An equivalence relation δ on a semigroup S is called a left congruence of S if (u, v) Ε λ implies that (su, sv) Ε λ for every s in S. With every set ℒ of pairwise disjoint left ideals (i. e. subsets L of S such that SL⊂EL), one can associate the left congruence {(u, v)|u = v or there exists an L in ℒ such that u Ε L and v Ε L}. Thus every nonempty left ideal is a left congruence class (i. e. an equivalence class of some left congruence). A left congruence has the form just described if and only if all its nontrivial classes (i. e. its classes containing at least two elements) are left ideals. Such a left congruence is called a Rees left congruence if there is at most one nontrivial class. The identity relation on S is a Rees left congruence since the empty set is a left ideal by definition.

Let X be a Banach space and K a convex subset of X. A mapping Tof K into K is called a nonexpansive mapping if | T(x) – T(y) | ≦ | x – y | for all x, yεK.

For any fixed value of x, let denote the set of all positive integers with exactly k prime factors counted according to multiplicity, each prime factor being ≦ x. In an earlier paper [1] in this journal we posed the following problem. Let

Show the existence or non-existence of an integer K such that, if

then

We now show that such a K exists, and that in (2) there is strict inequality in each case.

In this paper we give necessary and sufficient conditions on a commutative semigroup in order that it should have a maximal homomorphic image of one of the following types: (1) groups, (2) semigroups which are unions of groups and (3) pseudoinvertible semigroups, i. e. semigroups having the property that some power of each element lies in a subgroup of the semigroup.

Let Q(S) denote the maximal right quotient semigroup of the semigroup S as defined in [4]. In this paper, we initiate a study of Q(S) when S is a semilattice of groups. A structure theorem for such semigroups is given by Theorem 4.11 of [2].

Consider the system of difference equations

in which the unknown x(t) is a complex m-vector, t is a real variable and a1, …, an are complex m × m matrices whose elements are functions of t, x(t), x(t+1), …, x(t+n – 1). A positive definite hermitian form V(x1x2, …, xn), with constant coefficients, is called a strong autonomous quadratic Lyapunov function (written strong AQLF) of (1) if there exists a constant K > 1 such that K2v(t+1) < v(t) for all non-zero solutions x(t)of (1), where v(t) = V(x(t), x(t+ 1), …, x(t+n —1)). The existence of a strong AQLF is a sufficient condition for the trivial solution x =0 of (1) to be globally asymptotically stable. It is a necessary condition only in the special case of an equation

The following result is well known in the theory of analytic functions; see [1].

Theorem A. Suppose that f(z) is an entire function of a complex variable z. Then f(z) satisfies the functional equation

where z = x + iy (x, y real), if and only if f(z) = aexp(sz), where a is an arbitrary complex constant and s is an arbitrary real constant.

Let (χ Σ, μ) be a probability space, so that X is a non-empty set, Σ is a Boolean a-algebra of subsets of X, and μ is a probability measure defined on Σ. If D Ε S is such that μ(D)≠0, then one traditionally associates with D a new probability measure μD, called the conditional probability measure determined by D, and defined by μD(E)= μ(D∩E)/μ(D), for all EΕΣ.

A semigroup S with 0 and 1 is termed completely right injective provided every right unitary S-system is injective. A necessary condition for a semigroup to be com-pletely right injective is given in [2]; namely, every right ideal is generated by an idempotent. An example in section 3 of this paper shows the existence of semigroups with 0 and 1 satisfying this condition which are not completely right injective. In [3], it is shown that the condition that every right and left ideal is generated by an idempotent is necessary and sufficient in the case that S is both completely right and left injective (called completely injective). Such a semigroup is an inverse semigroup with 0 whose idempotents are dually well-ordered.

Using the definition of a Riemann surface, as given for example by Ahlfors and Sario, one can prove that all Riemann surfaces are orientable. However by modifying their definition one can obtain structures on non-orientable surfaces. In fact nonorientable Riemann surfaces have been considered by Klein and Teichmüller amongst others. The problem we consider here is to look for the largest possible groups of automorphisms of compact non-orientable Riemann surfaces and we find that this throws light on the corresponding problem for orientable Riemann surfaces, which was first considered by Hurwitz [1]. He showed that the order of a group of automorphisms of a compact orientable Riemann surface of genus g cannot be bigger than 84(g – 1). This bound he knew to be attained because Klein had exhibited a surface of genus 3 which admitted PSL (2, 7) as its automorphism group, and the order of PSL(2, 7) is 168 = 84(3–1). More recently Macbeath [5, 3] and Lehner and Newman [2] have found infinite families of compact orientable surfaces for which the Hurwitz bound is attained, and in this paper we shall exhibit some new families.

Cooke [1] modified a technique used by Erdelyi and Sneddon [2] to solve triple integral equations of a certain type. In this paper, we extend this method to solve the quadruple integral equations

where F1, G2, F3 and G4 are prescribed functions of p and ψ(ξ) is to be determined. With no loss of generality we shall assume that G2(p)≡0, G4(p)≡0.

In structures having high strength-to-weight ratios such as those used in aerospace applications, the presence of mechanical imperfections can reduce the capability of the structure to perform as intended. Thus, it becomes essential to account for the localized intensification of the stresses around through or surface cracks, which might trigger fracture under applied loads. This type of study is currently receiving great research emphasis.