Let {sn(z)} be a given sequence of linear fractional transformations (or simply l.f.t.'s) of the form

1.1

and let

1.2

The sequence of l.f.t.'s {Sn(z)} is called a continued fraction generating sequence (or simply a c.f.g. sequence).

Let H be a subgroup of Sn and let A be an n-square matrix over a field ƒ. Following Schur (7) we define the generalized matrix function dH(A) by

In this paper, the graphs under consideration may have multiple edges but they do not have loops. We enumerate the number N[H: n, p] of topologically distinct graphs with n vertices and p edges whose automorphism group is the permutation group H. As in (5), this enumeration is considered in the context of the theory of permutation representations of finite groups. We begin with some definitions and notation.

Given a set of vertices which have each been assigned one of the colours C1, C2, …, Cm, with nj vertices Cj, a formula is derived for the number of oriented trees on these vertices, having a designated root, and subject to any number of restrictions of the form “no arc goes from a vertex of colour Ci to a vertex of colour Cj”. The formula is based on a combinatorial construction which defines a correspondence between such trees and certain sequences.

Let be any topological space. In this paper, we show that there is a unique regular topology on X which is coarser than such that if Y is any regular space, the continuous maps are the same for and . We shall call the regular topology associated with and the regular space associated with .

Un espace topologique est dit quasi-compact s'il vérifie l'axiome de Borel-Lebesgue sans être nécessairement séparé. Un espace topologique est dit localement quasi-compact si chaque point admet un système fondamental de voisinages quasi-compacts.

Let p be a prime, and let S be a Sylow p-subgroup of a finite group G. J. Thompson (13; 14) has introduced a characteristic subgroup JR(S) and has proved the following results:

(1.1) Suppose that p is odd. Then G has a normal p-complement if and only if C(Z(S)) and N(JR(S)) have normal p-complements.

B. Z. Vulih has shown (13) how an essentially unique intrinsic multiplication can be defined in a Dedekind complete vector lattice L having a weak order unit. Since this work is available only in Russian, a brief outline is given in § 2 (cf. also the review by E. Hewitt (4), and for details, consult (13) or (11)).

Let X be a compact metric space with metric d. A complex-valued function ƒ on X is said to satisfy a Lipschitz condition if, for all points x and y of X, there exists a constant K such that

The smallest constant for which the above inequality holds is called the Lipschitz constant for ƒ and is denoted by ||ƒ||d, that is,

Suppose that G1 and G2 are two locally compact Hausdorff groups with identity elements e and e’ and with respective left Haar measures dx and dy. Let 1 ≦ p ≦ ∞, and Lp(Gi) be the usual Lebesgue space over Gi formed relative to left Haar measure on Gi. We denote by M(Gi) the space of Radon measures, and by Mbd(Gi) the space of bounded Radon measures on Gi. If a ϵ Gi we write ϵa for the Dirac measure at the point a. Cc(Gi) will denote the space of continuous, complex-valued functions on Gi with compact supports, whilst Cc+ (Gi) will denote that subset of Cc(Gi) consisting of those functions which are real-valued and non-negative.

A function ƒ(z) is said to be p-valent in a region if it is regular in if the equation

1

has p distinct roots in for some particular w0 , and if for each complex w0 , equation (1) does not have more than p roots in . The function ƒ(z) is also said to have valence p in . In the case when p = 1, the function is said to be univalent in .

In two previous papers (see 4; 5) O. T. O'Meara and I investigated the problem of generating the integral orthogonal group of a quadratic form by symmetries in the case where the underlying ring of integers was the integers of a dyadic local field of characteristic not 2. In this paper, the investigation is concerned with a local field of characteristic 2. As in (5), only the unimodular case is treated. As in (4) and (5), groups S(L), Xh(L), and O(L) are introduced for a unimodular lattice L and the relationship between S(L) and O(L) studied. As in the previously cited papers, generation by symmetries means that S(L) = O(L). The following result is obtained.

Except for using “algebra” rather than “ring” and “compact” rather than “bicompact”, we adopt the terminology used in (6). Every symmetric algebra, Ai, will have an identity, ei All representations will be cyclic and symmetric. Sets of functionals will carry the relative weak* topology.

In 1964, L. J. Herbst (3) introduced the generalized spectral density Function

1

for a non-stationary process {X(t)} denned by

1

where {η(t)} is a real Gaussian stationary process of discrete parameter and independent variates, the (a;)'s and (σj)'s being constants, the latter, which are ordered in time, having their moduli less than a positive number M.

Several years ago Kelley (2) showed that if X is a metric continuum then S(X), the space of non-null, closed subsets of X, and C(X), the space of non-null, closed, connected subsets of X, with the Vietoris topology, are arcwise connected continua. He further showed that S(X) is acyclic. In this note we extend these results to non-metric continua.

A sequence (xn) of real numbers is said to be well-distributed modulo 1 (abbreviated w.d.) if for each subinterval I = [a, b] of [0, 1] we have that

where χI is the characteristic function of I modulo 1. A sequence (r n ) of positive numbers is lacunary if

It is assumed that all topological spaces discussed in this paper are T1 spaces. A function ƒ mapping a topological space X into itself is a closed function if ƒ[H] is closed for each closed subset H of S. The semigroup, under composition, of all closed functions mapping X into X is denoted by Γ(X). These were among the semigroups under consideration in (4).

In this work we deal with algebraic properties of some fields of functions meromorphic in the complex plane with a view to determining the possible isomorphisms between two such fields. Interest in problems of this type began with a paper by Bers (2), in which it was shown that the algebraic structure of the ring of functions analytic on a plane region determines the conformai structure of the region to within conformai or anti-conformal equivalence, and this result was later extended to arbitrary non-compact Riemann surfaces by Nakai (7).

This paper is a sequel to (1), to which the reader is referred for definitions and known results. As before, £ is a compact Hausdorff space and C+(E) is the semi-algebra of all continuous non-negative functions defined on E.

Let μ be a vector measure (countably additive set function with values in a Banach space) on a field. If μ is of bounded variation, it extends to a vector measure on the generated σ-field (2; 5; 8). Arsene and Strătilă (1) have obtained a result, which when specialized somewhat in form and context, reads as follows: “A vector measure on a field, majorized in norm by a positive, finite, subadditive increasing set function defined on the generated σ-field, extends to a vector measure on the generated σ-field”.