The purpose of this note is to establish and exploit the fact that Utumi's maximal ring of right quotients (6) of an associative ring R (let us say with 1) is the bicommutator of the minimal injective extension of R regarded as a right R-module. Nothing new will be said about Johnson's ring of quotients (4), which is still the most important case.

The α-width ∊A(α) of a (0, 1)-matrix A is the minimal number of columns that can be selected from A in such a way that all row sums of the resulting submatrix of A are at least α. This notion was introduced in (2) and further studied in (3). In these papers the major emphasis was on the minimal α-width sequence for the class of (0, 1)-matrices generated from an arbitrary A by interchanges:

In an earlier paper (2) reflexive transitive binary relations were considered on a connected ordered space. These relations were topologically restricted and their minimal sets were either an end point of the space or empty. It was shown that these relations could be characterized as one of the two orders of the space. Viewing the situation somewhat differently as suggested by I. S. Krule, one could say that this class of relations was characterized in terms of the identity function on the space. In this case the relations are considered in their natural setting, the product of the space with itself.

Let

be an indefinite quadratic form in the integer variables x1, . . . , xn with real coefficients of determinant D = ||ars||(n) ≠ 0. The homogeneous minimum MH(Qn) and the inhomogeneous minimum MI(Qn) of Qn(x) are defined as follows :

In this paper the entropies of several sets of functions of bounded variation are calculated. The entropy of a metric set, a notion first introduced by Kolmogorov in (2), is a measure of its size in terms of the minimal number of sets of diameter not exceeding 2∊ necessary to cover it. Using this notion, Kolmogorov (4; p. 357) and Vituškin (7) have shown that not all functions of n variables can be represented by functions of fewer variables if only functions satisfying certain smoothness conditions are allowed.

When, for the generalized summation of series, we use A and B methods, giving A and B sums, respectively, we say that the A method is included in the B method, A ⊂ B, if the B sum exists and is equal to the A sum whenever the latter exists. A theorem proving such a result is called an Abelian theorem. For example, there is an Abelian theorem stating that if the A and B sums are the first Cesàro mean and the Abel mean, respectively, then A ⊂ B. If A ⊂ B and B ⊂ A, we say that A and B are equivalent, A = B. For example, the nth Hölder and nth. Cesàro means are equivalent.

The purpose of this note is to investigate the properties of a mapping α, of a Boolean algebra A into itself, which satisfies the functional equation α(p·aq) = ap·aq, where the multiplication is infimum. This is the so-called averaging identity of Kampé de Fériét. Garrett Birkhoff has noted (2) the occurrence of this identity both in the statistical theory of turbulence and in mathematical logic. That this is more than coincidence is shown by the existence of an explicit connection between the notion of quantification in logic and certain non-linear "extremal operators" in function algebras (16). Certain linear operators are associated in a natural way with these extremal operators.

It is well known that any semi-simple algebra over the real field R, or over the complex field C, is a direct sum (unique except for order) of simple algebras, and that a finite-dimensional simple algebra over a field is a total matrix algebra over a division algebra, or equivalently, a direct product of a division algebra over and a total matrix algebra over (1). The only finite division algebras over R are R, C, and , the algebra of real quaternions, while the only finite division algebra over C is C.

Let A be a simple ring with minimum condition, and B1, B2, and C be regular subrings of A such that Bi > C, i = 1, 2. A pair of isomorphisms σi of Bi into A such that σi|C is the identity, and that Biσi are regular subrings of A (i = 1, 2), is called compatible if σ1|B1 ∩ B2 = σ2|B1 ∩ B2. Here σ|X means the restriction of σ to X. Bialynicki-Birula has proved some necessary and sufficient conditions that every compatible pair (σ1, σ2) has a common extention to an automorphism σ of A (1 ). When A is a division ring, he shows that the linear disjointness of the division subrings B1 and B2 is necessary and almost sufficient for the existence of a common extension of any compatible pair.

Let us say that a function denned in the open unit disk D has the Montel property if the set of those points eiθ on the unit circle C where the radial limit exists coincides with the set where the angular limit exists. By a classical theorem of Montel (4), every bounded holomorphic function has this property. Meromorphic functions omitting at least three values and, more generally, the normal functions recently introduced by Lehto and Virtanen (3) also enjoy the Montel property (also see 1).

In (1), P. Erdös showed that the function

takes on irrational values whenever z = 1/t, t = 2 , 3 , 4 , 5 , . . . . The method of proof uses Lambert's identity,

where d(n) is the number of divisors of n; and it is shown that

We consider undirected graphs in which two vertices may be joined by more than one edge and in which a vertex may be joined to itself by one or more edges. G will always denote a graph. The set of vertices [edges] of G will be denoted by V(G) [E(G)]. G is finite or infinite according as V(G) ∪ E(G) is finite or infinite. The degree, d(ξ) or dG(ξ), of a vertex ξ of G is 2a + b, where a is the number of edges joining ξ to itself and b is the number of those joining ξ to other vertices.

In a previous paper (3) generalizations of M. Riesz's theorem by the method of asymptotic approximation have been given. The present paper is concerned with further generalizations for even and odd functions. In Section 3, we consider a generalization of a theorem due to Zygmund: If (10, p. 254). The results in this paper include generalizations of results by K. K. Chen (2) and T. M. Flett (4, Theorems 1, 2) as important special cases.

Two spaces of analytic functions are considered, each comprised of functions analytic on the open disk NR(0) of radius R (0 < R < +∞ ) centred at the origin. The first space consists of all analytic functions on NR(0) topologized according to the metric of uniform convergence on compact sets. As the second space we allow any Fréchet space of analytic functions on NR(0) for which the topology is stronger than that induced by . Our objective is then to present a scheme for constructing simultaneous automorphisms on and .

Let A = (ank) and {sn} (n, k = 0, 1, 2, . . .) be a matrix and a sequence of complex numbers, respectively. Let the members of the sequence {σn} be defined by

then we say {σn} is the A -transform of {sn}. The matrix A = (ank) is called regular if

In (2), Tutte has shown that the number, Bn, of rooted non-separable planar maps having n edges is [2(3n — 3)!]/[n! (2n — 1)!]. Rooting was accomplished by designating one edge as the root, orienting it, and distinguishing between its sides as left and right. We shall here compute the number, Bn,m, of rooted non-separable planar maps having n edges and such that the face to the left of the root is incident with exactly m edges, which maps will be said to be of type [n, m].

The purpose of this paper is the study of parabolically differentiable points of arcs in the real affine plane. In Section 2, two different definitions of convergence of a family of parabolas are given and it is observed (Theorem 1) that these are equivalent. In Section 3, tangent parabolas at a point p of an arc A are discussed and it is proved (Theorem 2) that all the non-degenerate non-tangent parabolas of A through p intersect A at p or that all of them support. In Section 4, osculating parabolas are introduced and the condition that an arc be twice parabolically differentiable at a point p is stated.

A Hilbert space, whose elements are entire functions, is of particular interest if it has these properties:

(H1) Whenever F(z) is in the space and has a non-real zero w, the function is in the space and has the same norm as F(z).

(H2) For each non-real number w, the linear functional defined on the space by F(z) —> F(w) is continuous.

(H3) Whenever F(z) is in the space, is in the space and has the same norm as F(z). If E(z) is an entire function satisfying

For a given class of transformations on graphs there arises naturally the problem of deciding when one graph can be transformed into another by a finite sequence of such transformations. We consider, in this paper, the special case of this problem when the graphs are finite trees and the transformations consist of rearranging the order of the segments from a point and replacing subtrees by other trees according to a given set of pairs of interchangeable trees. This decision problem is, in fact, equivalent to the word problem for one-generator algebras in a certain variety of algebraic systems and we exhibit a procedure for solving the word problem for these algebras.

A decomposition of a topological space X is a partitioning of X into non-empty, disjoint sets called elements of the decomposition. An element of a decomposition is non-degenerate if it contains more than one point. Associated with each decomposition D of X is a topological space D*, called the hyperspace of the decomposition. A classical problem on decompositions of topological spaces is to find conditions under which D* is homeomorphic to X. Often decompositions arise from mappings: if g is a mapping of a space X onto a space Y, then D = {g-1(y) |y ∊ Y} is a decomposition of X.