The theory of catalytic action recently advanced qualitatively by Pease, Taylor, Armstrong and Hilditch, Constable, which the author has endeavoured to treat quantitatively, is extended in this paper to the explanation of the effect of pressure and inert diluents on catalytic action on saturated surfaces. The results are shown to be in agreement with experiment, and further inferences may be made concerning the relative mean lives of the reactant and the diluent on the surface.

1. Let f(x) be a real function of period 2π, integrable L over (0, 2π), and let

By sn(x) and σn (x) we denote respectively the partial sums and the first arithmetic means of the series (1·1). Similarly, by and we denote the partial sums and the first arithmetic means of the series

conjugate to (1·1). By we mean the function conjugate to f(x), that is

where the integral is taken in the principal-value sense.

1. The cross-flow type of heat interchanger is shown diagrammatically in Fig. 1. One of the fluids involved is arranged to pass through a nest of small tubes, while the other streams past the tubes at right angles to them. To fix ideas we suppose that the tubes contain cooling water which enters at a temperature T1. The temperature of the water leaving a tube depends upon its position, but the water from all the tubes is mixed and is led to a single exit pipe in which the temperature is T2. In the same way the fluid to be cooled enters at a temperature t1 and leaves at a temperature t2. If the cooling water is not used again by being led through another nest of tubes also exposed to the fluid to be cooled, the arrangement is termed a single-pass heat interchanger.

Refined expressions are given for the error terms in the asymptotic expansion formulas for double zeta and double gamma functions, proved in the author's former paper [2]. Some inaccurate claims in [2] are corrected.

This paper continues the investigation into the short-wave asymptotic motion due to a cylinder heaving on water of finite constant depth, and we now present a rigorous method for the special case of a half-immersed circular cylinder. The aim is to justify, in one particular case, those assumptions made in a previous paper giving a non-rigorous method for an arbitrary smooth cylinder which intersects the free surface normally, concerning the asymptotic values on the cylinder of the two related auxiliary potentials which were introduced to enable an evaluation of the coefficients describing wave-making and virtual mass. This is done by constructing an integral equation of the second kind whose kernel is small and deducing the leading term in the iteration solution. The method is an extension of the method for infinite depth and it is found that there is a simple relationship between the kernels. In conclusion, computations for these asymptotic results are presented and compared with the trend of known non-asymptotic results.

We shall prove below part of a conjecture made by Shigefumi Mori, David Morrison and Ian Morrison in the course of their investigations into the properties of isolated terminal cyclic quotient singularities of prime Gorenstein index in dimension four [1]. The reader of the present paper need have no knowledge of algebraic geometry, because we quickly reduce the problem to one about the geometry of numbers that can be solved by elementary calculations. The calculations are very lengthy and not quite routine, so what the reader does need is either patience, if he intends to check them, or faith, if he does not. We give only part of the calculations below. Full details may be obtained from the author.*

Let θ be a Jordan homomorphism from an algebra A into an algebra B. We find various conditions under which the restriction of θ to the commutator ideal of A is the sum of a homomorphism and an antihomomorphism. Algebraic results, obtained in the first part of the paper, are applied to the second part dealing with the case where A and B are C*-algebras.

We give a complete set of finite type string link invariants of degree <5. In addition to Milnor invariants, these include several string link invariants constructed by evaluating knot invariants on certain closures of (cabled) string links. We show that finite type invariants classify string links up to Ck-moves for k ≤ 5, which proves, at low degree, a conjecture due to Goussarov and Habiro. We also give a similar classification of string links up to Ck-moves and concordance for k ≤ 6.

Let (R, m) be a d-dimensional Cohen–Macaulay local ring. In this paper we prove, in a very elementary way, an upper bound of the first normalized Hilbert coefficient of a m-primary ideal I ⊂ R that improves all known upper bounds unless for a finite number of cases, see Remark 2.3. We also provide new upper bounds of the Hilbert functions of I extending the known bounds for the maximal ideal.

In analogy with the lower Assouad dimensions of a set, we study the lower Assouad dimensions of a measure. As with the upper Assouad dimensions, the lower Assouad dimensions of a measure provide information about the extreme local behaviour of the measure. We study the connection with other dimensions and with regularity properties. In particular, the quasi-lower Assouad dimension is dominated by the infimum of the measure’s lower local dimensions. Although strict inequality is possible in general, equality holds for the class of self-similar measures of finite type. This class includes all self-similar, equicontractive measures satisfying the open set condition, as well as certain “overlapping” self-similar measures, such as Bernoulli convolutions with contraction factors that are inverses of Pisot numbers.

We give lower bounds for the lower Assouad dimension for measures arising from a Moran construction, prove that self-affine measures are uniformly perfect and have positive lower Assouad dimension, prove that the Assouad spectrum of a measure converges to its quasi-Assouad dimension and show that coincidence of the upper and lower Assouad dimension of a measure does not imply that the measure is s-regular.

A pair of variational principles are formulated for two-dimensional scattering by obstacles. The first of these is in terms of the obstacle boundary values, and it is shown that a simple ‘optical’ trial function leads to an incorrect frequency dependence for the scattering cross-section. In the second, the obstacle is viewed as the analogue of an aperture coupling two half spaces. The geometric optics part of the cross-section can then be made explicit and is split off to leave a stationary form for the frequency correction. The zero-order calculation for the cross-section of a circle, using corresponding ‘optical’ trial functions, is found to have the correct (Ka)−2/3 frequency dependence.

The main purpose of this paper is to describe the construction of an incidence table for the Hughes plane of order q2. To do this it is necessary first to construct a table of the same pattern for the Galois plane, and this requires the expression in terms of explicit matrices of the Singer cyclic group of the plane as the composition of cyclic groups of orders q2 + q + 1 and q2 − q + 1. The two tables constructed have some combinatorial properties of possible interest.

In the Galois plane of order q2 there are two types of polarities, one with q2 + 1 singular points on a conic, and the other with q3 + 1 singular points on the Hermitian analogue of a conic. Correspondingly in the Hughes plane there are polarities with ½(q3 + q + 2) and with ½(q3 + 2q2 − q + 2) singular points.

The paper concludes with the complete incidence table for the Hughes plane of order 25, together with the data necessary for its construction.

The Hughes plane and some of its properties are described in Hughes(1), Zappa(2), Rosati(3) and Ostrom(4). The property which enables the Hughes plane to be most easily constructed from the corresponding Galois plane, and which will be used in this paper, is to be found in Room (5).

Textbook proofs of the Riesz theorem on the representation of linear functionals on C(X) by measures tend to be self-contained, but consequently are rather long, and use ad hoc methods (see, for example (2, 4, 5)). The purpose of this note is to give a short proof by appealing to standard methods of modern analysis.

The aim of the paper is to provide a criterion for the stability of plane curves using log canonical thresholds. As an application, we compare stable pairs of degree $d$ with stable plane curves with degree $d$.

We establish some properties of Thurston's asymmetric metric L on the Teichmüller space of a surface of genus with punctures and with negative Euler characteristic. We study convergence of sequences of elements in in the sense of L, as well as sequences that tend to infinity in . We show that the topology that the asymmetric metric L induces on Teichmüller space is the same as the usual topology. Furthermore, we show that L satisfies the axioms of a (not necessarily symmetric) metric in the sense of Busemann and conclude that L is complete in the sense of Busemann.

A generalized m-manifold is a euclidean neighbourhood retract X that is a homol-ogy m-manifold, i.e. for which H*(X, X–x;Z) = H*(Rm, Rm–0;Z) for each point x of X. In a generalized m-manifold X, a pointy is called singular, or a singularity of X, if p has no neighbourhood homeomorphic to Rm. We shall denote the set of singular points of X by S(X).

This paper is a sequel to a recent paper (1) in which the author discussed gravity waves on a horizontal layer of conducting fluid with a normal electrostatic field at the free surface. In this work results are given for waves in an incompressible dielectric fluid, in a similar configuration. Treating the dielectric as an inviscid fluid the stability of the system is first described in terms of the changes in potential energy in a small static displacement. The result so obtained is then confirmed by a normal mode analysis in which a dispersion relation is obtained for the inviscid model. The paper gives finally a discussion of the results for a viscous dielectric fluid, the main point of which is that, as in (1), in the transition from stable to unstable disturbances viscosity plays no part, and that the stability characteristics are the same as those for an inviscid dielectric fluid.

In the usual notation for generalized hypergeometric functions we let

where

and (a) denotes the sequence of parameters

Throughout the present paper we shall suppose that there are A of the a parameters, Bof the b parameters, and so on. Thus ((a))m is to be interpreted as

and similar interpretations hold for ((b))m, etc.

We show that, under mild conditions, the (normalized) Frobenius splitting numbers of a local ring of prime characteristic are lower semicontinuous.