The object of the present paper is to evaluate m(k), the invariant measure (defined in §2) of the set of real matrices with determinant 1 and norm bounded by a positive real number k. The norm is that induced on the linear transformations of Euclidean n-space by the usual Euclidean norm in that space, and is also defined in § 2. We were led to this problem as a result of work arising from the Geometry of Numbers. A knowledge of the behaviour of m(k) was necessary in order to generalize a certain mean value formula (Macbeath and Rogers (9)) from Riemann to Lebesgue integrable functions.

The wreath product is a useful method for constructing new soluble groups from given ones (cf. P. Hall (3)). Now although the wreath product of one soluble group by another is (obviously) always soluble, the corresponding result is no longer true for nilpotent groups. It is the object of § 3 of this note to determine precisely when the wreath product W of a non-trivial nilpotent group A by a non-trivial nilpotent group B is nilpotent; in fact I prove that W is nilpotent if and only if both A and B are (nilpotent) p–groups with A of finite exponent and B finite.

If g is a set of generatore of an enumerably infinite Abelian group A, it is proved that the elements of A can be arranged in both a one-ended and an endless infinite sequence in which successive terms differ by ± an element of g, except that the one-ended arrangement is impossible if g is finite and the rank of A is 1. Let ν be a cardinal number. Consider an infinite ‘chessboard’ whose positions are those lattice points of ν-dimensional space which have only finitely many non-zero coordinates. A piece associated with this chessboard is a generalized knight if every vector obtainable from a move of the piece by permuting its components and changing the signs of a subset of them is itself a permitted move. It is ascertained which positions a given generalized knight can reach in a finite sequence of moves starting at the origin, and whether or not, if it can trace out the whole chessboard in (i) a one-ended, (ii) an endless infinite sequence of moves visiting each position exactly once.

Samuel (1) introduced a generalized Hilbert function, written Xq(r, a) and defined for arbitrary ideals a in a local ring Q with maximal ideai m. where q is m-primary.

Northcott(2) proved that for a homogeneous ideal ã in a polynomial ring A[X1, …, Xn], where A = Q/q, this is equal to the ordinary Hilbert function χ(r, ã).

We prove a theorem which facilitates homotopy classification of maps into a topological group G. Some information about homotopy groups of G is obtained, including the following two results. Consider the Samelson product, as defined in (7), which constitutes a bilinear pairing of πp(G) with πq(G) to πp+q(G). The product of a α ∈ πp(G) with β ∈ πq(G) is written in the form 〈α, β〉. There exist groups having Samelson products of infinite order. Homotopy-commutative groups have zero Samelson products. We shall prove

Theorem (1·1). If G is a connected Lie group then there exists a positive integer n such that n〈α, β〉 = 0 for every pair α, β of elements in the homotopy groups of G.

The present paper is concerned with particular cases, obtained by suitably restricting the spaces involved, of the following general problem.

Given a topological space X, we ask whether there exist integers n ≥ 2 and non-contractible spaces X1, …, Xn such that X has the homotopy type of the Cartesian product X1, × … × Xn or of the union X1, v … v Xn.

It is our purpose in this paper to present concise proofs of four theorems which take up the greater part of a paper due to Brudno (1). The proofs of Theorems 1 and 2 have appeared already ((2), (3)), but we present them again so that comparisons can be made with the other two.

Some years ago M. Jackson(5) obtained the sum of a particular 3H3 series which generalized the theorems of Whipple and Watson on sums of a 3F2 series, and later she(6) deduced the sum of a particular well-poised 6H6( – 1) series. In this note sums of certain particular bilateral hypergeometric series of the same type are given. They are believed to be new. In the sequel, the sum of a particular 8Ψ8 series has been obtained. The existence of a particular case of this sum was pointed out earlier by Slater and Lakin (11).

There is perhaps some methodological interest in developing the theory of quadratic forms over the rational field using only the methods of elementary arithmetic. Hitherto it has appeared necessary to use theorems of a fairly deep nature, most often Dirichlet's theorem about the existence of primes in arithmetic progressions (e.g. Minkowski(1), Hasse(2), Dickson(8), Skolem(9), Burton Jones(6)). Skolem(5) uses a weaker form of Dirichlet's theorem which is rather easier to prove and Siegel(4) uses instead the machinery of the Hardy-Littlewood circle method. In this note I indicate how it is possible to develop the theory of quadratic forms over the rationals without using extraneous resources. Pall(10) states that he has also found such a development of the theory but he does not appear to have published it.

The following result has recently been proved by Lewis (3), Davenport (2) and Birch (1).

There exists an integer n0 such that every cubic form with rational coefficients and at least n0 integral variables represents zero non-trivially.

The arguments of Lewis and Birch are simple, and yield also various generalizations of this result. Davenport's proof is complicated, but it shows that the minimal n0 satisfies

The asymptotic values of the variances and covariances for the distribution of the transition numbers AA, AB, BA and BB for a two-state Markoff chain have been given by Bartlett (l). Whittle (3) obtained the probability distribution and their factorial moments for the k-state chain under the restriction that the first and the last observations belong to the rth and the sth state. But the expressions for the factorial moments according to Whittle himself are of little immediate use. He obtains useful results for the low order factorial moments by approximating in a grosser fashion, and the evaluation of the cumulants from these factorial moments is rather cumbersome. The object of this note is to give the first four cumulants and product cumulants which are exact to the order nr (p11 – p21)n−r where r is the order of the cumulant, for the distribution of the transition numbers of a two-state Markoff chain. They have been calculated by using the method developed by Iyer and Kapur(2). This method can equally be used for evaluating the asymptotic values of the cumulants for the k-state also. This is done by differentiating directly the determinantal characteristic equation.