In 1936 Todd and Coxeter gave a method ((10), also described in (4), ch. 2) for establishing the order of a finite group defined by a set of relations

satisfied by its generators S1, S2,…, Sk. They enumerate systematically the cosets of a suitable subgroup whose order is evident from the defining relations for the whole group. They describe the method as being ‘purely mechanical’, and since that date the advent of electronic computers has led a number of people to programme the method for automatic execution. Most of this work has remained unpublished; an account is given here of work known to the author.

Let Q be a semi-regular local ring of dimension d, m be its maximal ideal, and q be an m-primary ideal. Then LQ(Q/qn+1), the length of Q-module Q/qn+1, is equal to the characteristic polynomial PQ(q,n) in n for a sufficiently large value of n:

where ei = ei(q), i = 0,1,2,…, d are integers uniquely determined by q, called normalized Hilbert coefficients of q according to (1). It was shown in (1) that e1(q) is a non-negative integer, and is equal to zero if and only if q is generated by a system of parameters. We shall prove, in this paper, that e2(q) is also a non-negative integer, and that this non-negativity is not necessarily true for other coefficients. We shall give an example with negative e3(q).

The topological theory of covering spaces may be used to prove results in group theory, for instance, the Kuros-Reidemeister-Schreier theorem (1). It seems likely that such methods can be applied to prove the Freiheitsatz (4) and the identity theorem (3), and also perhaps Lyndon's conjecture, that the normal closure in a free group F of a, single element r is freely generated by conjugates of r. However, although these problems may easily be stated in topological terms, no such proof is at present known. In this paper we prove a related result.

Hilton and Rees have proved (cf. (1), Theorem 1·3) that every natural map

is induced by a map from A to B (or, Hom (A, B) → Next1,1 (A, B) is surjective). It follows that Ext1 (B, −) and Ext1 (A, −) are naturally isomorphic if and only if A and B are quasi-isomorphic (loc. cit., Theorem 2·6), i.e. if there exist projective objects P, Q and an isomorphism . One can ask whether these theorems remain true for higher extension functors.

A normal preference system for a combination of goods is represented by an increasing function φ with convex levels. From Gossen's law, that preference and price directions coincide in equilibrium (a special consequence of his Second Law), it follows that, on the data that xr is the vector of quantities purchased at prices given by a vector pr (r = 1,…, k), the gradient gr = g(xr) of the function φ at the point xr is given by

for some positive multiplier μr;. There may be considered the class of preference functions thus satisfying Gossen's law in respect to the data, and thus with gradients taking prescribed directions at k prescribed points. In particular, there may be considered the subclass of these which are quadratic in some convex region containing the points xr. By choosing any multipliers μr, there is obtained a set of gradients gr associated with the points xr. It is asked if there exists a quadratic function which is increasing and has convex levels in a convex neighbourhood of the points xr, and whose gradient at xr is gr; also it is required to characterize the class of such functions, which, if any exist, form an infinite variety. This is the background of the questions which are going to be investigated, and which are of importance in a general method of empirical preference analysis in economics.

In an earlier paper (1) the idea of a c-homeomorphism on a 2-manifold was introduced; it can be described in the following way. Let the 2-manifold M contain an annulus, A, one of the boundary components of which is a simple closed curve c. There is a homeomorphism of A to itself, fixed on the boundary of A, which sends radial arcs onto arcs which spiral once around A (see Fig. 1). This can be extended to

a homeomorphism of M to itself, by the identity on M — A. Intuitively this homeomorphism can be thought of as the process of cutting M along c, twisting one of the now free ends, and then glueing together again. The c-homeomorphism has, in fact, only been defined up to isotopy, but as we shall only be considering homeomorphisms up to isotopy, this is sufficient. It was shown in (1) that any orientation preserving homeomorphism of a closed orientable 2-manifold can be expressed as a product of c-homeomorphisms. In that case, a c-homeomorphism could be associated with any simple closed path on such a manifold, but on a non-orientable 2-manifold a c-homeomorphism can only be associated with a simple closed orientation preserving path. It is this fact that invalidates many of the lemmas of (1) if they are regarded as applying to the non-orientable case. The purpose of this paper is to show that (Theorem 1) c-homeomorphisms do not generate the group of all homeomorphisms of a closed nonorientable 2-manifold, but that if another elementary type of homeomorphism, the Y-homeomorphism, is introduced, then c- and Y-homeomorphisms together do generate this group (Theorem 2). This result leads to a discussion of non-orientable 3-manifolds, to a result analogous to that obtained for orientable 3-manifolds in (1), and to a new proof of the fact that any 3-manifold bounds a 4-manifold. Throughout this paper all manifolds will be considered as possessing an underlying combinatorial structure, all paths will be polygonal with respect to some subdivision of the manifold containing them, and all homeomorphisms and isotopies will be piecewise linear.

Let X be a compact differentiable n-manifold of class C∞ and let f be a C∞ map of X into euclidean (n + k)-space. Then f is an immersion if its jacobian has rank n at each point of X. It is an embedding if it is also one-one. We write X ⊆ n + k to denote the existence of an immersion, X ⊂ n + k to denote the existence of an embedding. By a theorem of Whitney, we have X ⊆ 2n − 1, X ⊂ 2n.

The following properties of the real numbers are well known:

If (xn) is a sequence of real numbers, xn → x and xn ≥ 0 for each n then x > 0. (1)

If (xn) is a sequence of real numbers, xn → x and x > 0 then, for all sufficiently large n,xn > 0.

A copositive quadratic form is a real form which is non-negative for non-negative arguments. A completely positive quadratic form is a real form which can be written as a sum of squares of non-negative real forms. The completely positive forms are basic in the study of block designs arising in combinatorial analysis (3). The copositive forms arise in the theory of inequalities and have been considered in a paper by Mordell (4) and two papers by Diananda(1, 2).

Throughout this paper q1,q2,… are positive and x1,x2,… non-negative real numbers. When q1+…+ qn = 1, we define Δn, Sn and Σn as follows:

and

We use the term ‘comparable’ in Kober's(2) sense. Two functions f(x1, …, xn) and g(x1, …, xn) are comparable if and only if αg ≤ f ≤ βg for some positive constants α and β.

The object of the present note is to evaluate some integrals involving Meijer's G-function, in which the argument of the G-function contains a factor where m and n are positive integers and t is the variable of integration. Two different forms of the general result have been obtained, one for m > n and the other for m < n. The value of the corresponding integral when m = n is also obtained. For the definition, properties and the behaviour of the G-function, see (2), §§ 5·3, 5·31 and (5), § 18.

In this paper we first of all consider the dual integral equations

where f(ρ), g(ρ) are given, A(t) is unknown, and α is a given constant. This system, with g(ρ) = 0, was originally considered by Titchmarsh ((13), p. 337), and Busbridge (1), who obtained a solution by the use of Mellin transforms and analytic continuation in the complex plane. The method described in this paper involves the application of certain multiplying factors to the equations. In the present case it is relatively easy to guess the multiplying factors and then the method is essentially a real-variable technique. It is presented in this way in § 2 below.

In this paper first of all we find an exact solution of the dual series equations

where

and Jn is the Jacobi polynomial defined by

The Bn are unknown coefficients which are to be determined. The functions f, g and the parameters a, λ, μ, are given. Necessary restrictions on these quantities are indicated during the analysis.

The concept of Lyapunov's function is an important tool in studying various problems of ordinary differential equations. In the present paper we shall extend the Lyapunov's method to study some problems of differential equations in Banach spaces. Continuing the theory of one parameter semi-groups of linear and bounded operators founded by Hille and Yoshida, Kato(4) presented some uniqueness and existence theorems for the solutions of linear differential equations of the type

where A(t) is a given function whose values are linear operators in Banach space. Krasnoselskii, Krein and Soboleveskii (5,6) also considered such equations including non-linear differential equations of the type

Mlak (9) obtained some results concerning the limitations of solutions of the latter equation.

Dispersion relations are derived for the triangular, honeycomb and Kagomé lattices using theorems for the eigenvalues of block circulant matrices. Constant frequency contours are plotted for each of the spectral branches of the three lattices and the distribution functions for squared frequencies are sketched.

A new and general Monte Carlo technique is described for solving some well-known percolation and cluster-size problems on regular lattice networks. The method has been applied to ten two-dimensional structures.

Let Xν, ν= l, 2, …, n be n independent random variables in k-dimensional (real) Euclidean space Rk, which have, for each ν, finite fourth moments β4ii = l,…, k. In the case when the Xν are identically distributed, have zero means, and unit covariance matrices, Esseen(1) has discussed the rate of convergence of the distribution of the sums

If denotes the projection of on the ith coordinate axis, Esseen proves that if

and ψ(a) denotes the corresponding normal (radial) distribution function of the same first and second moments as μn(a), then

where and C is a constant depending only on k. (C, without a subscript, will denote everywhere a constant depending only on k.)

This paper presents a general theory of storage for dams subject to a steady release, and with continuous inputs having infinitely divisible distributions. The paper extends some earlier work by Downton, Lindley, Smith and Kendall ((10), see discussion), and gives a detailed account of results previously sketched by the authors (6).

The theory of group representations is explored with regard to statistical applications. Ease of analysis depends rather critically on the amount of symmetry present and this point is examined in detail. ‘ Stationarity’ assumptions are considered for a finite number of variates, with examples of their use in experimental situations.

Three distinct methods are used to obtain exact expressions for various characteristics of a particular asymmetric two-dimensional random walk. The results obtained include, for the transient unrestricted walk, the probability of return to the starting-point and the average number of arrivals at the general lattice point; and, for a walk restricted within a rectangular absorbing barrier, the average number of arrivals at any accessible point and the absorption probabilities on the boundary. Whilst there is some duplication of results by using the three different methods of analysis, this is not extensive and provides a useful check on the results. Also the methods are of some general interest in themselves.