Let S be a compact Hausdorff space; let C(S) be the algebra of all continuous complex valued functions on S; and let M(S) be the dual space of (S) (the space of all regular Borel measures on S). In [2] Grothendieck gave a description of weak sequential convergence in M(S) in terms of uniform convergence on sequences of disjoint open sets in S. In this note we give a condition on the carriers of measures to guarantee that weak zero convergent sequences are norm zero convergent. While this condition is interesting in its own right, it can also be used to obtain immediately some well-known results about compact operators from C(S) to c0.

In [1] it was shown if S is a stable semigroup and H an ℋ-class of S then there is a congruence (H) on S in which H is a single class. After considering some consequences of this result for abstract semigroups, we consider some analogous questions for compact semigroups.

If R is a commutative semiprime ring with identity Kist [4], [5] has shown that R can be embedded into a commutative Baer ring B(R), and has given some properties of this embedding. More recently Mewborn [7] has given a construction which embeds R into a commutative Baer ring with the stronger property that every annihilator is generated by an idempotent. Both of these constructions involve a representation of R as a ring of global sections of a sheaf over a Boolean space.

Let μ and ν be measures defined on some σ-algebrs with values in locally convex topological vector spaces X and Y, repectively. It is possible [1] to construct their product λ = μ × ν as a measure on a σ-algebra if λ is allowed to take its values in X ⊗ε Y, the completion of X ⊗ Y in the topology of bi-equicontinuous convergence. The reason is, roughly speaking, that the topology of biequicontinuous convergence on X ⊗ Y is coarse enough to make λ σ-additive and the completion X ⊗ε Y is big enough to accommodate all values of λ. Here we are going to improve the result by introducing a finer topology on X ⊗ Y in which λ will be σ-additive and such that all values of λ will belong to the completion of X ⊗ Y under that topology. The topology in question is obtained by a slight modification from a topology considered for the first time in the work [3] of Jacobs. Curiously enough, the proof of the improved result is simpler than that of [1] and reduces almost to a direct observation avoiding duality arguments.

Suppose S is a regular semigroup and E is its set of idempotents. If E is subsemigroup of S, then S has been called orthodox and studied recently by Hall [3], Meakin [6], and Yamada [8]. In this paper we assume that E is not (necessarily) a subsemigroup of S and consider the subsemigroup generated by E, denoted <E>. If E denotes the set of all elements of S which can be written E, denoted <E>. If E denotes the set of all elements of S which can be written as the product of n (not necessarily distinct) idempotents of S, then . We show that <E> is always a regular subsemigroup of S and investigate relationships between it and S. The case where <E> = S is of particular interest to us; such semigroups will be referred to as idempotent-generated regular semi- groups.

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.

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.

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).

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.

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.

In [6], Lorch and Newman proved the following lemma: If g(u) is continuous and of bounded variation, 0 ≦ u ≦ 1, then (1). This was extended more recently by Leviatan and Lorch ([5], Lemma 3) to functions which are of bounded variation on the positive real axis, where non the upper limit of integration on the inner integral is infinite.

Abstract. In this paper we develop necessary and sufficient conditions for a finite transformation semigroup to have a mean value which is invariant under the induced shift operators. The structure of such transformation semigroups is described and an explicit description of all possible invariant means given.

It is well known that the so-called reduced Borel algebra, that is, the Boolean algebra of all Borel subsets of the unit interval modulo the meager Borel sets of this interval, can be abstractly characterized as a complete, totally non-atomic Boolean algebra containing a countable join dense subset. (For an indication of the history of this result, see, for example, ([2], p. 483, footnote 12.) From this characterization, it easily follows that the reduced Borel algebra B is “homogeneous” in the sense that every non-trivial in B is isomorphic to B.

Loy and Miller [4] have studied tight Riesz groups (G, ≦), without pseudozeros, and with (G, ≼) an l-group. In this paper we study abelian l-groups (G, ≼) on which a tight Riesz order can the defined, such that ⋞ is precisely the associated order. Such an order we call a CTRO (compatible tight Riesz order).

Let α be an irrational number with simple continued fraction α = (a0, a1, a2,…). The problem studied is that of whether the sequence (qn) of denominators of the convergents pn/qn to α has a subsequence (Bn) = (qin} which is the sequence of denominators of convergents An/Bn to a different number α′. In other words, does there exist a subsequence {qin} which satisfies qio= 1 and For example, the sequence of denominators of convergents to ½(3 − e) is a subsequence of the sequence of denominators of convergents to ε.

The Dirac equation in its most usual form can be derived by the minimum action principle from the Lagrangian density where γk (k =1,2,3,4) are the Dirac 4 × 4 matrices satisfying relations; ψ(x) is a column matrix with 4 complex elements being functions of x1, x2, x3, x4, coordinates of the pseudo-Euclidean space with metric gik.(

By a topological semiring we mean a Hausdorff space S together with two continuous associative operations on S such that one (called multiplication) distributes across the other (called addition). That is, we insist that for all x, y and z in S. Note that, in contrast to the purely algebraic situation [1,2], we do not postulate the existence of an additive identity which is a multiplicative zero.