The purpose of this paper is to give conditions under which the absolute convergence of the subsequence of odd or of even approximants to a continued fraction implies convergence of the continued fraction. In § 2 we consider the problem in general, and in § 3 we impose a condition which gives absolute convergence of the odd or of the even part of the continued fraction and state conditions which imply convergence of the continued fraction.

Let X be a completely regular Hausdorff space, and let βX denote the Stone-Čech compactification of X. A point p ∈ βX is called a remote point of βX if p does not belong to the βX-closure of any discrete subspace of X. Remote points were first defined and studied by Fine and Gillman, who proved that if the continuum hypothesis is assumed then the set of remote points of βR((βQ) is dense in βR – R(βQ – Q ) (R denotes the space of reals, Q the space of rationals). Assuming the continuum hypothesis, Plank has proved that if X is a locally compact, non-compact, separable metric space without isolated points, then βX has a set of remote points that is dense in βX – X. Robinson has extended this result by dropping the assumption that X is separable.

We shall consider an irreducible, non-singular, totally geodesic holomorphic curve N in a compact quotient M = Γ\D of the unit ball D = {(z, w):|z|2 + |w|2 < 1} in C2 with the Kahler structure provided by the Bergman metric. The main result of this paper is an explicit construction of the harmonic form of type (1,1) which is dual to N. Our construction is as follows. Let p:D → Γ\D be the universal covering map. Choose a component D1 in the inverse image of N under p. The choice of D1 corresponds to choosing an embedding of the fundamental group of N into Γ. We denote the image by Γ1. Let π : D → D1 be the fiber bundle obtained by exponentiating the normal bundle of D1 in D. Let μ be the volume form of D1.

Let A = (ank) and x = {sn} (n,k = 0,1,2, … ) be a matrix and a sequence of complex numbers, respectively. We write formally

(1.1)

and say that the sequence x is summable A to the sum t or that the A matrix sums the sequence x to the value t if the series in (1.1) converges and

exists and equals t. We say that the matrix A is regular provided it sums every convergent sequence to its limit.

Let $X$ be a graph with vertex set $V$ and let $A$ be its adjacency matrix. If $E$ is the matrix representing orthogonal projection onto an eigenspace of $A$ with dimension $m$, then $E$ is positive semi-definite. Hence it is the Gram matrix of a set of $\left| V \right|$ vectors in ${{R}^{m}}$. We call the convex hull of a such a set of vectors an eigenpolytope of $X$. The connection between the properties of this polytope and the graph is strongest when $X$ is distance regular and, in this case, it is most natural to consider the eigenpolytope associated to the second largest eigenvalue of $A$. The main result of this paper is the characterisation of those distance regular graphs $X$ for which the 1-skeleton of this eigenpolytope is isomorphic to $X$.

Let G be a compact Lie group and a an action of G on a C*-algebra as *-automorphisms. Let denote the set of G-finite elements for this action, i.e., the set of those such that the orbit {αg(x):g ∊ G} spans a finite dimensional space. is a common core for all the *-derivations generating one-parameter subgroups of the action α. Now let δ be a *-derivation with domain such that Let us pose the following two problems:

1. Is δ closable, and is the closure of δ the generator of a strongly continuous one-parameter group of *-automorphisms?

2. If is simple or prime, under what conditions does δ have a decomposition

where is the generator of a one-parameter subgroup of α(G) and is a bounded, or approximately bounded derivation?

Our aim in this paper is to obtain some theorems concerning spaces of analytic functions on a finite open Riemann surface R which extend known results for the disc △ = {|z| < 1}. Suppose that R has a smooth boundary bR consisting of t closed curves, and that the interior genus of R is s. The first Betti number of R is then r = 2s + t — 1.

The theorem that every properly primitive binary quadratic form is capable of representing infinitely many prime numbers was first proved completely by H. Weber (5). The purpose of this paper is to give an elementary proof of the case where the form is ax2 + 2bxy + cy2, with a > 0, (a, 2b, c) = 1, and D = b2 — ac not a square. The cases where the form is ax2 + bxy + cy2 with b odd, and the case where the form is ax2+ 2bxy + cy2 with D a square, can be settled very simply once the first case is taken care of, and this is done in a page and a half in the Weber paper.

In [1], Bachmuth and Mochizuki conjecture, by analogy with a celebrated result of Tits on linear groups [8], that a finitely generated group of automorphisms of a finitely generated soluble group either contains a soluble subgroup of finite index (which may of course be taken to be normal) or contains a non-abelian free subgroup. They point out that their conjecture holds for nilpotent-by-abelian groups and in some other cases.

At the beginning of the chapter on induced representations in the treatise of Curtis and Reiner [8] on representation theory, they write “Most of the results have not yet found suitable generalization to rings with minimum condition or finite dimensional algebras, …”. The purpose of this paper is to indicate how some of the more basic theorems concerning induced representations can, in fact, be generalized to rings and algebras. In most cases we can do this by bringing together known results, so that in this sense this paper does not contain substantially new results.

Transferable and sharply transferable lattices were defined and characterized in [1]. Finite sublattices of free lattices were studied in [2] and a characterization of them given in [3]. In this paper, we will show that the class of finite sharply transferable lattices coincides with the class of finite sublattices of free lattices.

This paper is concerned with topological spaces whose continuous maps into a given space R are constant, as well as with spaces having this property locally. We call these spaces R-monolithic and locally R-monolithic, respectively.

Spaces with such properties have been considered in [1], [5]-[7], [10], [11], [22], [28], [31], where with the exception of [10], the given space R is the set of real-numbers with the usual topology. Obviously, for a countable space, connectedness is equivalent to the property that every continuous real-valued map is constant. Countable connected (locally connected) spaces have been constructed in papers [2]-[4], [8], [9], [11]-[21], [23]-[26], [30].

A quasigroup is an ordered pair (Q, •), where Q is a set and (•) is a binary operation on Q such that the equations ax — b and ya — b are uniquely solvable for every pair of elements a,b in Q. It is well-known (see, for example, [11]) that the multiplication table of a quasigroup defines a Latinsquare, that is, a Latin square can be viewed as the multiplication table of a quasigroup with the headline and sideline removed. We are concerned mainly with finite quasigroups in this paper. A quasigroup (Q, •) is called idempotent if the identity x2 = x holds for all x in Q.

We use the following notation throughout:

.

In R. L. Wilder's book [2] the open and closed generalized manifolds are extensively studied. However, no study is made of the generalized manifold with boundary nor is a definition of such a space given except in the case of the generalized closed n-cell. A definition of a generalized manifold with boundary was given by the author in his paper [1]. Before undertaking the study of further properties of these manifolds it seems appropriate to characterize the manifolds with boundary in terms of the open and closed manifolds of Wilder. It is to that purpose that this paper is directed and in particular the generalized closed n-cell of Wilder is characterized as a special manifold with boundary.

In § 1 we give a characterization of a lattice L that is freely α-generated by a given partially ordered set P. In § 2 we obtain a representation of an element of such a lattice as a sum (product) of additively (multiplicatively) irreducible elements which, although not unique, has some of the desirable features of the canonical representation, in Whitman (2), of an element of a free lattice. The usefulness of this representation is illustrated in § 3, where some further arithmetic properties of these lattices are derived.

We use + and . for the binary operations of lattice addition and multiplication, and Σ and II for the corresponding operations on arbitrary sets and sequences of lattice elements. In other respects the terminology will be the same as in Crawley and Dean (1).

Letƒ(x) be a measurable function denned in the interval (a, b), and let If the limit of exists and is finite at the point x, as h → 0, it is called the wth generalized Riemann derivative of ƒ(x) at the point x, Dnƒ(x).

0. Introduction. Simplicial quadratic forms (cf. Definition 1.4), and various equivalent forms, have occasionally been studied in geometry [8], and in number theory [9], [10], in connection with the extremal properties of integral quadratic forms. Our investigations, which employ simple techniques from graph theory and geometry, partly continue both those of Coxeter [5], who introduced the graphs described in Section 1, and Vinberg [20], [21], who described an algorithm for determining a fundamental region for a discrete group acting on spherical, Euclidean, or hyperbolic space. After a preliminary discussion of reflexible forms and the Caley-Klein model for (n − 1)-space (1.2), we define a simplicial form and its graph. Having enumerated them completely, we turn in Section 2 to their equivalence, which is related to a geometric dissection. The unit group for each simplicial form can then be determined from Theorem 3.7.

I wish to thank Professor H. S. M. Coxeter for many helpful ideas, and Professor G. Maxwell and the referee for suggesting numerous improvements.

It is shown here that, for $n\ge 2$, the $n$-torus is the only $n$-dimensional compact euclidean space-form which can admit a regular or chiral tessellation. Further, such a tessellation can only be chiral if $n\,=\,2$.

Let A = (Aij)l≦ij≦l be a Cartan matrix, i.e., Aii = 2 for all i and Aij is an integer ≦ 0 for i ≠ j, with Aij = 0 if Aji = 0. The size l of A is called its rank, for Lie-theoretic reasons, and may be larger than its matrix rank. We associate to A its Dynkin diagram, with vertices 1, 2, … , l, with AijAji lines joining i to j, and with an arrow pointing from i to j if Aij/Aji < 1, i.e., pointing toward the shorter root (see below). The Cartan matrix A is indecomposable if its diagram is connected, and symmetrizable if there exist positive rational numbers q1 … , ql with

qiAij = qjAji for all i and j.

Symmetrizability is automatic if the diagram contains no cycle.