Let k be a fixed integer ≧2. A positive integer n is called unitarily k-free, if the multiplicity of each prime divisor of n is not a multiple of k; or equivalently, if n is not divisible unitarily by the kth power of any integer > 1. By a unitary divisor, we mean as usual a divisor d > 0 of n such that (d,(n/d)) = 1. The integer 1 is also considered to be unitarily k-free. These integers were first defined by Cohen (1961; § 1). Let Q*k denote the set of unitarily k-free integers. When k = 2, the set Q*2 coincides with the set Q* of exponentially odd integers (that is, integers in whose canonical representation each exponent is odd) discussed by Cohen himself in an earlier paper (1961; § 1 and § 6). Let x denote a real variable 1 and let Q*k denote the number of unitarily k-free integers ≦ x. Cohen (1961; Theorem 3.2) established by purely elementary methods that , where , the product being extended over all primes p and ζ(k) denotes the Riemann Zeta function. In the same paper Cohen (1961; Theorem 4.2) improved the order estimate of the error term in (1.1) to O(x1/k), by making use of the properties of real Dirichiet series. Later, he (Cohen; 1964) proved the same result by purely elementary methods eliminating the use of Dirichlet series.

It is shown that every compact nonconnected semigroup (semiring) which has commuting congruences, has a nontrivial continous homomorphic image which is iseomorphic to a direct product of finite congruence free semigroups (semirings). (This extends parts of earlier work by Kaplansky (1947) on compact rings.) It is also shown that there is a possibly finer representation but onto a product of congruence free semigroups (semirings) known only to be compact Hausdorff. A number of the techniques used evolve from work of Professor Wallace, who retired in mid-1973, and to whom this paper is dedicated.

A simple diffusion-convection heat transfer model is formulated which leads to an axially symmetric partial differential equation. The equation is shown to be closely related to a second one which is adjoint to the original equation in one variable can and be interpreted as a description of another diffusion-convection model. Fundamental solutions of the original equation are constructed and interpreted with reference to both models. Some boundary value problems are solved in series form and integral representations of the solutions are also given. The boundary value problems are shown to be equivalent to an integral equation and the correspondence between the two formulations is understood in terms of the two diffusion-convection problems. A Péclet number is defined in one of the boundary value problems and the behaviour of the solutions is studied for large and small values of this parameter.

We construct all semirings with a given completely simple additive semiroup ℳ(G, I, ∧, P) by means of an associative and distributive multiplication on G, and associative multiplications on I and ∧ satisfying certain conditions.

We consider the problem of the existence of a left invariant measure in a class of topological semigroups. Several authors have considered this and related problems on semigroups satisfying similar conditions, but the invariance they considered is right invariance. This paper is different in that it deals with left invariance.

A linear mapping D of the algebra of polynomial functions P[0, 1] into the algebra of all continuous complex-valued functions C[0,1] is called a derivation provided D(fg) = fD(g) + gD(f) for all polynomials f and g. The derivations of P[0, 1] into C[0,1] are easily seen to be all mappings of the form Dw where w is a continuous function on [0, 1] and Dw (f) = wf' (f' denotes the ordinary derivative of f). In fact, w = D(x) where x is the coordinate function. Let Dw be such a derivation, and let ∥ · ∥ denote the supremum norm on C[0,1]. Then Dw gives rise to an algebra norm ∥ · ∥w on P[0,1] denned by .

The tensor product G ⊕ H of graphs G and H is the graph with point set V(G) × V(H) where (υ1, ν1) adj (υ2, ν2) if, and only if, u1 adj υ2 and ν1 adj ν2. We obtain a characterization of graphs of the form G ⊕ H where G or H is K2.

The purpose of this paper is an extension of a theorem of Hughes (1973). He showed: Let R be a ring which has a right artinian right quotient ring and let G be a group which has a (transfinite) ascending normal series with each factor either finite or cyclic, but only a finite number of finite factors. Then the group ring RG has a right artinian right quotient ring.

Banaschewski (1963) and Frink (1964) generalized the compactification procedure of Wallman to obtain Hausdorff compactifications of Tychonoff spaces. Numerous papers have been devoted to the problem whether all Hausdorff compactifications may be obtained in this way, and for many classes of compactifications an affirmative answer has been given. This note is a contribution in this direction. We show that if a (Hausdorff) compactification αX of X is the quotient space of a Wallman compactification γX in such a way that the set of multiple points of αX with respect to γX is not too large, then αX too is a Wallman compactification. The results are generalizations of earlier results of Steiner and Steiner (1968) and by the author (1966) for the special case that γX is the Stone Čech-compactification.

Let be an infinite r.e. repère, an infinite dimensional r.e. space such that ≦ L(). A condition is derived that is both necessary and sufficient for the existence of an infinite subset β ⊂ such that L(β)∪ is not an α-space. Examples which satisfy this condition are exhibited, proving that the class of α-spaces is not closed under intersections.

Fitting classes of finite solvable groups were first considered by Fischer, who with Gäschutz and Hartley (1967) showed in that in each finite solvable group there is a unique conjugacy class of “-injectors”, for a Fitting class. In general the behaviour of Fitting classes and injectors seems somewhat mysterious and hard to determine. This is in contrast to the situation for saturated formations and -projectors of finite solvable groups which, because of the equivalence saturated formations and locally defined formations, can be studied in a much more detailed way. However for those Fitting classes that are “locally defined” the theory of -injectors can be made more explicit by considering various centralizers involving the local definition of , giving results analogous to some of those concerning locally defined formations. Particular attention will be given to the subgroup B() defined by where the set {(p)} of Fitting classes locally defines , and the Sp are the Sylow p-subgroups associated with a given Sylow system − B() plays a role very much like that of Graddon's -reducer in Graddon (1971). An -injector of B() is an -injector of G, and for certain simple B() is an -injector of G.

The notion of a recursive density type (R.D.T.) was introduced by Medvedev and developed by Pavlova (1961). More recently the algebra of R.D.T.'s was initiated by Gonshor and Rice (1969). The R.D.T.'s are equivalence classes of sets of integers, similar in many respects to the R.E.T.'s. They may both be thought of as effective analogues of the cardinal numbers. While the equivalence relationfor R.E.T.'s is defined in terms of partial recursive functions, that for R.D.T.'s may be characterized in terms of recursively bounded partial functions (see 4.22a).

Let Q, R be rational numbers and real numbers respectively. We use V(F) and W(F) to denote finite dimensional inner product spaces over F. Given V(Q), we use V(R) for the smallest inner space over R containing V(Q). It is known that an R-homomorphism of V(R) to W(R) is continous. We prove that if a Q-homomorphism f: V(R) → W(R), then f is dispersive, i.e., given any v0 ∈ V(Q) and ε > 0, the image set f[D(v0, ε)], where D(v0, ε) = [v: v ∈ V(Q), ¦v – v0¦ < ε], is not bounded. It is also shown that some Q-homomorphism f: V(Q) → W(Q) can be explosive in the sense that for any v0 ∈ V(Q) and ε > 0, the set f[D[v0, ε)] is dense in W(Q). As a particular case of dispersive and explosive Q-homomorphisms, we show that the algebraic number field isomorphism f: Q(a) → Q(β), where f(a) = β and α ≠ β or βmacr; (βmacr; being complex conjugates of β) is explosive.

Let G be an abelian lattice ordered group (an l-group). If G is, in fact, totally ordered, we say that G is an 0–group. A subgroup and a sublattice of G is an l-subgroup. A subgroup C of G is called convex if 0 ≦ g ≦ c ∈ C and g ∈ G imply g ∈ C, C is an l-ideal if C is a convex l-subgroup of G. If C is an l-ideal of G, then G/C is also an l-group under the canonical ordering inherited from G. If, in fact, G/C is an 0–group, then C is said to be a prime subgroup of G.

Holton (1973) introduced the following concept. A graph G is semi-stable if there exists a point v in G for which Γ(Gv) = Γ(G)v: where Γ(G) is the automorphism group of G, Gv is the graph G with v deleted and Γ(G)v is the subgroup of Γ(G) that fixes v. We say G is semi-stable at v. A partial stabilising sequence in Gs1, v2,…vk of its points such that Γ(G)v1v2…vi ═ Γ(Gv1v2…v1) for i = 1,2,…,k. If there exists a partial stabilising sequence in G for which k equals the number of points of G then G is said to be stable (Holton (1973a)). Most natation and terminology in what follows is explained in Harary (1969).

For any collection of closed subspaces of a complex Hilbert space the set of bounded operators that leave invariant all the members of the collection is a weakly-closed algebra. The class of such algebras is precisely the class of reflexive algebras as defined for example in Radjavi and Rosenthal (1969) and contains the class of von Neumann algebras.In this paper we consider the problem of when such algebras are finitely generated as weakly-closed algebras. It is to be hoped that analysis of this problem may shed some light on the famous unsolved problem of whether every von Neumann algebra on a separable Hilbert space is finitely generated. The case where the underlying space is separable and the collection of subspaces is totally ordered is dealt with in Longstaff (1974). In the present paper the result of Longstaff (1974) is generalized to the case of a direct product of countably many totally ordered collections each on a separable space. Also a method of obtaining non-finitely generated reflexive algebras is given.

Our purpose is to improve the Gale-type multifunction Ascoli theorem of Mancuso (1971; page 470). This latter supposes the range space to be normal and Hausdorff, and therefore does not contain Gale's theorem (1950; page 304). To obtain a multifunction theorem containing Gale's theorem (also Mancuso's theorem), we return to Gale's essential hypotheses. Thus, we assume the regularity of the range space in the sufficiency direction, and, in the necessity direction, weassume the domain to be a k-space and the range to be a regular Hausdorff space. We dispense with the “point-like” condition imposed by Mancuso. Unexplained terminology and notation is that of Mancusopos;s paper.

Two generalized mean value theorems, for functions with values in a linear locally convex topological space, are proved, as consequences of two theorems for real valued functions of real variable.

Burkill (1957) introduced extended Riemann-Stieltjes integrals of the form ∫baƒ(x)dkg(x)/dxk-1 to provide an alternative approach to the theory of distributions. We will denote Burkill's integral by (B)∫baƒ(x)dkg(x)/dxk-1 Burkill's paper (1957) partially motivated the study of the Riemann-Stieltjes integral in Russell (1970). He showed that the two integrals are not equivalent.

We sometimes think of medial (that is, infinite Dedekind-finite) sets as being “small” infinite sets. Medial cardinals can be defined as those cardinals that are incomparable to ℵℴ; hence we tend to think of them as being spread out on a plane “just above” the natural numbers, which seems to lend support to the view expressed above that medial sets are “small”.