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

The errors occurring in a large number of Jeffreys's phases for neutral chlorine have been determined. A discussion of the source of these errors is given, from which it appears that errors in the phases of zero order and of order unity are due to a cumulative error over the range of integration. A method of correcting for this error is obtained. This method is of easy practical application, and it reduces the error almost to zero for phases of these orders.

The error occurring in higher order phases is found to be mainly due to errors in the two π/4 terms, which take into account the exponential tails of the waves in the totally reflecting region. For phases of high order these errors in the two π/4 terms cancel each other, and consequently the total error in phases of high order is practically zero.

A correction graph for the zero-order phase is given from which the error in this phase for atoms of atomic number up to 36 can be read off. The graph is such that an extrapolation to higher atomic number would not involve much error.

A notion of good behavior is introduced for a definable subcategory of left R-modules. It is proved that every finitely presented left R-module has a pure projective left -approximation if and only if the associated torsion class of finite type in the functor category (mod-R, Ab) is coherent, i.e., the torsion subobject of every finitely presented object is finitely presented. This yields a bijective correspondence between such well-behaved definable subcategories and preenveloping subcategories of the category Add(R-mod) of pure projective left R-modules. An example is given of a preenveloping subcategory ⊆ Add(R-mod) that does not arise from a covariantly finite subcategory of finitely presented left R-modules. As a general example of this good behavior, it is shown that if R is a ring over which every left cotorsion R-module is pure injective, then the smallest definable subcategory (R-proj) containing every finitely generated projective module is well-behaved.

In this paper we consider the problem of characterizing the variation of the spectrum of a holomorphic family of compact operators ƒ:G → KB(X), where G is an open subset of ℂ and X is a Banach space. The natural conjecture, which the author first heard in a lecture by Professor B. Aupetit, is that these spectra are characterized as those analytic multivalued functions which have as values null sequences. This is obviously a necessary condition, and we prove that this is also sufficient. It will be convenient to use the notation K(ℂ) for the set of compact non-empty subsets of the plane and K0(ℂ) for the subset of K(ℂ) consisting of null sequences.

Given a relational structure M and a cardinal λ < |M|, let øλ denote the number of isomorphism types of substructures of M of size λ. It is shown that if μ < λ are cardinals, and |M| is sufficiently larger than λ, then øμ ≤ øλ. A description is also given for structures with few substructures of given infinite cardinality.

A solution is given of a problem already solved by Inglis, a rectangular plate stressed by the uniform longitudinal relative displacement of two opposite edges, the other pair of sides remaining free. The method used here yields a more accurate solution with greater rapidity than did the previous paper. The general form of the solution is given for a plate of any dimensions, and a plate of sides in the ratio 2:1 is considered in more detail. An approximate solution is obtained by taking four terms of an infinite series, and the consequent errors discussed and compared with those of Inglis.

In the present paper application of the work of paper I is made to a specific problem, namely the excitation of an adsorbed atom by the surface electrons of the underlying metal. It is shown that the corresponding mean life-time in a typical excited level is 3·7 × 10−11 sec., which is comparable with the value 1·7 × 10−12 sec. for deactivation by the conduction electrons when the Brillouin zone is only partly filled. Thus, though this is an essentially surface phenomenon, the effect of the surface electrons is begligible. This suggests that their effect can, in general, only become appreciable in semi-conductors or insulators.

Some results are presented which establish connections between shape theory and the theory of multivalued maps. It is shown how to associate an upper-semi-continuous multivalued map F: X → Y to every approximative map f = {fk, X → Y} in the sense of K. Borsuk and it is proved that, in certain circumstances, if F is ‘small’ and admits a selection, then the shape morphism S(f) is generated by a map, and if F admits a coselection then S(f) is a shape domination.

Let A = {Aq}q≥0 be an anti-commutative graded ring with 1. Let X be a space and let G(X; A) be the group of multiplicative units of the ring Πi≥0Hi(X; Ai) of the form l + x1 + x2 +…, where xi∊ Hi (X; Ai), In [6] Segal showed that there is a connective cohomology theory G*(X; A), with G0(X;A) = G(X; A). This was improved by Steiner, who in [7] showed that A→G*(X; A) is a functor from anticommutative graded algebras to cohomology theories.

Bieri and Strebel [2] have established a striking criterion that determines whether a metabelian group is finitely presented. This criterion is of a sort that implies that all homomorphic images of a finitely presented metabelian group are finitely presented. Another of their results is that the class of nilpotent-of-class-two-by-abelian groups has the property that all homomorphic images of finitely presented groups in this class are finitely presented. However, they point out that an example due to Abels [1] shows that the class of nilpotent-of-class-three-by-abelian groups does not have this property. One could not therefore expect that a criterion of the same type as that of Bieri and Strebel would distinguish the finitely presented groups in general classes of nilpotent-by-abelian groups. One might hope, however, to find such a criterion for nilpotent-of-class-two-by-abelian groups. The aim of this paper is to make some steps in this direction, by finding a sufficient condition for such groups to be finitely presented.

It is a well-known result in one-dimensional dynamics that if a continuous map of the interval has positive topological entropy, then it has a periodic orbit of period 2i for each integer i ≥ 0 [15] (see also [12]). In fact, one can say rather more: such a map has a sequence of periodic orbits (P)i ≥ 0 with per (Pi) = 2i which form a period-doubling cascade (that is, whose points are ordered and permuted in the way which would occur had the orbits been created in a sequence of period-doubling bifurcations starting from a single fixed point). This result reflects the central role played by period-doubling in transitions to positive entropy in a one-dimensional setting. In this paper we prove an analogous result for positive-entropy orientation-preserving diffeomorphisms of the disc. Using the notion [9] of a two-dimensional cascade, we shall show that such diffeomorphisms always have infinitely many ‘zero-entropy’ cascades of periodic orbits (including a period-doubling cascade, though this need not begin from a fixed point).

Suppose that t is a group homomorphism from a topological group E into a topological group F. When is it true, that the closure of t−1(V) in E is a neighbourhood of the identity in E for every neighbourhood V of the identity in F? This question arises naturally in the study of the closed graph theorem in the context of topological groups; for example, see (1) and ((3), p. 213). The concept of a g-ultrabarrelled space introduced in this paper is the result of an investigation aimed at answering this question.

Consider the second order nonlinear equation

where, r, p, f ε C ([a, + ∞), R), R = (–∞, +∞) with r > 0 and λ > 0 is the quotient of odd integers. One of the prototypes of this class of equations is the generalized Thomas-Fermi of Emden-Fowler equation

The simplest exact classical theory, of a particle possessing charge and dipole moment proportional to the spin leads to a wave equation with an explicit spin interaction with the field. This wave equation is used in this paper to calculate the scattering probability coefficient in the Compton effect by two methods. The first method uses the unquantized radiation field; the second uses Dirac's scheme of field quantization. The scattering probability coefficient is given in the general case in terms of the charge e, the constant of proportionality in the dipole moment C, and the quantities k, l specifying the spin. Some special results are considered for particles of spin ½ and 1. Formulae are given for the scattered intensity as a function of the angle of scattering, and the total scattering cross-section as a function of the energy of the unpolarized incident radiation. Graphs are given for these formulae when the magnetic part of the dipole moment takes the empirical values in the case of the electron, proton and neutron. In the case of the meson, normal and abnormal values of the dipole moment are used.

Let be a commutative normed algebra whose elements are functions on a Banach space, and in which the product is of convolution type. We show how , under reasonable assumptions upon its semi-group of translation operators and rather severe restrictions upon its semi-simplicity, can be renormed so that its completion contains mean Gâteaux or Fréchet differentials of all its members. In this way the function algebra can be embedded in an algebra of generalized functions defined as strong limits of functions, which admits strong differentiation as an everywhere-defined operation. The differentials show most of the properties of point-wise Gâteaux differentials. The completion of is an example of what I have called in (3) an ‘inflation’.

This work is consequent on a forthcoming joint paper by A. C. Stevenson and the present writer on the use of the torsion function in the solution of flexure problems. The treatment is general, but special attention is paid to doubly-connected cross-sections which can be mapped conformally on a circular annulus. The eccentric circles cross-section (4) is solved in order to illustrate the power of the method, and results are obtained for the cross-section bounded by confocal limacons. This is of interest, since it has been customary to regard these problems as largely determined by the cross-section of the cylinder; but these results show that the terms involving the elastic constants are now no longer negligible. The method is capable of giving useful information to aeronautical engineers, and the writer hopes to present in a later communication a treatment of an aerofoil-like profile.

We present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus ≥ 2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is the computation of these cocycles on a particular element of the mapping class group, which is topologically a hyperelliptic involution.

In supersymmetric field theories, the use of Berezin-like integrals for curved superspace presupposes the existence of a smooth manifold structure on the underlying curved space (the body). The aim of this paper is to address the following question: when does a Rogers supermanifold have a body manifold?

Elementary techniques from the theory of foliations are used to derive some necessary and sufficient conditions. The main result is the necessity of regularity of the soul foliation in the sense of Catenacci, Reina and Teofilatto. Regularity is not sufficient (for example, punctured superspace is regular with non-Hausdorff body). A sufficient condition (albeit highly sensitive to the choice of charts) is the transitivity of the body relation ˜, a property enjoyed by De Witt supermanifolds.

We show that the body of a quotient supermanifold S/Γ is the quotient of the body (under the induced action). Several examples in the text illustrate that this is extremely useful.

New characterizations ρ-supermanifolds and their saturations are given and the saturation construction is extended to all bodied supermanifolds.

Let M be a compact Riemannian manifold equipped with Laplace–Beltrami operator Δ. We use the Rademacher-Menchoff theorem and the asymptotics of the eigenvalues of Δ to show that if a function belongs to an L2 Sobolev space of positive index then its expansion, in terms of eigenfunctions of Δ, converges almost everywhere on M.

The extension of Mellin's inversion formula expressed by the equations has been considered by Fowler who shows that some form of Stieltjes integral is essential to Poincaré's proof of the necessity of the quantum hypothesis. Fowler confines his discussion to a restricted type of function φ (y) which is sufficient for the physical problem. It will be proved here that the formulae hold with a general Stieltjes integral in the first equation.