Einleitung. Man verdankt Richard Dedekind (siehe etwa(1)) das erste Beispiel einer Umkehrungsformel einer endlichen Reihe; dabei spielt, wie man weiβ, die sogenannte Möbius'sche Funktion (der Zahlentheorie) eine groβe Rolle. Auch die sogenannten Aufzählungsmethoden der Gruppentheorie, usw., die mit der soeben erwähnten Dedekindschen Umkehrungsformel eng verbunden sind, stützen sich, wie hauptsächlich von Herren Louis Weisner(2) und Philip Hall (3) gezeigt wurde, auf eine passende Verallgemeinerung der gewöhnlichen Möbius'schen Funktion, die für alle teilweise geordneten Mengen einen Sinn hat, deren Quotienten endlich sind. Siehe auch Ward (4).

The determination of a pair of fundamental units in a totally real cubic field involves two operations—finding a pair of independent units (i.e. such that neither is a power of the other) and from these a pair of fundamental units (i.e. a pair ε1; ε2 such that every unit of the field is of the form with rational integral m, n). The first operation may be accomplished by exploring regions of the integral lattice in which two conjugates are small or else by factorizing small primes and comparing different factorizations—a trial-and-error method, but often a quick one. The second operation is accomplished by obtaining inequalities which must be satisfied by a fundamental unit and its conjugates and finding whether or not a unit exists satisfying these inequalities. Recently Billevitch ((1), (2)) has given a method, of the nature of an extension of the first method mentioned above, which involves less work on the second operation but no less on the first.

Let be a free additive abelian group, and let be a basis of , so that every element of can be expressed in a unique way as a (finite) linear combination with integral coefficients of elements of . We shall be concerned with the ring of endomorphisms of , the sum and product of the endomorphisms φ, χ being defined, in the usual manner, by the equations

A permutation of a set will be called restricted if it moves only a finite number of elements. We call an endomorphism of a permutation endomorphism if it induces a restricted permutation of the basis .

In papers I(1) and II(2), the author has investigated the Stokes phenomenon associated with the arbitrary constants that occur in the asymptotic expressions of solutions of the nth-order differential equation

and has provided rules for tracing a given linear combination of n such asymptotic expressions across certain defined lines, designated as Stokes lines, anti-Stokes lines and branch-cuts. The object of these investigations is to apply the results to the approximate solution of more complicated linear differential equations of the nth order, and to this end it is an advantage to develop the theory using matrix techniques. In the present paper, the equations are recast into a special matrix form, and it is shown how the results obtained in II may be used in a systematic manner.

In a recent series of papers (10), (11), (12), I. M. James has made an illuminating study of Stiefel manifolds. We shall begin by describing his results (for the complex case). Let Wn, k, for k > 1, denote the complex Stiefel manifold U(n)/U(n − k), where U(n) is the unitary group in n variables. Then we have a natural fibre map Wn, k → Wn, 1 = S2n−1, where Sr denotes the r-dimensional sphere. Let Pn, k, for k ≥ 1, denote the ‘stunted complex projective space’ obtained from the (n − 1)-dimensional complex projective space† Pn by identifying to a point a subspace Pn−k. Then we have a natural ‘cofibre map’ Pn, k → Pn, 1 = S2n−2. The space Pn, k is said to be S-reducible if some suspension of the map Pn, k → S2n−2 has a right homotopy inverse. The results of James can then be summarized as follows.

The Hilbert transform, Hf, of a function f is defined by Hf = g, where

P denoting the Cauchy principal value and the integral being assumed to exist in some sense. When f is suitably restricted, Hf exists and

In the first part of Theorem 1 sufficient conditions are given for the validity of (1·2) rather more general than those of Wood ((6), p. 31). The present proof is based on the well-known condition of Riesz for the validity of (1·2), namely, that f is Lp(−∞, ∞) for some p > 1, and on the ‘Parseval’ relation (Lemma 3, (b)), which was used in a similar way by Hardy ((3), p. 110).

A generalization to two independent variables of Lagrange's expansion of an inverse function was given by Stieltjes and proved rigorously by Poincaré. A new method of proof is given here that also provides a new and sometimes more convenient form of the generalization. The results are given for an arbitrary number of independent variables. Applications are pointed out to random branching processes, to queues with various types of customers, and to some enumeration problems.

In this paper we shall be concerned with the differential equation

in which a and b are constants, p(t) is a continuous periodic function of t with a least period ω, and dots indicate differentiation with respect to t. The function h(x) is assumed continuous for all x considered, so that solutions of (1) exist satisfying any assigned initial conditions. In an earlier paper (2) explicit hypotheses on (1) were established, in the two distinct cases:

under which every solution x(t) of (1) satisfies

where t0 depends on the particular x chosen, and D is a constant depending only on a, b, h and p. These hypotheses are, in the case (2),

or, in the case (3),

In what follows here we shall refer to (2) and (H1) collectively as the (boundedness) hypotheses (BH1), and to (3) and (H2) as the hypotheses (BH2). Our object is to examine whether periodic solutions of (1) exist under the hypotheses (BH1), (BH2).

We consider a random walk on a two-dimensional rectangular lattice in which steps are strictly between nearest neighbour points. The conditions of the walk are that the walker must, at each step, turn either to the right or to the left of his previous step with respective probabilities ½(1+α), ½(1−α), (≤ α ≤ 1). To fix the ideas it is assumed that he starts from the origin and the probability of each of the four possible starting directions is ¼. If Ar denotes the probability of return to the origin after r steps we shall show that

where β = ½(α + α−1) and Pn is the nth Legendre polynomial. It is clear that Ar is zero for r ≢ 0(mod 4).

The theorems of Pitman and Koopman on laws of chance that yield sufficient statistics are extended to laws where the parameter or the observable, or both, may admit only a discrete set of values. The method depends on making changes, not necessarily small, within the values permitted for the variables, but it is still necessary to assume a solubility condition.

Field equations in general relativity theory have sometimes been generated by subjecting, in an invariant action integral, the components of linear connexion and the components of a covariant tensor of valence 2 to independent variation. The conceptual objections to this process, and some of the manifold formal difficulties inherent in it, are discussed in some detail. At the same time certain results obtained elsewhere are strengthened and in part corrected.

An attempt is made to relate such diverse phenomena as kinetic energy, rest mass energy, the size of the universe, the velocity of propagation both of light and gravitation, and the recession of the galaxies. The correlation is made by considering an extension of the Newtonian gravitational law which covers a particular simple case when the two attracting bodies are in relative motion. The extension is treated as postulatory, although in the last section using the idea of gravitational flux, the assumed gravitational law will be shown to be the simplest of a number of possible extensions to the Newtonian law. The paper implies a new approach to special relativity, and it is therefore hoped to treat aspects of the work in greater detail subsequently.

The problem is considered as to whether, in accordance with Newtonian theory, energy can be transferred from one system to another across empty space by gravitational interaction alone. Familiar examples of apparent energy transfer by this means do not give an unambiguous answer since they involve some net change of gravitational potential energy and this is not localized in the theory. Two examples are given here of systems in which the potential energy is the same at the beginning and end of an operation that does produce a resultant energy transfer. The establishment of this result is significant as a preliminary to the discussion of energy transfer according to general relativity theory. The appendix gives a particular illustration of one of the examples that admits exact mathematical treatment.

In a previous paper solutions have been derived for the deflexion of a circular annular plate under various combinations of boundary conditions and subject to certain line loadings spread along the circumference of a concentric circle. These solutions, which were used in the analysis of circular ring plates and sectorial plates acted upon by a concentrated load or a concentrated couple at any point, are utilized in this paper to obtain solutions appropriate to lateral loadings of the type distributed over the entire circular ring plate. Solutions corresponding to the case φ(r) = rm are then explicitly found, and the principle of superposition can be applied to get the solution when the load on the annulus can be expanded into a Laurent-Fourierseries in the polar coordinates r, θ. The method of images is used to deduce solutions for an isotropic plate bounded by two arcs of concentric circles and two radii, and loaded by the normal pressure p = p0rm when the plate is simply supported along the straight edges and is subject to various conditions along the circular edges.

The following example arose out of an unsuccessful attempt to improve a result by Ganea and Hilton (5), according to which

implies that for each field of coefficients F one at least of the factors Xi is acyclic; here cat X is the Lusternik-Schnirelmann category of X. It was natural to ask whether this assumption also implies that one at least of the Xi is simply connected. This could be proved if it were possible to obtain spaces of arbitrarily large 1-dimensional category ((3), (4)) by forming Cartesian products of sufficiently many 2-dimensional pseudoprojective spaces (1) corresponding to different primes. However, we prove that this is impossible:

Theorem. If P1, …, Pn are 2-dimensional pseudo-projective spaces corresponding to different primes p1, …, pn, then

Before proving this theorem, we recall the definitions involved. The 1-dimensional category cat1X of a space X is the least number of open sets covering X and such that each closed path in any of them is contractible in X. A 2-dimensional pseudo-projective space P corresponding to the prime p is the complex obtained from the disk |z| ≤ 1 by identifying the points ei[θ+(2mπ|p)], m = 0, 1, …, p − 1. We have (1)

for k ≥ 2 and cat1P = 3 since π1(P) is not free and cat1P ≤ 1 + dim P (4).