A well-known theorem of Jacobson (1) states that if every element x of a ring R satisfies xn(x) = x where n(x) > 1 is an integer, then R is commutative. A series of generalizations of this theorem have been proved by Herstein (2; 3; 4; 5; 6), his last result in this direction (6) being that a ring R is commutative provided every commutator u of R satisfies un(u) = u. We now define a γ-ring to be a ring R in which un(u) — u is central for every commutator u of R (where n(u) > 1 is an integer). In the present paper we verify the following conjecture of Herstein: every commutator of a γ-ring is central.

Let A, B, and X be n-square matrices over an algebraically closed field F of characteristic 0. Let [A, B] = AB — BA and set (A, B) = [A, [A, B]]. Recently several proofs (1; 3; 5) of the following result have appeared: if det (AB) ≠ 0 and (A,B) = 0 then A-1B-1AB - I is nilpotent. In (2) McCoy determined the general form of any X satisfying

1.1

in the case that A has a single elementary divisor corresponding to each eigenvalue, that is, A is non-derogatory. In Theorem 1 we determine the structure of any matrix X satisfying (1.1) and also give a formula for the dimension of the linear space of all such X in terms of the degrees of the elementary divisors of A.

Throughout this paper, the symbol P = [Pij] will represent the transition probability matrix of an irreducible, null-recurrent Markov process in discrete time. Explanation of this terminology and basic facts about such chains may be found in (6, ch. 15). It is known (3) that for each such matrix P there is a unique (except for a positive scalar multiple) positive vector Q = {qi} such that QP = Q, or

1

this vector is often called the "invariant measure" of the Markov chain.

The first problem to be considered in this paper is that of determining for which vectors U(0) = {μi(0)} the vectors U(n) converge, or are summable, to the invariant measure Q, where U(n) = U(0)Pn has components

2

In § 2, this problem is attacked for general P. The main result is a negative one, and shows how to form U(0) for which U(n) will not be (termwise) Abel summable.

Although we possess a fairly complete knowledge of the abelian subrings of rings of operators in a Hilbert space which are algebraically isomorphic to the ring of all bounded operators of a finite or infinite dimensional unitary space, that is of factors of Type I, very little is known of abelian subrings of factors of Type II1. In (1), Dixmier investigated several properties of maximal abelian subrings of factors of Type II. It turned out that their structure differs essentially from that of maximal abelian subrings of factors of Type I. He showed the existence of maximal abelian subrings in approximately finite factors, possessing the property that every inner*-automorphism carrying this subring into itself is necessarily implemented by a unitary operator of this subring. These maximal abelian subrings are called singular. In addition, he constructed a IIi factor containing two singular abelian subrings which cannot be connected by an inner *automorphism of this ring.

Let Ai, A2 , … , An be n linearly independent points in n-dimensional Euclidean space of a lattice Λ. The points ± A1, ±A2 , . . , ±An define a closed n-dimensional octahedron (or “cross poly tope“) K with centre at the origin O. Our problem is to find a basis for the lattices Λ which have no points in K except ±A1, ±A2 , … , ±An.

Let the position of a point P in space be defined vectorially by

1

where the p are real numbers. We have the following results.

When n = 2, it is well known that a basis is

2

When n = 3, Minkowski (1) proved that there are two types of lattices, with respective bases

3

When n = 4, there are six essentially different bases typified by A1, A2, A3 and one of

4

In all expressions of this kind, the signs are independent of each other and of any other signs. This result is a restatement of a result by Brunngraber (2) and a proof is given by Wolff (3).

Any real number y leads to a continued fraction of the type

(1)

where ai, bi are integers which satisfy the inequalities

(2)

by means of the algorithm

(3)

the a's being assigned positive integers. The process terminates for rational y; the last denominator bk satisfying bk ≥ ak + 1. For irrational y, the process does not terminate. For a preassigned set of numerators ai ≥ 1, this C.F. development of y is unique; its value being y.

Bankier and Leighton (1) call such fractions (1), which satisfy (2), proper continued fractions. Among other questions, they studied the problem of expanding quadratic surds in periodic continued fractions. They state that “it is well-known that not only does every periodic regular continued fraction represent a quadratic irrational, but the regular continued fraction expansion of a quadratic irrational is periodic.

Let L0 be a differential operator of even order n = 2v on the half open interval 0 ≤ t < ∞ which is formally self adjoint and satisfies the conditions of Kodaira (5, p. 503). We consider a perturbed operator of the form L∈ = Lo + ∈q where q(t) is a real-valued bounded function and ∈ is a real parameter. The object of this paper is to set up conditions on the operator L0 and the function q(t) such that L∈ determines a self-adjoint operator H∈ and such that the spectral resolution operator E∈(Δ) corresponding to H∈ is analytic in a neighbourhood of ∈ = 0, where Δ is a closed bounded interval.

Our conditions are a natural generalization of conditions considered by Moser for the case n = 2(6). Moser has given a number of examples showing that when his conditions do not hold E∈(Δ) need not be analytic.

Soient Γ un groupe localement compact, l'ensemble des classes de représentations unitaires irréductibles de Γ. Dans (4), Godement a défini une topologie sur que nous appellerons "topologie canonique." Cette topologie est facile à étudier lorsque Γ est abélien ou compact, mais fort mal connue en général. Dans un article récent (3), Fell a repris l'étude de la topologie canonique de , et réussi à la calculer lorsque Γ est le groupe spécial linéaire complexe à n variables. L'espace est alors "presque" séparé, et, par suite, "presque" localement compact.

Il semble nécessaire d'étudier complètement quelques cas particuliers avant d'entreprendre une théorie générale de .

A problem which has generated considerable interest during the past couple of decades is that of characterizing abstractly systems of realvalued continuous functions with various algebraic or topological-algebraic structures. With few exceptions known characterizations are of systems of bounded continuous functions on compact or locally compact spaces. Only recently have characterizations been given of the systems C(X) of all realvalued continuous functions on an arbitrary completely regular space X (1). One of the main objects of this paper is to provide, by using certain special techniques, a characterization of C(X) for a particular class of (not necessarily compact) completely regular spaces.

Let f1 , …, fk be linearly independent real functions on a space X, such that the range R of (f1, …, fk) is a compact set in k dimensional Euclidean space. (This will happen, for example, if the fi are continuous and X is a compact topological space.) Let S be any Borel field of subsets of X which includes X and all sets which consist of a finite number of points, and let C = {ε} be any class of probability measures on S which includes all probability measures with finite support (that is, which assign probability one to a set consisting of a finite number of points), and which are such that

is defined. In all that follows we consider only probability measures ε which are in C.

In this note we will discuss the question of the measurability of the multiplier function of a relatively invariant measure on a group. That is, for a group G, σ-ring S, and a measure μ defined on the sets of S, we assume: E in S, x in G implies xE is in S and μ(XE) = σ(x)μ(E) and study the measurability of the function σ(x).

The problem was discussed by Halmos (1, p. 265), on locally compact groups and there the situation proved to be as nice as it could be, that is, if the measure is a non-trivial, relatively invariant Baire measure then the multiplier function is continuous. We prove two theorems for groups in which no topology is assumed. In the first theorem we assume a shearing condition and answer the question completely. The second theorem places a condition on the measure and weakens the shearing assumption. Its proof is complicated and occupies the major portion of this paper.

We generalize in several directions a paper by Porges (2) who considered the integer F(A) obtained from the positive integer .1 by taking the sum of the squares of the digits of A. Porges showed that if A > 99, then F(A) < A, so that under iteration of F(A) all the positive integers are divided into a finite number of classes, called orbits in the terminology of Isaacs (1), each containing a finite cycle. For his F(A) Porges showed there are only two orbits: one with the 1-cycle: 1 → 1 ; and the other with the interesting 8-cycle: 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4.

Consider the set Z of non-negative integers and choose as a base of enumeration any desired integer B ≧ 2 (not necessarily B = 10). Then only the “digits” 0, 1, 2, … , B — 1 are needed, in suitable multiplicity, to represent any A of Z.

Let a1, a2, … , at be a set of groupwise relatively prime positive integers. Several authors, (2; 3; 5; 6), have determined bounds for the function F(a1 , …, at) defined by the property that the equation

1

has a solution in positive integers X1, …, xt for n > F(a1, ..., at). If F(a1 , …, at) is a function of this type, it is easy to see that

2

is the corresponding function for the solvability of (1) in non-negative x's.

It is well known that a1a2 is the best bound for F(a1 , a2) and a1a2 — a,1 — a2 for G(a1 , a2). Otherwise only in very special cases have the best bounds been found, even for t = 3.

In the present paper a symmetric expression is developed for the best bound for F(a1 , a2, a3) which solves that problem and gives insight on the general problem for larger values of t. In addition, some relations are developed which may be of interest in themselves.

Let n and r be integers, r positive, and define the coreγ(r) of r to be the product of the distinct prime factors of r (γ(1) = 1). Let f(n,r) be a complex-valued, arithmetical function of n and r. If for all n,f﹛n,r) = f((n,r), r) then f(n, r) is called an even function (mod r), and if f(n,r) = f(γ(n, r), r) for all n, γ(n, r) = γ((n, r)), then f(n, r) is said to be a primitive function (mod r). Clearly, both classes of functions are subclasses of the periodic functions (mod r), while the primitive functions form a subclass of the even functions (mod r).

In a series of three papers (3; 5; 6) the author developed parallel, though interrelated, trigonometric and arithmetical theories of the even and primitive functions (mod r).

By a labelled graph we shall mean a set of “nodes,“ distinguishable from one another and denoted by A1, A2 , …, and a collection of “edges” viz., pairs of nodes. We say that an edge “joins” the pair of nodes which specifies it. We further stipulate that at most one edge joins any two nodes, and that no edge joins a node to itself.

By a “colouring” of a graph in k colours we shall mean a mapping of the nodes of the graph onto a set of k colours C1, C2, …, Ck such that no two nodes which are joined by an edge are mapped onto the same colour. A graph so coloured in exactly k colours will be called a k-coloured graph. Since it is usually possible to colour a graph in more than one way, there will, in general, be many k-coloured graphs corresponding to a given graph.

This paper deals with the discrete groups of rigid motions of the hyperbolic plane. It is known (12) that the finitely generated, orientation-preserving groups have the following presentations:

Generators: .

Defining relations:

where km = ambmam-1bm-1 . We shall denote this group by F(p; n1 , … , nd; r).

In particular, the finitely generated free groups are contained among these. Indeed, one purpose of this paper is to indicate some geometrical methods for investigating free groups.

Let G be a locally compact and (σ-compact topological group and let H be a discrete subgroup of G. We shall use G/H to denote the space of right cosets Hx of H with the usual topology (cf. (8, pp. 26-28)). Let μ be the left Haar measure in G. μ induces a measure in the space G/H3; this measure will, without ambiguity in this paper, also be denoted by μ. If μ(G/H) is finite, the group H is called a lattice. If the space G/H is compact, then H is certainly a lattice and is called a bounded lattice. These terms are an extension of the usage of the Geometry of Numbers, where G is the real n-dimensional vector space Rn . In this case any lattice is generated by n linearly independent vectors, all lattices are bounded, and the whole family of lattices is permuted transitively by the automorphisms of G (which are the non-singular linear transformations).

If A is a matrix with complex elements and if A = AT (where AT denotes the transpose of A), there exists a non-singular matrix P such that PAPT = D is a diagonal matrix (see (3), for example). It is also true (see the principal result of (5)) that for such an A there exists a unitary matrix U such that UAUT = D is a real diagonal matrix with nonnegative elements which is a canonical form for A relative to the given U, UT transformation.

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.

The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

This paper continues the study appearing in (9) and (10) of the combinatorial properties of a matrix A of m rows and n columns, all of whose entries are 0's and l's. Let the sum of row i of A be denoted by ri and let the sum of column j of A be denoted by Sj . We call R = (r1, … , rm) the row sum vector and S = (s1 . . , sn) the column sum vector of A. The vectors R and S determine a class

1.1

consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. The majorization concept yields simple necessary and sufficient conditions on R and S in order that the class 21 be non-empty (4; 9). Generalizations of this result and a critical survey of a wide variety of related problems are available in (6).