Let Ωn denote the convex set of all n × n doubly stochastic matrices: chat is, the set of all n × n matrices with non-negative entries and row and column sums 1. If A = (aij) is an arbitrary n × n matrix, then the permanent of A is the scalar valued function of A defined by

where the subscripts i1, i2, …, in run over all permutations of 1, 2, …, n. The permanent function has been studied extensively of late (see, for example, (1), (2), (3), (4), (6)) and it is known that if A ∈ Ωn then 0 < cn ≤ per (A) ≤ 1, where the constant cn depends only on n. It is natural to inquire if per (A) is a convex function of A for A ∈ Ωn. That this is not the case was shown by a counter-example given by Marcus and quoted by Perfect in her paper ((5)). In this paper, however, she shows that per (½I + ½A) ≤ ½ + ½ per (A) for all A ∈ Ωn. Here I = In is the identity matrix of order n.

In a previous paper ((8)), the joint probability generating function was written down for the sizes of the generations in a tree consisting of c distinct species or colours. It was pointed out in (12) that a generalization of Lagrange's expansion could be applied in order to obtain explicit formulae in some circumstances. The techniques have received new application, to polymer chemistry, in (16), (15), and (17). This application should not be confused with the application of the enumeration of trees to that of isomers ((4), (2), (26)). It was further pointed out in (12) that the technique could be applied to some problems concerning the enumeration of ‘ordered’ (planar) trees. This idea is here developed further, and is applied also to ‘labelled’ trees ‘ordered within colours’. One of our objectives is to specify some circumstances in which iterated generating functions, and consequently also the generalization of Lagrange's expansion, are applicable. The examples show how the techniques can be applied almost automatically in order to deduce results, both old and new. The number of labelled ‘chromatic’ trees with c colours is found: Scoins gave the result for c = 2.

In the integral the functions g(z), f(z, α) are analytic functions of their arguments, and N is a large positive parameter. When N tends to ∞, asymptotic expansions can usually be found by the method of steepest descents, which shows that the principal contributions arise from the saddle-points, i.e. the values of z at which ∂f/∂z = 0. The position of the saddle-points varies with α, and if for some α (say α = 0) two saddle-points z1(α), z2(α) coincide (say at z = 0) the ordinary method of steepest descents gives expansions which are not uniformly valid for small α. In an earlier paper (Chester, Friedman and Ursell (1)), a uniform asymptotic expansion of the form

was obtained, where Ai and Ai' are the Airy function and its derivative respectively, and where the regular functions A(α) and ζ(α) are given by

The coefficient functions As(α), Bs(α) are also regular functions of a for which, however, no explicit expressions are known. This Airy-function expansion was shown to be valid in a circle |α| ≥ Rα, independent of N. Airy-function expansions of the same form but involving slightly different argument functions instead of our A(α) and ζ(α) have long been known but are valid only in a region which shrinks to the point α = 0 as N → ∞. This improvement in the region of validity greatly simplifies the matching of steepest-descents and Airy-function expansions across the common region of validity.

In the present paper a further improvement is obtained. The validity of our Airyfunction expansion is extended to a still larger region which may be unbounded and which in many practical cases covers the whole region of interest, so that no matching with other expansions is needed. For this purpose the relation between steepestdescents and Airy-function expansions is investigated.

It is easy to see that by a process of matching the steepest-descents coefficients can be expressed in terms of the Airy-function coefficients. It is now shown that conversely the Airy-function coefficients can be expressed in terms of the steepestdescents coefficients, and that they involve the two saddle-points symmetrically.It is thus possible to infer desired properties of the Airy-function coefficients (e.g.analytic continuation and boundedness) from the corresponding properties of the steepest-descents coefficients, and hence to infer the equivalence of the two expansions (except near α = 0).

The following result is typical. Suppose that the steepest-descents expansion can be shown to be valid in a region (excluding a neighbourhood of α = 0) of the α-plane, and suppose further that the steepest-descents coefficients and the functions ζ(α) and A (α) can be shown to satisfy certain simple conditions of regularity and boundedness.(In practice it is usually not difficult to verify these.) Then it is shown that our Airy-function expansions can be continued into the same region and that it is a valid asymptotic expansion of the integral there.

These conditions of boundedness are not satisfied in a certain integral arising in the study of Kelvin's ship-wave pattern, where the coefficients at one saddle-point become unbounded near the track of the disturbance. The argument is modified to show that, nevertheless, the Airy-function expansion holds uniformly up to the track of the disturbance.

In this paper we consider the problem of distribution of stress in an infinite wedge of homogeneous elastic isotropic solid under the usual assumptions pertaining to plane strain in classical (infinitesimal) theory of elasticity. The wedge is supposed to occupy the region 0 ≤ ρ ≤ ∞, − α ≤ θ ≤ α (in plane polar coordinates) where the pole is taken on the apex of the wedge with the line bisecting the wedge angle as the initial line. We report here the investigations of two types of stress fields: (i) the stress field which is set up by the application of known pressure to inner surfaces of a crack situated on the bisector of the wedge angle, and (ii) the stress field generated by the indentation of the plane faces of the wedge by the rigid punch. The corresponding boundary-value problems are shown to be equivalent to the problem of solving dual integral equations involving inverse Mellin-transforms. Similar equations have been discussed by Srivastav ((l)) in a recent paper. The method of (1) is easily modified to reduce the dual equations to single Fredholm equations of the second kind which are best solved numerically. The boundary-value problems discussed here are of the mixed type and appear to have been considered only recently by Matczynski ((2)) who has investigated a contact problem. He identifies the problem with a Wiener-Hopf integral equation and solves the integral equation approximately using a method due to Koiter ((3)). The method presented here seems to have the advantage that it requires no elaborate tools of analysis which are necessary for resolving the problem of Wiener-Hopf and numerical results may be obtained comparatively easily.

Bickley and McNamee (1) describe techniques for obtaining the solution of finite difference equations, arising from partial differential equations, making extensive use of matrix methods. In all cases solutions are obtained by solving algebraic equations as distinct from differential equations. For example, in order to solve

the second space derivative is replaced by finite differences and the time derivative is replaced either by substituting the backward finite difference form or by using the Laplace transformation.

Renewal processes in discrete time (or as they are commonly termed, recurrent events) are appropriately described by renewal sequences {un} which are generated by discrete distributions , according to the equation

Any two renewal sequences {u′n}, {u″n} define another renewal sequence {un} by means of their term-by-term product {un} = {u′nu″n}, for the joint occurrence of two independent recurrent events ℰ′ and ℰ″ is also a recurrent event. Considering a renewal process in continuous time for which we shall suppose a frequency function f(x) of the lifetime distribution exists, so that a renewal density exists, the analogous property would be that for two renewal density functions h1(x) and h2(x), the function h(x) = h1(x) h2(x) is a renewal density function. A little intuitive reflexion shows that while h(x) dx has a probability density interpretation, this is not in general true of h1(x) h2(x) dx. It is not surprising therefore to find in example 1 a case where the product of two renewal densities is not a renewal density. Example 2, on the other hand, shows that in some cases it is true, and taken together with example 1, there is suggested the problem of characterizing the class of renewal densities h(x) for which αh(x) is a renewal density for all finite positive α and not merely α in 0 < α ≤ A < ∞. In turn this characterization enables us to define a class of renewal densities for which h1(x) and imply that .

Suppose a particle executes a random walk on a two-dimensional square lattice, starting at the origin. The position of the particle after n steps of the walk is Xn = (Xl, n, X2n), where

and we will assume that the Yi are independent bivariate discrete random variables with common moment generating function (m.g.f.)

where a, b, c and d are non-negative. We assume further that (i) pi, j is non-zero for some finite positive and negative i, and some finite positive and negative j (− a ≤ i ≤ b, − c ≤ jd), such values of i and j including – a, b and – c, d, respectively, whenever a, b, c or d is finite, and (ii) the double series defining Φ(α, β) is convergent at least in some finite region D, of the real (α, β) plane, which includes the origin.

The forced torsional motion of a semi-infinite isotropic right-circular elastic cylinder is studied in this paper, the vibrations being excited by a harmonically oscillating rigid disc, securely attached to the plane face of the cylinder. Two distinctsituations are considered, namely, when the curved surface of the cylinder is clamped and when it is stress free. In both cases the problem is reduced to the solution of a Fredholm integral equation of the second kind which can be solved iteratively when the frequency of oscillation is small and the cylinder radius large. Using the iterative solutions, approximate expressions are deduced for the distribution of shear stress under the disc and the driving couple required. In an appendix a discussion is given of a mathematically similar problem in the theory of the swirling flow of a perfect fluid.

The propagation of type N gravitational radiation in homogeneous cosmological models with space-sections of constant negative curvature is considered. It is found that the propagation of radiation distorts the isotropy of the fluid pressure in the models permanently.

It is shown that, when a plane harmonic P or S wave is incident upon a two-or three-dimensional obstacle in an infinite elastic solid, the scattering cross-section of the obstacle can be calculated from an appropriate far-field scattering amplitude.

In what follows we shall derive some properties of the gravitational field of an isolated, axially symmetric, uniformly rotating mass of perfect fluid in a steady state, according to the general theory of relativity. Several exact models describing rotating fluids are known in Newtonian mechanics, the Maclaurin and Jacobi ellipsoids ((6)) being perhaps the most interesting. In general relativity, no such exact solution is known in its entirety, although Kerr ((4)) has exhibited a certain vacuum solution possessing features that one might expect of a space-time exterior to some rotating body. Throughout this paper we shall have Kerr's solution in mind. The question that we shall keep before us is whether a perfect fluid interior can be matched to any given exterior field. Our main results exhibit the class of all possible fluid boundaries, given the exterior field, and some relations between the pressure, density, 4-velocity, and interior metric tensor.

The most general differential equation to which the dynamical description of the title applies is

where dots denote differentiation with respect to t. The essential problem for this equation is to determine the behaviour of solutions as t → ∞. When we attack this problem, the most obvious question is whether, under reasonable conditions on p(t), every solution is bounded as t → ∞ this question is open except when g(x) is linear. In the special case when p(t) is periodic, (1·1) may have periodic solutions; it is clear that any such solution is bounded, and it is worth mentioning that finding periodic solutions is the easiest way of finding particular bounded ones. So long as the bounded-ness problem is unsolved, there is a special interest in finding a large class of particular bounded solutions: if we know such a class then, although we cannot say whether the general solution is bounded or not, we can make the imprecise comment that either the general solution is in fact bounded or the structure of the whole set of solutions is quite complicated.

The problem considered is the diffraction of an electromagnetic wave by a perfectly conducting wedge embedded in a plasma on which a uniform magnetic field is impressed. The plasma is assumed to behave as an anisotropic dielectric and the problem is reduced, by employing a contour integral representation for the solution, to solving a difference equation. Surface waves are found to be excited on the wedge and expressions are given for their amplitudes.

The preceding paper ((1)) dealt with some general properties of the gravitational field of a rotating fluid mass. An interesting example of a vacuum solution that might be the exterior field of some rotating body was recently found by Kerr ((4)). It was natural to apply the preceding theory to the Kerr solution. This paper deals with other aspects of that solution, particularly the behaviour of its bounded geodesics (planetary orbits). It would seem desirable to know what sort of rotating body could be a source of the Kerr field. It will appear that one of the parameters in Kerr's solution can plausibly be related to the angular momentum per unit mass of a uniformly rotating sphere, the other parameter being a measure of the mass of the sphere.

An important unsolved problem in the theory of non-linear oscillations is to establish the boundedness or unboundedness of the general solution of

where dots denote differentiation with respect to t. When p(t) is periodic, we may seek periodic solutions. This search is interesting for its own sake, and of course leads us to special bounded solutions. In three previous papers (2, 3, 4) I have exhibited the equation

as tractable: on the assumption that e(t) is even and periodic, it was shown that the equation has an infinity of periodic solutions.

In a recent communication ((5)), Skal'skaya has considered the interesting two-dimensional problem of the diffraction of a plane E-polarized electromagnetic wave by two perfectly conducting half-planes, inclined with respect to each other. For simplicity, a symmetric case was studied; thus the current densities on the two half-planes were identical, and it was necessary to determine only one function. By expressing the scattered field in terms of the induced current density, and by imposing the (Dirichlet) boundary condition satisfied on the half-planes, Skal'skaya derived an integral equation of the first kind for the unknown density. This was transformed into an equation of the second kind (a more suitable starting point for iterative procedures) by an application of the Kontorovich–Lebedev integral transform. The free term in the new equation was the solution for an isolated half-plane and, by iteration, an approximate solution was obtained.

The following equation arose in a study of the evolution of a one-dimensional quantum system:

where α and κ are positive constants, and

It will be shown that this system provides an example of exactly exponential decay. This is of interest in view of recent papers discussing the validity of the exponential decay law (see, for example, Khalfin (1)and Martin (2)).

1. A topological game is a game in which the positions are the points of a topological space; the set of possible moves from any such point varies continuously with the point. Play can start from any point of the space and at each point it is specified which of the players has the initiative. Play ends when a position is encountered at which the set of positions from which the player with the initiative can choose is empty. The payoff to each player depends on the set of positions met in the play. The notion of topological game is due to Berge (1), (2).