Let

be a quadratic form in n variables (n ≥ 3) with integral coefficients. One of us gave recently (2) a simple proof of a result of Cassels (l), in the following form: if the equation f = 0 is properly soluble in integers, then it has a solution satisfying

where γn–1 denotes Hermite's constant.

1. Introduction. In this paper we consider certain general properties of propositional calculi. Two forms of the definition of a finite model of such a calculus are discussed, these forms differing in the manner prescribed for the satisfaction of the rules of the calculus. The methods of definition are shown to be equivalent for the application of the finite model method for the demonstration of the unprovability of formulae in the calculus. It is further proved that, although the decidability of a calculus follows from the existence of a finite model counter-example for each unprovable formula of the calculus, the converse result is not true.

This paper is in two parts. In Part I we are concerned with one or more linear series on an algebraic curve; we consider a set of points on the curve which are contained with assigned multiplicities in a set of each of the linear series and, by persistent use of Severi's equivalence relation for the united points of an algebraic correspondence with valency, we derive formulae for the number of such sets of points when the constants involved are such as to make this number finite. All this is essentially a generalization of the formula for the number of points in the Jacobian set of a linear series of freedom 1, and the main result is Theorem 3.

McCallum and Smith (1) have developed a logical computer which solves problems which are essentially of the following kind.

Our main object is to give an exposition of a simplified version of Siegel's proof of his mean value formula in the Geometry of Numbers, also proved by A. Weil in a more general context ((1), (2), (3), (4)). In deriving the particular case which is Siegel's formula, however, certain difficulties of convergence arise, which were resolved by Siegel through rather tedious calculations, and by Weil through the use of Fourier analysis in topological groups. A new method of dealing with these difficulties, depending on Lebesgue's bounded convergence theorem and the relatively simple Lemma 8, seems to be our major contribution. In other respects our proof is very similar to Siegel's. At some points we have simplified his treatment; at others, where his work seems rather condensed, we have expanded it. We are indebted to Mahler (9) for some of the simplifications and grateful to him for letting one of us see his unpublished work. By using Lemma 8 and giving a separate proof of the existence of a fundamental domain for the unimodular group, we give a proof which is independent of Minkowski's reduction theory.

Let

be a real quadratic form in n variables (n ≥ 2) with integral coefficients and determinant D = |fij| ≠ 0. Cassels ((1),(2)) has recently proved that if the equation f = 0 is properly soluble in integers x1, …, xn, then there is a solution satisfying

where F = max | fij and cn depends only on n. An example given by Kneser (see (2)) shows that the exponent ½(n – 1) is best possible. A simpler proof of Cassels's result has since been given by Davenport(3), and the theorem has been improved in certain cases by Watson(4). Here I consider the inequality f(x1, …, xn) > 0, where f is an indefinite form, and obtain a result analogous to that of Cassels.

The object of this note is to prove the following generalization (from Riemann to Lebesgue integrable functions) of a theorem already proved by the authors (1). The notation and terminology of the earlier paper are used.

In this note we show that if a real square matrix P fulfils certain rather general conditions then there exists a real diagonal matrix D such that the characteristic equation of DP is stable and, furthermore, aperiodic. (A characteristic equation is called stable if the real parts of its roots are all negative. If the roots are all real and simple the equation is said to be aperiodic; see Fuller(3).)

Let G denote a group of order a power of the prime p, and let G′ be the derived group of G. The lower central series of G will be written

For any subgroup H of G we denote by P(H) the subgroup of H generated by all elements xp as x runs through H, and by Φ(H) the Frattini subgroup of H. We write (H:Φ(H)) = pd(H); thus d(H) is the minimal number of generators of H.

In this paper we bring together two aspects of homological algebra, one modern and the other classical. MacLane(6) has suggested the desirability of avoiding superfluous references to the objects of the category in proving fundamental results in homology theory. Accordingly, we consider categories whose maps form a ringoid in the sense of Barratt(1) and impose on the ringoid axioms, expressed in the language of ring theory, but having a close affinity to the axioms of Buchsbaum(2) for an exact category. (See also Cartan and Eilenberg(3); Appendix.) The purpose of these axioms is to ensure that the standard concepts of homology theory are applicable to the ringoid. Consequently, we call such a ringoid a homological ringoid.

In a previous paper of the same title (1) on

where e(t) is even and has least period 2π, I have shown that the equation has a family of periodic solutions or, equivalently, that a topological transformation T has a family of fixed points. Since much of the detail in (1) is irrelevant here, it will be convenient if the results we must quote about are put into the following form:

Theoerm A. There is a constant integer p such that, whenever k ≥ p, there is at least one solution xk(t) of (1·1) for which

(i) xk(0) > 0, ẋk(0) = 0 and xk(t) has period 2π;

(ii) xk(t) = xk(2π–t);

(iii) xk(t) has, in 0 ≤ t < 2π, k positive maxima, k negative minima and no other stationary points;

(iv)

This except (ii), is a special case of Theorem 2 with the notation x*(t|k, 1, 0) simplified to xk(t) and the constant p introduced to guarantee that (vi) of Theorem 2 implies (iv) above. Finally Lemma 1 of (1), in its original form or as specialized in Theorem B (i) below, gives (ii).

We shall write ak = xk(0) and note, though we do not need to use it here, that, by (v) of Theorem 2, ak = kw + O(k−2) as k→ ∞. If Fk denotes the point (ak, 0) of the (a, b)-plane then Fk is a fixed point of T and we can best state this paper's results in terms of the segment joining a pair of such points.

1. Introduction. The class of finite groups in which any two subgroups of the same order are isomorphic will be denoted by (C), and ‘G ∈ (C)’ will mean ‘G belongs to the class (C)’.

In studying the structure of perfect sets, an important method would be to establish homoeomorphisms in general classes of sets. For then the general structural features of sets could be more easily seen by considering the simplest sets of a homoeomorphic class.

In this paper we establish the homoeomorphism in two general classes of plane perfect sets (we consider only bounded sets).

Let R be a ring, not necessarily associative. A derivation of R is a mapping D:R → R, not identically zero, such that

for all x, y ε R. If R is a linear algebra over a field F, then D also satisfies

for all x ε R and α ε F.

For a long time there has been some interest in obtaining spherical harmonics with the symmetry of the regular polyhedrons, particularly for electrostatic problema involving polyhedral conductors. Work on the icosahedral group has been done, among others, by Meyer (7), Laporte (6), Hodgkinson (3) and Poole (8), apart from the classic work of Klein (4). In the several approaches of these authors, only spherical harmonics for the totally symmetric representation were obtained, the most complete table being that of Laporte who obtained the spherical harmonics up to l = 21. New interest in the icosahedral group has arisen in connexion with the structureof some proteins (5), and we obtain here, by the recently developed methodof Altmann(1), expansions in spherical harmonics for ali the representations of this group. This has been done up to and including l = 14. (For the totally symmetric representation we have also included l = 15.) In what follows we shall use Altmann's formulae and notation.

The distribution of the total (or ‘second’) curvature of a stationary random Gaussian surface is derived on the assumption that the squares of the surface slopes are negligible. The distribution is found to depend on only two parameters, derivable from the fourth moments of the energy spectrum of the surface. Each distribution function satisfies a linear differential equation of the third order, and the distribution is asymmetrical with positive skewness, in general. A special case of zero skewness occurs when the surface consists of two intersecting systems of long-crested waves.

The first two papers of Reifenberg ((4), (5)) under the general heading ‘Parametric Surfaces’ contain a detailed and profound study of the tangential properties of these surfaces. Since their publication one of the fundamental problems in the subject, that of obtaining a convenient representation for the surface, has been solved by Cesari (1). In this paper we obtain direct proofs of Reifenberg's results from Cesari's theorem. Whereas Reifenberg had to contend with both topological and real-variable problems combined the effect of Cesari's theorem is to remove the topological difficulties and to leave a straightforward real variable problem. The definition of approximate tangential plane used here is not the same as either of the two employed by Reifenberg, but the differences between it and one of Reifenberg's definitions ((5), definition 2) are not very important.

Numerical values of the above integral are given for 3/2λ√3 = 0(0·1)1, to six decimals for |ω| = 0(0·l)2·0(0·2)4·8 and to seven decimals for |ω| = 5(0·5)10(1)15.

The integral satisfies a third-order differential equation; this equation is integrated numerically for ω ≥ 0 and ω ≤ 0, and in each case the method of isolating the solution which satisfies f(ω) → 0 as ω → ∞ is described.

The main purpose of this note is to show some connexions between Legendre polynomials and random walks. More specifically we shall be concerned with what may be called ‘trinomial’ random walks i.e. walks on the points 0, ± 1, ± 2, …, such that the probabilities of ‘currently’ taking steps of – 1, 0, + 1 are p–1, p0, p1.

We shall be concerned with the differentiation of an arbitrary set function ψ from certain classes of set functions with respect to another fixed set function φ. In (1), § 2·5, a halo evanescence condition is shown to be necessary and sufficient for the differentiation of a fixed measure ψ with respect to φ. Here we formulate a type of halo condition which ensures the differentiability of entire classes of set functions. Although the necessity of this condition is not shown, we demonstrate by means of an example that the results cannot be improved upon in certain respects.