In a recent paper (1) Nagata proved that a (linear associative) algebra, not necessarily of finite dimension, over a field of characteristic 0 which satisfies the identical relation xn = 0 satisfies also the relation x1x2 … xN = 0, where N is an integer depending only on n. He remarked further that it is a corollary that the result remains true if the ground field is of prime characteristic p, provided that p is large enough compared with n; and he conjectured that the obviously necessary condition p > n is in fact sufficient. The object of this note is to prove Nagata's conjecture. To do this, we give a new proof of his theorem, and as a by-product we obtain a rather better bound for N than his, showing, namely, that we can take N = 2n − 1. The determination of the best possible value of N, or even of its order of magnitude, seems not to be easy; at any rate, the best I have been able to do in the opposite direction is to show that for large n we cannot take N as small as n2/e2, where e is the base of natural logarithms.

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.

The purpose of the present note is to prove a basis theorem for polynomial modules (Theorem 1 below). As an application, a proof of the Hilbert function theorem for homogeneous modules (Snapper (1)) is obtained. Both these proofs are elementary in the sense that they do not depend on the decomposition theory for modules over a Noether ring.

In an earlier paper (4) it was shown how to define for any matrix a unique generalization of the inverse of a non-singular matrix. The purpose of the present note is to give a further application which has relevance to the statistical problem of finding ‘best’ approximate solutions of inconsistent systems of equations by the method of least squares. Some suggestions for computing this generalized inverse are also given.

Given a plane set E of positive and finite linear measure, the tangent at a point x at which the upper density of E is positive (which is so at almost all points of E and is not so at almost all points outside E) is defined in the following way. The line l through x is said to be the tangent to the set at the point x if in any angle vertex x that leaves the line outside the density of E is zero. This definition when applied to sets of infinite linear measure leads often to the conclusion that no tangent exists in cases when the structure of the set singles out some lines that have strong claim to be tangents to the set.

From the fact that Hausdorff s-dimensional measure is a regular Carathéodory outer measure follows (see Saks (3), ch. II, §§ 6, 8) the standard result:Theorem A. If {En} is any increasing sequence of sets, then ∧sEn → ∧s(ΣEn) as n → ∞.

Since ∧s X is denned (for every set X) as , the problem arises whether for every positive δ and every increasing sequence of sets one can prove

Now there are four possible ways of defining ; it is the lower bound of taken over coverings ΣUr of X by convex sets, but in most applications it is not important whether the restriction on the convex sets is that they shall be(i) open and of diameter less than δ,(ii) closed and of diameter less than δ,(iii) open and of diameter not exceeding δ, or(iv) closed and of diameter not exceeding δ.

Let Z, Q, C denote respectively the ring of rational integers, the field of rational numbers and the field of complex numbers. Minkowski (4) solved the problem of minimizing

for x, y ∈ Z(i) or Z(ρ), where a, b, c, d ∈ C have fixed determinant Δ ≠ 0. Here ρ = exp 2/3πi, and Z(i) and Z(p) are the rings of integers in Q(i) and Q(ρ) respectively. In fact he found the best possible results

for Z(i), and

for Z(ρ), where

while Buchner (1) used Minkowski's method to show that

for Z(i√2). Hlawka(3) has also proved (1·2), and Cassels, Ledermann and Mahler (2) have proved both (1·2) and (1·3). In a paper being prepared jointly by H. P. F. Swinnerton-Dyer and the author, general problems of the geometry of numbers in complex space are discussed and a systematic method given for solving the above problem for all complex quadratic fields Q(ϑ). Here, ϑ is a non-real number satisfying. an irreduc7ible quadratic equation with rational coefficients. The above problem is solved in detail for Q(i√5), for which

and the ‘critical forms’ can be reduced to

Let F(z) be a rational function of z which is regular at z = 0 and so possesses a convergent power series

The problem arises of characterizing those rational functions F(z) that have infinitely many vanishing Taylor coefficientsfh. After earlier and more special results by Siegel(2) and Ward(4) I applied in 1934(1) a p-adic method due to Skolem(3) to the problem and obtained the following partial solution.

Let

be a function regular for | z | < 1. With the hypotheses f(0) = 0 and

for some positive constant α, Cartwright(1) has deduced upper bounds for |f(z) | in the unit circle. Three cases have arisen and according as (1) holds with α < 1, α = 1 or α > 1, the bounds on each circle | z | = r are given respectively by

K(α) being a constant which depends only on the corresponding value of α which occurs in (1). We shall always use the symbols K and A to represent constants dependent on certain parameters such as α, not necessarily having the same value at each occurrence.

Allowance is made for the error terms in the cases where χ has a simple zero at x = 0, or two simple zeros at a; = ± 1 (with χ either positive or negative for − 1 < x < 1). The errors of the approximations given by neglecting the error terms are found to be small in all cases, sometimes much smaller than there was any previous reason to expect.

The paper deals with the behaviour of finite non-homogeneous Markov chains having regular transition matrices. Two types of ergodic behaviour are distinguished, and sufficient conditions are given for each type. Chains with commutative transition matrices are specially investigated. By means of expansions in terms of the characteristic roots several types of behaviour of chains which are not ergodic are studied.

1. Pólya's walk in the Euclidean plane is the motion of a particle, which takes successive independent random steps of unit length parallel to the coordinate axes, the probabilities of each of the four possible directions at each step being ¼. We imagine the particle starts at the origin, and that, in taking the ith step (i = 1,2,…) from (xi-1, yi-1) to (xi, yi), it leaves a straight track of unit length between these two points. Let Tn denote the track thus marked out in the first n steps; so that Tn is a connected chain of n straight segments each of unit length. Define the interior of Tn to be the set of points, which neither lie on Tn nor can be connected to the point at infinity without crossing Tn at least once; and let An denote the area of the interior of Tn. Associated with each walk ω, there is a sequence of random numbers {An(ω)}; what can be said about the behaviour of this sequence, especially for large n? We shall proveTheorem 1. For all ω,0 ≤ An(ω) ≤ 1/16 n2.

In a previous paper (3) the method of conformai transformations was employed to study the behaviour of a universe assuming the perfect cosmological principle, i.e. that apart from local irregularities the general aspect of the universe is the same at all points independent of place and time. To conform with Einstein's general theory an anisotropic cosmological term had to be postulated.

The properties of centre-of-mass systems at high energies are discussed, and the single time form of field theory is used to set up state vectors that describe the simpler centre-of-mass states of fields. These states, which are labelled by the eigenvalues of total angular momentum, are expanded in terms of the spherical harmonic resolution of the fields about a fixed point. A discussion of the application of the method to pion-nucleon scattering is given.

The calculation of the strengths of forbidden lines in the 1s22s22p2 configuration is extended to include the effect of interaction with the 1s22p4 configuration. For this case it is shown that if observed energy levels are used instead of theoretical energies in line-strength formulae which neglect configuration interaction the major part of the effect of configuration interaction is nevertheless taken into account. The results are illustrated by detailed computations on the Ca xv spectrum. The 1s22s22p4 configuration is also discussed.

In Part I of this series it was shown that for CCl4 the approximation of smoothing out the Cl nuclei over the surface of a sphere did not lead to reasonable results for the Thomas–Fermi distribution of electrons in the molecule. An alternative method in which the inner-shell Cl electrons were first compressed into the nuclei was not theoretically satisfactory because of the neglect of the exclusion principle. In this paper therefore a general scheme is presented in which we describe the inner-shell electrons of the outer atoms wave mechanically, assuming them to be undisturbed by molecular binding, but still use a Thomas–Fermi description for the valence electrons. A method by means of which the exclusion principle can be imposed is formulated. The following paper of the series deals with the application of the method described here to obtain an improved valence electron distribution in CCl4.

In this paper a precise distribution-theoretic formalism for the description of point charges interacting through a classical electromagnetic field is given. The distribution solutions of the Maxwell equations are shown to reduce to the Lienard–Wiechert fields, which are summable over the whole of space-time. It is not possible to obtain directly a unique field intensity, on the world lines, from these distribution solutions. A brief analysis shows that analytic continuation methods also do not give a unique field intensity on the world lines. The energy tensor for the field is a distribution which can only be rigorously defined in terms of the motions of the charges. Thus only the value of the field energy or momentum residing in any finite region of space-time can be given any meaning. Equations of motion for the charges may be derived from the field energy tensor, using the principle of conservation of total energy and momentum for the system. These equations agree with those obtained previously by Dirac (2). Also any canonical formalism in which the field potentials and the particle variables enter in an equivalent manner is not possible. The difficulty of the definition of products of distributions in the stress tensor does not occur.

In this paper, a study is made of the properties of the motion due to a general localized disturbance in an infinite fluid initially at rest. Two invariants of the motion are found, which are identified with the net linear momentum, or effective dipole strength, and the net angular momentum imparted to the fluid, and an equation is derived expressing the rate of change of the effective quadrupole strength of the motion. Solutions are obtained for the final period of decay of such a motion, and it is shown that, when the net linear momentum of the fluid is non-zero, the motion corresponds to a type of viscous vortex ring. The length scale increases with time t as [ν(t – t0)]½, and the total energy decays as [v(t–t0)]-1, where ν is the kinematic viscosity and t0 is a virtual time origin. If the net momentum is zero, the total energy decays as , and expressions are given for the velocity field in this case also.

It is shown that, if it is attained, the final period of any turbulent motion can be found from the vortex ring type of solution described above. The final period of homogeneous turbulence is clearly one particular example of such a motion. The final period of an axially symmetrical turbulent wake is treated in this way and is shown to consist generally of the superposition of two types of motion: (i) a mean streaming motion whose energy per unit length of the wake decreases as [ν(t – t0)]-1, and (ii) a superimposed fluctuating motion whose energy per unit length decays as [v(t–t0)]-1. Expressions are given for the distribution of mean-square vorticity and the energy spectrum tensor in the wake.

Let X be a set and let f be a mapping of P(X), the set of all subsets of X, into P(X). Conditions are given which are necessary and sufficient for there to exist a topology on the set X with respect to which, for all S ∈ P(X), the frontier of S is fS.

By the measure of a set of directions in the plane we mean the linear measure of the set of points P on a unit circle for which the direction from the centre to P belongs to the set.

The ray issuing from a point P in direction θ will be denoted by P(θ).