In this paper we analyse the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes. We consider first the case of compact spacelike hypersurfaces, completing some previous results given in [2]. We next extend these results to the complete noncompact case. In that case, our approach is based on the use of a generalized version of the Omori–Yau maximum principle for trace type differential operators, recently given by the authors in [3].

In a recent paper ((2)) Saffman gave a representation of the flow of a dusty gas in which the dust was described in terms of a number density of small particles with very small volume concentration but appreciable mass concentration. In the same paper the problem of the stability of plane parallel flows to small disturbances was formulated, and subsequently the author ((1)) gave some approximate results for the stability of plane Poiseuille flow.

In this paper, we prove that, for any prime p, a Bol loop of order 2p or of order p2 is necessarily a group, and we show that there exist exactly six non-associative Bol loops of order 8.

1. Let

be a positive definite binary quadratic form with determinant αβ − δ2 = 1. A special form of this kind is

We consider the Epstein zeta-function

the series converging for . The function Zh(s) can be analytically continued over the whole s-plane and it is regular except for a simple pole with residue π at s = 1.

The aim of this paper is to generalize a result of Carter and Lusztig(1) proving the existence of certain non-zero homomorphisms between Weyl modules for the special linear group SLn(K) over an infinite field of characteristic p, and between Specht modules for the symmetric group Sr over K.

Consider the finite set Xn = {1,2, …,n} ordered in the standard way. Let Tn denote the full transformation semigroup on Xn, that is, the semigroup of all mappings α: Xn→Xn under composition. We shall call α order-preserving if i ≤ j implies iα ≤ jα for i,j∈Xn, and α is decreasing if iα ≤ i for all i∈Xn. This paper investigates combinatorial properties of the semigroup O of all order-preserving mappings on Xn, and of its subsemigroup C which consists of all decreasing and order-preserving mappings.

1. In this paper we find generating functions and asymptotic expressions for the number of partitions of a positive integer n into two sets of positive integers satisfying the conditions

The set ‘b’ can be empty. Such partitions are considered by Temperley (1) in a forth-coming paper on the roughness of crystal surfaces. We shall consider them in more detail and under different sets of conditions on the a's and b's.

The relativistic field theories of elementary particles are extended to cases where the field equations are derived from Lagrangians containing all derivatives of the field quantities. Expressions for the current, the energy-momentum tensor, the angular-momentum tensor, and the symmetrized energy-momentum tensor are given. When the field interacts with an electromagnetic field, we introduce a subtraction procedure, by which all the above expressions are made gauge-invariant. The Hamiltonian formulation of the equations of motion in a gauge-invariant form is also given.

After considering the Lagrangian L as a scalar in a general relativity transformation and thus a function of gμν and their derivatives, the functional derivative of

with respect to gμν (x) at a point where the space time is flat is worked out. It is shown that this differs from the symmetrized energy-momentum tensor given in the above sections by a term which vanishes when certain operators Sij are antisymmetrical or when the Lagrangian contains the first derivatives of the field quantities only and whose divergence to either μ or ν vanishes.

In 1897 Rayleigh(14) pointed out that it is possible to obtain a family of solutions to certain problems in diffraction theory, notably those involving plane obstacles, by the simple operation of differentiation of some one solution. This, of course, alters the nature of the solution, especially so in the neighbourhood of a sharp edge. For, at an edge the parent solution may be finite and yet its derivative can become infinite. Further, such a differentiated solution may produce an infinite total energy in the neighbourhood of the edge. It is, therefore, natural to ask what conditions are required to make the solutions of these problems physically and mathematically acceptable and thus define the boundary-value problem uniquely.

In this paper A is a commutative noetherian ring, a an ideal of A and the A- modules are given the a-adic topology.

It is a general feeling that completeness is a kind of finiteness condition. We make precise that feeling and, after a result concerning the homology of a complex of complete modules which can be used in place of Nakayama's Lemma, we establish analogies between complete modules and finitely generated ones, with respect to flat dimension, injective dimension, Bass numbers and the Koszul complex. This is particularly clear in the local case, where we have also some partial information on the support of a complete module. With respect to dimension however, the analogy fails, as shown by an example.

The statistical treatment of order-disorder transformations, as developed by Bethe and Peierls, has been extended to alloys having superlattices of the AB and AB3 types but compositions which differ slightly from the ideal compositions required for the formation of perfect superlattices. If the composition of the alloy is specified by the concentration c of one component, which we have chosen to be the A component, then the results show that for alloys of the AB type the critical temperature should have a maximum value when c = ½, that is, for the ideal composition. This is in agreement with experiment. For alloys of the AB3 type it is found that the maximum value should occur for some value of c > ¼. This result disagrees with that obtained experimentally, but both results agree, at least qualitatively, with those obtained by the theory of Bragg and Williams.

An almost tangent structure on a manifold N is a type (1,1) tensor field S onN with the property that at each point y ∊N the kernel of Sy (regarded as a linear endo-morphism of TyN) coincides with its image. An almost tangent structure is said to be integrable if its Nijenhuis tensor vanishes.

Let X be a closed surface, i.e. a compact connected 2-manifold without boundary. If Gx denotes the group of all homeomorphisms of X to itself, and Nx is the normal subgroup consisting of homeomorphisms which are isotopic to the identity, then the quotient group Gx/Nx is called the homeotopy group of X and is denoted by ∧x.

The results of the bombardment of mercury, lead and thallium by 9 M.e.V. deuterons are reported. The following radioactive isotopes have been detected: 5·5 min., 48 min., 36 hr., 60 day mercury isotopes; 4·4 min., 10·5 hr., 44 hr., and 13 day thallium isotopes; 10·25 min., 2·75 hr., and 54 hr. lead isotopes; 18 hr. and 6·35 day bismuth isotopes. The 10·25 min. lead isotope is positron emitting, an interesting result in an element of high atomic number. Absorption measurement have been made of the radiations emitted by many of these isotopes and assignments have been made in most cases.

In conclusion we wish to thank Dr N. Feather for valuable discussions, and also for making for us a Ra E source. We are indebted to Dr Lewis for advice in setting up the thyratron scale of eight counter. This paper would be incomplete without a sincere acknowledgement of our indebtedness to the hard work of past members of this laboratory who have been mainly responsible for setting up the cyclotron.

One of us (R. S. K.) is grateful to the Royal Commissioners for the Exhibition of 1851 for the grant of an overseas scholarship which made this work possible.

The two-dimensional incompressible fluid flow problem along an infinite flat plate with no heat transfer between the fluid and the plate is discussed when the suction velocity normal to the plate, as well as the external flow velocity, varies periodically with time. Expressions for the velocity and temperature fields in the boundary layer have been obtained in a non-dimensional form. It has been found that the effect of increasing the suction velocity fluctuations is that the amplitude of the skin-friction fluctuations is increased while the phase is decreased and it may be negative for very small values of the frequency. Also the plate temperature decreases with increasing the suction velocity fluctuations for small values of the frequency while there is no effect for very large frequencies.

Let f: H1 → H2be any pair of quasitriangular Hopf algebras over k with a Hopf algebra map f between them. We construct in this situation a quasitriangular Hopf algebra B(H1, f, H2) in the braided monoidal category of H1-modules. It consists in the same algebra as H2 with a modified comultiplication and has a quasitriangular structure given by the ratio of those of H1 and H2. This transmutation procedure trades a non-cocommutative Hopf algebra in the category of k-modules for a more cocommutative object in a more non-commutative category. As an application, every Hopf algebra containing the group algebra of ℤ2 becomes transmuted to a super-Hopf algebra.

The use of even-order isotropic tensors in non-linear elasticity theory is discussed in this paper. A notation is adopted through which these tensors can be represented conveniently so that their interdependence is clearly shown. Information about the number of independent elastic constants required is then readily available for use in an expansion of the stress to various orders in the strain relative to the undistorted configuration of the elastic material in question.

For an incompressible isotropic hyperelastic solid, it is shown that each principal component of the distortional part of the stress is expressible as a function only of the corresponding principal component of strain to the fourth order. Under certain conditions, which are not too restrictive, this result can be extended to higher orders.

In mathematics, analysis on fractals was originated by the works of Kusuoka [17] and Goldstein[8]. They constructed the ‘Brownian motion on the Sierpinski gasket’ as a scaling limit of random walks on the pre-gaskets. Since then, analytical structures such as diffusion processes, Laplacians and Dirichlet forms on self-similar sets have been studied from both probabilistic and analytical viewpoints by many authors, see [4], [20], [10], [22] and [7]. As far as finitely ramified fractals, represented by the Sierpinski gasket, are concerned, we now know how to construct analytical structures on them due to the results in [20], [18] and [11]. In particular, for the nested fractals introduced by Lindstrøm [20], one can study detailed features of analytical structures such as the spectral dimensions and various exponents of heat kernels by virtue of the strong symmetry of nested fractals, cf. [6] and [15]. Furthermore in [11], Kigami proposed a notion of post critically finite (p.c.f. for short) self-similar sets, which was a pure topological description of finitely ramified self-similar sets. Also it was shown that we can construct Dirichlet forms and Laplacians on a p.c.f. self-similar set if there exists a difference operator that is invariant under a kind of renormalization. This invariant difference operator was called a harmonic structure. In Section 2, we will give a review of the results in [11].

1. Introduction. Let A be a subset of a Hausdorff topological linear space. By a convex series of elements of A we mean a series of the form where an∈A and λn ≥ 0 for each n, and . We say that A is:

(i) CS-closed if it contains the sum of every convergent convex series of its elements;

(ii) CS-compact if every convex series of its elements converges to a point of A (this bold terminology is chosen because sets satisfying this condition turn out to have properties analogous to those of compact sets).

In many technical problems on conduction of heat involving convection, radiation, or evaporation at the surface of a body, the flux of heat at the surface is known empirically as a function of the surface temperature with reasonable accuracy. The thermal properties of the body also vary with the temperature, but in many cases the nature of this variation is completely unknown, and in others it is slight over the range of temperature involved. Thus it seems worth while studying problems on conduction of heat in a medium with constant thermal properties and with heat transfer at its surface a given function of the surface temperature. Mathematically such problems occupy an interesting position between the classical linear theory and the general case in which both the differential equation and the boundary conditions are non-linear.