The algebraical foundation of this representation is very simple: it depends on the theorem

If a coordinate system in [m] is given, then in a general [k] of the [m] a unique set of k + 1 points can be found whose coordinates form the scheme

where h = m − k − 1, and n = (m − k)(k + 1) = kh + m.

§ 1. The six Plücker coordinates of a straight line in three dimensional space satisfy an identical quadratic relation

which immediately shows that a one-one correspondence may be set up between lines in three dimensional space, λ, and points on a quadric manifold of four dimensions in five dimensional space, S5. For these six numbers pij may be considered to be six homogeneous coordinates of such a point.

The multiconfiguration method of Grein[3] based on the generalized Brillouin theorem is discussed and analysed, with particular emphasis on the use of molecular symmetry to reduce the amount of computing. Some results concerning the energy decrease in each iteration are also included.

Let G be a finite group and p be a prime. We investigate isomorphism invariants of $\mathbb{Z}_p$[G]-lattices whose extension of scalars to $\mathbb{Q}_p$ is self-dual, called regulator constants. These were originally introduced by Dokchitser–Dokchitser in the context of elliptic curves. Regulator constants canonically yield a pairing between the space of Brauer relations for G and the subspace of the representation ring for which regulator constants are defined. For all G, we show that this pairing is never identically zero. For formal reasons, this pairing will, in general, have non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to considering permutation lattices, then we show that the pairing is non-degenerate modulo the formal kernel. Using this we can show that, for certain groups, including dihedral groups of order 2p for p odd, the isomorphism class of any $\mathbb{Z}_p$[G]-lattice whose extension of scalars to $\mathbb{Q}_p$ is self-dual, is determined by its regulator constants, its extension of scalars to $\mathbb{Q}_p$, and a cohomological invariant of Yakovlev.

Let K be a number field or a function field of characteristic 0, let φ ∈ K(z) with deg(φ) ⩾ 2, and let α ∈ ℙ1(K). Let S be a finite set of places of K containing all the archimedean ones and the primes where φ has bad reduction. After excluding all the natural counterexamples, we define a subset A(φ, α) of ℤ⩾0 × ℤ>0 and show that for all but finitely many (m, n) ∈ A(φ, α) there is a prime 𝔭 ∉ S such that ord𝔭(φm+n(α)−φm(α)) = 1 and α has portrait (m, n) under the action of φ modulo 𝔭. This latter condition implies ord𝔭(φu+v(α)−φu(α)) ⩽ 0 for (u, v) ∈ ℤ⩾0 × ℤ>0 satisfying u < m or v < n. Our proof assumes a conjecture of Vojta for ℙ1 × ℙ1 in the number field case and is unconditional in the function field case thanks to a deep theorem of Yamanoi. This paper extends earlier work of Ingram–Silverman, Faber–Granville and the authors.

This paper is devoted chiefly to the consideration of the surface oscillations of water contained in a vessel in the shape of a circular cylinder with its axis vertical, when the motion is slightly disturbed from a uniform rotation about the axis of the vessel. The work was undertaken with the hope of finding some indication of the effect of the depth of the water in the vessel on the period of the surface waves, and for the purpose a vessel of circular cross-section was naturally chosen. It is shown that a slight change of shape does not affect the periods of the oscillations. The solution of the corresponding problem when the surface oscillations take the form of “long waves” or “tidal waves” is well known, and the present paper deals only with “short waves,” for which the horizontal velocity is not the same at all depths.

The subject of squared rectangles and squared squares was developed by Brooks, Smith, Stone and Tutte(1) using techniques based on the theory of electrical networks. In this note we shall treat the (apparently) more general topic of rectangulations using purely algebraic techniques. The starting point is the observation that the incidence matrices of the pair of dual networks associated with a squared rectangle in (l) can be more naturally derived as incidence matrices of the squaring (or rectangulation) itself. Further, the topological information contained in the networks is a simple consequence of Euler's formula applied directly to the rectangulation.

The Fourier series of the function defined by

is

and if

then

while

The expression (1) exceeds (2); this is the Gibbs phenomenon.

We study the second Chinburg invariant of a Galois extension of number fields. The Chinburg invariant lies in the class-group of the integral group-ring of the Galois group of the extension. A procedure is given whereby to evaluate the invariant in the case of the real cyclotomic case of regular prime power conductor and their subextensions of p-power degree. The invariant is shown to be zero in the latter cases, which yields new examples giving an affirmative answer to a question of Chinburg ([1], p. 358) which has come to be known as ‘Chinburg's Second Conjecture’ ([3], §4·2).

Let p(x) be periodic with mean value c and let α > 0. Then the investigation of the asymptotic solution of the second-order equation

can be split into four cases: (i) α > 2, (ii) α > 1 and c = 0, (iii) 0 < α ≤ 1 and c = 0, (iv) 0 < α ≤ 2 and c ≠ 0. The first two cases are straightforward because then there are solutions y1 and y2 of (1·1) such that y1(x) ~ x and y2(x) → 1 as x → ∞ [5, theorems 2·7·2 and 2·7·5; 7, p. 375; 8]. Thus the coefficient of y in (1·1) represents a perturbation which has no effect on the dominant asymptotic behaviour of solutions. Case (iii) is less simple because here the nature of p(x) does influence the asymptotic form of solutions and, indeed, when α < 1 the solutions of (1·1) are oscillatory. This case has been fully covered by a number of authors: Atkinson, Eastham and McLeod[2, §3], Cassell[3, 4], Eastham[5, §4·13], Mahoney[10], Willett[13] and Wong[14, 15]. In addition, corresponding results for a higher-order self-adjoint analogue of (1·1) have been obtained recently by Al-Hammadi and Eastham [1, 6] as an application of general asymptotic methods for differential systems.

A linear vector m-space Rm defined along a curve (C) in a Vn and lying in the complete osculating space of (C) will be called a characteristic Rm of (C) if it is auto-parallel along the curve and makes constant angles with its tangent and the principal normals. Curves admitting a characteristic R1 have been studied by Hayden under the name of generalized helices and generalized by me§. In this paper we give a complete determination of the curves with a characteristic R2. Curves whose curvatures are proportional to a set of constants, which have been considered by Syptak for the particular case when Vn is an Rn, form one of the classes of curves of this type. As a consequence, the existence of the curves admitting a characteristic Rm (m > 2) is partly established, but the problem has not been completely solved. At the end we prove two theorems in connexion with two particular types of characteristic Rm's.

In many radio investigations of the ionosphere it is desirable to use senders working on different wave-lengths and situated in different positions with respect to the receiver. For this purpose it is very convenient to make use of commercial senders which work more or less continuously at high power. It so happens, however, that these senders usually emit Morse signals consisting of dots and dashes of variable lengths and with variable intervals between them, so that an instrument which records the mean signal will give an indication which depends on the speed of sending and on the spacing and length of the signs. In order to obtain a reading which depends only on the signal amplitude, it is possible to employ a quick moving recorder such as a string galvanometer or a cathode ray oscillograph, which will follow the individual Morse signs. With both these types of recorder the record must be made photographically so that changes in the signal strength are not noticed until the record is developed. This is a serious disadvantage in investigations of short-lived disturbances of the “catastrophic” type.

P. M. Cohn has given a characterization of pseudovaluation on a field in (1). There he defines the notion of an integral element over a multiplicatively closed set and states that the association of the set A to the set A* of all elements that are integral over it is a closure operator. There is an error in the statement that A* = (A*)*. In this note we show, by an example, that A* and (A*)* are not equal in general. We also give an example to show that even for the special type of sets that Cohn needs, namely the gauge sets, this property fails to hold. However, since the other conditions for a closure operator (A ⊆ A* and A ⊆ B implies A* ⊆ B*) are satisfied, one obtains a Kuratowski closure operator by iterating *. If we denote by Ā the closure of A by iterating * then theorem 10·3, its corollary and theorem 10·4 in (1) that have a bearing on this error, are valid as stated with the same proof replacing only K* by . Further theorem 13·3 is valid without modification since Cohn considers only a special class of pseudovaluations namely regular pseudo valuations for which K* = .

We have defined the generalized hypergeometric polynomial ((6), eqn. (2·1), p. 79) by means of

where δ and n are positive integers and the symbol Δ(δ, − n) represents the set of δ-parameters

The polynomial is in a generalized form which yields many known polynomials with proper choice of parameters and therefore the results are of general character.

In the paper entitled ‘Asymptotic formula in a generalized Waring's problem’, I established an asymptotic formula for the number of representations of a large natural number N in the form

where x1, x2, …, x7 and k are natural numbers with k ≥ 4 (see (2) Theorem 2).

The theory developed in our first paper is applied here to the case of algebras A such that every pair of congruences on A commute. The general refinement theorem which we obtain is already known when the ascending and descending chain conditions hold for the lattice of congruences. This result is due to O. Ore. We here weaken this finiteness assumption to a point comparable with that of group theory when both chain conditions are assumed for subgroups of the centre. Our path to the theorem is obtained by adapting to a different situation methods developed by Kurosch, Baer, and Jónsson and Tarski. From Kurosch (4) we take the concept of the centre of a pair of direct decompositions, which we extend and formulate as a pair of mappings π [α, β; γ, δ] over the set of congruences. In this connexion, Theorems 12·5 and 12·6 appear to be new, even for groups. From Baer (1) we take the concept of canonical refinements and generalize it to our case. The final approach (§ 14) to the refinement theorem is essentially following Jónsson-Tarski (3). Possibly our most interesting conclusion is that the elegant methods using endomorphisms, introduced by Fitting into group theory, generalize naturally to our case.

The purpose of this note is to count almost complex structures on products of some lens spaces.

In a previous paper I proved that the density of the positive integers of the form where the letters p, q, and later P, Q, r, denote primes, is positive. As indicated in the Introduction of I, I now give proofs of the following results:

The density of each of the sets of integers

is positive.