The double integral equation, which takes the place of the Milne equation in the one-dimensional problem, is derived from the governing partial differentio-integral equations. An analytical solution of the problem of a distribution of point sources on a plane, when the other boundaries are at infinity, is then found. The possibility of more complicated boundary conditions is discussed.

The aim of this paper is to analyse Riemannian Kähler–Einstein metrics g in real dimension four admitting an isometric action of SU(2) with generically three-dimensional orbits. The Kähler condition means that there is a complex structure I, with respect to which the metric is hermitian, such that the two-form Ωdefined by

is closed. It is well-known that if this condition holds then Ω is in fact covariant constant.

Before the war, self-consistent field calculations for the Au+ ion had been carried out by W. Hartree but were left still unpublished at his death (see prefatory note in (5)). These results have been used by Brenner and Brown (1) in a relativistic calculation of the K-absorption edge for gold, and they were also used in obtaining initial estimates for the partial self-consistent field calculations for thallium of which results are given in §§3–5 of the present paper. In the meantime an independent calculation for Au+ has been carried out by Henry (6), and his results agree closely with those of W. Hartree. However, it still seems desirable to publish the latter, since they give directly the radial wave function P(nl; r) at exact values of r, whereas Henry used log r as independent variable, as had been done for similar calculations for Hg(4), and has tabulated r½P(nl; r) which is the natural dependent variable to use with log r as independent variable (2); in some applications it is more convenient to have the radial wave functions themselves.

Several forms of the lower-bound variational method for the calculation of the eigenvalues in a wave-mechanical problem are considered, and compared; the particular case of the harmonic oscillator being chosen. All forms have certain unsatisfactory features, but some of them are considerably worse than others. One reason why calculations of lower bounds are in general less satisfactory than Ritz-type calculations of an upper bound is shown to be that whereas, in the presence of a scale factor, this latter wave-function satisfies the virial theorem, in none of the lower-bound wave-functions is this true. Similar calculations are reported for the ground state of the helium atom.

We identify the class of smooth boundary conditions that yield an initial-boundary value problem admitting a unique smooth solution for the case of a dispersive linear evolution PDE of arbitrary order, in one spatial dimension, defined on a finite interval.

This result is obtained by an application of a spectral transform method, introduced by Fokas, which allows us to reduce the problem to the study of the singularities of the set of functions arising as the unique solution of a certain linear system.

We establish a maximin characterisation of the linear escape rate of the orbits of a non-expansive mapping on a complete (hemi-)metric space, under a mild form of Busemann's non-positive curvature condition (we require a distinguished family of geodesics with a common origin to satisfy a convexity inequality). This characterisation, which involves horofunctions, generalises the Collatz–Wielandt characterisation of the spectral radius of a non-negative matrix. It yields as corollaries a theorem of Kohlberg and Neyman (1981), concerning non-expansive maps in Banach spaces, a variant of a Denjoy–Wolff type theorem of Karlsson (2001), together with a refinement of a theorem of Gunawardena and Walsh (2003), concerning order-preserving positively homogeneous self-maps of symmetric cones. An application to zero-sum stochastic games is also given.

1. A steady motion of viscous liquid of constant density is considered in §§ 2–7; full allowance is made for the inertia of the motion, and it is assumed that the stream-function can be expanded in a double series in the coordinates x, y.

In any valuated ring R, more specifically, a ring with a degree-function d(a) as defined in §2, the notion of right R-dependence may be denned as follows: (i) The elements a1, …, ar of R are right R-dependent if there exist b1, …, br ∈ R such that . (ii) An element a of R is said to be right R-dependent on the elements a1, …, ar of R if a = 0 or if there exist c1, …, cr ∈ R such that .

The aim of this note is to prove an analogue of P. Hall's theorem on distinct representatives (1) for a family of measurable sets.

Let G be an infinite partially directed graph of finite outgoing degree. Thus G consists of an infinite set of vertices, together with a set of edges between certain prescribed pairs of vertices. Each edge may be directed or undirected, and the number of edges from (but not necessarily to) any given vertex is always finite (though possibly unbounded). A path on G from a vertex V1 to a vertex Vn (if such a path exists) is a finite sequence of alternate edges and vertices of the form E12, V2, E23, V3, …, En − 1, n, Vn such that Ei, i + 1 is an edge connecting Vi and Vi + 1 (and in the direction from Vi to Vi + 1 if that edge happens to be directed). In mixed Bernoulli percolation, each vertex Vi carries a random variable di, and each edge Eij carries a random variable dij. All these random variables di and dij are mutually independent, and take only the values 0 or 1; the di take the value 1 with probability p, while the dij take the value 1 with probability p. A path is said to be open if and only if all the random variables carried by all its edges and all its vertices assume the value 1. Let S be a given finite set of vertices, called the source set; and let T be the set of all vertices such that there exists at least one open path from some vertex of S to each vertex of T. (We imagine that fluid, supplied to all the source vertices, can flow along any open path; and thus T is the random set of vertices eventually wetted by the fluid). The percolation probability

is defined to be the probability that T is an infinite set.

The structure theory of separable complex L*-algebras was given by Schue in [10] and [11]. In [3] Balachandran makes a study of infinite-dimensional complex topologically simple L*-algebras of classical type and poses the question whether these algebras exhaust the class of all infinite-dimensional complex topologically simple L*-algebras. In this paper we give an affirmative answer by determining all the complex topologically simple infinite-dimensional L*-algebras. The case of the real L*-algebras was studied previously in [4], [9] and [12] also under the separability condition. Applying the result of Balachandran, our result yields the structure theory for real L*-algebras. The main tool used here is the ‘approximation’ of the L*-algebra by topologically simple separable L*-algebras via an ultraproduct construction.

It is a well known result of Y. André (a basic special case of the André-Oort conjecture) that an irreducible algebraic plane curve containing infinitely many points whose coordinates are CM-invariants is either a horizontal or vertical line, or a modular curve Y0(n). André's proof was partially ineffective, due to the use of (Siegel's) class-number estimates. Here we observe that his arguments may be modified to yield an effective proof. For example, with the diagonal line X1+X2=1 or the hyperbola X1X2=1 it may be shown quite quickly that there are no imaginary quadratic τ1,τ2 with j(τ1)+j(τ2)=1 or j(τ1)j(τ2)=1, where j is the classical modular function.

In this paper we develop a method which may be used to compute the Chern-Simons invariants of a large class of representations on a large class of manifolds. This class includes all representations on all Seifert fiber spaces, all graph manifolds, and some hyperbolic manifolds. I owe many thanks to Peter Scott, John Harer, Frank Raymond, Ron Fintushel, Paul Kirk and Eric Klassen, without whose help and support this paper could not have been written.

The present note is concerned with the proof and applications of the following theorem: Let A be a commutative ring, N be an A-module, and g1, …, gk be elements of A such that (g1,…, gi−1) A:gi = (g1,…, gi−1) N:gi = (g1,…, gi−1) N (i = 1,…, k), so that 0:g1 = 0 in N. Let g denote the ideal (g1 …, gk), let B = A/g and let M be any A-module such that gM = 0. then

where, on the right-hand side of (ii), M, N/gN are considered as B-modules.

The purpose of this note is to extend in two respects some previous work of the writer on the wave equation of a particle in a central field of force which is a Coulomb field for sufficiently large r.

Let $M$ be a compact, connected surface without boundary and different from $\rp$, and let $B_n(M)$ and $P_n(M)$ denote its braid group and pure braid group on $n$ strings respectively. In this paper, we study the roots of the ‘full twist’ braid in $P_n(M)$ and $B_n(M){\setminus} P_n(M)$. Our main results may be summarised as follows: first, the full twist has no non-trivial root in $P_n(M)$. Further, if $M\ne \St$ and $k\geq 2$, it has a $k\th$ root in $B_n(M){\setminus} P_n(M)$ if and only if $k$ divides either $n$ or $n-1$. This generalises results concerning the sphere of Gillette, Van Buskirk and Murasugi. We also show that the Artin pure braid groups and the pure braid groups of the sphere admit a (non-trivial) splitting as a direct product of which one of the factors is the cyclic group generated by the full twist.

We consider random monic polynomials of degree n over a finite field of q elements, chosen with all qn possibilities equally likely, factored into monic irreducible factors. More generally, relaxing the restriction that q be a prime power, we consider that multiset construction in which the total number of possibilities of weight n is qn. We establish various approximations for the joint distribution of factors, by giving upper bounds on the total variation distance to simpler discrete distributions. For example, the counts for particular factors are approximately independent and geometrically distributed, and the counts for all factors of sizes 1, 2, …, b, where b = O(n/log n), are approximated by independent negative binomial random variables. As another example, the joint distribution of the large factors is close to the joint distribution of the large cycles in a random permutation. We show how these discrete approximations imply a Brownian motion functional central limit theorem and a Poisson-Dirichiet limit theorem, together with appropriate error estimates. We also give Poisson approximations, with error bounds, for the distribution of the total number of factors.

Let C be a set with at least two, and at most ℵ0, members, and for any set X let [X]2 denote the set of its 2-element subsets. If Γ is a countable set, and Fc is a function from [Γ]2 into C, then the structure Γc = (Γ, Fc) is called the countable universal C-coloured graph if the following condition is satisfied:

Whenever α is a map from a finite subset of Γ into C there is xεΓ–dom α such that (∀yεdom α) Fc {x, y} = α(y).

1. The form of the absorption curves of homogeneous β-rays in aluminium has been investigated by means of a Geiger counter. It is found, in agreement with Schonland, that the main portion of the curve is linear, with pronounced initial and final flattenings. Similar initial flattenings were found for copper and silver, but not for gold.

2. The range of β-rays emergent at different angles has been determined, and it appears that the loss of range is due to the increased path consequent upon scattering.

3. Some evidence has been obtained of the angular distribution of plurally scattered particles, and of the existence of a most probable angle of scattering, similar to that found for α-rays by Geiger.