1. Good (4) has proved the formula

where Pn(x) is the Legendre polynomial of degree n and t is any integer greater than n. The form of the right-hand side suggests that (1) may be of use in deriving arithmetic properties of Pn(x).

Elsewhere (1) the writer indicated a connexion between divisibility properties of Pm(a) for special values of a and the complex multiplication of elliptic functions. If p = 2m + 1 is an odd prime, put

It is well known ((1), pp. 120–1) that if G is a finite p–group then the factor group G/G′ is cyclic if and only if G is itself cyclic. A trivial consequence of this fact is that, in such a case, the factor group G′/G″ is the unit group. (As usual, G(r) denotes the rth derived group of the group G.)

We prove the following theorem.

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.

Van der Corput has shown (2), using a general criterion of Weyl (1), that a necessary and sufficient condition, that a sequence of points Pn = (αn, βn) (n = 1, 2,…) in two-dimensional space be uniformly distributed modulo 1, is that for all pairs of integers (u, v) other than u = v = 0 the one-dimensional sequence (uαn + vβn) (n = 1, 2,…) is uniformly distributed modulo 1. The object of this paper is to give a quantitative form to the sufficiency part of this qualitative criterion.

In a recent note (1) I sharpened existing theorems, which state, roughly speaking, that unless a symmetric convex body K has an excessively small homogeneous minimum, it must have a reasonably small inhomogeneous minimum; in other words, if the translates of K centred at points of the integer lattice form an efficient packing, then by expanding them we may derive an efficient covering. In this note I will prove the converse, that a bad packing leads to a bad covering; the result is a much less useful one, as the worst case occurs when we foolishly try to pack rather spiky bodies point to point. A result of this kind has previously been proved by Mahler (4), but his result is less precise from our present point of view; a full account is given by Cassels(3).

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.

Throughout this paper, the concept of a ‘freely generated algebra’ will be meant in the sense of (3), Definition 2·1. It is understood that all conventions concerning terminology and notation introduced in (3) hold in this paper. In particular, it is understood that the sign denotes a freely generated algebra, and that the definition-set and the free basis of are denoted by C and D respectively. The letter q will denote a cardinal number.

Let R* be a ring with unit element I. For every A ∈ R*, let the mapping A → A′ be an involutory anti-automorphism, i.e. a one-to-one mapping of R* onto itself with the following properties:

Let C* denote the set of all symmetric elements of R*, i.e. C′ = C for all C ∈ C*.

An element A ∈ R* is said to be skew-symmetric if A′ + A = 0 (the zero element of the ring).

It is well known that if K is a commutative field then the polynomial ring K[x1, x2, …, xn] has global dimension (in the sense of homology theory) equal to the number of variables. This, of course, is not an isolated result but, in view of the special interest which attaches to such a familiar object as a polynomial ring, a short and comparatively elementary proof of this particular result may be of interest. The present paper is devoted to such a proof and therefore it is proper that this first section should indicate, in some detail, what knowledge of homological algebra is presupposed.

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.

Asymptotic theory of partitions of unipartite numbers has been extensively investigated after the classic work of Hardy and Ramanujan on the subject. The more complicated problem of bipartite partitions has been only recently broached. Auluck(1) has deduced asymptotic formulae for p(m, n), the number of partitions of the bipartite number (m, n), both for m fixed and n → ∞, and for both m and n tending to infinity in such a way that m/n = 0(1) and n/m = 0(1). It will be shown here that the formula obtained for the first case is actually valid for m = 0(n¼). This proves the conjecture made earlier by the author (3).

This paper is concerned with what, for lack of a better name, the author proposes to call geometric modules over a field k. These modules are defined as follows. A geometric module M consists of a vector space, also denoted by M, over k, together with a ring of linear transformations A of M into itself subject to the following conditions:

(i) A is a homomorphic image of the ring Sn = k[X1, …, Xn] of polynomials in n indeterminates X1, …, Xn for some value of n;

(ii) M, considered as an A-module, is finitely generated.

It is well known that Hubert's function of a homogeneous ideal in the ring of polynomials K[x0, …, xm], where K is a field and x0, …, xm are independent indeterminates over K, is, for large values of r, a polynomial in r of degree equal to the projective dimension of (1). Samuel (4) and Northcott (2) have both shown that if the field K is replaced by an Artin ring A, is still a polynomial in r for large values of r. Applying this generalization Samuel (4) has shown that in a local ring Q the length of an ideal qρ, where q is a primary ideal belonging to the maximal ideal m of Q, is, for sufficiently large values of ρ, a polynomial in ρ whose degree is equal to the dimension of Q.

The statistical operation of multiple linear regression by least squares is equivalent to the orthogonal projection of vectors of observations on a space spanned by vectors of observations; and a partial regression can be similarly represented as an oblique projection. This connexion between a statistical and a formal algebraical operation gives the main source of interest for this investigation. Its object is to develop algebraical theory which supplies terms necessary for a unified algebraical and geometrical formulation of concepts in multivariate analysis. An exposition of the statistical theory is to be made in a separate paper.

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.

1. Discussion of results

1·1. Introduction: The classical theorem of Lagrange states that the order of a subgroup of a finite group G divides the order, (G), of G. More generally, if H and K are subgroups of G, and H ≥ K, then (G:K) = (G:H)(H:K), where (G:K) denotes the index of K in G, etc. We call a number a possible order of a subgroup of G if it is a divisor of (G), and a possible order of a subgroup of G containing a subgroup H if it is a divisor of (G) and a multiple of (H). In this paper we discuss conditions on G for the existence of subgroups of every possible order, the existence of subgroups of every possible order containing arbitrary subgroups, and similar properties.

In the theory of vector-spaces an ordered, ortho-normal set of r vectors is called an r-frame. Let Sn denote the unit sphere in euclidean (n+ 1)-space, where n ≥ 1. By an r-field on Sn is meant a continuous function which assigns to each point of Sn an r-frame in the tangent space at that point. If q < r we obtain a q-field from an r-field by suppressing the first r – q vectors of each r-frame. Certainly Sn admits a 0-field, and does not admit an (n+ 1)-field, since the tangent space is n-dimensional. An n-field on Sn is called a parallelism. Notice that an (n − 1)-field on Sn can always be extended to an n-field, since spheres are orientable. The problem is to determine the greatest value of r such that Sn admits an r-field.

By an iteration of a regular summation matrix B = (bmn) with a regular matrix A = (amn), or B.A, we shall mean the summation method defined by the matrix B operating on the sequence {tn} of A transforms of a sequence {sn}. Thus the sequence {τn} of B. A is given by

The purpose of this note is to present two characterizations of conjugate Banach spaces. More precisely, we present two conditions, each necessary and sufficient for a (real or complex) Banach space to be isomorphic to the conjugate space of a Banach space, and two corresponding conditions for to be equivalent to the conjugate space of a Banach space. Other characterizations, in terms of weak topologies, have been given by Alaoglu ((1), Theorem 2:1, p. 256, and Corollary 2:1, p. 257) and Bourbaki ((4), Chap, IV, §5, exerc. 15c, p. 122). Here, by the conjugate space* of a Banach space we mean ((2), p. 188) the space of continuous linear functionals over . Two Banach spaces and are said to be isomorphic if there is a one-one continuous linear mapping of onto (its inverse is necessarily continuous by the inversion theorem ((2), Théorème 5, p. 41; (6), Theorem 2·13·7, Corollary, p. 29)); they are said to be equivalent if there is a norm-preserving linear mapping of onto .((2), p. 180).

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.