The present discussion of the conduction levels of copper and silver was undertaken in view of the experimental investigations of the absorption and emission bands of metals, in the soft X-ray region, which have been carried out in this laboratory and elsewhere*. It is a well-known result of the electron theory of metals that, for cubic crystals, of lattice constant a, if k is the wave vector of an electron moving in the crystal lattice, there are discontinuities in the energy E of the electron for values of k such that

(n1, n2, n3) being the Miller indices of the planes for which Bragg reflexion of the electron waves takes place. If we draw the energy E as a function of k for a particular direction in the lattice, then we obtain, using the usual notation, curves such as those shown in fig. 1, d being the half-distance between the corresponding planes for which Bragg reflexion takes place.

Two simple laws connecting the amplitude and phase functions of a monochromatic electromagnetic wave of arbitrary form are derived, holding in the case when one of the field vectors is linearly polarized. The first is a generalized Fermat's principle which enables determination of the phase when the amplitude is known; the second expresses the propagation of the (vector) amplitude along the curves orthogonal to the co-phasal surfaces. Some other general properties of linearly polarized fields are also discussed, and illustrative examples are given.

Consider the infinite system of linear ‘inequations’,

in which the coefficients pij have the two properties

We define as properly divergent (p.d.) a solution, {xi} (i = 0, 1, 2, …), such that

and such that |xi| < ∞ for i < ∞. Since {k + xi} is also a solution for any non-negative constant k, there is no loss in generality in assuming (as we always shall) in place of this last condition that

We are concerned with conditions for the existence of p.d. solutions.

In this paper we consider the slow two-dimensional motion of viscous liquid past a sharp edge projecting into and normal to the undisturbed direction of the stream. The liquid is supposed bounded by rigid planes represented by ABCDE in Fig. 1, and, apart from the disturbance caused by the projection, is assumed to be in uniform shearing motion. The stream function is then a bi-harmonic function that must vanish together with its normal derivative at all points of the boundary, and must be proportional to y2 at a great distance from the projection.

This note describes a simple method of solving the matrix forms of the customary families of finite difference equations which approximate to Poisson's equation (in three dimensions as well as two) and to the bi-harmonic equation. It is intended for use where the boundary of the region over which a solution is wanted is comparatively simple, that is to say, can be subdivided into a comparatively few rectangular areas or rectangular parallelepipeds.

1. A slow two-dimensional steady motion of liquid caused by a pressure gradient in a semi-infinite channel is considered. The medium is bounded by two parallel semi-infinite planes represented in Fig. 1 by the straight lines AB, DE. The stream-function ψ is a biharmonic function of x, y which exactly satisfies the condition that AB, DE must be stream-lines, but the condition that there must be no velocity of slip on these boundaries is satisfied only approximately, and the calculated velocity of slip gives a measure of the accuracy of the solution.

We investigate a generalization of the method introduced by Kouchnirenko to compute the codimension (colength) of an ideal under a certain non-degeneracy condition on a given system of generators of I. We also discuss Newton non-degenerate ideals and give characterizations using the notion of reductions and Newton polyhedra of ideals.

F.K.C. Rankin and H.P.F. Swinnerton–Dyer proved that all the zeros of the Eisenstein Series $E_k$ contained in the standard fundamental domain $\mathcal{F}$ lie on the arc $A\,{=}\, \{ e^{i\theta}| {\pi}/{3} \,{\le}\, \theta \,{\le}\, {\pi}/{2}\}$. Recently, J. Getz has generalized the method of Rankin and Swinnerton–Dyer to show that modular forms under certain conditions have similar properties. In this paper we prove similar results for certain types of cusp forms, motivated by the work of R.A. Rankin. Further, we give a closed formula for the zeros of a class of cusp forms in terms of the Fourier coefficients following the method of Kohnen.

Some models for spatial point processes are difficult to simulate directly and are most easily realized as the equilibrium distribution of certain spatial-temporal Markov processes. This paper examines the convergence of such processes, concentrating mainly on the ‘hard core’ case when the points represent the centres of non-overlapping discs. Coupling methods from the theory of Markov chains are used to establish sufficient conditions for the processes to converge to the required equilibrium, and to give a lower bound on the rate of convergence. One technique used is to couple processes when they become close in a suitable metric.

We study Δ(x; ϕ), the error term in the asymptotic formula for [sum ]n[les ]xcn, where the cns are generated by the Rankin–Selberg series. Our main tools are Voronoï-type formulae. First we reduce the evaluation of Δ(x; ϕ) to that of Δ1(x; ϕ), the error term of the weighted sum [sum ]n[les ]x(x−n)cn. Then we prove an upper bound and a sharp mean square formula for Δ1(x; ϕ), by applying the Voronoï formula of Meurman's type. We also prove that an improvement of the error term in the mean square formula would imply an improvement of the upper bound of Δ(x; ϕ). Some other related topics are also discussed.

In all methods of measuring excitation and ionisation potentials which are in common use the current-voltage characteristic curves of the apparatus are plotted. In the absence of gas these curves are smooth, but on the admission of gas at a suitable pressure sharp upward bends occur at definite voltages, which are interpreted as the critical potentials of the gas. In a variation by Franck and Hertz, the so-called inelastic impact method, these bends are downward, the critical points being represented by maxima.

A (connected) Riemannian manifold M is geodesically complete (i.e. each geodesic may be extended to a geodesic with domain ( −∞, + ∞)) iff, as a metric space under the induced Riemannian distance function, M is Cauchy complete ([12], p. 138). Furthermore, all compact Riemannian manifolds are complete. On the other hand, compact pseudo-Riemannian manifolds exist which are not geodesically complete. For example Fierz and Jost[8] have constructed incomplete metrics of the form 2dx dy + h dy2 on T2. Furthermore, Williams [16] has shown that geodesic completeness may fail to be stable for pseudo-Riemannian manifolds. Here a property is said to be stable if the set of metrics for M with this property is open. This failure of stability may occur for both compact and non-compact manifolds. In particular, Williams has found that M = S1 x S1 = ℝ/(2πZ x 2πZ), which may be given the complete Lorentzian metric g = 2dx dy, also admits Lorentzian metrics gn = 2dx dy+(sin(x)/n) dy2 which are geodesically incomplete yet which for large n are arbitrarily close to g in the Whitney fine Cr topologies. In this example the set S = {(0, y)|0 ≤ y ≤ 2π} represents the image of a closed null geodesic for (M, g) and also for all (M, gn). However, for the metrics (M, gn) this closed null geodesic has affine parametrizations which are incomplete because the velocity vector of this null geodesic returns to a scalar multiple of itself after each complete circuit of S. In general, a closed null geodesic will be complete iff its velocity vector returns exactly to itself after each trip around its image.

The electromagnetic energy & of the field of a charged particle is calculated, using Møller's theory summarized in the previous paper. The contribution of & to the gravitational mass of the charged particle is discussed, by studying the behaviour of a neutral test particle in its field. The conclusion is that & gives rise to an ‘effective’ gravitational mass of the charged particle, which is equal to the (Newtonian) gravitational mass of the charged particle, plus the mass-equivalence of &. This is contrary to the currently accepted theory, that what we have called the ‘effective’ gravitational mass is equal to the ‘Newtonian’ gravitational mass of the charged particle.

Two methods of solution are given for the flow of a viscous liquid in a circular pipe whose radius has a slight sinusoidal variation. Numerical examples are given, and some related problems are briefly discussed.

Let Γ be a Kleinian group of finite covolume and denote by Γ(2) the subgroup generated by {γ2:γ ∈ Γ}. In [9] the trace field of Γ(2) was shown to be an invariant of the commensurability class of Γ. In [8] this field was termed the invariant trace field of Γ and further properties of this field were studied. Following the notation of [8] we denote the invariant trace field of Γ by k(Γ).

A method is described for finding the frequency functions of bivariate random variables which are the products or ratios of other bivariate random variables. If (ξ, n) are a pair of bivariate random variables with joint frequency function f(x, y) then the method depends upon the fact that the expectation of │ ξ │r–1│η│s–1 is related to the Mellin transform of f(x, y) in two dimensions. Knowing the expectation we can then recover the frequency function by means of the inverse Mellin transform. Some examples are given to illustrate the theory.

The transition 20F → 20Ne is shown to occur with the emission of a β-particle of maximum energy 5·0 × 106e. V., followed by a γ-quantum of energy 2·2 × 106e.V.

The transition from the excitation level of 20Ne at 2·2 × 106 e.V. sometimes occurs in two stages with the emission of two quanta of smaller energy.

In this paper we prove that if n>30 030, then the nth element of any Lucas or Lehmer sequence has a primitive divisor.

In this paper, we prove that for any odd prime p, there are exactly two non-associative, non-Moufang, (right) Bol loops of order 4p.

In the study of highest weight categories, the class of Weyl modules Δ(λ) and their duals ∇(λ) are of central interest; this is for example motivated by the problem of finding the characters of the simple modules. Weyl modules form the building blocks for the category [Fscr ](Δ), whose objects have a filtration 0 = M0 [les ] M1 [les ] … [les ] Mi−1 [les ] Mi = M with quotients isomorphic to Δ(λ) for various λ. Knowing Extr(Δ(λ), Δ(μ)) is essential for the understanding of this category.

In [3, 13], we determined Ext1 for Weyl modules of SL(2, k) and q-GL(2, k) over an infinite field k of characteristic p > 0. Here we are able to extend these results to determine Ext2 for Weyl modules in both these cases (see Theorem 4 · 6). Moreover, this also gives Ext2 between any pair of Weyl modules Δ(λ), Δ(μ) for q-GL(n, k) (where n [ges ] 2) such that both λ and μ have at most two rows or two columns, or where they differ by some multiple of a simple root (see Section 7).

Consider (for simplicity) polynomial representations of degree d for GL(n, k). A partition of d which has at most two rows is uniquely determined by the difference in the row lengths, which we use as a label for the partition. In this case our main result is