In a previous contribution to these Proceedings (Ramakrishnan(1)) the concept of product density was introduced to describe the statistical distribution of a discrete number of particles in a continuous space E, corresponding to a single point t, where t is the parameter with respect to which the stochastic process evolves. This is extended to densities corresponding to n points on the t axis and correlation problems associated with these density functions are studied with particular reference to electron-photon cascades.

In an attempt to give extensions of certain results in the theory of Mac-Robert's E-function and Meijer's (G-function, the integrals

and

are evaluated, for positive integral values of n, in terms of Agarwal's and their numerous interesting particular cases are deduced. The scope of a further generalization of (i) and (ii), with the aid of the Mellin inversion formula, is also discussed.

It is observed that the integral (ii) provides an elegant generalization of some of the recent results of Srivastava which, in turn, incorporate a well-known formula due to Slater.

We show that the mapping class group of a closed, connected, oriented surface of genus at least three is generated by 3 elements of order 3. Moreover, we show that the mapping class group of a closed, connected, oriented surface of genus at least three is generated and by 4 elements of order 4.

Complex variable methods are applied to obtain exact solutions for the complex potentials and deflexions of thin isotropic slabs bounded by regular curvilinear polygonal contours with n sides and subject to symmetrical loading distributed over a concentric circle. The supported boundary is either clamped or has equal boundary cross-couples. The plates taken in the z-plane are conformally mapped on the unit circle in the ζ-plane by the mapping function . Polynomial approximations to the Schwarz—Christoffel transformations are then used to discuss the bending of clamped and simply supported rectilinear plates symmetrically loaded over a concentric circle or acted upon by a central point load.

Lejeune–Jalabert and Reguera computed the geometric Poincaré series $P_{geom}(T)$ for toric surface singularities. They raised the question whether this series equals the arithmetic Poincaré series. We prove this equality for a class of toric varieties including the surfaces, and construct a counterexample in the general case. We also compute the motivic Igusa Poincaré series $Q_{geom}(T)$ for toric surface singularities, using the change of variables formula for motivic integrals, thus answering a second question of Lejeune–Jalabert and Reguera's. The series $Q_{geom}(T)$ contains more information than the geometric series, since it determines the dimension of the tangent space at the singularity. In some sense, this is the only difference between $Q_{geom}(T)$ and $P_{geom}(T)$.

Recent papers by Atkinson and Reuter(1), Brown and Carr(5), and Barbour(2) have proved several important results concerning wave solutions of the usual deterministic model for the spatial spread of an epidemic, such as measles or influenza. In particular Atkinson and Reuter showed that non-trivial wave solutions, in a population along a line, only exist provided the contact distribution function has an exponentially bounded tail, and that the speed of propagation c must be at least some critical value c0. They constructed a solution for c > c0, which Barbour later showed to be the unique solution, modulo translation, at that speed. Brown and Carr showed that a solution was also possible at speed c0, though it has not been possible to show uniqueness at this speed.

Let q be a prime power, which will be fixed throughout the paper, let k be a field, and let be the field with q elements. Let Gn(k) be the general linear group GL(n, k), and Sn(k) the special linear group SL(n, k). The corresponding groups over will be denoted simply by Gn and Sn. We may embed Gn(k) in Gn+1(k) via the map

Forming the direct limit of the resulting system, we obtain the stable general linear groupG∞(k) over k.

We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized integral operators can be composed unrestrictedly. This leads to the definition of the exponential, and more generally entire functions, of a subclass of such operators.

Let X be a compact differentiable manifold of dimension 2m. A differentiable map from X to euclidean (2m + t)-space is an immersion if its Jacobian has rank 2m at each point of X; it is an embedding if it is also one–one. The existence of such an embedding or immersion implies that the characteristic classes of X satisfy certain integrality conditions; these can be used to obtain lower bounds for the integer t. In a similar way many other geometric properties of X can be deduced from a single integrality theorem involving characteristic classes of various vector bundles over X (see for instance (5)).

In (1) we showed that a subgroup closed Fitting formation is a primitive saturated formation, and in (2) we showed that a subgroup closed and metanilpotent Fitting class is a formation. Whether or not a subgroup closed Fitting class is always a formation is a question that has plagued us ever since. The purpose of this paper is to prove

It is known [14] that every topological complex vector bundle on a smooth rational surface admits an algebraic structure. In [10] one constructs algebraic vector bundles of rank 2 on with arbitrary Chern classes c1, c2 subject to the necessary topological condition c1 c2 = 0 (mod 2). However, in dimensions greater than 2 the Chern classes don't determine the topological type of a vector bundle. In [2] one classifies the topological complex vector bundles of rank 2 on and one proves that any such bundle admits an algebraic structure.

If V is a symplectic space, each affine contact transformation is shown to define an automorphism of a certain algebra of Schwartz functions on V. This automorphism turns out to be a generalized inner automorphism and allows one to associate to the contact transformation a tempered distribution which can be found explicitly. It is shown by means of various examples that these distributions contain much information about the corresponding quantized system. For example the energy levels and stationary states of the harmonic oscillator can be deduced from the general formulae. Generalizations to other locally compact abelian groups V are described. Some connections with the theory of Fourier integral operators and with the characters of certain projective representations are also outlined.

We study the dynamics of non-entire transcendental meromorphic functions with a finite asymptotic value mapped after some iterations onto a pole. This situation does not appear in the case of rational or entire functions. We consider the family of non-entire functions \[ f(z)=\frac{a\exp(z^p)+b\exp(-z^p)}{c\exp(z^p)+d\exp(-z^p)} \] with this property, i.e. there exists a finite asymptotic value $\xi$ and a positive natural number $q$ such that $f^q(\xi)\,{=}\,\infty$, where $p\,{\geq}\,1$, $a,b,c,d\,{\in}\,\mathbb{C}$. We show that the Hausdorff dimension of the set of points in the Julia set with bounded trajectory is strictly greater than one. We then impose two conditions on functions from this family. If the first condition holds, then the Lebesgue measure of the Julia set is zero, but the Hausdorff dimension is equal to two. On the other hand, if a function $f$ satisfies the second condition, then the Lebesgue measure of the Julia set is positive. Finally, we describe the measurable dynamics of these functions and prove that there does not exist an invariant measure on the Julia set absolutely continuous with respect to the Lebesgue measure and finite on compact subsets of $\mathbb{C}$.

It was shown by Birkhoff ((1), p. 253) that every spherically symmetric solution of the field equations of general relativity for empty space,

may be reduced, by suitable coordinate transformations, to the static Schwarzschild form:

where m is a constant. This result is known as Birkhoff's theorem and excludes the possibility of spherically symmetric gravitational radiation. Different proofs of the theorem have been given by Eiesland(2), Tolman(3), and Bonnor ((4), p. 167).

Let be a maximal measure on a compact convex set K, K* be the state space of the space of all continuous functions f: KK ℝ which are affine in the first variable, 1 be the -algebra on K generated by the Baire sets and the compact extremal subsets of K, and = {BeK1}. Then

(i) For any fixed continuous function g:K ℝ and -almost all x in K, there is a closed face of K containing x on which g is constant.

(ii) The image of under the map :KK* defined by f, (x) = f(x, x) is the unique maximal measure on K* representing its barycentre

(iii) induces a measure on (eK) satisfying certain regularity conditions.

In continuation of earlier work [7], we study in this paper the structure of cocommutative Hopf algebras of finite representation type. Since every such algebra is the group algebra H([Gscr ]) of a finite algebraic group [Gscr ], this problem has two interrelated aspects. On the one hand, one has to understand the structure of the group scheme [Gscr ], while on the other the Morita equivalence class of H([Gscr ]) has to be determined. Although we shall here be mainly concerned with the latter, structural information on the underlying group schemes enters crucially at several points.

Solutions of Fock's equations for the self-consistent field with exchange have been carried out for Ne+4, using the EDSAC at the Mathematical Laboratory, Cambridge. The initial estimates for the calculation were made by a simplified version of a method previously suggested for interpolating atomic wave functions with respect to atomic number. This gave good estimates for Ne+4, and it is probable that estimates for Ne+3, obtained similarly, are already accurate enough for practical use. Such wave functions for Ne+3 are given. The results have application in astrophysical contexts.

Background (1·1). Ordered Abelian groups have been studied for nearly a century. Since the early 1950's, it has been recognized that orderings in locally compact Abelian groups can play an important rôle in harmonic analysis on such groups. In this paper we study orderings, especially in topological Abelian groups with either topological or measure-theoretic properties, obtaining nearly a complete classification of such orderings. We then apply these results to determine the limitations of the celebrated theorem of F. and M. Riesz on such groups.

We prove a robust version of Freiman's 3k – 4 theorem on the restricted sumset A+ΓB, which applies when the doubling constant is at most (3+$\sqrt{5}$)/2 in general and at most 3 in the special case when A = −B. As applications, we derive robust results with other types of assumptions on popular sums, and structure theorems for sets satisfying almost equalities in discrete and continuous versions of the Riesz–Sobolev inequality.

All manifolds in this paper are piecewise linear (or smooth if one wishes).