We prove the existence of a (unique) S1-invariant Ricci-flat Kähler metric on a neighbourhood of the zero section in the canonical bundle of a real-analytic Kähler manifold X, extending the metric on X.

It is a familiar fact that an important part is played in the Analytic Theory of Numbers by Fourier series. There are, for example, applications to Gauss' sums, to the zeta functions, to lattice point problems, and to formulae for the class number of quadratic fields.

For a complete, finite volume real hyperbolic n-manifold M, we investigate the map between homology of the cusps of M and the homology of M. Our main result provides a proof of a result required in a recent paper of Frigerio, Lafont and Sisto.

The following concept of (not necessarily commutative) Noetherian unique factorization domain (UFD) was introduced recently by A. W. Chatters. Recall that an ideal P of a ring R is called completely prime if R/P is a domain. The element p∊R will be called a prime element if pR =Rp is a completely prime, height one prime of R. C(P) denotes the set of elements of R which are regular modulo P. Set C = ∩ C(P) where P ranges over the height one primes of R.

The study of solutions to special examples of elliptic differential equations, using Clifford algebras, has been developed by a number of authors [4–12, 14–16, 18, 19, 21–30, 32]. This study has also been applied [3, 13, 17, 28] within a number of areas of theoretical physics, including Yang–Mills field theory, and the Kähler equation. Most of the function theories associated to the solutions of these elliptic differential equations, referred to as generalized Cauchy–Riemann equations [21,32], generalize in a natural manner many aspects of classical one-variable complex analysis [1]. For instance, each of these function theories contain analogues of the Cauchy theorem, Cauchy integral formula and Laurent expansion theorem. However, no information has yet been obtained on the local topological behaviour of a general solution to any of these equations.

All rings considered in this paper are non-trivial commutative algebras over a field k; unless indicated otherwise, all tensor products are understood to be taken over k. (Some of the results and concepts extend to general rings, but it is not worth while noting such generalizations.)

Let $\mathscr{S}$ be an irreducible topological Markov shift, and let μ be a shift-invariant Gibbs measure on $\mathscr{S}$. Let (Xn)n ≥ 1 be a sequence of i.i.d. random variables with common law μ. In this paper, we focus on the size of the covering of $\mathscr{S}$ by the balls B(Xn, n−s). This generalises the original Dvoretzky problem by considering random coverings of fractal sets by non-homogeneously distributed balls. We compute the almost sure dimension of lim supn →+∞B(Xn, n−s) for every s ≥ 0, which depends on s and the multifractal features of μ. Our results include the inhomogeneous covering of $\mathbb{T}^d$ and Sierpinski carpets.

When Rossby number is small but Ekman number is very much smaller, study of the flow field far from a body moving at right angles to the axis of rotation of a large body of fluid indicates that the region of influence should not be a Taylor column parallel to the axis, but a trailing Taylor column, bent backwards on both sides of the body at a small angle (proportional to Rossby number) to the axis. The paper reviews the physical significance of, and experimental evidence for, this conclusion.

We define torsors for sheaves of simplicial groups and sheaves of groupoids enriched in simplicial sets, and give classification results for these torsors in terms of the homotopy theory of simplicial sheaves. The proofs of the classification results use a new, general approach to cocycles taking values in simplicial sheaves. We prove a homotopy classification result for gerbes locally isomorphic to a fixed sheaf of groups.

When a diode thermionic tube, having cylindrical symmetry, is placed in a magnetic field parallel to its axis it is commonly called a magnetron. If there is a given potential difference between the anode and cathode of the tube, and if the magnetic field is steadily increased, a sharp and pronounced decrease of anode current occurs when the field reaches a certain value. It is easy to show that, if electrons leave without velocity from a cathode of radius b, they will just graze a concentric anode of radius a at potential V when the magnetic field H has the value given by

In a recent paper, Smith ((l)), considering a statement or conjecture of Isbell ((2)), proved that the Hausdorff uniformities arising from two different uniformities on the same set must induce different topologies on the ‘hyperspace’ of subsets, except possibly in one case which remained open, namely, when the two given uniformities induce the same proximity and neither is precompact. I now give an example to show that different uniformities can (in this case) induce the same topology on the hyperspace (in the example one of the uniformities is metric). This disproves Isbell's conjecture.

The aim of this paper is to reformulate the work of Massey(6), and Spanier(9, 10), on higher order cohomology operations, in terms of vector bundles in such a way as to produce geometrically some higher order operations in K-theory, which we will call Massey products.

In this paper we study the arithmetically Cohen-Macaulay (ACM) property for sets of points in multiprojective spaces. Most of what is known is for ℙ1 × ℙ1 and, more recently, in (ℙ1)r. In ℙ1 × ℙ1 the so called inclusion property characterises the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in ℙm × ℙn. In such an ambient space it is equivalent to the so-called (⋆)-property. Moreover, we start an investigation of the ACM property in ℙ1 × ℙn. We give a new construction that highlights how different the behavior of the ACM property is in this setting.

We establish Manin's conjecture for a cubic surface split over ℚ and whose singularity type is 2A2 + A1. For this, we make use of a deep result about the equidistribution of the values of a certain restricted divisor function in three variables in arithmetic progressions. This result is due to Friedlander and Iwaniec (and was later improved by Heath–Brown) and draws on the work of Deligne.

An improved circuit is described for a two thyratron counting unit such as is used in the “scale of two” thyratron counter. The advantages which result are that the minimum time of resolution is reduced and a source of occasional failure (“jamming”) of the counter is removed. Any type of impulse may be counted and, with selected thyratrons, impulses separated by only 0.00015 sec. may be reliably counted as separate. This makes possible a counting speed of 8000 random impulses per minute with a probable loss of only 2%.

The modification introduced into the circuit used in the original “scale of two” counter is that small condensers (0·0015 μF max) are cross connected between the grids and anodes of the pair of thyratrons, and the extinguishing condenser connected between the anodes is made small (0·02 μF max). The anode resistances are 10,000 ohms and grid leaks 100,000 ohms for the values of capacity mentioned. A valve amplifying stage is introduced, to pass the impulses from the modified circuit of the first pair to the second pair of thyratrons. For greater convenience the mechanical counting meter operated by the third pair is modified to count in multiples of eight so that no multiplication of its reading is necessary.

E. Trefftz has discussed the problem of the torsion of a beam whose cross-section is bounded by a polygon with the help of the Schwarz-Christoffel transformation given by

where a1, a2, …, an are external angles of the polygon in the w-plane, and ξ1, ξ2, …, ξn are the points on the real ξ-axis in the t-plane that correspond to the angular points of the polygon in the w-plane. In the case of regular polygons a further transformation of the upper half of the t-plane into the interior of a circle in the z-plane with the help of the transformation

greatly simplifies the problem, and some definite results can be obtained.

Extremum principles are obtained for the laminar flow of a conducting liquid in a straight pipe of arbitrary cross-section with conducting walls of arbitrary thickness, which is subject to a uniform applied magnetic field. These extrema provide upper and lower bounds for the mass-flow rate in the pipe. These bounds are particularly useful in the construction of asymptotic estimates for the flow rate at high Hartmann number. The method is illustrated by its application to the pipe of circular cross-section and to pipes with thin conducting walls.

Let be a plane closed convex curve, and regard as the boundary of a billiard table. If a perfectly elastic billiard ball is set moving on the table, its path will be a polygon inscribed in ; and this will be closed if and only if the motion is periodic. We seek periodic paths which are triangles.

In this paper, we study families of singular surfaces in ℂ3 parametrised by $\mathcal {A}$-finitely determined map germs. We consider the topological triviality and Whitney equisingularity of an unfolding F of a finitely determined map germ f : (ℂ2, 0) → (ℂ3, 0). We investigate the following question: topological triviality implies Whitney equisingularity of the unfolding F? We provide a complete answer to this question, by giving counterexamples showing how the conjecture can be false.