It has been shown that every general recursive function is definable by application of the five schemata for primitive recursive functions together with the schema

subject to the condition that, for each n–tuple of natural numbers x1,…, xn there exists a natural number xn+1 such that

1. Introduction and recapitulation. In a previous paper (4) we have defined a ringoid (with identities) as a set of elements α, β,… in which addition and multiplication are defined for certain pairs of elements in such a way that, whenever operations are defined, the usual ring axioms are satisfied. Precisely, is an abstract category ((3), p. 108) with an addition operation such that the following axioms hold:

(R 1) For any two identity elements ι, ι′ of, let ℋ(ι, ι′) be the set of elements ofwhose left identity is ι and right identity is ι′; then ℋ(ι, ι′) is empty or an Abelian group under addition.

1. Introduction. The present paper is chiefly concerned with a generalization, to be known as a ‘D-interval’, of the notion of interval or segment in an arbitrary partially ordered group. This idea is originally due to Duthie (2), but was developed by him only in a lattice. In analogy with the use of the interval in the normal sense, notions of ‘D-distributivity’ and ‘D-modularity’ are defined in terms of the D-interval, and analogues of known properties of lattice-groups or ‘l– groups’ can be formulated which might be valid when a lattice structure is no longer assumed to exist; in particular, an attempt is made to provide such a generalization of the result of Freudenthal (3) that every Z-group is a distributive lattice, but, for an arbitrary partially ordered group, it is shown that only an ‘approximation’ (in terms of non-Archimedean elements) to the desired result actually holds, although any Archimedean partially ordered group is necessarily D-distributive.

1. An elegant theorem due to Tarski states that a completely distributive complete Boolean algebra is isomorphic with a lattice of sets, and in fact the lattice of all the subsets of some aggregate. The obvious generalization of the question underlying this theorem is to ask whether one can pick out by means of a distributivity condition those lattices (not necessarily Boolean algebras) which are isomorphs of lattices of sets. The answer is no. The real numbers with their natural order form a complete lattice which satisfies the strongest possible distributivity conditions and yet is not iso-morphic with any lattice of sets.

1. Introduction. Ledermann(1) has treated the problem of calculating the asymptotic probabilities that a system will be found in any one of a finite number N of possible states if transitions between these states occur as Markov processes with a continuous time parameter t. If we denote by pi(t) the probability that at time t the system is in the ith state and by aij ( ≥ 0) the constant probability per unit time for transitions from the jth to the ith state, the rate of change of pi is given by

where the sum is to be taken over all j ≠ i. This set of equations can be written in matrix form as

where P(t) is the vector with components pi(t) and the constant matrix A has elements

Let G be a locally finite connected graph and c be a positive real-valued function defined on its edges. Let D(ξ) denote the sum of the values of c on the edges incident with a vertex ξ. A particle starts at some vertex α and performs an infinite random walk

in which (i) the ξj are vertices of G, (ii), λj. is an edge joining ξj–1 to ξj (j = 1, 2, 3, …), (iii) if λ is any edge incident with ξj, then

Let υ be a set of vertices of G such that the complementary set of vertices is finite and includes α. A geometrical characterization is given of the probability (τ, say) that the particle will visit some element of υ without first returning to α. An essentially equivalent problem is obtained by regarding G as an electrical network and c(λ) as the conductance of an edge λ; the current flowing through the network from α to υ when an external agency maintains α at potential I and all elements of υ at potential 0 is found to be τD(α).

A necessary and sufficient condition (of a geometrical character) for the particle to be certain to return to α. is obtained; and, as an application, a new proof is given of a conjecture of Gillis (3) regarding centrally biased random walk on an n–dimensional lattice.

A new and exact solution is given for the diffraction of the field of an electric line current by an imperfectly conducting wedge of exterior angle . Previous attempts at solving this problem assumed that the solution could be expressed as a series in inverse powers of the conductivity of the wedge, but this approach can only be applied to wedges of exterior angle less than π as the solution obtained is singular at the edge of the wedge for angles exceeding π. It is shown that in the present case no such expansion is uniformly valid; at large distances from the wedge, however, the present solution agrees with that obtained by assuming an expansion in inverse powers of the conductivity, even though this latter solution is singular at the edge, to order (conductivity of the wedge)−1; the next term in the expansion of the present exact solution is O(conductivity). The case of an incident plane wave may be obtained by letting the source tend to infinity.