Let ℙ denote the weighted projective space with weights (1, 1, 1, 3) over the rationals, with coordinates x, y, z and w; let $\mathcal{X}$ be the generic element of the family of surfaces in ℙ given by

\begin{equation*} X\colon w^2=x^6+y^6+z^6+tx^2y^2z^2. \end{equation*}

$\mathcal{X}$

$\mathcal{X}$

$\mathcal{X}$

In this paper we present two versions in several variables of the following result:

Theorem 1([2, 3]). Let f be a function in the disc algebra (more generally in H1). Then for every point z0 in the open unit disc, there is an interval I on the unit circle T such that f(z0) = 1/|I| ∫Ifdσ, where 0 < |I| ≤ 2π denotes the length of I and σ the Lebesgue measure on T.

We propose a smooth correction for skewness in an asymptotically normal statistic. Unlike the case of approximation by Edgeworth expansion, our correction applies uniformly in all values of the level. The correction is based on a ' comparison statistic', and, in the case of the Studentized mean, it enables removal of all effects of skewness up to terms of order .

We study quotients of mapping class groups ${\Gamma _{g,1}}$ of oriented surfaces with one boundary component by the subgroups ${{\cal I}_{g,1}}(k)$ in the Johnson filtrations, and we show that the stable classifying spaces ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(k))^ + }$ after plus-construction are infinite loop spaces, fitting into a tower of infinite loop space maps that interpolates between the infinite loop spaces ${\mathbb {Z}} \times B\Gamma _\infty ^ + $ and ${\mathbb {Z}} \times B{({\Gamma _\infty }/{{\cal I}_\infty }(1))^ + } \simeq {\mathbb {Z}} \times B{\rm{Sp}}{({\mathbb {Z}})^ + }$ . We also show that for each level k of the Johnson filtration, the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity.

We study the Hausdorff dimensions of certain sets of non-normal numbers defined in terms of the exact rate of convergence of digits in their N-adic expansions. As an application of our results we analyse the rate of convergence of local dimensions of multinomial measures.

If a light quantum has an electric moment, we should expect its axis to coincide with the direction of the electric vector, and therefore to be perpendicular to the plane of polarisation. Now an electric doublet of moment μ on entering a uniform electric field E with its doubletaxis parallel or antiparallel to the field will have its energy changed by an amount ∓ μE. For a light quantum this is equivalent to a change in wave-length

when in the field. The method adopted to detect this change was to set up a grating with a strong electric field normal to its surface, and then have a plane polarised parallel beam of light fall on the ruled surface with the electric vector parallel to the direction of the field. The spectra formed near the normal to the grating were then examined both in the presence and absence of the field.

Let X be a set and let f be a mapping of P(X), the set of all subsets of X, into P(X). Conditions are given which are necessary and sufficient for there to exist a topology on the set X with respect to which, for all S ∈ P(X), the frontier of S is fS.

In this paper we obtain a cohomological splitting criterion for locally free sheaves on arithmetically Cohen–Macaulay surfaces with cyclic Picard group, which is similar to Horrocks' splitting criterion for locally free sheaves on projective spaces. We also recover a duality property which identifies a general K3 surface with a certain moduli space of stable sheaves on it, and obtain examples of stable, arithmetically Cohen–Macaulay, locally free sheaves of rank two on general surfaces of degree at least five in ${\mathbb P}^3$.

Let E(s, Q) be the Epstein zeta function attached to a positive definite quadratic form of discriminant D < 0, such that h(D) ≥ 2, where h(D) is the class number of the imaginary quadratic field ${\mathbb{Q}} (\sqrt D)$. We denote by ${N_E}({\sigma _1},{\sigma _2},T)$ the number of zeros of $[E(s,Q)$ in the rectangle ${\sigma _1} < {\mathop{\rm Re}\nolimits} (s) \le {\sigma _2}$ and $T \le {\mathop{\rm Im}\nolimits} (s) \le 2T$, where $1/2 < {\sigma _1} < {\sigma _2} < 1$ are fixed real numbers. In this paper, we improve the asymptotic formula of Gonek and Lee for ${N_E}({\sigma _1},{\sigma _2},T)$, obtaining a saving of a power of log T in the error term.

An ‘inner measure’ analogous to Lebesgue inner measure and associated with a von Neumann algebra is constructed on the linear sets of a Hilbert space. We supposed to be of finite type and countably decomposable, as these restrictions will be necessary for some of the results obtained. As remarked by Dye in (2), Type II1 algebras have a structure analogous to that of a finite non-atomic measure space, the trace corresponding to the measure. We define a class of ‘measurable sets’ and obtain some of its properties. The development of the theory is indicated also from the starting point of a function-valued dimension function. The idea for the construction of such an inner measure comes from a remark in (3), paragraph 16·2, and the terminology of (3) and of (5) is used throughout.

Let f (z) be meromorphic and non-rational in the domain |z| < R ≤ ∞, and let a be an arbitrary complex number, which may be infinite. The deficiency δ(a) of the value a is defined by

where m(r, a), N(r, a) and T(r) are defined as usual (cf. (10), pp. 156 ff.). For the class of functions considered in this paper the characteristic function T(r) is unbounded, and this will be assumed throughout. The upper (or Valiron) deficiency (16) of the value a is denned by

from which it follows that 0 ≤ δ(a) ≤ Δ(a) ≤ 1. A value a for which Δ(a) > 0 is called exceptional or deficient, and a value for which Δ(a) = 0 is called normal. We shall denote by G[a, σ] the open set of all values z in | z | < R for which | f(z) – a | < σ, where σ is a given positive number; we shall say that a component Gn[a, σ] of G[a, a] is bounded if the closure G¯n[a, σ] is contained in | z | < R, otherwise Gn[a, σ] will be called unbounded. In the case a = ∞, it is natural to define Gn[∞, σ] as the set of all z for which | f(z) | > 1/σ.

H. Herrlich ((2), p. 193) defines, for a given nearness ξ, a property ξ̃ (which by analogy with the case of filters on a proximity space may be called compression) and remarks that ξ is determined by ξ̃. He continues: ‘Consequently, each of the axioms (Ni) can be translated into a condition concerning ξ̃, thus providing an axiomatization of the concept of collections of sets containing arbitrary small members.’ This last remark seems slightly misleading; it might reasonably be taken to mean that the reciprocity between ξ and ξ̃ is a mere set-theoretic tautology, independent of the nearness axioms themselves. (Compare, for example, the relation between the ideas of ‘closure-point of a set’ and ‘neighbourhood of a point’ in the axiomatics of topology.) This is not in fact the case; however, one cannot select from Herrlich's axioms a subset which is necessary and sufficient for ξ to be determined by ξ̃. Moreover, the relation between ξ and ξ̃ appears to be asymmetrical. We shall exhibit, in terms of stacks ((4), p. 36), an elegant set of mutually independent axioms, first for the more general ‘Čech nearness’ discussed by Naimpally in (3) and then for a Herrlich nearness (called in (3) a LO-nearness, with a compatible topology as there defined); these axioms make the reciprocity between nearness and compression, and its relationship with the axioms, explicit and obvious.

Chinburg and Reid have recently constructed examples of hyperbolic 3-manifolds in which every closed geodesic is simple. These examples are constructed in a highly non-generic way and it is of interest to understand in the general case the geometry of and structure of the set of closed geodesics in hyperbolic 3-manifolds. For hyperbolic 3-manifolds which contain immersed totally geodesic surfaces there are always non-simple closed geodesics. Here we construct examples of manifolds with non-simple closed geodesics and no totally geodesic surfaces.

When any metal film is deposited on the surface of a massive copper oxide slab and illuminated with visible light, a current flows in such a direction that the electrons pass from the Cu2O to the metal in greater numbers than from the metal to the Cu2O. This is the reverse direction to the easier flow of electrons under an applied voltage when the contact is allowed to act as a rectifier. This effect is sometimes called the “Vorderwandeffekt” to distinguish it from the similar group of phenomena which occur at the natural interface between massive copper and Cu2O when the surface of the copper has been oxidized in situ (“Hinterwandeffekt”).

The theory of large elastic deformation of reinforced elastic materials is applied to discuss the rotation of a circular tube of incompressible elastic material, reinforced with inextensible cords lying along helical paths.

There is a theorem for three circles in a plane, that if three tangents, each of two of these circles, either all transverse or one transverse and two direct, can be drawn to meet in a point, then the three tangents, each of two of the circles, respectively conjugate to those first taken, likewise meet in a point. The theorem was stated by Quidde, with a proof for the necessity of the condition as to the tangents to be taken, in a paper designed to establish Steiner's solution of Malfatti's problem. Casey gives the theorem with omission of the condition for the character of the tangents, as does Salmon, who, however, gives a proof depending on the right choice of certain square roots which enter. Quidde's theorem is stated, accurately, in the Nouvelles Annales, and a simple metrical proof, from the diagram drawn (essentially Quidde's, see 6 below) is given later in the same Journal by Mannheim; this is practically repeated by Hart. Recently, Prof. Neville, emphasizing the necessity of the condition for the character of the tangents, has called attention to Quidde's paper.

The classical Adams spectral sequence [1] has been an important tool in the computation of the stable homotopy groups of spheres . In this paper we make another contribution to this computation.

For a gas with a general equation of state the stream function for the flow past a flat plate is determined when the plate is held at an angle ½π–α to the stream. At infinity the Mach number M1 = 1 and free streamlines, on which M1 = 1, extend from the edges of the plate to infinity downstream. The method of solution requires the determination of the correct singularity corresponding to the free stream and the subsequent satisfying of the boundary conditions. The drag coefficient is determined in the form of a contour integral and the computational problems in evaluating the drag are discussed. The integral is evaluated for small values of α for a gas with an equation of state as given by Tomotika and Tamada(6).

In [1], J. Bruna studied the Beurling classes EM(I) of infinitely differentiable functions f defined on an interval I such that for every positive ε, there exists a constant Cε > 0 with the property that

where M = {Mn} is a given sequence of positive numbers. With the hypothesis that the class EM(ℝ) is differentiable, he proved that it is inverse-closed (in the sense that if fεEM(ℝ) and if f is bounded away from zero on ℝ, then its inverse f-1 lies in EM(ℝ)) if and only if the associated sequence A = {An = (Mn/n!)1/n} is almost increasing (i.e. Am ≤ KAn for all m ≤ n, where K > 0 is a constant independent of m and n).

A strong approximation for randomly indexed U-statistics is derived. The special case when the indices are stopping times based upon the jackknife estimator of the variance is discussed.