A table of totally complex quartic fields, with small discriminants, which possess quadratic subfields was given by Delone and Faddeev(1); the purpose of the present paper is to give a table of such fields which have no subfield. The method used is based on an idea used recently (2) in the case of totally real quartic fields and depends on a special case of the following theorem, which is stated in full generality.

In a previous paper, the author gave necessary and sufficient conditions for the algebra of a finite semigroup S over a field of suitable characteristic to be semisimple. When this algebra is semisimple, the matrix representations of S over are completely reducible.

There are two natural analogues of the symmetric group on n symbols in the theory of semigroups, namely, the set of all mappings of a set of n symbols into itself, and the set of all partial transformations of such a set, with the obvious definitions of multiplication. We are concerned here with the latter system. This is an inverse semigroup, and accordingly we call it the ‘symmetric inverse semigroup’. It gives rise to a semisimple algebra over a field of characteristic zero or a prime greater than n, and its matrix representations over such a field are thus completely reducible.

Burnside ((1), p. 241) has proved the following theorem:

If G is a non-metabelian p-group, then the centre of the derived group of G cannot be cyclic. In particular, a non-Abelian group of order p3 cannot be the derived group of a p-group.

In (6), the author introduced a numerical character of an ideal of a Noether ring A, called the grade of . This can be defined as the least integer k such that (the definition given in (6) differed from this, but, as will be seen below, is equivalent). The purpose of the present paper is to study this character in more detail.

In an earlier paper (3) the author developed a theory of the neighbourhoods of a local domain, in which the concept of the first neighbourhood ring played a central role. This notion has since proved useful in connexion with certain one-dimensional problems, but it has emerged, in the process, that a considerable advantage would be gained if the theory could be freed from the assumption that the basic ring was to be without zero-divisors. This parallels the situation in the geometry of plane curves, where it is desirable, so far as is possible, that results and methods should apply with equal facility to reducible as well as to irreducible curves. Accordingly Part I of the present paper is devoted to a fresh account of the first neighbourhood ring from a more general standpoint than that used previously. Besides the greater generality thus obtained, the theory is extended by the addition of some new results. Furthermore, the proof of one of the main results of the original paper ((3), Theorem 10) has been considerably simplified. Of course, the ideas of (3) reappear here in a modified form, but, to spare the reader the irritating fragmentation of the subject which would otherwise be necessary, the revised account has been made independent of the earlier one. To this extent, Part I is self-contained.

The comparison theorem is an algebraic theorem, which corresponds to the following topological situation. A map between two fibre spaces induces a homomorphism between the two corresponding spectral sequences relating the homologies of the base, fibre and fibre space; if the map induces an isomorphism on any two of these quantities, then it also does on the third. J. C. Moore (l) proved the theorem using a mapping cylinder, which requires that the spectral sequences arise from filtered differential groups. The present proof assumes only the existence of the spectral sequences from the E2 terms onwards. Moreover, we generalize the theorem, assuming that the homomorphisms on the two given quantities are isomorphic only up to given dimensions, and deducing the same for the third quantity. For algebraic translucence we use a simpler hypothesis (property (iii)) than Moore's concerning the E2 terms. However, if his hypothesis is assumed, together with the existence of E1 terms, then his theorem can be deduced as a corollary. The analogous result for cohomology is similar, and is stated at the end.

Consider a topological n–sphere Sn, where n ≥ 1. By a commutative product I mean a continuous function f: Sn × Sn → Sn such that

In a recent paper (1) I have studied tangential properties of sets of infinite measure. This note represents a complementary study of sets and, in particular of arcs, of σ-finite linear measure. Suppose we have a linearly measurable set E of infinite measure that can be represented as the sum of sets of finite measure. Write En = En, 1 + En, 2, where En, 1 is the set of regular points of En and En, 2 that of irregular ones. The set F = ΣEn, 1 is a regular component of E and the set G = ΣEn, 2 an irregular one. A tangent toEn exists at almost all points of En, 1 and at almost no points of En, 2. Denote by Tn the set of all tangents to En and by T the sum-set ΣTn. T will be called a tangent-set to E. Lines of T are defined corresponding to almost all points of F and to the points of a subset of G of measure zero.

The boundary of a simply-connected plane domain in the simplest case is a Jordan curve, but in the general case it is of a very complicated nature. An interesting problem arises about the nature of the boundary, when we consider not a single domain but some extensive family of domains. The question is whether only exceptional domains of the family have a complicated boundary, the rest having simple ones.

The theory of distributions of L. Schwartz (3) provides a unified and rigorous foundation for special methods used in various branches of mathematics. Schwartz's treatment is on the most general level, and presupposes an understanding of modern abstract analysis. Several alternative approaches to distributions have been developed, all of them ‘elementary’ in one sense or another. We follow here the approach of Mikusiński (2) and Temple (4), in which distributions are defined as generalized limits of sequences of continuous functions. We find that, with this approach, it is possible to prove the basic theorem: every distribution is (locally) a derivative. The property of continuity of a distribution does not enter into the arguments establishing this result, but instead follows from it. Hence we are able to reduce the ‘regular sequence’ definition of a distribution to its simplest form. In a later paper we shall study convolution products of distributions, defined in the natural manner by regular sequences.

Let f (z) be meromorphic and non-rational in the domain |z| < R ≤ ∞, and let a be an arbitrary complex number, which may be infinite. The deficiency δ(a) of the value a is defined by

where m(r, a), N(r, a) and T(r) are defined as usual (cf. (10), pp. 156 ff.). For the class of functions considered in this paper the characteristic function T(r) is unbounded, and this will be assumed throughout. The upper (or Valiron) deficiency (16) of the value a is denned by

from which it follows that 0 ≤ δ(a) ≤ Δ(a) ≤ 1. A value a for which Δ(a) > 0 is called exceptional or deficient, and a value for which Δ(a) = 0 is called normal. We shall denote by G[a, σ] the open set of all values z in | z | < R for which | f(z) – a | < σ, where σ is a given positive number; we shall say that a component Gn[a, σ] of G[a, a] is bounded if the closure G¯n[a, σ] is contained in | z | < R, otherwise Gn[a, σ] will be called unbounded. In the case a = ∞, it is natural to define Gn[∞, σ] as the set of all z for which | f(z) | > 1/σ.

The formula to be established is

where α > ½, β > ½and b is real and positive. In § 3 we shall generalize this by proving

where α > (p−l)/p, β > (p−l)/p, b > 0 and p = 2,3,….

A new analysis is made of the coefficients appearing in periodic Mathieu functions, and this is applied to establish the limiting properties of certain solutions of the equation

The theory of time-periodic wave problems falls into two parts. On the one hand there is the rigorous formulation in terms of differential wave equations, on the other there are approximate theories like geometrical optics. It should be possible, in principle, to deduce the latter from the former by a logical process; but this has been done only for a few simple configurations, e.g. the circle. A possible approach to the solution of the general problem is suggested here, and is applied to a typical two-dimensional acoustical example. An arbitrary closed convex curve (satisfying certain regularity conditions) is emitting short sound waves towards infinity, the normal velocity V(s) exp (− iωt) is prescribed on the curve as a function of the arc-length s, and the potential is to be found, first on the curve and then at any point in the sound field. (Only the first part of the problem is treated in detail.) The potential ø(s) exp (− iωt) on the curve satisfies all the integral equations

where G(s, s′) is any Green's function of the problem, and V(a) is prescribed. All the equations corresponding to different Green's functions have the same solution. An asymptotic and convergent short-wave solution can be found by iteration if G can be chosen explicitly so that the integral equation has a small kernel for high frequencies. At any point of the curve draw the local circle of curvature; then the explicit known solution for a source on this circle is (with slight modifications) a possible Green's function, and the equation formed with it has a small kernel and can be solved rigorously by iteration. If V(a) is independent of the frequency, the leading term in the resulting asymptotic expansion is

where c is the velocity of sound and 2πk−1 is the (short) wavelength corresponding to the frequency ω/2π. If V(a) varies rapidly, as in diffraction theory, the iterative solution still gives a convergent asymptotic expansion, but the first approximation is then practically useless in the shadow region. Diffraction problems are not treated in the present paper.

The present work appears to be the first practical and rigorous solution of a short-wave problem in optics or acoustics when a solution in closed form is not available. It is suggested that the technique (suitably combined with formal expansions) may be applicable to a wider class of radiation and diffraction problems.

This paper describes a method for computing the coefficients in the Chebyshev expansion of a solution of an ordinary linear differential equation. The method is valid when the solution required is bounded and possesses a finite number of maxima and minima in the finite range of integration. The essence of the method is that an expansion in Chebyshev polynomials is assumed for the highest derivative occurring in the equation; the coefficients are then determined by integrating this series, substituting in the original equation and equating coefficients.

Comparison is made with the Fourier series method of Dennis and Foots, and with the polynomial approximation method of Lanczos. Examples are given of the application of the method to some first and second order equations, including one eigenvalue problem.

A discussion is given of the convergence of the eigenvalues Λ (N) of two-dimensional finite-difference equations towards the eigenvalues Λ of the corresponding second-order differential equation, and it is shown that

where h = N−1 and ν2 is a constant. As in our previous paper (4), this can be used to make an accurate estimate of λ by extrapolating to h = 0. After an account of the relaxation method used for computing Λ(N) and a discussion of the residual vector, results are presented for an approximation to the lowest spatially symmetric and antisymmetric states of two electrons in a sphere, interacting through their Coulomb potential.

A scheme is proposed for solving a class of integral equations by electronic analogue computing techniques in times as short as one-tenth of a second. The scheme utilizes a recently developed high-speed analogue function store for carrying out a special iterative procedure which is shown to be more efficient than the classical Neumann process. The problem of the kernel generation at high repetition rates is considered and a novel method based on pivotal function generators is described. Likely errors are analysed and an overall accuracy of the order of 1% is shown to be attainable with known techniques.

The power and appropriateness of renewal theory as a tool for the solution of general problems concerning counters has been amply demonstrated by Feller (7), who considered a variety of counter problems and reduced them to special renewal processes. The use of what may be called renewal-type arguments had certainly been made by authors other than Feller (e.g. in § 3 of Domb (3)), but it was only in (7) that the simplicity of the renewal approach to counter problems was recognized and systematically applied. More recently, Hammersley (8) was concerned with the generalization of a counter problem previously studied by Domb (2). This problem may be introduced, mathematically, as follows. Let {xi}, {yi} be two independent sequences of independent non-negative random variables which are non-zero with probability one (i.e. two independent renewal processes). The {xi}, are distributed in a negative-exponential distribution with mean λ-1, and we write Eλ for their distribution function and say ≡ {xi} is a Poisson process to imply this special property of ; the {yi} have a distribution function ‡ B(x) with mean b1 ≤ ∞. Form the partial sums and define ni to be the greatest integer k such that Xk ≥ t, taking X0 0 and nt = 0 if x1 > t. Then define the stochastic process

Hammersley'sx counter problem concerns the stochastic process

Various definitions of uniform acceleration are possible in relativity theory. Two of these are considered with particular reference to their physical realizations.