For graphs on a finite set of vertices with arbitrary probabilities of independently occurring edges, the reliability is defined as the probability that the graph is connected, and the redundancy as the expected number of spanning trees of the graph. Analogous measures of connectivity are defined for random finite directed graphs with arbitrary probabilities of independently occurring directed edges. Recursive formulas for computing the reliability are known. Determinantal formulas, based on matrix-tree theorems, for computing the redundancy are given here. Among random graphs with a given sum of edge probabilities, the more evenly the probabilities are distributed over potential edges, the larger the redundancy. This inequality, proved using the theory of majorization, in combination with examples shows unexpectedly that conflicts between reliability and redundancy can arise in the design of communication networks modelled by such random graphs. The significance of these calculations for the command and control of nuclear forces is sketched.

Introduction. In a recent paper, Borwein(1) constructed a new method of summability which would read: Let

and let {sn} be any sequence of numbers. If, for λ > − 1,

is convergent for all x in the open interval (0,1) and tends to a finite limit s as x → 1 in (0,1), we say that the sequence {sn} is Aλ convergent to s and write sn → s(Aλ). The A0 method is the ordinary Abel method.

A rubber molecule containing n + 1 carbon atoms may be represented by a chain of n links of equal length such that successive links are at a fixed angle to each other but are otherwise at random. The statistical distribution of the length of the molecule, that is, the distance between the first and last carbon atoms, has been considered by various authors (Treloar (1) gives references). In particular, if the first atom is kept fixed at the origin of a system of coordinates and the chain is otherwise at random, it has been conjectured that the distribution of the (n + 1)th atom will tend, as n increases, towards a three-dimensional normal distribution of the form

where σ depends on n. Thus r2 (= x2 + y2 + z2) will be approximately distributed as σ2χ2 with three degrees of freedom.

In this paper some elastic stability problems are solved for infinitely long, rectangular, isotropic plates, simply supported at the long edges. The problem is solved when the same arbitrarily prescribed distribution of normal stress is applied to each long edge, with no stresses and normal displacement at infinity. The extensional deformation of the middle surface during buckling is neglected; this leads to an underestimation (at most of the order of 10%) of the critical forces, but is unlikely to influence the general conclusions. A strain-energy method of analysis is used. The critical stress distributions are shown to be given by the eigenvalues of an ordinary differential equation, or, alternatively, of an integral equation. The general problem is then specialized to that of sets of concentrated normal forces, and to that of a normal force distributed uniformly over a finite length of each long edge; some numerical results are given for these cases. In the first of these problems there is negligible error in treating pairs of concentrated forces individually when they are separated by distances greater than twice the plate width. In the second problem there is little change in the critical stress when the length of edge stressed exceeds four times the plate width; also, if the length of edge stressed is less than one-quarter of the plate width, the distributed force required to produce instability does not differ appreciably from the concentrated force required to produce instability.

All sets considered in this paper will be subsets of n-dimensional Eucidean space En. In this paper, we shall consider the ‘total edge-lengths’ of polytopes which are inscribed in a given sphere and which contain its centre. We first, however, mention some related problems considered by various authors which may be of interest to the reader.

Dipolar fluids have been discussed by Bleustein and Green in quite a recent paper (3). They have obtained explicit constitutive equations for homogeneous incompressible linear dipolar fluids. An application of this theory is made here to the problem of Couette flow. Monopolar and dipolar stress fields are determined. Numerical values for velocity and rate of shear for different values of radius in the annulus are given. Graphs are also drawn to compare our results with the classical ones. It is found that the rate of shear does not change sign within the annulus.

The paper considers the application of Dirac's classical theory of radiating electrons to examine the motion of an electron in a uniform magnetic field. The non-relativistic equations of motion are solved exactly. It is shown that for the particular case when the motion is confined to a plane, the physical motion is such that the electron describes an equiangular spiral with velocity which steadily decreases as

exp {negative constant × t};

and in the non-physical motion the electron spirals outwards with velocity which steadily increases as

exp {positive constant × t},

and ultimately the electron escapes to infinity. The relativistic equations of motion are solved approximately as a series in ascending powers of the field. It is shown that in the physical motion the velocity begins to decrease, and hence after a time the motion is given correctly by the non-relativistic solution.

I take this opportunity to express my deepest gratitude to Prof. Dirac, who suggested this problem, for his patient guidance and supervision.

Levinson's theorem with all its ramifications is a well-established tool in the spectral analysis of ordinary differential operators (see in particular §§3·10, 11 and 4·3, 4 of Eastham's book [5]). In this note we should like to draw attention to a result that describes the solutions of asymptotically constant linear systems under weaker assumptions less precisely than the Levinson theorem. This result can be called the Perron–Lettenmeyer–Hartman–Wintner theorem after the contributions of these authors in [11, 10, 6, 8]. (Note that the Hartman–Wintner theorem that is discussed in [5, p. 17] is a substantially different version of this theorem.)

It is shown that, if $\Sigma$ is a closed orientable surface and $\varphi\!: \Sigma \to \Sigma$ a homeomorphism, then one can find an ordering of $\pi_1(\Sigma)$ which is invariant under left- and right-multiplication, as well as under $\varphi_* \colon \pi_1(\Sigma) \to \pi_1(\Sigma)$, provided all the eigenvalues of the map induced by $\varphi$ on the integral first homology groups of $\Sigma$ are real and positive. As an application, if $M^3$ is a closed orientable 3-manifold which fibres over the circle, then its fundamental group is bi-orderable if the associated homology monodromy has all eigenvalues real and positive. This holds, in particular, if the monodromy is in the Torelli subgroup of the mapping class group of $\Sigma$.

While a number of special properties of differential forms on a Kähler manifold have been mentioned in the literature on complex manifolds, no systematic account has yet been given of the theory of differential forms on a compact Kähler manifold. The purpose of this paper is to show how a general theory of these forms can be developed. It follows the general plan of de Rham's paper (2) on differential forms on real manifolds, and frequent use will be made of results contained in that paper. For convenience we begin by giving a brief account of the theory of complex tensors on a complex manifold, and of the differential geometry associated with a Hermitian, and in particular a Kählerian, metric on such a manifold.

We discuss the origin, an improved definition and the key reciprocity property of the trilinear symbol introduced by Rédei [16] in the study of 8-ranks of narrow class groups of quadratic number fields. It can be used to show that such 8-ranks are ‘governed’ by Frobenius conditions on the primes dividing the discriminant, a fact used in the recent work of A. Smith [18, 19]. In addition, we explain its impact in the progress towards proving my conjectural density for solvability of the negative Pell equation $x^2-dy^2=-1$ .

We study embedded spheres in 4–manifolds (2–knots) via doubly pointed trisection diagrams, showing that such descriptions are unique up to stabilisation and handleslides, and we describe how to obtain trisection diagrams for certain cut-and-paste operations along 2–knots directly from doubly pointed trisection diagrams. The operations described are classical surgery, Gluck surgery, blowdown, and (±4)–rational blowdown, and we illustrate our techniques and results with many examples.

In this article we study the convergence behaviour of least squares approximations of various types by radial basis functions, i.e. least squares approximations from spaces spanned by radially symmetric functions φ (‖· −xj‖). Here the xj are given ‘centres’ in ℝn which we assume to lie on a grid. The inner products with respect to which the least squares problem is considered are discrete and Sobolev, i.e. may involve derivative information. Favourable estimates for the least squares errors are found that are shown to decrease as powers of the gridspacing. The work on discrete least squares approximations also gives rise to the construction of prewavelets from radial function spaces with respect to a discrete Sobolev inner product, which are discussed in the paper as well.

In this paper, a free-boundary problem of the steady plane flow of an ideal fluid through a slot is solved. The fluid flows under the effect of gravity between rigid boundaries, and then out of a slot as a jet which becomes horizontal at infinity downstream. A constructive proof of the existence of solutions to a non-linear integral equation is given for a parameter range 0 < ε < ε* ≃ 1·02 (where ε, the inverse Froude number, measures the effect of gravity). Approximations to the solution are then found for ε → 0.

The object of this paper is to evaluate an infinite integral involving the product of Meijer's G-function (5) and Kampé de Fériet function (1) in terms of Kampé de Fériet function. A number of papers of Bailey (3,4), Ragab (7,8), Slater (9), and Srivastava (10) have appeared, evaluating an integral in terms of a hypergeometric function of two variables or in terms of an E-function. Their results are obviously the particular cases of my result. Since Meijer's G-function is the most general function of one variable which can be expressed in terms of special functions (5) and Kampé de Fériet's function being the most general hypergeometric function of two variables, the integral given by me is the most general integral ever obtained and generalizes most of the results obtained so far for the integral of Mellin type in terms of generalized hypergeometric series. This is because the Kampé de Fériet function reduces to the product of two generalized hypergeometric functions by choosing parameters suitably.

We investigate spherically symmetric equilibrium states of the Vlasov–Poisson system, relevant in galactic dynamics. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a region with compact closure. Based on a dynamical systems analysis we derive theorems that guarantee that the steady state solutions have finite mass and compact support.

1. Introduction. We are concerned with measures µ, in the measure algebra M(G) of a locally compact Abelian group G, which have independent (mutually singular) powers. In (6), Williamson showed that the spectrum, σ(µ) of a Hermitian independent power measure µ satisfying ∥µ∥n=∥µn∥ for positive integer n, contains an infinity of points on the real axis. He conjectured that, in fact, σ(µ) is the disc {λ:|λ|≤∥µ∥}. Taylor (5), has recently proved that, in the case G = R, any positive continuous independent power µ has σ(µ) = {λ:|λ|≤∥µ∥}. His methods depend on his deep and beautiful theory of critical points. Here we verify Williamson's conjecture, give an elementary proof of Taylor's result, and give a simple characterization of the class of LCA groups for which the natural generalization is valid.

Let α > − 1. For a given series write

it being tacitly assumed that the series defining a(t) and A(t) converge for all t > 0. Σan is said to be summable (Aα) to A if

In a previous paper the values for some lines of the β-ray spectrum of the natural L radiation of radium B have been given. The purpose of the present paper is to show that many of these lines can be accounted for by assuming that rays corresponding to L X-rays, but generated within the atom, act on various atomic levels. Before proceeding to do so, however, it will be advisable to give a fuller account of the experimental methods.

In a recent series of papers analytical methods have been introduced which allow of a mathematically simple treatment of the theorems of statistical mechanics for the usual assemblies of isolated or effectively isolated systems. By this we mean that the individual component systems may be treated for energy content as if they were never interfered with. It is only then that energy can be assigned to systems rather than to the assembly as a whole, and it is on this partition of energy among the systems that the analysis is based. When this independence, for example between separate atoms, breaks down as in a molecule, and still more in a crystal, we can take the whole complex to be a system. The analysis will still apply, and if we can formulate the dynamical motions of the complex system, we can still make progress. The essential step for any system is to construct its partition function. Examples of such constructions for molecules and crystals will be found in the papers quoted, and are of course otherwise well known.