Let X1, X2, … be a sequence of random variables and let {ajk}, j,k = 1, 2, …, be a matrix of real numbers. Write We establish the following result.

If V is a variety of metabelian Lie algebras then V has a finite basis for its laws [3]. The proof of this result is similar to Cohen's proof that varieties of metabelian groups have the finite basis property [1]. However there are centre-by-metabelian Lie algebras of characteristic 2 which do not have a finite basis for their laws [4] this contrasts with McKay's recent result that varieties of centre-by-metabelian groups do have the finite basis property [2]. The rollowing theorem shows that once again “2” is the odd man out.

In this paper, we show how to generate orthomodular posets from sesquilinear forms on a vector space.

Let Hn denote the set of complex n-square positive semidefinite hermtian matrices. We partially order Hn: If A, B, A–B єHn, write A > B. For A єHn, Write P(A) for the permanental adjoint of A, i.e., P(A) is the n-suare matrix whose i, j entry is per A(j/i), where A(j/i) is the submatrix of A obtained by deleting row j and column i. Now, P(A) is a principal submatrix of t he (n–1)st induced power matrix of AT. Hence, P(A) ∈Hn. Also D(A), the classical adjoing, is in Hn.

Let n be a positive integer and let T be any nonempty set of positive integers. By (n, T) = 1 we men n is relatively prime to each element of T. Hence n can be written as n = n1n2 where n1 is the largest divisor of n such that (n1, T)= 1. Let P be a property associated with positive integers. We shall say that a positive integer n is a P-number if it satisfies the property P. If in the above representation of n, the integer n1 is a P-number. then we shall say that n is a quasi P-number relative to T, or, simply, a quasi P-number. In particular, for k a positive integer >1, n is quasi k-free for given set T) if n1 is k-free.

This article is devoted to studying maximal π spaces where π = Lindelöf, countably compact, connected, lightly compact or pseudocompact. Necessary and sufficient conditions for Lindelöf or countably compact spaces to be maximal Lindelöf or maximal countably compact have been obtained. On the other hand only necessary conditions for maximal π spaces have been deduced where π = connected, lightly compact or pseudocompact.

Consider a k-variable normal distribution Ν (μ,Σ where mgr; = (μ1,μ2, … μk)' and Σ is diagonal matrix of unknown elements >0,i = 1,2, … k. The problem of sequential estimation of = 1 αiμi is considered. The stopping rule is shown to have some interesting limiting properties when the σi's become infinite.

It is our purpose to show that the constructible orthomodular lattices defined by Janowitz in [2], are embeddable into Boolean lattices. In fact they are subdirect products of Boolean lattices, where the subdirect products are taken in the class of orthomodular posets. We shall make these notions precise. Other concepts, such as disjoint sum and constructible lattice, are defined in [1] and [2].

The homology functor from the category of free abelian chain complexes and homotopy classes of maps to that of graded abelian groups is full and replete (surjective on objects up to isomorphism) and reflects isomorphisms. Thus such a complex is determined to within homotopy equivalence (although not a unique homotopy equivalence) by its homology. The homotopy classes of maps between two such complexes should therefore be expressible in terms of the homology groups, and such an expression is in fact provided by the Künneth formula for Hom, sometimes called ‘the homotopy classification theorem’.

In [1] Van Est and Freudenthal introduced and studied several new separation axioms for a topological space. One of these was the pτsq axiom: Given distinct points p and q of X, there exists a real continuous function f on X with f(p) ≠ f(q). They observed that the pτsq axiom lies strictly between the Hausdorff and completely regular axioms, and that it neither implies nor is implied by the T3 axiom.

With R the set of real numbers and S a Banach space, let ℒ be the class of functions A from R × S to S which have following properties: (1) if B is a bounded subset of S then the family {A(·, P): P is in B} is equicontinuous; i.e., if t is a number and ε > 0 then there is a positive number δ such that if |s – t | < δ and P is in B then | A(s, P) – A(t, P)| < ε.

First, we introduce the following condition (C1) for semigroups S. Let n, k be fixed positive integers such that n > k. (C1) For any given sequence of n elements of S, there existK elements, where 1 ≦ i1 < i2…, < ik ≦ n, such that where

In many respects the theory of semi-prime rings (i.e. rings without proper nilpotents) is similar to that for lattice-ordered groups. In this paper semi-prime rings are faithfully represented as subrings of continuous global sections of sheaves of integral domains with Boolean base spaces. This representation allows a simple description of a particular extension of a semi-prime ring as the corresponding ring of all continuous global sections. The ideals in a semi-prime ring R that give rise to the stalks in the sheaf representation are then characterized when R is projectable. Finally equivalent conditions are given for a semi-prime ring R to satisfy a condition, that in the case of lattice-groups, was termed “weak projectability” by Spirason and Strzelecki [8]. Some of the results that are common to semi-prime rings and lattice-groups (and semi-prime semigroups) have been extended to certain universal algebras by Davey [3].

An “elementary” proof of Peano's existence theorem is given that, in addition to avoiding the Ascoli lemma, relies neither on Dini's theorem, nor on uniform continuity of the right hand side of φ' = f(t,φ). It is based on superfunctions. Also, another standard proof of that theorem, based on approximation of the right hand side, is made elementary.

We are concerned with the flow of a viscous incompressible electrically conducting fluid of constant properties past a semi-infinite rigid plate. The governing boundary layer equations were derived by Greenspan and Carrier [2] in 1959. Numerical solutions of these equations subject to different boundary conditions have been considered by Stewartson and Wilson [5], Wilson [8], and recently by Bramley [1].