1. Let A be an associative algebra over a commutative field K; A is then a vector space over K, and multiplication is defined in A in such a way that

for x, y, z in A and λ in K. We suppose in addition that A does not contain a unit, and we denote by Ae the algebra obtained by adjoining a unit e to A.

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 results and methods of algebraic geometry, when analysed in terms of modern algebra, have revealed on several occasions algebraic principles of surprising generality. Recently it has become apparent that the geometric theory of infinitely near points has, as it were, an abstract form which forms part of the ideal theory of commutative rings, but there are many details which have yet to be worked out. Roughly speaking, one may say that what corresponds to the theory of the sequence of points on a curve branch is now known in some detail, and forms a substantial addition to our knowledge of the properties of one-dimensional local rings†; but the construction of an abstract theory similarly related to the theory of neighbourhoods in n-dimensional projective space can hardly be said to have been started. A number of necessary preliminary steps were taken by the author in (3)—in the process of providing algebraic foundations for certain applications of dilatation theory—and later some applications were made to 2-dimensional problems. However, the present paper should be regarded as an attempt to initiate a dilatation theory of regular local rings to run parallel to the general theory of infinitely near points in n-dimensional space.

It has been shown that every general recursive function is definable by application of the five schemata for primitive recursive functions together with the schema

subject to the condition that, for each n–tuple of natural numbers x1,…, xn there exists a natural number xn+1 such that

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.

1. Introduction and recapitulation. In a previous paper (4) we have defined a ringoid (with identities) as a set of elements α, β,… in which addition and multiplication are defined for certain pairs of elements in such a way that, whenever operations are defined, the usual ring axioms are satisfied. Precisely, is an abstract category ((3), p. 108) with an addition operation such that the following axioms hold:

(R 1) For any two identity elements ι, ι′ of, let ℋ(ι, ι′) be the set of elements ofwhose left identity is ι and right identity is ι′; then ℋ(ι, ι′) is empty or an Abelian group under addition.

Introduction. By the content of a polynomial with coefficients in a ring R, one understands either the R-ideal or (better) the additive group which the coefficients generate. There are important connexions between the contents of two polynomials and the content of their product, and results of this kind are to be found in the work of Gauss. Here we shall give a wide generalization of a theorem of this type not involving polynomials at all; however, as the reader will observe, the proof is essentially a reduction to the polynomial case.

Eilenberg and Steenrod(1) give a set of axioms for the homology theory of pairs of spaces and their maps, and prove that these axioms are categorical on triangulable pairs.

Here we give a set of axioms for the homology theory of single spaces and their maps, that is, for absolute rather than relative homology. This axiomatization is shown to be essentially equivalent to that of Eilenberg and Steenrod, the relative homology groups being suitably denned in terms of the absolute groups.

1. Introduction. The present paper is chiefly concerned with a generalization, to be known as a ‘D-interval’, of the notion of interval or segment in an arbitrary partially ordered group. This idea is originally due to Duthie (2), but was developed by him only in a lattice. In analogy with the use of the interval in the normal sense, notions of ‘D-distributivity’ and ‘D-modularity’ are defined in terms of the D-interval, and analogues of known properties of lattice-groups or ‘l– groups’ can be formulated which might be valid when a lattice structure is no longer assumed to exist; in particular, an attempt is made to provide such a generalization of the result of Freudenthal (3) that every Z-group is a distributive lattice, but, for an arbitrary partially ordered group, it is shown that only an ‘approximation’ (in terms of non-Archimedean elements) to the desired result actually holds, although any Archimedean partially ordered group is necessarily D-distributive.

In this note I prove the following theorem:

Theorem 1. Given 3N points in a plane, we can divide them into N triads such that, when we form a triangle with the points of each triad, the N triangles will all have a common point.

The proof depends on three lemmas; the first two are well known, and are quoted without proof; the third I believe to be new, and leads directly to the proof of the theorem.

Suppose throughout that

and that

is an integral function. Suppose also that l, sn(n = 0,1,…) are arbitrary complex numbers and denote by ρ(ps) the radius of convergence of the series

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.

1. Introduction. We say that a smooth compact manifold is embedded in q–space when there is given a regular one-one map of the manifold into q–dimensional Euclidean space. Whitney (5) has shown that embeddings always exist when q is not less than twice the dimension of the manifold.

1. An elegant theorem due to Tarski states that a completely distributive complete Boolean algebra is isomorphic with a lattice of sets, and in fact the lattice of all the subsets of some aggregate. The obvious generalization of the question underlying this theorem is to ask whether one can pick out by means of a distributivity condition those lattices (not necessarily Boolean algebras) which are isomorphs of lattices of sets. The answer is no. The real numbers with their natural order form a complete lattice which satisfies the strongest possible distributivity conditions and yet is not iso-morphic with any lattice of sets.

1. Let φ(z) be regular in │z│ < 1, let

and for any real α let

where (in)−α = n−α exp(— ½απi). The function φα is an adaptation to power series of the Weyl fractional integral of order a of α function of a real variable.† It exists and is regular in │z│ < 1 for all α. Let also

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.

1. Introduction. Ledermann(1) has treated the problem of calculating the asymptotic probabilities that a system will be found in any one of a finite number N of possible states if transitions between these states occur as Markov processes with a continuous time parameter t. If we denote by pi(t) the probability that at time t the system is in the ith state and by aij ( ≥ 0) the constant probability per unit time for transitions from the jth to the ith state, the rate of change of pi is given by

where the sum is to be taken over all j ≠ i. This set of equations can be written in matrix form as

where P(t) is the vector with components pi(t) and the constant matrix A has elements

We prove that the length of the shortest closed path through n points in a bounded plane region of area v is ‘almost always’ asymptotically proportional to √(nv) for large n; and we extend this result to bounded Lebesgue sets in k–dimensional Euclidean space. The constants of proportionality depend only upon the dimensionality of the space, and are independent of the shape of the region. We give numerical bounds for these constants for various values of k; and we estimate the constant in the particular case k = 2. The results are relevant to the travelling-salesman problem, Steiner's street network problem, and the Loberman—Weinberger wiring problem. They have possible generalizations in the direction of Plateau's problem and Douglas' problem.

Since the first introduction of the concept of a matrix, questions about functions of matrices have had the attention of many writers, starting with Cayley(i) in 1858, and Laguerre(2) in 1867. In 1883, Sylvester(3) defined a general function φ(a) of a matrix a with simple characteristic roots, by use of Lagrange's interpolation formula, and Buchheim (4), in 1886, extended his definition to the case of multiple characteristic roots. Then Weyr(5) showed in 1887 that, for a matrix a with characteristic roots lying inside the circle of convergence of a power series φ(ζ), the power series φ(a) is convergent; and in 1900 Poincaré (6) obtained the formulae

for the sum, where C is a circle lying in and concentric with the circle of convergence, and containing all the characteristic roots in its ulterior, such a formula having effectively been suggested by Frobenius(7) in 1896 for defining a general function of a matrix. Phillips (8), in 1919, discovered the analogue, for power series in matrices, of Taylor's theorem. In 1926 Hensel(9) completed the result of Weyr by showing that a necessary and sufficient condition for the convergence of φ(a) is the convergence of the derived series φ(r)(α) (0 ≼ r < mα; α) at each characteristic root α of a, of order r at most the multiplicity mα of α. In 1928 Giorgi(10) gave a definition, depending on the classical canonical decomposition of a matrix, which is equivalent to the contour integral formula, and Fantappie (11) developed the theory of this formula, and obtained the expression

for the characteristic projectors.