In 1967, Singer (11) gave 3 classes of n-valued two-place functors and proved that all these functors were Sheffer functions. Out of the n possible assignments needed to define a functor completely, Singer showed that it was sufficient to define 3n − 2, 3n − 2, and 2n assignments respectivelyfor the 3 classes. We shall enlarge Singer's classes to give functors of type Ia, type II and type III. For types Ia and III, it will be shown that it is sufficient to define 2n − 1 assignments and for type II we require 2n − 1 assignments to be defined and conditions on a further n/p1 assignments (where P1 is the least prime factor of n). These classes of functors include all of Singer's classes. We also introduce functors of type Ib, similar to those of type Ia, and show that for these itis sufficient to define 2n − 1 assignments to ensure the functor is a Sheffer function.

Penrose (4) gave a necessary and sufficient condition for the consistency of the linear matrix equation AXB = C and also its complete class of solutions. A necessary and sufficient condition for the equations AX = C, XB = D to have a common solution was given by Cecioni (3) and an expression for the general common solution by Rao and Mitra ((6), p. 25). In the present paper, we obtain a necessary and sufficient condition for the equations A1XB1 = C1 and A2XB2 = C2 to have a common solution and also an expression for the general common solution. This result isuseful in computing a constrained inverse of a matrix, a concept originallyintroduced by Bott and Duffin(2) and recently extended by Rao and Mitra(7) who consider more general constraints with the object of bringing together the various generalized inverses and pseudoinverses under a common classification scheme.

René Thom(5) has shown that if the state of a system is determined by the local minimization of a potential – that is, if the system is so highly dissipative that transients can safely be ignored–then, though a smooth change in the potential function may give rise to a discontinuous change of state, the ways in which this can happen are quite limited. Infact, if we have at most a four-parameter family of potentials, discontinuities of this kind can occur in only seven ways up to local diffeotype, if they are to be structurally stable. (This latter condition is the requirement that it be im-possible to alter the discontinuity type by an arbitrarily small perturbation of the family of potentials: roughly it requires that the behaviour of the system, considered as a function of possible families ofpotentials, be ‘continuous’ at the family concerned. It bears the same relationship as does continuity to computability: a computer given approximately correct data for which to compute the value of a function or the behaviour of a system will give an approximately correct answer if the function is continuous, the system structurally stable. If not, only analytic methods will serve, and the physical significance of the result will be dubious. Fortunately, in the dimensions with which we are concerned, structural stability is an open dense property in the space of possible families of potentials; thus unstable systems can be ignored for almost all purposes, just as we ignore the ‘possibility’ of balancing a pin on its point.)

In this paper, we shall first describe the theory of distance-regular graphs and then apply it to the classification of Moore graphs. The object of the paper is to prove that there are no Moore graphs (other than polygons) of diameter ≥ 3. An independent proof of this result has been given by Barmai and Ito(1). Taken with the result of (4), this shows that the only possible Moore graphs are the following:

The Hughes–Zassenhaus plane, , of order 25 is the simplest of the exceptional Hughes planes (see, e.g. (1), p. 391).

In this paper, an incidence table is constructed analogous to tables for the field plane and the regular Hughes plane, and the following rather surprising properties of the plane emerge:

(i) The table has much greater internal symmetry than either of the analogous tables,

(ii) any polarity with regard to a conic in the central subplane Δ0 of extends to two pairs of polarities in one pair with 30 and the other with 60 singular points each in addition to the six in Δ0.

1. It was shown by Barlotti(1) that the number, k, of points on a (k, n)-arc in a Galois plane S2, q, of order q, where n and q are coprime, satisfies

Regular arcs, in which all the points are of the same type have been studied by Basile and Brutti(2) and, for n = 3 by d'Orgeval(4). By means of an electronic computer Lunelli and Sce(5) have enumerated many arcs in Galois planes of low order. The object of this note is to show how the (11, 3)-arcs of S2, 5, none of which is regular, may be described using only geometrical properties.

In this paper, some theorems of the Borsuk-Ulam type (1) are given. One of these can be applied to show that certain homotopy classes in manifolds cannot be realized by embedded spheres. The n-dimensional sphere Sn is the subset of the euclidean space

Rn+l consisting of all points (x1, …,xn+1) satisfying . Let be a piecewise linear (PL) involution on Sn without fixed points.

Segal(4) defines the algebra of measurable operators affiliated with a von Neumann algebra, and convergence nearly everywhere of a sequence of measurable operators, and shows that addition is jointly sequentially continuous and multiplication separately sequentially continuous in the star topology associated with convergence nearly everywhere.

In this paper we derive the conditions which have to be satisfied in order to obtain some classes of zeros of the generalized hypergeometric series of the type

These conditions read:

A necessary and sufficient condition for operator equations Tϕ = M(u, ϕ), T*u = N(u, ϕ) to arise from a variational principle is presented.

It is shown that in the four-dimensional, source-free Einstein–Maxwell theory, four distinct orthogonal eigenvectors of the energy-momentum tensor may be obtained formally by the action of four projection operators on an almost arbitrary vector.

The extremal field, of the unified theory of Maxwell, Einstein and Rainich, is obtained explicitly in terms of eigenvectors of the energy-momentum tensor and the energy-momentum tensor itself, expressed in terms of eigenvectors, is seen to take on two equivalent forms. From this, a two-parameter group of Lorentz rotations, which leave the tetrad of eigenvectors unchanged, is deduced.

The asymptotic treatment of high-frequency scalar wave problems has in the past been rather unsatisfactory. Typically, the integral representations which arose were evaluated by stationary phase, or as a series of residues. The justification of these methods was usually heuristic and formal. In this paper, a method is advanced which, it is claimed, may be applied to any one-parameter separation of variables problem. The method assumes an integral representation whose contour of integration is the real axis. It is then only necessary to deform this contour in the neighbour-hood of the real axis to derive rigorous asymptotic expansions of the field in both the illumination and shadow. The method is applied to the particular example of scattering by a plane boundary in a general stratified medium with monotonically increasing refractive index.

This is the second part of a two-part paper in which a systematic method is advanced for treating separation of variables problems asymptotically as the frequency becomes large. The method assumes an integral representation in which the contour of integration is the real axis. The contour is then deformed in the neighbourhood of the real axis to derive rigorous asymptotic expansions of the field. In this paper the method is applied to scattering in homogeneous media by the circular and parabolic cylinders.

The to method of matched asymptotic expansions is employed to calculate the mass tre sport velocity due to combinations of small amplitude oscillatory waves propagatir, at a single frequency in fluid systems with density and viscosity dis-continuities. Interfacial boundary layers are considered in terms of the curvilinear coordinate system described by Longuet-Higgins(1). The order of magnitude of the mass transport velocity calculated for a general oscillatory disturbance is the same as that calculated for interfacial progressive waves by Dore(2). For standing waves, the time-averaged motion of the fluid particles forms a cellular structure in each fluid layer; the mass transport velocity due to modal interactions is associated with a similar structure.

This paper deals with the influence of a vortex sheet separating two fluids in relative motion on the radiation from a point source of sound. Both the harmonic and impulsive sources are considered and it is found that waves due to Helmholtz instability must be included in order to ensure that there is no field before the source is excited. The instability wave is confined to a finite region and dominates other disturbances in that region. It is suggested that the instabifity wave is initiated by the unrestricted growth of the specularly reflected wave.

In a Newtonian fluid, in which the body forces are conservative, in which the pressure is a function only of density and in which the kinematic viscosity v is uniform, the vorticity ω satisfies the equation

where u is the velocity field, so that ω = ▿ × u. This equation is normally derived by taking the curl of the Navier–Stokes equations. However, the vorticity has many interpretations in terms of the angular velocity of elements of fluid and it is natural to expect that (1) can be derived by equating the rate of change of the angular momentum of a small material volume element about its centre of mass with the moment of the forces acting on the element. Such a derivation is presented here in the hope that it may be of pedagogic interest.

A radiating point source is abruptly switched on and, thereupon, constantly maintained while immersed within a multi-dimensional, non-dispersive, originally uniform, unbounded, propagative medium with transverse isotropy, equivalent to an intrinsic axisymmetry. The number of spatial dimensions concerned is odd. The pertinent radiation problem is posed as a partial differential equation coupled with a preactivation zero condition. This problem is, first, resolved by a Fourier integral, which is then ‘frozen’ onto (i.e. evaluated at points on) the axis of symmetry. Postulating appropriate hyperbolic-type as well as elliptic-type conditions, a systematic reduction of the ‘frozen’ integral is carried out, enabling an analytic exact solution to be constructed with reference to certain phase curves in a phase plane. Away from the source, this solution is valid throughout the symmetry axis except at certain discrete, moving singularities. These fall into three different categories. The behaviour near each typical singularity is estimated. Three significant physical features are demonstrated (as corollaries); namely, the implicit compliance with a radiation principle, the linear interception of the symmetry axis by a Petrow sky's lacuna (associated with the instantaneous point impulse), and, the eventual attainment of either a steady or quiet state after an infinite period. Interpretations are provided in terms of the group velocity. To assist in probing into these various aspects, two appendices are separately included. An application is made to consider magnetically aligned, radiation reception within a field permeated, electrically conducting gas.