This paper answers two questions posed by P. M. Neumann and K. Hickin on embeddings of infinite permutation groups. We first give some definitions that would make their questions understood.

In this paper we shall introduce a certain class of operators from a Banach lattice X into a Banach space B (see Definition 1) which is closely related to p-absolutely summing operators defined by Pietsch [8].

These operators, called positive p-summing, have already been considered in [9] in the case p = 1 (there they are called cone absolutely summing, c.a.s.) and in [1] by the author who found this space to be the space of boundary values of harmonic B-valued functions in .

Here we shall use these spaces and the space of majorizing operators to characterize the space of bounded p-variation measures and to endow the tensor product with a norm in order to get as its completion in this norm.

We will assume throughout this paper that polynomials are nonconstant. Let P be any complex polynomial and let p denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and multiplication is defined by fg = f ο P ο g for all f,g∈p. The near-ring p is referred to as a laminated near-ring and P is referred to as the laminating element or laminator. In [1] the problem was posed of determining Aut p the automorphism group of p. It was shown that exactly three infinite groups occur as automorphism groups of the laminated near-rings p and for each of the three groups those polynomials P were characterized such that Aut p is isomorphic to that particular group. The infinite groups turn out to be GL(2), the full linear group of all 2×2 nonsingular real matrices and two of its subgroups.

Let ξ be an irrational number with simple continued fraction expansion ξ= [a0;a1,a2,…], Pn/qn be its nth convergent, . The following two theorems were proved by Müller [9] and rediscovered by Bagemihl and McLaughlin [1]:

Theorem 1.For n>1,

We obtain a series representation for ζ(s) valid in Res> – k(<0). This representation is obtained by a sequence of regrouping of the series 1–2−s + 3−s–4−s + …( = (1—21−s)ζ(s), Re s > 0). We can obtain asymptotic relations like

as an application of our series representation for ζ(s).

Let G be a group and K a field. We shall denote by U(KG) the group of units of the group ring of G over K. Also, if X is a group, T(X) will denote the torsion subset of X, i.e., the set of all elements of finite order in X.

Group theoretical properties of U(KG) have been studied intensively in recent years and it has been found that some conditions about U(KG) imply that T = T(G) must be a subgroup of G and that every idempotent of KT must be central in KG.

We show that if X is a Banach space of infinite dimension and μh is a Hausdorff measure, where h is continuous, then there exists a measurable set K ⊂ X such that 0<μh(K)< + ∞. We also characterize the normed spaces in which the unit ball can be covered by a sequence of balls whose radii rn < 1 converge to zero as n → ∞.

A group G is called a Bn-group, if all its subnormal subgroups are of defect n. If all infinite subnormal subgroups of G have defect n, we say that G is an IBn group. DeGiovanni and Franciosi [2] have considered soluble IBn groups and characterized them.

In an earlier paper [4] we considered the question of whether an injective module E over a noncommutative ring R remains injective after localization with respect to a denominator set X in R. A related question is whether, given an essential extension N of an R-module M, the localization N[X–1] must be an essential extension of M[X–1]. In [1] it is shown that if R is left noetherian and X is central in R, then localization at X preserves both injectivity and essential extensions of left R-modules and, hence, preserves injective hulls and minimal injective resolutions.

A group G is said to be quasi-injective if, for each subgroup H of G and homomorphism θ:H→G, there is an endomorphism such that . It is of course well known that the category of groups does not possess non-trivial injective objects and so we consider groups satisfying the weaker condition of quasi-injectivity.

Let H be any group. We call a cardinal number r the rank r(H) of H if H can be generated by a generating system X with cardinal number r but not by a generating system Y with cardinal number s less than r. Let r(H) be the rank of H.

We call a generating system X of H a minimal generating system (M.G.S.) of H if X has the cardinal number r(H).

In this note we prove the following.

In [1], Calderón proved that, if u is a harmonic function on Rn × ]0, ∞[, and at each point ξ of a subset E of Rn, u is bounded in some cone with vertex (ξ, 0), then u has a nontangential limit at almost every point of E × {0}. The main result of this note is a stronger version of this theorem, in which the hypotheses remain unchanged but the nontangential limits in the conclusion are replaced by limits through the more general approach regions first considered by Nagel and Stein in [7].

We shall concern ourselves with the class of continuous, four-parameter, one-sided probability distributions which can be characterized by the probability density function (pdf) class

It depends on the four parameters: shift c ∈ R, scale b > 0, initial shape p < 1, and terminal shape β > 0. For p ≦ 0, the definition of f(x) can be completed by setting f(c) = β/bΓ(β−1)>0 if p = 0, and f(c) = 0 if p < 0. For 0 < p < 1, f(x) remains undefined at x = c; f(x)↑ + ∞ as x↓c.

The Bernstein polynomials are algebraic polynomial approximation operators which possess shape preserving properties. These polynomial operators have been extended to spline approximation operators, the Bernstein-Schoenberg spline approximation operators, which are also shape preserving like the Bernstein polynomials [8].

The lattice of varieties of bands was constructed in [1] by providing a simple system of invariants yielding a solution of the world problem for varieties of bands including a new system of inequivalent identities for these varieties. References [3] and [5] contain characterizations of varieties of bands determined by identities with up to three variables in terms of Green's relations and the functions figuring in a construction of a general band. In this construction, the band is expressed as a semilattice of rectangular bands and the multiplication is written in terms of functions among these rectangular band components and transformation semigroups on the corresponding left zero and right zero direct factors.

The Takai duality theorem has proved to be a fundamental tool in the theory of crossed products of C*-algebras. It was inspired by Takesaki's duality theorem for crossed products of von Neumann algebras [7], so it is not surprising that the original proof [6] depended heavily on spatial techniques. Here we shall prove Takai's theorem by exploiting the universal properties of crossed products.