We begin with the following definitions.

Let f(z) be regular near z=0 and let f(0)=0, f'(0)≠0. Let α and λ be two real numbers such that |α|<π/2 and 0≤λ1. Then Rα,λ is the largest value of r such that the following conditions are satisfied for |z|<r

An m-permutation of n, repetitions allowed, is an m-sequence

(1)

A w-rise is a pair (ei, ei+1) such that ei+1-ei≥w>0. In this note we find an expression for Tk, w(n, m), the number of m-sequences having precisely k w-rises. The case w = 1 is given in [1] [2]. Also, when w = 1 we give the number when each of the integers 1, 2, …, r must appear at least once.

An order type α is said to be weakly homogeneous (ℵ0 homogeneous) if for any x1 < x2 and y1 < y2 there exists an order preserving bijection f on α such that f(xi)= y i for i = 1, 2. The reverse of an order type a is denoted, as usual, by α*. We say that α is order invertible if α*≤α. J. Q. Longyear [5] has asked whether for a weakly homogeneous order type α such that no (non-trivial) interval of α is order invertible we may deduce that every interval of α contains a copy of ηω1 or (ηω1)*.

The main result of this paper is little more than a juxtaposition of a remark in [Leroux] (§5, part A), with a theorem of [Banaschewski]. However, the results are individually little known and their juxtaposition not at all. More explicitly, Leroux remarks, in somewhat different form,

Let the category X be complete and cocomplete, and have a set of cogenerators. Then a necessary and sufficient condition that there exist infective effacements (with respect to the class of regular monomorphisms) is that X be coregular and satisfy, in addition, condition COEX 4*.

Let (Ω, α, μ) be a σ-finite measure space. By Lp(Ω, α, μ) or Lp for short we denote the usual Banach space of pth power μ-integrable functions on Ω if 1≤p<+ ∞ and μ-essentially bounded functions on Ω, if p= +∞. In section (2) we characterize conditional expectation, by a method different than those used previously. Modulus of a given contraction is discussed in section (3). If the given contraction has a fixed point, then its modulus has a simple form (theorem 3.2).

Several years ago, C. Long wrote two papers ([3], [4]) that related to F(n) the number of ordered factorizations of n. The second of these papers [4] was devoted entirely to a discussion of conjectured formula for F(n).

In this note our principal interest is in using nets to give spaces of non-absolutely convergent integrals as extensions of the spaces of absolutely convergent Riemann and Lebesgue integrals. For this purpose we develop a general theory of extensions, by nets, of functions defined on the open intervals with closures in the complement of a fixed closed set, the nets being directed by inclusion for finite disjoint collections of such intervals. Two cases are considered leading to open extension (OE-) and conditional open extension (COE-) nets, the latter being subnets of the former. Necessary and sufficient conditions for the convergence of the OE- and COE-nets are given, those for the COE-nets being similar to conditions that arise in the definition of the restricted Denjoy integral. Properties of inner continuity, weak additivity and the existence of a continuous integral are defined and studied. These relate to the more specialized nets that are suitable for the extension of integrals.

The purpose of this paper is to extend to almost periodic (a.p.) solutions a stability result on the periodic solutions of the autonomous equation

x′ = F(x),

(cf. Coppel [1], p. 82 or Coddington and Levinson [2], p. 323.)

The twisted wreath product of two groups was first defined by B. H. Neumann ([1]) who used this construction to present a group-theoretic proof of a theorem due to Auslander and Lyndon. In this paper we present a complete characterization of the direct product decompositions of a restricted twisted wreath product of two groups A and B provided this product is not simply a semi-direct product of A by B.

In this note we give a generalization of a spectral representation theorem for self adjoint operators in a Hilbert space recently obtained by the author [1]. Our interest here is to develop corresponding results for a k-tuple (T1, …, Tk) of commuting self adjoint operators.

The relationship between properties of the generalized inverse of A, A†, and of the adjoint of A, A*, are studied. The property that A†A and AA† commute, called (E4), is investigated. (E4) generalizes the property of A being EPr. A canonical form and a formula for A† are given if a matrix A is (E4). Results are in a Hilbert space setting whenever possible. Examples are given.

Let X and Y be isomorphic normed linear spaces and let i ℒ denote the set of isomorphisms from X to Y (i.e. bounded linear mappings from X onto Y which have bounded inverses).

It is known that a Hilbert space, L2(μij), can be constructed from an n × n positive matrix measure (μij), [5, pp. 1337–1346]. The aim of this note is to show that Banach spaces, corresponding to the usual LP spaces, can also be constructed and to investigate their properties.

Let Fn be the free group on {ai:i ∊ ℤ n} where the set of congruence classes mod n is used as an index set for the generators. The permutation (1, 2, 3, …, n) of ℤn induces an automorphism θ of Fn by permuting the subscripts of the generators. Suppose w is a word in Fn and let N(w) denote the normal closure of {wθi-1:l ≤i≤n}. Define the group Gn(w) by Gn(w)=Fn/N(w) and call wdi-1=l the relation (i) of Gn(w).

In 1967, Powers [7] proved that there exists a one-parameter family of pairwise non-isomorphic hyperfinite factors of type III. Powers′ result on hyperfinite factors has been extended by Araki and Woods [1]. Connes [4], and Williams [11], with different proofs, showed that there exists a continuous family of mutually non-isomorphic non-hyperfinite factors of type III.

Recently, Dunham has given examples for 1-parameter and 2-parameter Chebyshev sets which are not suns. In this note 2-parameter sets with these properties are described.

When studying the old problem whether Chebyshev sets are always convex, Klee [10] introduced certain sets which were called suns by Efimov and Stechkin [7]. Recently, in two shorts notes Dunham [4, 5] has given examples of 1-parameter- and 2-parameter-sets which are Chebyshev sets but not suns (cf. also [3]). The examples refer to Chebyshev sets in containing an isolated point.

Let T, P denote two continuous operators from E into F, where E and F are locally convex spaces. It is proved by L. Schwartz [8] and G. KÖthe [6] that if E and F are Fréchet spaces, T is a ϕ_-operator and P a compact operator, then T+P is a ϕ_-operator.

Throughout, topological spaces are assumed to be completely regular. C(X) (resp. C*(X)) will denote the ring of all (resp. all bounded) continuous real-valued functions. βX will denote the Stone-Cech compactification of X. In [2], Nel and Riorden defined C≠(X) to be the set of all f ∊ C(X) such that M(f) is real in the residue class ring C(X)/M for every maximal ideal M in C(X). C≠(X) is a subalgebra as well as a sublattice of C*(X).

In this paper the following result is proved.

Let n > 0 and k ≥ 0 be integers with n — 2k ≥ 1. Given any n — k integers

there is a non-empty subset of indices I ⊂ {1, 2,…, n — k} such that the sum Σi∊I ≡ 0(mod n) if at most n — 2k of the integers (1) lie in the same residue class modulo n.

The result is best possible if n ≥ 3k — 2 in the sense that if "at most n — 2k" is replaced by "at most n — 2k + 1" the result becomes false. This can be seen by taking aj = 1 for 1 ≤ j ≤ n — 2k + 1 and aj = 2 for n —2k + 2 ≤ j ≤ n — k, noting that the number of 2's here is n — k — (n — 2k + 1)= k — 1 ≤ n — 2k + 1.

Let Rp(p≥2) denote p-dimensional Euclidean space, D the half space defined by {P = (x1, x2, …, xp) ∊ Rp: xp > 0} and ∂D the frontier of D in Rp. The Martin boundary (see [2]) of D can be identified with ∂D∪{∞}.