A general form of the covering principle and relative differentiation of additive functions. II*.

By A. S. Besicovitch

P. 5, line 7 from the bottom, after ‘F-singular’ add ‘in the class of B-sets’.

P. 5, the bottom line, after ‘function Θ(X)’ add ‘in the class of B-sets’.

P. 7, line 9 from the bottom, forG⊃Xnread.

P. 8, line 4, for.

P. 8, line 5, for.

* Vol. 42 (1946), pp. 1–10.

By a twisted quartic of the first kind is intended the curve in space of three dimensions defined by two quadratic equations, i.e. the complete intersection of two quadric surfaces. If this intersection consist of the aggregate of two or more curves of lower order than the fourth, it will be called a degenerate quartic. In general, a twisted quartic of the first kind will be referred to for brevity as a quartic.

In hypergeometry loci defined by parametric equations

have been frequently discussed from various points of view. Among other properties these loci possess two systems of generating spaces obtained by keeping the α's (or β's) fixed while the β's (or α's) vary, the simplest example being the generators of a quadric in S3. Another method of obtaining loci having these properties is by taking the flat spaces through the sets of a linear series of sets of points on an elliptic curve and the flat spaces through the sets of the residual series (the curve being supposed normal. In this way we shall obtain a configuration of ∞PSP's and ∞qSq's, the dual of which gives a locus included among those represented by (1).

The two fourth-order partial differential equations giving the transverse displacement and the stress functions for shallow shells are reduced to a sixth-order differential equation in one variable which has been solved, and the values of displacements, stress resultants and couples are all expressed in terms of Bessel Functions of imaginary argument. Numerical values of the displacement presented for points on the shell on different meridian lines, showed that on the symmetrical line deflexion is maximum at the point of loading but the point of maximum deflexion shifts gradually towards the pole.

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfying

where H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).

In discussing Hecke's function

where {t} = t − [t] − ½ Hardy and Littlewood* made use of the identity

Here

θ is an irrational number between 0 and 1, and pν/qν is the νth convergent to θ (beginning with p0 = 0, q0 = 1). The proof of (1) given by Hardy and Littlewood is of an analytic character.

An account is given of experiments to determine the polarisation of downcoming waves of long wavelength, using the method devised by Ratcliffe and White. The results show that on the whole the polarisation is left-handed, although there is a tendency for right-handed polarisation in the region of 1500 metres.

We consider self-similar iterated function systems in the sub-Riemannian setting of Carnot groups. We estimate the Hausdorff dimension of the exceptional set of translation parameters for which the Hausdorff dimension in terms of the Carnot–Carathéodory metric is strictly less than the similarity dimension. This extends a recent result of Falconer and Miao from Euclidean space to Carnot groups.

We shall prove the following Theorem. There are exactlyisomorphism classes of residually finite groups with every proper quotient finite.

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.

Starting from an arbitrary algebra in place of the algebra of smooth functions on a differentiable manifold M, we construct an algebra which generalizes the algebra of polynomial functions on T*M. We define a Poisson bracket on this algebra and show that it can be obtained from a natural one-form which generalizes the fundamental one-form on T*M.

MacLane(1) has introduced the class of functions with a dense set of asymptotic values on {|z| = 1} in the following sense:

Definition. Let f(z) be regular and not constant in the unit disc and let ξ be a point of {|z| = 1}. Suppose that there exists a curve Γξ in {|z| < 1} and a in the closed complex plane such that f(z) → a as z → ξ along Γξ. Then we say ξ ∈ Aa.

Schweizer and Sklar, in their book Probabilistic Metric Spaces, raise the question of whether every triangle function is causal. This note provides a negative answer, by observing certain properties of a suitable partial quaternary operation on the unit interval.

Let [sum ]∞an denote the set of cluster points of the sequence of partial sums of a series [sum ]an with terms in ℝd. For any permutation f of the set ℕ of positive integers, [Cscr ]f (ℝd) denotes the set of all sets [sum ]∞af(n) arising from series [sum ]an with terms in ℝd and sum 0. For each f, we use the Max-Flow Min-Cut Theorem to determine all convex sets in [Cscr ]f(ℝd) which are symmetric about a point. These sets depend only on a parameter w(f) ∈ ℕ ∪ {0, ∞}, called the width of f. We show that w(f), when it is a positive integer, falls far short of completely determining [Cscr ]f(ℝd) but, for each q ∈ ℕ, we find the largest of the sets [Cscr ]f(ℝd) arising from permutations f of width q. We also describe the smallest of these sets when q = 1.

A phase-space anisotropic operator in =L2(ℝn) is a self-adjoint operator whose resolvent family belongs to a natural C*-completion of the space of Hörmander symbols of order zero. Equivalently, each member of the resolvent family is norm-continuous under conjugation with the Schrödinger unitary representation of the Heisenberg group. The essential spectrum of such a phase-space anisotropic operator is the closure of the union of usual spectra of all its “phase-space asymptotic localizations”, obtained as limits over diverging ultrafilters of ℝn×ℝn-translations of the operator. The result extends previous analysis of the purely configurational anisotropic operators, for which only the behavior at infinity in ℝn was allowed to be non-trivial.

A well-known theorem, due to Castelnuovo and Enriques(1), states that a surface on which the process of successive adjunction, applied to any curve system, terminates, must be rational or scrollar; actually the result shows that, if the property in question holds for any one system of sufficiently general type, then it must hold for all systems; but this has not been established directly.

We refer for general background to N. Bourbaki, Intégration, chapters iv and v. We consider a locally compact Hausdorff space R and denote the set of continuous functions with compact support by The Riesz-Markov theorem shows that there is a 1−1 correspondence between the set of regular Borel measures on R and the set of positive elements of the topological dual of . Let {μn}, μ be regular Borel probability measures on R. The sequence of measures {μn} is said to converge vaguely to μ if, for all . Thus the vague topology is that of simple convergence in . We shall call a μ-measurable set μ-quarrable if its boundary is a μ-null set.

The so-called AF + BΦ theorem of Max Noether refers to a whole collection of results. In most of these, the object is to determine conditions under which a form H(x, y, z) will belong to the ideal generated by two given forms F(x, y, z) and Φ(x, y, z). This problem is of obvious importance in the theory of plane curves, and therefore it is customary to state the conditions in geometric language. To give one example, a particularly important form of the theorem may be stated as follows.*

Since the first introduction of the concept of a matrix, questions about functions of matrices have had the attention of many writers, starting with Cayley(i) in 1858, and Laguerre(2) in 1867. In 1883, Sylvester(3) defined a general function φ(a) of a matrix a with simple characteristic roots, by use of Lagrange's interpolation formula, and Buchheim (4), in 1886, extended his definition to the case of multiple characteristic roots. Then Weyr(5) showed in 1887 that, for a matrix a with characteristic roots lying inside the circle of convergence of a power series φ(ζ), the power series φ(a) is convergent; and in 1900 Poincaré (6) obtained the formulae

for the sum, where C is a circle lying in and concentric with the circle of convergence, and containing all the characteristic roots in its ulterior, such a formula having effectively been suggested by Frobenius(7) in 1896 for defining a general function of a matrix. Phillips (8), in 1919, discovered the analogue, for power series in matrices, of Taylor's theorem. In 1926 Hensel(9) completed the result of Weyr by showing that a necessary and sufficient condition for the convergence of φ(a) is the convergence of the derived series φ(r)(α) (0 ≼ r < mα; α) at each characteristic root α of a, of order r at most the multiplicity mα of α. In 1928 Giorgi(10) gave a definition, depending on the classical canonical decomposition of a matrix, which is equivalent to the contour integral formula, and Fantappie (11) developed the theory of this formula, and obtained the expression

for the characteristic projectors.