The method of applying the scattering correction in accurate γ ray absorption measurements is discussed, and it is shown that the scattering correction only diminishes, as the solid angle subtended by the chamber at the source is reduced, if the absorber diameter is reduced correspondingly. The method of calculation of the scattering correction in cases in which the absorber is placed very close to the chamber is explained, and the result of the calculation for one particular chamber is shown to be in satisfactory agreement with the experimental values. The scattering correction allows the ionization function of chambers to be determined and an (ionization function)/(wave-length) curve is shown for a steel thick-walled high-pressure ionization chamber.

Maximum and minimum principles are developed for boundary-value problems with field equations T*Tφ + f(φ) = 0 in some region V, subject to φ = h(Tφ) on the boundary of V.

Measurements of the reflection coefficient at normal incidence on lead and tantalum mirrors in the supraconducting and non-supraconducting state did not reveal any change within the limits of accuracy (0·2−0·5% in R). This does not agree with a suggestion of R. de L. Kronig, according to which an increase in reflectivity of about 35% could have been expected.

Introduction. This paper gives some embedding and unknotting theorems for bounded manifolds in the PL and differential categories. The theorems are of a similar nature to the embedding and unknotting theorems of Irwin and Zeeman (9, 18), except that the boundaries are allowed to move and the appropriate connectivity conditions become conditions on the relative homotopy groups of the manifolds modulo the boundary, or some suitable part of the boundary.

Let Qm be the HNN extension of Z/m × Z/m where the stable letter conjugates the first factor to the second. We explore small presentations of the groups Γm,n = Qm* Qn. We show that for certain choices of (m,n), for example (2,3), the group Γm,n has a relation gap unless it admits a presentation with at most 3 defining relations, and we establish restrictions on the possible form of such a presentation. We then associate to each (m,n) a 3-complex with 16 cells. This 3-complex is a counterexample to the D(2) conjecture if Γm,n has a relation gap.

Weak precompactness in spaces of vector measures and in the space of Bochner integrable functions is studied. Uniform countable additivity and uniform integrability are characterized in terms of weak precompactness. Through this, a connection between strongly bounded operators and operators having weakly precompact adjoints on abstract continuous function spaces is established. These operators are compared with weakly completely continuous operators.

In this paper we study Iwasawa modules arising from Galois cohomology over general $p$-adic Lie extensions both in the local and global case. In particular we calculate their $\Lambda$-ranks. We then apply the results to abelian varieties.

Introduction. Two previous papers (1,2) have dealt with additive processes defined on finite Markov chains. Such a process in discrete time may be treated as a bivariate Markov process {R(k), X(k)}. The process R(k) is an irreducible Markov chain on states r = 1, 2, …, R governed by a stochastic transition matrix B0 with components brs. The marginal process X(k) ‘defined’ on the chain R(k) is a sum of random increments ξ(i) dependent on the chain, i.e. if the ith transition takes the chain from state r to state s, ξ,(i) is chosen from a distribution function Drs(x) indexed by r and s. The distribution of the bivariate process may be represented by a vector F(x, k) with components

These are generated recursively by the relation

where the increment matrix distribution B(x) has components brsDrs(x). We denote the nth moment of B(x) by Bn = ∫xndB(x), so that B0 = B(∞).

We investigate homomorphisms between finite oriented paths. We demonstrate the surprising richness of this perhaps simplest case of homomorphism between graphs by proving the density theorem for oriented paths. As a consequence every two dimensional countable poset is represented finite paths and their homomorphisms, and every finite dimensional poset is represented finite oriented trees and their homomorphisms. We then consider related problems of universal representability and extendability and on-line representability.

1. The gliding of bodies on the surface of water has received mathematical treatment by Wagner for both two- and three-dimensional cases. In this note, attention is confined to the two-dimensional gliding of a plane plate on a stream of infinite depth. Wagner's method is approximate and only applies when the angle of incidence of the gliding plate to the stream is small. He expresses such quantities as the total lift on the plate, not in terms of the total length of the plate, but in terms of the length of the pressure surface, which is the distance between the trailing edge of the plate and the point of stagnation. Further, his solution requires that the direction of the jet or spray which is formed should be parallel to the length of the plate. If Lis the total normal lift on the plate, β the small angle of incidence of the plate to the stream, V the velocity of the plate, δ the thickness of the spray and c the length of the pressure surface, then Wagner gives

In this paper we introduce and investigate the notions of diagrams and discrete extensions in the study of finitary 2-representations of finitary 2-categories.

We introduce new formulations of aperiodicity and cofinality for finitely aligned higher-rank graphs Λ, and prove that C*(Λ) is simple if and only if Λ is aperiodic and cofinal. The main advantage of our versions of aperiodicity and cofinality over existing ones is that ours are stated in terms of finite paths. To prove our main result, we first characterise each of aperiodicity and cofinality of Λ in terms of the ideal structure of C*(Λ). In an appendix we show how our new cofinality condition simplifies in a number of special cases which have been treated previously in the literature; even in these settings our results are new.

Let be a free additive abelian group, and let be a basis of , so that every element of can be expressed in a unique way as a (finite) linear combination with integral coefficients of elements of . We shall be concerned with the ring of endomorphisms of , the sum and product of the endomorphisms φ, χ being defined, in the usual manner, by the equations

A permutation of a set will be called restricted if it moves only a finite number of elements. We call an endomorphism of a permutation endomorphism if it induces a restricted permutation of the basis .

A short solution is given for the urn problem proposed by Paul and Tatiana Ehrenfest in 1907.

In 1907 P. and T. Ehrenfest(3) proposed an urn model for the resolution of the apparent discrepancy between irreversibility and recurrence in Boltzmann's theory of gases (2). In this model it is assumed that m balls numbered 1, 2, …, m are distributed in two boxes. We perform a series of trials. In each trial we choose a number at random among 1, 2, …, m in such a way that each number has probability 1/m. If we choose j, then we transfer the ball numbered j from one box to the other. Denote by ξn the number of balls in the first box at the end of the nth trial. Initially there are ξ0 balls in the first box. If the trials are independent, then the sequence {ξn;n = 0, 1, 2,…} forms a homogeneous Markov chain with state space I = {0, 1, 2,…, m} and transition probabilities pi,i+1 = (m − i)/m for i = 0, l,…, m − 1, Pi,i−1 = i/m for i = 1, 2,…, m, and pi,k = 0 otherwise. The problem is to determine the transition probabilities

for i∈I, k∈I and n = 0,1, 2,….

A geodesic in space-time is complete if it can be extended to infinite values of its affine parameter: it is shown that all strongly causal spaces are conformal to space-times in which all null geodesies are complete, and that a wide class of space-times are conformal to ones in which almost no null geodesies are complete.

Diagrams have been used in group theory by numerous authors, and have led to significant results (see [4] and the references cited there). The idea of applying diagrams to semigroups seems to be more recent [3, 7, 8]. In the present paper we discuss semi group diagrams and use them to obtain results concerning the word problem for one-relator semigroups. The word problem for one-relator groups has been solved by Magnus [6], but the analogous question for semigroups remains open. We are not able to solve the problem in full generality, but have obtained some partial results.

Given a knot K in a 3-manifold M, we use N(K) to denote a regular neighbourhood of K. Suppose γ is a slope (i.e. an isotopy class of essential simple closed curves) on ∂N(K). The surgered manifold along γ is denoted by (H, K; γ), which by definition is the manifold obtained by gluing a solid torus to H – Int N(K) so that γ bounds a meridional disc. We say that M is ∂-reducible if ∂M is compressible in M, and we call γ a ∂-reducing slope of K if (H, K; γ) is ∂-reducible. Since incompressible surfaces play an important rôle in 3-manifold theory, it is important to know what slopes of a given knot are ∂-reducing. In the generic case there are at most three ∂-reducing slopes for a given knot [12], but there is no known algorithm to find these slopes. An exceptional case is when M is a solid torus, which has been well studied by Berge, Gabai and Scharlemann [1, 4, 5, 10]. It is now known that a knot in a solid torus has ∂-reducing slopes only if it is a 1-bridge braid. Moreover, all such knots and their corresponding ∂-reducing slopes are classified in [1]. For 1-bridge braids with small bridge width, a geometric method of detecting ∂-reducing slopes has also been given in [5]. It was conjectured that a similar result holds for handlebodies, i.e. if K is a knot in a handlebody with H – K ∂-irreducible, then K has ∂-reducing slopes only if K is a 1-bridge knot (see below for definitions). One is referred to [13] for some discussion of this conjecture and related problems.

Certain physical theories are short-wave limits of more general theories. Thus ray optics is the short-wave limit of wave optics, and classical mechanics is the short-wave limit of wave mechanics. In principle it must be possible to deduce the former from the latter theories by a rigorous mathematical limiting process; in fact the arguments found in the literature are formal, plausible and non-rigorous. (We are here concerned with linear wave equations and time-periodic phenomena.) For some wave equations there are, however, a few explicit rigorous canonical solutions relating to simple geometrical configurations, e.g. to conics in two dimensions for the equations of acoustics, and for these the asymptotics can be found rigorously. For more general configurations the solution of a typical boundary-value problem can be reduced to the solution of a Fredholm integral equation of the second kind.