We let X be an arbitrary non-empty set throughout. Many papers have been written describing the automorphisms of various transformation semigroups defined on X: total (Lyapin (1955), Magill (1967), Malcev (1952), Schreier (1936)), partial (Gluskin (1959), Magill (1967)), partial and 1–1 (Liber (1953)), partial and shifting at most a finite number of elements (Subov (1961a)). In all these cases the automoprhisms are shown to be “inner”, and using the fact this authors deduce that the automorphism group of the given transformation semigroup is isomorphic to the group gx of all permutations defined on X.

An arithmetic function f is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n. Furthermore an arithmetic function is said to be linear (or completely multiplicative) if f(1) = 1 and f(mn) = f(m)f(n) for all positive integers m and n.The Dirichlet convolution of two arithmetic functions f and g is defined by for all n∈Z+. Recall that the set of all multiplicative functions, denoted by M, with this operation is an abelian group.

Let X be an arbitrary set and θ a transformation of X. One may use θ to induce an associative operation in Jx, the set of all mappings of X to itself as follows: . We denote the resulting semigroup by {Jx;θ) Magill (1967) introduced this structure and it has been studied by Sullivan and by myself.

Denote by P the Euclidean plane with a rectangular Cartesian coordinate system where the x-axis is horizontal and the y-axis is vertical. An arc in P shall mean a simple continuous curve Λ:{t: 0 ≦ t < 1} P having the properties that limitt→1 Λ(t) exists and limitt→1Λ(t) ≠ Λ(t0) for 0 ≦t0 < 1. An arc at a pointζ in P shall be an arc Λ where limt→1 Λ(t) = ζ. If S is an arbitrary subset of the plane, ζ is termed an ambiguous point relative to S provided there are arcs Λ and Γ at ζ with Λ ⊆ S and Γ ⊆ P–S; such arcs are referred to as arcs of ambiguity at ζ. If A is a set of arcs we say a point ζ in P is accessible via A provided there is an arc at ζ which is an element of A. If B is also a collection of arcs, then A and B are said to be pointwise disjoint if whenever α∈A and β ∈ B, α ∩ β = Ø. The collections A and B are said to be terminally arcwise disjoint if whenever α ∈ A and β ∈ B and both α and β are arcs at a point ζ in P, then a ∩ β contains no arc at ζ. If S is a planar set, we let A(S) denote the set of all arcs contained in S. Note that if S ∩ T = Ø then A(S) and A(T) are pointwise disjoint collections of arcs.

Several authors have studied various types of rings of continuous functions on Tychonoff spaces and have used them to study various types of compactifications (See for example Hager (1969), Isbell (1958), Mrowka (1973), Steiner and Steiner (1970)). However many important results and properties pertaining the Stone-Čech compactification and the Hewitt realcompactification can be extended to a more general setting by considering appropriate lattices of sets, generalizing that of the lattice of zero sets in a Tychonoff space. This program was first considered by Wallman (1938) and Alexandroff (1940) and has more recently appeared in Alo and Shapiro (1970), Banachewski (1962), Brooks (1967), Frolik (1972), Sultan (to appear) and others.

The aim of this paper is to give solutions of some of the problems raised in Wiegold (1974). The main content is the proof of Theorems A and B following, which are improvements of Theorems 4.5 and 3.4 respectively of Wiegold (1974). All unexplained notation will be as in that paper, which would perhaps best be viewed as a preliminary to this one.

Rudin's synthesis method for investigating closed subalgebras of L1(G), where G is an infinite compact abelian group, is extended to the study of closed subalgebras in homogeneous Banach algebras and Segal algebras. Necessary and sufficient conditions are given for the synthesis to hold in certain classes of homogeneous Banach algebras and it is proved that in the Ap(G) algebras the synthesis holds for 1 ≦ p 2 but fails for Ap(T), 2 < p < ∞.

Recently, attention has been drawn to the problem of estimation of a k-variate probability density and its partial derivatives of various orders. Specifically, let X1, …, Xn be i.i.d. k-variate random variables with common density f wrt Lebesgue measure μ on the k-dimensional σ-field Bk. Parzen (1962) in the k = 1 case and Cacoullos (1966) in the k ≧ 1 case gave the asymptotic properties of a class of kernel estimates fn(x), x ∈ Rk, of f(x) based on X1, …, Xn. The asymptotic properties given in the above two papers concern consistency, asymp-totic unbiasedness, bounds for the mean squared error and asymptotic normality of fn. Also in the context of an empirical Bayes two-action problem, Johns and Van Ryzin (1972) introduced kernel estimates for f(x) and the derivative f'(x)for x∈R1 when f is a mixture of univariate exponential densities wrt Lebesgue measure on B1. They also investigated the asymptotic unbiasedness and themean squared error convergence properties of these estimates. Lin (1968) statedsome generalizations of the results of Johns and Van Ryzin, with applicationsto empirical Bayes decision problems.

A semigroup is said to be congruence-free if it has only two congruences, the identity congruence and the universal congruence. It is almost immediate that a congruence-free semigroup of order greater than two must either be simple or 0-simple. In this paper we describe the semilattices of congruence-free inverse semi-groups with zero. Further, congruence-free inverse semigroups with zero are characterized in terms of partial isomorphisms of their semilattices. A general discussion of congruence-free inverse semigroups, with and without zero, is given by Munn (to appear).

We show that a graph G is semi-stable at vertex v if and only if the set of vertices of G adjacent to v is fixed by the automorphism group of Gv, the subgraph of G obtained by deleting v and its incident edges. This result leads to a neat proof that regular graphs are semi-stable at each vertex. We then investigate stable regular graphs, concentrating mainly on stable vertex-transitive graphs. We conjecture that if G is a non-trivial vertex-transitive graph, then G is stable if and only if γ(G) contains a transposition, offering some evidence for its truth.

In this paper, some methods are developed for obtaining explicitly a basis for the integral closure of a class of coordinate rings of algebraic space curves.

The investigation of this problem was motivated by a need for examples of integrally closed rings with specified subrings with a view toward examining questions of unique factorization in them. The principal result, giving the elements to be adjoined to a ring of the form k[x1, …,xn] to obtain its integral closure, is limited to the rather special case of the coordinate ring of a space curve all of whose singularities are normal. But in numerous examples where the curve has nonnormal singularities, the same method, which is essentially a modification of the method of locally quadratic transformations, also gives the integral closure.

We consider linear recursive sequences of integers not all zero such that , where the aj are rational integers.

Throughout this note, let E and F be locally convex Hausdorff spaces over the real number field R. We denote real numbers by Greek letters. The sets of all continuous semi-norms on E and F will be denoted by P(E) and P(F) respectively, and A will always stand for an open subset of E.

Denote by Nn(α, β) the number of distinct fractions p/q, where 1 ≦ q ≦ n and α < p/q < β. Let .

Let G be a group and αn the mapping which takes every element of G to its nth power, where n is an integer. It is well known that if αn is an automorphism then G is Abelian in the cases n = -1,2, and 3. For any other integer n(≠ 0) there exists a non-Abelian group which admits αn as the identity automorphism. Indeed Miller (1929) has shown that if n ≠ 0, ±1, 2, 3 then there exist non-Abelian groups which admit αn as a non-trivial automorphism.