Cordially dedicated to Professor Alexander Weinstein, on the occasion of his Seventyseventh birthday, January 21, 1974.

This note contains the proof of an extension of Alexander Weinstein's mean value theorem for Generalized Axially Symmetric Potential Theory.

A homological characterization is given of when a twisted group ring relative to an automorphism of an arbitrary field has all of its simple right modules injective (= a right V-ring). This answers a question raised by Osofsky. A “Hilbert Theorem 90” type theorem determines the cardinality of the isomorphism classes of one-dimensional simple modules.

The structure of inverse semigroup extensions of one inverse semigroup R by any other is analyzed in the case where R is a semilattice. Both a representation and method of construction are given. A brief preliminary examination is made of a certain class of congruences, on inverse semigroups, which are intimately related to such extensions.

A class of near rings which generalizes the class of integral domains is defined. The definition of near domain is derived from the desirability of embedding near domains in near fields. The near domains presented here are shown to contain the D-rings of Berman and Siverman.

We prove some new theorems and reprove some old ones about finitely generated soluble groups and Lie algebras by a uniform method. Among the applications are Gruenberg's Theorem on Engel groups, for which we obtain a very short proof; and the Milnor and Wolf polynomial growth theorem. It is shown that a finitely generated soluble group with all 2-generator subgroups polycyclic is itself polycyclic, and that a finitely generated soluble Lie algebra, all of whose inner derivations are algebraic, is finite-dimensional. This last result enables us to give a partial answer to a question of Jacobson.

Let R be a metrization with distance xy of an open convex set D in the 2-dimensional real affine plane such that xy + yz = xz whenever x, y, z lie on an affine line with y between x and z and such that all the balls px ≤ ρ are compact. The study of such metrics, called open plane projective metrics falls under the topic of Hilbert's Problem IV of his famous mathematical problems. In this paper it is proved that if in R the sets of points equidistant from lines lie again on lines then D must be the entire affine plane and the distance must in fact be a norm. The paper contributes to and gives extensions of similar results proved earlier. The novel features of the present result are that in the space collinearity of points x, y, z is taken only as a sufficient condition for the equality xy + yz = xz. Consequently the solution encompasses all normed linear planes, that is, norms whose unit circles are not necessarily strictly convex are also admitted.

The proliferation of classifying spaces in recent years owes much to the theorem of Edgar H. Brown, Jr on the representability of homotopy functors. Since the theorem only gives a representation for functors defined on the category of spaces having the homotopy type of a CW complex, there is some interest in finding conditions under which the domain category may be enlarged. It appears that a version of the theorem holds for any small full subcategory of Htp, the category of topological spaces and homotopy classes of continuous maps, but that the resulting classifying space is generally intractable.

A semigroup S is called categorical if every ideal I of S has the property that abc є I (a, b, c є S) implies ab є I or be bc є I Necessary and sufficient conditions for an orthodox semigroup to be categorical are found and then used to characterize those bands which appear as an isomorphic copy of the band of idempotents of some orthodox categorical semigroup and to simplify the proof of a theorem of Mario Petrich. The structure of commutative categorical semigroups is found modulo the structure of abelian groups and categorical semilattices.

Using a theorem of Jacques Lewin, the zero divisor question is solved for group rings of supersolvable groups.

Let G be a locally compact abelian group and Ĝ be its dual group. For 1 ≤ p < ∞, let Ap (G) denote the set of all those functions in L1(G) whose Fourier transforms belong to Lp (Ĝ). Let M(Ap (G)) denote the set of all functions φ belonging to L∞(Ĝ) such that is Fourier transform of an L1-function on G whenever f belongs to Ap (G). For 1 ≤ p < q < ∞, we prove that Ap (G) Aq(G) provided G is nondiscrete. As an application of this result we prove that if G is an infinite compact abelian group and 1 ≤ p ≤ 4 then lp (Ĝ) M(Ap(G)), and if p > 4 then there exists ψ є lp (Ĝ) such that ψ does not belong to M(Ap (G)).

For a completely regular space X, G(X) denotes the free topological group on X in the sense of Graev. Graev proves the existence of G(X) by showing that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). It is natural to ask if the topology given by these two-sided invariant pseudo-metrics on G(X) is precisely the free topological group topology on G(X). If X has the discrete topology the answer is clearly in the affirmative. It is shown here that if X is not totally disconnected then the answer is always in the negative.

Conditions are given, under which a quasi-proximally continuous function is quasi-uniformly continuous, or a continuous function is quasi-proximally continuous. Thus, basic results on uniform and proximal continuity are extended and some new results are obtained. Three results in the literature are shown to be false.

A relationship is presented which greatly simplifies the application of the Schwarz-Christoffel formula in mapping a quadrilateral having one axis of symmetry. The relationship has been verified numerically, and it is hoped that its evident simplicity and usefulness will stimulate the development of an analytical proof.

In this paper we introduce the concept of a δθ-cover to generalize Aquaro's Theorem that every point countable open cover of a topological space such that every discrete closed family of sets is countable has a countable subcover. A δθ-cover of a space X is defined to be a family of open sets where each Vn covers X and for x є X there exists n such that Vn is of countable order at x. We replace point countable open cover by a δθ-cover in Aquaro's Theorem and also generalize the result of Worrell and Wicke that a θ-refinable countably compact space is compact and Jones′ result that ℵ1-compact Moore space is Lindelöf which was used to prove his classic result that a normal separable Moore space is metrizable, using the continuum hypothesis.

Let S be a topological semigroup with separately continuous multiplication and H a uniformly closed invariant subspace of LUC(S) (the space of left uniformly continuous bounded functions on S ) that contains the constants. It is shown that if H is left introverted and H admits a tight two-sided invariant mean m, then for each h ∈ H, m(h) is the unique constant function in the norm closed convex hull of the left orbit of h; consequently, H has a unique left invariant mean. (In fact, it is enough for H to admit a tight right invariant mean and a left invariant mean. ) For certain S, a similar result is obtained when H is a left compact-open introverted subspace of LCC(S) (the space of left compact-open continuous functions on S ).

We give the representation formula of a periodic function with period one in terms of its translated means on the unit interval. We also give an application of this formula to a boundary value problem.

Let d(G) denote the minimum of the cardinalities of the generating sets of the group G. Call a generating set of cardinality d(G) a minimal generating set for G. If A is a finitely generated nilpotent group, B a non-trivial finitely generated abelian group and A wr B is their (restricted, standard) wreath product, then it is proved (by explicitly constructing a minimal generating set for A wr B ) that d(AwrB) = max{l+d(A), d(A×B)} where A × B is their direct product.

The remark is that the recent work in this Bulletin dealing with the extension of a partial to a total strict order on a module over a partially ordered ring falls under standard order extension results for operator groups once it is noted that the added requirement of strictness just comes to injective operation of the positive scalars. These standard results are in turn generalized to a universal algebra setting.

Results of Tutte and of Anderson giving conditions for a simple graph G to have a perfect matching are generalized to give conditions for G to have a matching of defect d. A corollary to one of these results is a theorem of Berge on the size of a maximum matching in G.