Let Z2 denote the integer lattice in the plane, let A be a non-singular 2 x 2 matrix and let c ∈ R2. Then G = AZ2 + c is called a grid, and its determinant det G is defined to be det A. The grid G is called a subgrid if G ⊂ Z2.

In 1966, Hewitt and Zuckerman(3,4) proved that if G is a non-discrete locally compact Abelian group with Haar measure λ, then there exists a non-negative, continuous regular measure μon G that is singular to λ(μ ┴ λ) such that μ(G)= 1, μ * μ is absolutely continuous with respect to λ(μ * μ ≪ λ), and the Lebesgue-Radon-Nikodym derivative of μ * μ with respect to λ is in (G, λ) for all real p > 1. They showed also that such a μ can be chosen so that the support of μ * μ contains any preassigned σ-compact subset of G. It is the purpose of the present paper to extend this result to obtain large independent sets of such measures. Among other things the present results show that, for such groups, the radical of the measure algebra modulo the -algebra has large dimension. This answers a question (6.4) left open in (3).

The exact distribution of the simple correlation coefficient was established by Dr R. A. Fisher in 1915 (Biometrika, X, p. 507). If ρ is the true correlation in the population, and r the sample value, then the frequency element is given by

where θ = cos− (−ρr).

We give examples of closed hyperbolic 3-manifolds with first Betti number 2 and 3 for which no sequence of finite abelian covering spaces increases the first Betti number. For 3-manifolds $M$ with first Betti number 2 we give a characterization in terms of some generalized self-linking numbers of $M$, for there to exist a family of $\mathbb{Z}_n$ covering spaces, $M_n$, in which $\beta_1(M_n)$ increases linearly with $n$. The latter generalizes work of M. Katz and C. Lescop, by showing that the non-vanishing of any one of these invariants of $M$ is sufficient to guarantee certain optimal systolic inequalities for $M$ (by work of Ivanov and Katz).

In the first part of this paper, tests were described for determining which points of any line in a Galois plane II of order q2 are to be transferred to the conjugate line in order to transmute II into the corresponding Hughes plane Ω. In this part of the paper the tests are refined to provide, in relation to some fixed point in the central subplane δ0 of Ω (i) a simple geometrical condition of transfer for a certain set of ½q(q2−1) points of II and (ii) a simple aglebraic condition for the remaining points of II – δ0. These tests eliminate from the computation (for a given value of q) the necessity of calculating the third coordinates of ½q2 (q2−1) points in order to determine which are not-squares.

Suppose that R → S is an extension of local domains and ν* is a valuation dominating S. We consider the natural extension of associated graded rings along the valuation grν*(R) → grν*(S). We give examples showing that in general, this extension does not share good properties of the extension R → S, but after enough blow ups above the valuations, good properties of the extension R → S are reflected in the extension of associated graded rings. Stable properties of this extension (after blowing up) are much better in characteristic zero than in positive characteristic. Our main result is a generalisation of the Abhyankar–Jung theorem which holds for extensions of associated graded rings along the valuation, after enough blowing up.

The integral

where n is a positive integer, was obtained recently by Van der Pol by operational methods.

The quantum-mechanical energy-level density g(E) is given as a functional of the quantum-mechanical kernel K(q″, q′, t″ −t′). On taking the kernel K in the Feynman's form, one obtains the function g(E), without solving a Schrödinger equation. As an example, the embedding of a particle in the one-dimensional square well with infinitely high walls is analysed. The functions K(x″, x′, t−t′) and g(E) are represented as sums of terms corresponding to classical paths of different types. By an adequate choice of some terms due to the ‘most important’ paths, one may construct partial sums giving approximations of the function g(E). The utilization of such approximations for estimation of energy levels is demonstrated.

We establish convergence results involving the nth Cesaro mean σβn(θ) of order β of f(eiθ) = [sum ]aneinθ where the coefficients (an) satisfy a Dirichlet-type condition [sum ]nα[mid ]an[mid ]2 < ∞ for a fixed α in (0, 1]. When α = 1, for example, we prove that [sum ][mid ]σβn(θ)−f(eiθ) [mid ]2 < ∞ for every β > −½ at almost all points θ in [−π,π]. We also derive weighted convergence results of this type outside exceptional sets of capacity zero, and show by examples that these results are, in a certain sense, sharp with respect to the size of the exceptional sets.

Istrăţescu (2) has introduced a class of operators on a Banach space called operators of class (N, k). An operator T (a bounded linear transformation) on a Banach space X is said to be an operator of class (N, k), k = 2, 3,…, if ‖Tx‖k ≤ ‖Tkx‖ for all x ∈ X such that ‖x‖ = 1. If k = 2, such an operator is called an operator of class (N)(3). If X is a Hilbert space, then the class of operators of class (N) on X is an extension of the class of hyponormal operators on X (3). The object of this note is to generalize to operators of class (N, k) on a Banach space X, some results which are known to be true for normal operators on a Hilbert space, particularly with regard to their ascent and descent.

In recent seismological work several questions have arisen concerning the construction of tables from data containing a certain amount of accidental error, and it seems that some of the methods may be of more general interest. The first is the problem of smoothing. A method suggested by Dr L. J. Comrie was as follows. Suppose that we have values of y for five equally spaced values of x, and that we wish to make the second differences vary as smoothly as possible. We try to find a cubic polynomial such that the sum of the squares of the deviations of y from it will be a minimum. It is found that the polynomial is less than the observed value of y at the centre of the range by of the central fourth difference; or approximately by of the fourth difference. I have since found that the method had previously been suggested by Sir G. H. Darwin. It gives a great improvement in the steadiness of the differences and thereby makes interpolation much easier.

1. Introduction Born and Infeld originally proposed their non-linear theory of electromagnetism (1) as one derivable from a Lorentz invariant Lagrangian and, in contradistinction to the usual Maxwell theory, possessing solutions of the ‘point charge’ type without singularities at the position of the charge. Subsequently, examination of the theory by means of iterative methods (2) has demonstrated the self-scattering of outgoing radiation and the existence of wave tails. In the present paper it is shown that the theory, under suitable asymptotic conditions, gives rise to peeling of the type exhibited by Maxwell's theory as well as by the asymptotically flat space-times of general relativity (3, 4, 5). It is also shown that certain conserved quantities exist paralleling those exhibited in the above theories. The Maxwell theory exhibits a sequence of such conserved quantities, the truncation of which is examined in detail for the Born-Infeld theory.

The Kuhn–Schwartz non-realizability theorem states the following: if the mod p cohomology of a topological space is finitely generated as a module over the Steenrod algebra [Ascr ] then it is finite. We generalize this result to the category of G-spaces, where G is a compact Lie group, by considering the equivariant cohomology of a G-space as an object in the category of all [Ascr ]-modules with a compatible H*(BG; [ ]p)-module structure.

For the quantitative examination of the behaviour of two-dimensional insoluble films on water surfaces the trough apparatus devised originally by Langmuir has been hitherto employed. With the aid of this apparatus it has been possible to demonstrate the existence of two-dimensional films in the solid, smectic, liquid and vaporous states, to observe the phenomenon of two-dimensional dimorphism, to study the conditions of phase equilibria and to measure the entropy changes associated with the process of two-dimensional fusion, vaporisation and sublimation. No very definite concept of molecular orientation has been obtained from the force area diagrams. Apart from the fact that the polar groups adhere to the surface and that the hydrocarbon chains in long chain compounds adhere to one another, our picture of all forms of films except possibly the solid and the highly attenuated or vaporous states is by no means clear and the various suggestions made as to the molecular architecture are somewhat problematical. Two other methods suggest themselves for gaining a further insight into molecular structure: optical methods on the lines outlined by Rayleigh and by Raman and measurements of the interphasic potential differences. While it may be possible in the future to develop the former into a method of precision, we have succeeded in rendering the latter as accurate as the method of the Langmuir trough. That interphasic potentials between air and liquid exist and are of not inconsiderable magnitude has long been known, although the first systematic investigations on changes in these potentials on the addition of capillary active materials to the liquid phase were made by Kenrick (Z.P.C. xix, 625,1896). The technique was considerably improved by Guyot (C.R. clix, 307, 1914, Ann. d. Phys. x, 2501, 1924) and by Frumkin (Z.P.C. cix, 34, 1924, cxi, 196, 1924, cxv, 485, 1925).

In a recent note the writer has examined the varieties whose generic curve sections are canonical curves of genus p, of general character, and whose surface sections contain only complete intersections with primals; following Fano's classification, we call these varieties of the first species. Such varieties are all rational provided that r > 3 and p > 6. In the present paper we consider their representations on linear spaces for the case r = 4, from which, in conjunction with the previous results, we conclude that fourfolds of the first species exist if, and only if, p ≤ 10; this agrees with the conjecture made by Fano in the case r = 3. It will be seen that the representation of these varieties on [4] provides interesting illustrations of Semple's formulae for composite surfaces in higher space.

Throughout this paper we shall denote the space Lp(− ∞, ∞) by Lp, and the numbers p and p′ will be related by 1/p + 1/p′ = 1.

The Poisson operator Pa is defined by

We provide a non-trivial measure of irrationality for a class of Mahler numbers defined by infinite products. This class includes the Thue–Morse constant. Among other things, our results imply a generalisation to [12].

We provide a new and concise proof of the existence of suprema in the Cuntz semigroup using the open projection picture of the Cuntz semigroup initiated in [12]. Our argument is based on the observation that the supremum of a countable set of open projections in the bidual of a C*-algebra A is again open and corresponds to the generated hereditary C*-subalgebra of A.

We show, from a topological viewpoint, that most numbers are not normal in a strong sense. More precisely, the set of numbers $x \in (0,1]$ with the following property is comeager: for all integers $b\ge 2$ and $k\ge 1$ , the sequence of vectors made by the frequencies of all possibile strings of length k in the b-adic representation of x has a maximal subset of accumulation points, and each of them is the limit of a subsequence with an index set of nonzero asymptotic density. This extends and provides a streamlined proof of the main result given by Olsen (2004) in this Journal. We provide analogues in the context of analytic P-ideals and regular matrices.