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.

Let ε, Λ, Λ*, ΛR stand for the set of non-negative integers, isols, isolic integers and regressive isols respectively, and let P(τ) be the ω-group of Gödel numbers of permutations of the set τ ε ε which move only finitely many elements of τ. The concept of an ω-group was studied by Hassett [5]. He proved in P12 of [5] that for an isolated set τ, the decomposition of P(τ) into conjugacy sets is a gc-decomposition if and only if τ is regressive. For the finite symmetric group on n elements, Sn it is known that the order of the conjugacy class is p(n), where p(n) is the partition function. The author shows in this paper, using a result of Barback [1], that if pΛ(T) is Nerode's canonical extension of p(n) to Λ and Req (τ) = T, then pΛ(T) = Req Cτ, where Cτ is the decomposition of P(τ) into conjugacy sets. The reader is assumed to be familiar with the contents of [2], [4] and [5].

In this paper we consider two contact problems for an anisotropic elastic half-space in which the stress is independent of one of the Cartesian co-ordinates. The results hold for the most general anisotropy in which no symmetry elements of the material are assumed. Problems of this type have been considered by Brilla [1], Clements [2], Galin [3], Green and Zerna [4] and Milne-Thomson [5], but the work of these authors is only applicable to a restricted class of anisotropic materials. We begin in section 1 by deriving some fundamental equations for the stress and displacement. It will be noted that at an early stage (equation (7)) the roots of a sextic polynomial are required. Since these cannot be obtained explicitly any application of the general theory must, of necessity, be numerical. However the calculation of the stress and displacement is simplified if certain symmetry elements of the material are assumed and the way in which these simplifications occur is indicated in section 2. In sections 3 and 4 we consider contact problems in which the elastic half-space is indented by a rigid body. For the problem considered in section 3, the rigid body is assumed to be able to move relative to the surface of the half-space, while for the problem considered in section 4 it is assumed to be linked to the half-space.

In 1964 Frink [8] generalized Wallman's method [17] of compactification and asked the question: “Is every Hausdorff compactification of a Tychonoff space a Wallman compactification?”. This problem, which is as yet unsolved, has led to the discovery of a number of necessary and/or sufficient conditions for a Hausdorff compactification to be Wallman; (see Alò and Shapiro [2], [3], Banasĉhewski [6], Njåstad [13] and Steiner [15]). Recently Alò and Shapiro [4], [5] have generalized the Wallman procedure to discuss what they call Z*-realcompact spaces and Z*-realcompactification n(Z) corresponding to a countably productive (c.p.) normal base Z on X. Some further work has been done by Steiner and Steiner [14].

If A, B, H, K are abelian group and φ: A → H and ψ: B → K are epimorphisms, then a given central group extension G of H by K is not necessarily a homomorphic image of a group extension of A by B. Take for instance A = Z(2), B = Z ⊕ Z, H = Z(2), K = V4 (Klein's fourgroup). Then the dihedral group D8 is a central extension of H by K but it is not a homomorphic image of Z ⊕ Z ⊕ Z(2), the only group extension of A by the free group B.

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.

Let T denote one of the nn−2 trees with n labelled nodes that is rooted at a given node x (see [6] or [8] as a general reference on trees). If i and j are any two nodes of T, we write i ∼ j if they are joined by an edge in T. We want to consider random walks on T; we assume that when we are at a node i of degree d the probability that we proceed to node j at the next step is di–1 if i ∼ j and zero otherwise. Our object here is to determine the first two moments of the first return and first passage times for random walks on T when T is a specific tree and when T is chosen at random from the set of all labelled trees with certain properties.

In this note I follow the same notation adopted in [2]. Let Qk(x, n) denote the number of k-free integers ≦x which are prime to n. In [2], I proved the following: For o≦s>1/k uniformly, where σ*−s(n) is the sum of the reciprocals of the sth powers of the square-free divisor of n.

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].

Geometries of one, two and three dimensional spaces over GF(3) have been considered in [2]. A line in a space over GF(3) contains 4 points and every pair of points is harmonic to its complementary pair. This feature makes the transformation of harmonic inversion interesting in a space over GF(3).

The Stokes-Beltrami equations, being applicable to many classes of physical problems, have for a long time received a great deal of attention from authors investigating their various aspects. One may cite, for example, Weinstein [1] and more recently Ranger [2] as just two of many research papers into this topic. Baecklund transformations have, in previous work, been applied to hodograph-type equations (Loowrner ], Power, Rogers and Onborn [4], Rogers [5, 6]) and reduction to appropriate canonical forms in elliptic, parabolic, and hyprebolic regimes has been achieved.

Let k and r be fixed integers such that 1 < r < k. It is well-known that a positive integer is called r-free if it is not divisible by the r-th power of any integer > 1. We call a positive integer n, a (k, r)-integer, if n is of the form n = a kb, where a is a positive integer and b is a r-free integer. In the limiting case, when k becomes infinite, a (k, r)-integer becomes a r-free integer and so one might consider the (k, i) integers as generalized r-free integers.

One of the still unsolved problems posed by Fuchs in his well-known book “Abelian Groups” [2] is Problem 45: characterize the rings R for which . I present here a partial solution.

Let S be a bisimple semigroup, let Es denote the set of idempotents of S, and let ≦ denote the natural partial order relation on Es. Let ≤ * denote the inverse of ≦. The idempotents of S are said to be well-ordered if (Es, ≦ *) is a well-ordered set.

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’.

I can indicate the type of refinement mentioned in the title by referring to Kirchberger's theorem [4]. Its picturesque form in the plane is: if sheep and goats are grazing in a field and for every four animals there exists a line separating the sheep from the goats then there exists such a line for all the animals. The refinement is that the words ‘every four animals’ may be replaced by ‘every four animals including an arbitraily chosen animal’; this reduces the ‘Kirchberger number’ from four to, effectively, three.

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.

This paper describes a one-step method besed upon the Lobatto four-point quadreture formula for the numerical integration of differential: y″(x) = f(x, y(x), y'(x)); y(x0)=y0, y'(x0)=y'0. The method has a local truncation error 0(h6) in y(x) and 0(h5) in y′(x). In the case of linear second-order differential equation, a stability criterion has been developed. Theoretical and computational comparisons of the new method existing method is discussed.

In a distributive lattice L with 0 the set of all ideals of the form (x]* can be made into a lattice A0(L) called the lattice of annulets of L. A 0(L) is a sublattice of the Boolean algebra of all annihilator ideals in L. While the lattice of annulets is no more than the dual of the so-called lattice of filets (carriers) as studied in the theory of l-groups and abstractly for distributive lattices in [1, section4] it is a useful notion in its own right. For example, from the basic theorem of [3] it follows that A 0(L) is a sublattice of the lattice of all ideals of L if and only if each prime ideal in L contains a unique minimal prime ideal.

Let G be a finite 2-group having a minimal generating set {x1, …, xr} so that r = d (G) is an invariant of G. Suppose further that G has a presentation then.