A (round-robin) tournament Tn consists n of nodes u1, u2, …, un such that each pair of distinct nodes ui and uj is joined by one of the (oriented) arcs or The arcs in some set S are said to be consistent if it is possible to relabel the nodes of the tournament in such a way that if the arc is in S then i>j. (This is easily seen to be equivalent to requiring that the tournament contains no oriented cycles composed entirely of arcs of S.) Sets of consistent arcs are of interest, for example, when the tournament represents the outcome of a paired-comparison experiment [1]. The object in this note is to obtain bounds for f(n), the greatest integer k such that every tournament Tn contains a set of k consistent arcs.

Packing problems of this kind are obviously equivalent to the problems of placing k (here 9) points in a unit square such that the minimum distance between any two of them be as large as possible. The solutions of these problems are known for 2 ≤ k ≤ 9. The largest possible minimum distances mk are given in table 1, and the corresponding "best" configurations shown in figure 1.

This paper is largely an expository account of known facts, but it contains at least one result believed to be new, Proposition 6.

Our main technique is the method of lifting idempotents developed in Part I. This has been treated in the literature, but not quite in the generality required here. It turns out that much of classical artinian ring theory can be done for the semi-perfect rings introduced by Bass, as will have been noticed by many other people.

Let R be a commutative ring with unit. Let (Aα, φ βα) (α ≤ β) (resp. (Bα, ψαβ) (α ≤ β)) be an infective (resp. projective) system of R-algebras indexed by a directed set I; let ((Xα, dα), fβα) (α ≤ β) (resp. ((Yα, δα), gαβ) (α≤β)) be an injective (resp. projective) system of complexes, indexed by the same set I, such that for each αϵI, (Xα, dα) (resp. (Yα, δα)) is a complex over Aα (resp. over Bα). The purpose of this paper is to show that the covariant functor from the category of all such injective systems of complexes and complex homomorphisms over the R-algebra Aα is such that it associates with an injective system ((Uα, dα), hβα) of universal complexes a universal complex over Aα whereas the same is not true of the covariant functor the category of all such projective systems of complexes and their maps.

It is well known that the Laplacian operator is the infinitesimal generator of Brownian motion in Rn. Moreover, the classical harmonic measures are the hitting measures of Brownian motion. In other words, there is a natural correspondence between the Brownian motion and the classical harmonic functions. In this paper we will show that any family of abstract harmonic functions satisfying the axioms of M. Brelot [4] are annihilated by the infinitesimal generator of a diffusion, and that the corresponding harmonic measures are the hitting measures of the diffusion. This answers a question raised by P. A. Meyer [11] who has proved the existence of a Markov semi-group associated with a harmonic sheaf.

It is well known [1] that the variational problem of minimizing

1

where F is positive homogeneous of degree one in (henceforth abbreviated to "plus-one" in ) leads to a Hamiltonian H{xi, pi} and corresponding Hamilton-Jacobi equation

2

In his paper [1] V. L. Klee gave an example of a smooth bounded convex body C, in En, with the property that extC s closed and extC - expC is dense in extC. As in [1] extC and expC denote the sets of extreme and exposed points of C respectively. It is the purpose of this note to exhibit a similar example in a general separable Banach space using a direct construction which involves Minkowski summation of convex sets.

In this note real valued functions, defined on a linear sublattice S of a linear lattice R and satisfying the two order conditions (M1) and (M2), are studied from the point of view of the existence and uniqueness of extensions to R. The paper is partly expository and supplements and extends §3 of [4] where S was assumed to be an ℓ-ideal.

The purpose of this paper is to present a geometric theorem which provides a proof of a fundamental theorem of finite Markov chains.

The theorem, stated in matrix theoretic terms, concerns the asymptotic behaviour of the powers of an n by n stochastic matrix, that is, a matrix of non-negative entries each of whose row sums is 1. The matrix might arise from a repeated physical process which goes from one of n possible states to another at each iteration and whose probability of going to a state depends only on the state it is in at present and not on its more distant history.

This paper is a continuation of two papers [4], [5] and brings out the solution of the ballot problem in its general form.

In [5], Narayana has considered a generalised occupancy problem which can be viewed as a problem in compositions of integers. In what follows, we use the definitions of [6].

A collineation is a one-one mapping of a projective plane onto itself, taking points into points, lines into lines and preserving incidence, ([1], p. 247). A perspective collineation (sometimes called a central collineation) is a collineation which leaves invariant all points on a line h called the axis, and all lines through a point H called the centre. The perspective collineation is an elation if H is incident with h; otherwise it is a homology.