A random rooted labelled tree on n vertices has asymptotically the same shape as a branching-type process, in which each generation of a branching process with Poisson family sizes, parameter one, is supplemented by a single additional member added at random to one of the families in that generation. In this note we use this probabilistic representation to deduce the asymptotic distribution of the distance from the root to the nearest endertex other than itself.

Semiconvex sets are objects in the algebraic variety generated by convex subsets of real linear spaces. It is shown that the fundamental notions of convex geometry may be derived from an entirely algebraic approach, and that conceptual advantages result from applying notions derived from algebra, such as ideals, to convex sets. Some structural decomposition results for semiconvex sets are obtained. An algebraic proof of the algebraic Hahn-Banach theorem is presented.

Epstein and Horn, in their paper ‘Chain based lattices’, characterized P1-lattices, and P2-lattices in terms of their prime ideals. But no such prime ideal characterization for P0-lattices was given. Our main aim in this paper is to characterize P0-lattices in terms of their prime ideals. We also give a necessary and sufficient condition for a P-algebra to be a P0-lattice (and hence a P2-lattice).

In this paper we study the abstract convex program

where S is an arbitrary convex cone in a finite dimensional space, Ω is a convex set and p and g are respectively convex and S (on Ω). We use the concept of a minimal cone for (P) to correct and strengthen a previous characterization of optimality for (P), see Theorem 3.2. The results presented here are used in a sequel to provide a Lagrange multiplier theorem for (P) which holds without any constraint qualification.

An example is given to show that a class of finite soluble groups that is both a Fitting class and a Schunck class need not be a formation. The novel feature of this class is that it is defined by imposing conditions on complemented chief factors of groups in it: this technique usually does not give rise to Fitting classes that are not formations.

We extend the results obtained by Hines and Thompson for a Markov chain which has a single reflecting barrier at the origin, nearest neighbour transitions and which moves from {j} to {j + l} with probability j/(j + 1). Martingale limit theorems are used to work out an asymptotic theory for a general class of such chains for which the probability above has the form l – λ(j) = O>λ(j)>1 (j ∈N),λ(j)→ O (j →∞)and Σλ(j)=∞ We discuss the case where the last sum is finite and some alternative versions of the general case.

Given a semigroup S, we define {N(S), +, ·} to be the ‘free’ distributively generated near-ring. Since all words in N(S) can be expressed as the sum and difference of elements of S, we are able to define a length function on the words of N(S). The following theorems then follow: Theorem 1. N(S) contains a multiplicatice identity e if and only if e ∈ S. Theorem 2. If S is the free semigroup in the cariety of all semi—groups then N(S) is left cancellatice.

Let Q be the ring of quotients of the f-ring R with respect to a positive hereditary torsion theory and suppose Q is a right f-ring. It is shown that if the finitely-generated right ideals of R are principal, then Q is an f-ring. Also, if QR is injective, Q is an f-ring if and only if its Jacobson radical is convex. Moreover, a class of po-rings is introduced (which includes the classes of commutative po-rings and right convex f-rings) over which Q(M) is an f-module for each f-module M.

An associative ring R with identity is said to be c-commutative for c ∈ R if a, b ∈ R and ab = c implies ba = c. Taft has shown that if R is c-commutative where c is a central nonzero divisor]can be omitted. We show that in R[x] is h(x)-commutative for any h(x) ∈ R [x] then so is R with any finite number of (commuting) indeterminates adjoined. Examples adjoined. Examples are given to show that R [[x]] need not be c-commutative even if R[x] is, Finally, examples are given to answer Taft's question for the special case of a zero-commutative ring.