J. S. Ratti

The diameters of the graphs of semirings

Let be a family of sets, {F a |α∈Λ}. By the graph G of the system , we mean the graph whose set of vertices is and in which the vertices F β ∈, are adjacent (that is, are joined by an edge) if and only if F α ≠ F β and F α ∩ F β □, where □denotes the empty set.

On high indices theorems

which satisfy the condition ni+llni _ q > 1, i = 1, 2, Throughout, let { Xn } be a sequence of positive numbers such that 1 ? 0 and lim f(x) as x-*O exists and is finite. The Dirichlet series (1.1) is called lacunary if the Xn satisfy the condition (1.2) XAn+ 1/n > q > 1, n = 1, 2, The series Jan is called A, XI summable if the series (1.1) converges for x> 0 and f(x) is of bounded variation in (0, co). We write

The graphs of semirings. II

In this paper the authors continue their investigation of the connectivity of the graphs of semirings. We give some sufficient conditions under which the graph of a semiring is connected. The paper also contains some open problems.

Tauberian theorems for absolute summability

[8, Theorem 21. It is easily seen that for m =1, summability I A, V m and summability f R, X, k | m are the same as summability I A, X J [9] and summability fR, X, kf [21 respectively. Borwein [1] has shown that for An-n summability I R, X, k I m of EaZ is equivalent to its absolute Cesaro summability with index m. 1.2. Hyslop [6] has established the following Tauberian theorem for absolute summability.

ON ANTI-COMMUTATIVE SEMIRINGS

An anticommutative semiring is completely characterized by the types of multiplications that are permitted. It is shown that a semiring is anticommutative if and only if it is a product of two semirings R1 and R2 such that R1 is left multiplicative and R2 is right multiplicative.

Probability measures on 2 × 2 stochastic matrices: some open problems

Let u be a probability measure on 2 × 2 stochastic matrices with finite support such that the sequence μn, the nth convolution power of μ, weakly converges to a probability measure λ whose support consists of 2 × 2 stochastic matrices with identical rows. The probability measure λ can, therefore, be regarded as a measure on the unit interval [0,1]. In this paper, we discuss some open problems regarding when λ is continuous singular or absolutely continuous with respect to the Lebesgue measure on [0,1], and when λ determines μ uniquely.