It is proved that there exists a universal good inner function in the open unit polydisc Un, that is its non Euclidean translates are dense in the closed unit ball of H∞ (Un) and that there exists a universal function in the open unit ball Bn of Cn. These generalize Heins' result on universal Blaschke products.

1980 Mathematics subject classification (Amer. Math. Soc.): primary 32 A 10.

Let (Хζ,λ) be a σ-finite measure space, and let ϕ be a non-singular measurable transformation from X into itself. Then a composition transformation Cϕ on L2(λ) is defined by Cϕf = f ∘ ϕ. If Cϕ is a bounded operator, then it is called a composition operator. The space L2(λ) is said to admit compact composition operators if there exists a ϕ such that Cϕ is compact. This note is a report on the spaces which admit or which do not admit compact composition operators.

It is shown, using the author's construction for ‘labelled semilattices’, that every partially ordered set, in which every two elements have a common lower bound, is isomorphic to the partiallyordered set of 𝒥-classes of some completely semi-simple inverse semigroup.

1980 Mathematics subject classification (Amer. Math. Soc): primary 20 M 10, secondary 04 A 05, 08 A 05.

This paper deals with solid topologies and convergence structures on the vector-lattice CX (the set of all continuous real-valued functions on a space X): the closed ideals and locally convex topologies associated with such structures are studied in particular. The work stems from E. Hewitt's paper on bounded linear functionals, touches on the classical theorems of L. Nachbin, T. Shirota and others (determining when the topology of compact convergence is barrelled or bornological), and extends some recent results on the duality between x and CX.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 54 C 35; secondary 54 A 20.

The formal power series (fps) f(z) = Σ∞i=1ai zi is homozygous mod k if ai ≠ 0 implies i ≡ j mod k. This generalizes even and odd fps. If f is homozygous mod k then all iterates of f (fn = f ο fn−1) are also homozygous mod k, but the converse is false–there are many non-odd fps f for which f(f(z)) = z. It is shown that if f is not homozygous mod k but fn is homozygous, then fnr(z) = z for some r. If all coefficienrs ar real then, in fact, f(f(z)) = z.

Subject classification (Amer. Math Soc. (MOS) 1970): 39 A 05.

For certain problems in the weak convergence theory of probability measures on non-separable metric spaces it is convenient to consider measures defined on a σ-field B0 smaller than the Borel σ-field. A suitable theory has been developed by Dudley and Wichura. In this paper it is shown that some of the key results in that theory can be deduced directly from the better known weak convergence theory for Borel measures. This is achieved by a remetrization of the underlying space to make it separable. The σ-field B0 contains the new Borel σ-field and weak convergence in Dudley's sense becomes equivalent to convergence of restrictions of the measures to the latter σ-field.

1980 Mathematics Subject Classification (Amer. Math. Soc.): primary 60 B 10, 60 B 05.

N(G) denotes the near-ring of all continuous selfmaps of the topological group G (under composition and the pointwise induced operation) and N0(G) is the subnear-ring of N(G) consisting of all functions having the identity element of G fixed. It is known that if G is discrete then (a) N0(G) is simple and (b) N(G) is simple if and only if G is not of order 2. We begin a study of the ideal structure of these near-rings when G is a disconnected group.

Let G/G' be finitely generated and let G = B1 x A1 = B2 x A2 = … = Bi x Ai = … with each Bi isomorphic to a fixed group B which obeys the maximal condition for normal subgroups. Then the Ai represent only finitely many isomorphism classes. We give an example with B infinite cyclic, G/G' free abelian of infinite (countable) rank and such that G is decomposed as above with no two Ai isomorphic.

The concept of very weak local uniform rotundity (very WLUR) is introduced. It is shown that if a Banach space E is WLUR, very WLUR, or LUR then its dual E* is smooth, very smooth, or Fréchet differentiable, respectively, on a norm dense subset. This leads to examples of nonreflexive spaces which are Fréchet differentiable at every nonzero point of a (relatively) norm dense subset of the embedding of E** in the fourth dual. When E is reflexive, necessary and sufficient conditions for E and E* to be WLUR and LUR are given.

Subject classification (Amer. Math. Soc. (MOS) 1970): 46 B 99.

This paper is concerned with graphs of order n and diameter at most 3 having the property that by deleting any s or fewer vertices (edges) the resulting subgraphs (partial graphs) have duameter at most. λ. A graph satisfying the above constraints and having minimum number of edges is said to be extramal. A characterization of extremal graphs is presented for the case s = 1.

Subject classification (Amer. Math. Soc. (MOS) 1970): 05 C 35.

Let 〈fn≧0 be nonnegative real numbers with generating function f(x) = Σfnxn. Assume f(x) has the following properties: it has a finite nonzero radius of convergence x0 with its only singularity on the circle of convergence at x = x0 and f(x0) converges to y0; y = f(x) satisfies an analytic identity F(x, y) = 0 near (x0, y0); Fy(l) (x0, y0)= 0, 0 ≦ i < k and Fy(k) (x0, y0) ≠ 0. There are constants γ, a positive rational, and c such that fn~cx0−n n−(1 +ggr;). Furthermore, we show (i) in all cases how to determine γ and c from f(x) and (ii) in certain cases how to determine them from F(x, y).

Let k be an algebraically closed field of characteristic p, and G a finite group. Let M be an indecomposable kG-module with vertex V and source X, and let P be a Sylow p-subgroup of G containing V. Theorem: If dimkX is prime to p and if NG(V) is p-solvable, then the p-part of dimkM equals [P:V]; dimkX is prime to p if V is cyclic.

A veristy is called solutionally complete if any solvable system of algebraic equations over an algebra A in which has at most one solution in every extension of A in has the solution in A. A necessary and sufficient condition for solutional completeness is given which is a weaker form of the strong amalgamation property.

Subject classification (Amer. Math. Soc. (MOS) 1970): 08 A 15.

The classes of prestarlike functions Rα, α ≧ – 1, were studied recently by St. Ruscheweyh. The author generalizes and extends these classes. In particular the author obtains the radius of Ra+1 for the class Rα, α ≧ –1.

Let Zn be the factor ring of the integers mod n and t be a positive integer. In this paper some results are given on the structure of the semigroup of all mappings from Zn into Zn and on the structure of the group of all permutations on Zn, which, for any t elements, can be represented by a polynomial function. If n = ab and a, b are relatively prime, then this (semi)group is isomorphic to the direct product of the respective (semi)groups for a and b. Thus it is sufficient to consider only the case where n = pe, p being a prime. In this case it is proved, that the (semi)group is isomorphic to the wreath product of a certain sub(semi)group of the symmetric (semi)group on Zpe−1 by the symmetric (semi)group on Zp. Some remarks on these sub(semi)groups are given.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 B 99, 13 B 25.

The completions of certain nilpotent groups with respect to some ascending sequences of integral domains are constructed. These completions are generalizations of Lazard completions for the groups under consideration and they are Lie algebras over the first integral domain in the sequence. The construction is possible in particular for finite p-groups of exponent p and class < p.

It is well known that in any (v, b, r, k, λ) resolvable balanced incomplete block design that b≧ ν + r − l with equality if and only if the design is affine resolvable. In this paper, we show that a similar inequality holds for resolvable regular pairwise balanced designs ((ρ, λ)-designs) and we characterize those designs for which equality holds. From this characterization, we deduce certain results about block intersections in (ρ, λ)-designs.

Let R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

The paper examines the classes K1 and Γ1 of Hausdorff uniform spaces which are Gδ-closed in their Samuel compactifications, or completions. It is shown that the classes are epi-reflective, the reflections K1 and Γ are described, K1 and Γ1 are represented as epi-reflective hulls, membership in the classes is described by fixation of certain zero-set ultrafilters, and it is shown that k1 = Γ1 exactly on spaces without discrete sets of measurable power. The results include familiar facts about realcompact and topologically complete topological spaces and are closely connected with the theory of metric-fine uniform spaces.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 54 C 50, 54 E 15, 18 A 40; secondary 54 B 05, 54 B 10, 54 C 10, 54 C 30.

In a previous paper, ‘The concept of “largeness” in group theory’, a partial order was defined on the class of infinite groups, and this partial order was seen to give some precision to our intuitive notions of what it means for one infinite group to be ‘larger’ than another. The aim of this paper is to look more closely at groups which are ‘low down’ in this partial order, and to examine the interplay between properties of groups and finiteness conditions in group theory.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 20 E 15, 20 F 15, 20 K 10, 20 K 25; secondary 20 E 10.