is not known whether T ≅ M2(W) for some ring W. The significance of p being an odd prime is that H/pH is a full 2 x 2 matrix ring, whereas H/2H is commutative. Whether or not a tiled matrix ring such as T above can be re-written as a full matrix ring depends on the sizes of the matrices involved in T and H/pH. To be precise, let H be a local integral domain with unique maximal ideal M and suppose that every one-sided ideal of H is principal. Then H/M ≅ Mk(D) for some positive integer k and division ring D. Given a positive integer n. let T be the tiled matrix ring consisting of all n x n matrices with elements of H on and below the diagonal and elements of M above the diagonal. We shall show in Theorem 2.5 that there is a ring W such that T ≅ Mn(W) if and only if n divides k. An important step in the proof is to show that certain idempotents in T/J(T) can be lifted to idempotents in T, where J(T) is the Jacobson radical of T. This technique for lifting idempotents also makes it possible to show that there are (k + n − 1)!/ k!(n−1)! isomorphism types of finitely generated indecomposable projective right T-modules (Theorem 2·10).

We define a metric filtration of the Gordian graph by an infinite family of 1-dense subgraphs. The nth subgraph of this family is generated by all knots whose fundamental groups surject to a symmetric group with parameter at least n, where all meridians are mapped to transpositions. Incidentally, we verify the Meridional Rank Conjecture for a family of knots with unknotting number one yet arbitrarily high bridge number.

In this paper we prove that for any odd prime p, a Moufang loop of order 2p2 is a group and there is exactly one non-associative right Bol loop of order 2p2.

1. A theorem of Montel (14), states that if f(z) is analytic and bounded in the half-strip

and if there exists an x0 in (a, b) such that

as y → ∞, then

as y → ∞ l. u. on (a, b)(locally uniformly on (a, b), i.e. uniformly on every compact subset of (a, b)). Bohr (3), has proved a version of this result applicable to functions analytic, but not necessarily bounded, in S(a, b) and Hardy, Ingham and Pólya (10), have considered whether or not f(z) can be replaced by |f(z)| in (1·1) and (1·2). They show that this is not so but prove that if f(z) is analytic and bounded in S(a, b) and if

as y → ∞ for three distinct values x1, x2 and z3 in (a, b) then there exist constants A and B such that

as y → ∞, l.u. on (a, b). Cartwright (Theorem 5, (6)) has proved that if f(z) is analytic and satisfies |f(z)| < 1 in, S(a, b) and if for some x0 in (a, b)

asy → ∞, then

as y → ∞ co, l.u. on (a, b) and Bowen (5), has shown that if (1·4) is replaced by the weaker condition

as n → ∞ for some suitable sequence of points Zn in S(a, b) then (1·5) is still valid. In sections 2–5 of this paper we shall consider whether or not these results are valid |f(z)| is replaced by a subharmonic function.

In [6] the first author proved that for any β ∈ (1, βKL) every x ∈ (0, 1/(β − 1)) has a simply normal β-expansion, where βKL ≈ 1.78723 is the Komornik–Loreti constant. This result is complemented by an observation made in [22], where it was shown that whenever β ∈ (βT, 2] there exists an x ∈ (0, 1/(β − 1)) with a unique β-expansion, and this expansion is not simply normal. Here βT ≈ 1.80194 is the unique zero in (1, 2] of the polynomial x3 − x2 − 2x + 1. This leaves a gap in our understanding within the interval [βKL, βT]. In this paper we fill this gap and prove that for any β ∈ (1, βT], every x ∈ (0, 1/(β − 1)) has a simply normal β-expansion. For completion, we provide a proof that for any β ∈ (1, 2), Lebesgue almost every x has a simply normal β-expansion. We also give examples of x with multiple β-expansions, none of which are simply normal.

Our proofs rely on ideas from combinatorics on words and dynamical systems.

The previous paper dealt first with the shot effect without secondary emission. We considered a stream of electrons arriving at random, with a fixed probable density N, at the anode of the resistance-capacity coupled first valve of a linear amplifier. This does not allow for the effect whereby the arrival of electrons alters the anode potential, and consequently the probability of arrival of succeeding electrons. The valve must be working with what is conventionally called an infinite internal resistance, and without space-charge effects. Each electron arriving at time tr is supposed to produce at time t an output effect f(t − tr) from the amplifier. We considered the hypotheses, first, that electrons may, with specific probabilities, have lives of any length on the anode system, during which they add − ε/C to the anode potential, and alternatively that each electron adds to the anode potential of the valve whose anode to earth capacity is C and whose feed resistance is R. These hypotheses determine the form of f(t − tr) when the behaviour of the main part of the amplifier is known. Since the amplifier is linear, the effects of various electrons are additive, so that the total output is ∂(t) = ∑rf(t − tr).

An extension of Bellman's analogue of Wald's identity is proved for Markov processes with a finite number of states, in which (a) the diagonal matrices of the state functions and the transition matrices may be heterogeneous but are so related that the product of the two matrices which apply at any stage has at least one modal vector in common with all of the corresponding products which apply at other stages, and (b) the probability distribution of the duration of any one realization of the process satisfies certain conditions. The condition (a) is called equimodality. The conditions for (b) are satisfied when there is asymptotically a sufficiently abrupt exponential upper bound to the probability distribution of the duration, and extends simply to validate analogues involving latent roots other than the one which appears in Bellman's formula. A certain generalization of Wald's stopping rule produces a bound of this kind.

In this paper we will be concerned with the problem of describing the Jacobson radical of the semigroup algebra K[S] of an arbitrary semigroup S over a field K in the case where this algebra satisfies a polynomial identity. Recently, Munn characterized the radical of commutative semigroup algebras [9]. The key to his result was to show that, in this situation, the radical must be a nilideal. We are going to extend the latter to the case of PI-semigroup algebras. Further, we characterize the radical by means of the properties of S or, more precisely, by some groups derived from S. For this purpose we will exploit earlier results leading towards a characterization of semigroup algebras satisfying polynomial identities [5], [15], which generalized the well known case of group algebras (cf. [13], chap. 5).

Two related issues are explored for bond percolation on ${\mathbb{Z}^d$ (with d ≥ 3) and its dual plaquette process. Firstly, for what values of the parameter p does the complement of the infinite open cluster possess an infinite component? The corresponding critical point pfin satisfies pfin ≥ pc, and strict inequality is proved when either d is sufficiently large, or d ≥ 7 and the model is sufficiently spread out. It is not known whether d ≥ 3 suffices. Secondly, for what p does there exist an infinite dual surface of plaquettes? The associated critical point psurf satisfies psurf ≥ pfin.

In our previous consideration in (1) of the constant-coefficient fifth-order differential equation:

an attempt was made to identify (though not exhaustively) different sufficient conditions on a1,…,a5 for the instability of the trivial solution x = 0 of (1·1). It was our expectation that the conditions so identified could be generalized in some form or other to equations (1·1) in which a1,…,a5 were not necessarily constants, thereby giving rise to instability theorems for some non-linear fifth-order differential equations; and this turned out in fact to be so except only for the case:

with R0 = R0(a1, a2, a3, a4) > 0 sufficiently large, about which we were unable at the time to derive any worthwhile generalization to any equation (1·1) in which a1, …,a5 are not all constants.

In the famous paper in which he developed his theory of radiation, Einstein investigated two problems. He investigated first the problem of what distribution of radiant energy in the steady state would be set up by an enclosed assembly of atoms in thermodynamic equilibrium radiating and absorbing according to the assumptions of his theory; he showed that the distribution so set up was given by Planck's law. He investigated secondly the velocities set up amongst the atoms in consequence of the random variations in direction of the emissions and absorptions; provided that emissions are assumed to be directed, so giving rise to recoil momentum, he showed that the mean square velocity-component in a given direction, , is equal to its equipartition value, given by .