Let L be the linear, elliptic, self-adjoint partial differential operator given by

where Dj denotes partial differentiation with respect to xj, 1 ≤ j ≤ n, b is a positive, continuous real-valued function of x = (x1,…,xn) in n-dimensional Euclidean space En, the aij are real-valued functions possessing uniformly continuous first partial derivatives in En and the matrix {aij} is everywhere positive definite. A solution u of Lu = 0 is assumed to be of class C1.

In this paper, we consider real linear spaces. By (V:‖ ‖) we mean a normed (real) linear space V with norm ‖ ‖. By the statement "V has the (Y, X) norm preserving (Hahn-Banach) extension property" we mean the following: Y is a subspace of the normed linear space X, V is a normed linear space, and any bounded linear function f: Y → V has a linear extension F: X → V such that ‖F‖ = ‖f‖. By the statement "V has the unrestricted norm preserving (Hahn-Banach) extension property" we mean that V has the (Y, X) norm preserving extension property for all Y and X with Y ⊂ X.

A. M. Macbeath, in November 1965, communicated the following theorem to me which he proved with the aid of the Lefschetz fixed point formula.

Theorem. If Γ is a Fuchsian group and N a torsion free normal subgroup, then the rank of N/[Γ, N] is twice the genus of the orbit space D/Γ where D denotes the hyperbolic plane which Γ acts.

This theorem will follow from a consideration of the exact sequence

*

T. Evans in [4] introduced a concept generalizing the associative law for groups. Certain very restricted cases of this generalization had occurred, somewhat as side results, in the considerations of A. K. Suškevič [7], D. C. Murdoch [5], [6], and R. H. Bruck [2] associated with quasigroups which obey theorems generalizing certain results in group theory.

In this paper we consider quasigroups obeying an instance of Evans' generalization. We show two means of constructing such quasigroups, and we demonstrate these constructions with several examples. The principal result of this paper is a theorem which reveals a peculiar trait of quasigroups obeying an instance of the generalized associative law we are considering. Such a quasigroup obeys a law expressed with permutations in which the nontrivial permutations involved either are both automorphisms of the quasigroup or else both fail to distribute over any product from the quasigroup.

In a recent study of a specific class of quasi-Frobenius rings, Feller has found it useful to introduce the X-rings ([3]). He suggested among others the following topics:

1. (A) Determine the properties of completely indecomposable rings and matrix rings over completely indecomposable rings.

2. (B) Determine the properties of modules over quasi-Frobenius X-rings.

We point out that the completely indecomposable rings are the local quasi-Frobenius rings. Problems (A) and (B) then lead naturally to semi-local quasi-Frobenius rings, and to matrix algebra over local quasi-Frobenius rings. These types of rings are discussed in sections 1 and 2.

In this paper we consider the types of pairs of multiple zeros which a solution to the differential equation

can possess on an interval I of the real line. The results obtained generalize those in [2] and (for n = 3) in [3].

I. Let f satisfy the condition

1.1

for all t ∊ I, u0 ≠ 0, and all u1, … un-1.

Cordes and Labrousse ([2] p. 697), and Kaniel and Schechter ([6] p. 429) showed that if S and T are domain-dense closed linear operators on a Hilbert space H into itself, the range of S is closed in H and the codimension of the range of S is finite, then, (TS)* = S*T*. With a somewhat different approach and more restricted condition on S, the same assertion was obtained by Holland [5] recently, that S is a bounded everywhere-defined linear operator whose range is a closed subspace of finite codimension in H.

If is a universal algebra, the set of endomorphisms of forms a monoid (i.e., semigroup with identity) under composition. We denote it by End (). For definitions and notations, see [1]. It is well known (e.g., [1], Theorem 12.3) that for any monoid M there is a unary algebra with M ≅ End (). E. Mendelsohn and Z. Hedrlin [3] have proved that the monoid of a subalgebra of an algebra is independent of the monoid of . In [2], Hedrlin proves the same for the monoid of a homomorphic image of . The proofs of these depend heavily on graph-theoretical and category-theoretical considerations. In this note considerably shorter direct algebraic proofs are given of these results.

Suppose there are N not necessarily distinct points on a plane in such a position that any triangle (degenerate or non-degenerate) determined by these points has perimeter length at most 1. Denote by m the number of triangles with maximal perimeter length, (briefly, the number of maximal triangles), and put f(N) = max m where the maximum is taken over all permissible configurations. At the Colloquim on Graph Theory in Calgary, 1969, P. Erdös proposed the problem of determining f(N). He conjectured that the following construction gives the maximal number: place approximately half of the points at a point A and the others at B where AB = 1/2. The aim of this note is to prove this conjecture.

A tournament Tn is a set of n nodes a1a2, …, an such that every pair (ai, aj) of distinct nodes is joined by exactly one of the oriented edges or . If is in Tn, then we say that ai dominates aj and write ai → aj.

The (automorphism) group G(Tn) of a tournament Tn is the group of all permutations ϕ of the nodes of Tn such that ϕ(a)→ϕ(b) if and only if a → b. It is known [9] that there exist tournaments whose group is abstractly isomorphic to a given group H if and only if H has odd order; thus all tournament groups are solvable, by the Feit-Thompson Theorem [7].

The main aim of this paper is to study the convergence of a sequence of contractions in a generalized metric space. More specifically, we investigate the following question:

"If a sequence of contractions {fr} with fixed points ur (r = 1,2,...) converges to a mapping f with a fixed point u, under what conditions will the sequence ur converge to u?"

A partial answer to the above question has been given in metric spaces by Bonsall [1]. This result has since been improved by Russell and Singh [6]. Further results will now be given in a generalized metric space.

If U is an open set in Euclidean N-space EN which has finite Lebesgue measure |U| then a complete packing of U by open spheres is a collection C={Sn} of pairwise disjoint open spheres contained in U and such that Σ∞n=1|Sn| = |U|. Such packings exist by Vitali's theorem. An osculatory packing is one in which the spheres Sn are chosen recursively so that from a certain point on Sn+1 is the largest possible sphere contained in (Here S- will denote the closure of a set S). We give here a simple proof of the "well-known" fact that an osculatory packing is a complete packing. Our method of proof shows also that for osculatory packings, the Hausdorff dimension of the residual set is dominated by the exponent of convergence of the radii of the Sn.

By an (incomplete) r × s latin rectangle is meant an r × s array such that (in some subset of the rs cells of the array) each of the cells is occupied by an integer from the set 1, 2, …, s and such that no integer from the set 1,2, …, s occurs in any row or column more than once. This definition requires that r≦s. If r=s we will replace the word rectangle by square. It is easy to see that for any n≧2 there is an incomplete n × 2n latin rectangle with 2n cells occupied which cannot be completed to a n × 2n latin rectangle. In this paper we prove the following theorem.

Theorem 1. An incomplete n × 2n latin rectangle with 2n—1 cells occupied can be completed to a n × 2n latin rectangle.

The following lemma, which appears as Lemma 4 in [5], was used to determine certain multicoherence properties of inverse limits of continua.

Lemma. Let X denote the inverse limit of an inverse system {Xλ, fλμ, Λ} of compact Hausdorff spaces Xλ. If Xλ has no more than k components (where k < ∞ is fixed) for each λ ∊ Λ, then X has no more than k components.

In this paper we give a set theoretic analogue of this lemma and an extension which was suggested to the author by Professor F. W. Lawvere. An application to inverse limits of finite groups is then given.

P. M. Cohn [7] calls a submodule P of the left A-module M pure iff 0 → E ⊗ P → E ⊗ M is exact for all right modules E. This concept has been studied in [11] and [12]. We will call a non-zero module pure simple iff its only pure submodules are 0 and itself, and the ring A left pure simple iff it is pure simple as a left A-module. We relate these concepts to the PP and PF rings of Hattori [13], and give several new characterizations of these rings. In order to establish these, we use the following known result: the Jacobson radical of any module is the sum of all its small submodules.

Parts of this paper are contained in the author's doctoral thesis [10] at McGill University.

Let In be the graph of the unit n-dimensional cube. Its 2n vertices are all the n-tuples of zeros and ones, two vertices being adjacent (joined by an edge) if and only if they differ in exactly one coordinate. A path P in In is a sequence x1, …, xm of distinct vertices in In where xi is adjacent to xi+1 for 1 ≤ i ≤ m-1; P is a circuit if it is also true that xm and x1 are adjacent. A path is Hamiltonian if it passes through all the vertices of In. Finally, for vertices x and y in In, we define d(x, y) to be the graph theorectic distance between x and y, i.e., the number of coordinates in which x and y differ.

Let Rm denote a m dimensional Euclidean space. When x ∊ Rm will write x = (x1, x2,..., xm). Let R+m ={x: x ∊ Rm, xi < 0 for all i} and R-m ={x: x ∊ Rm, xi < 0 for all i}. In this paper we consider a class of functions which consists of mappings, Er(K) and Hr(K) of Rm into R which are indexed by K ∊ R+m and K ∊ R-m respectively, and defined at any point α ∊ Rm by

1.1

Etant données deux variétés différentiables compactes M et N, nous avons la formule de Künneth

qui établit la relation entre la cohomologie réelle de la variété produit avec celle des facteurs. Cette formule est bien connue en topologie algébrique, et elle est valable dans le contexte plus général de la catégorie des espaces topologiques. Nous voulons donner ici une démonstration de cette formule par la théorie harmonique de Hodge-Kodaira [1]. Une telle démonstration n'est pas explicite, autant que je sache, dans la littérature; elle est désirable, par exemple, dans un cours de géométrie différentielle sur les formes harmoniques.

It is well known that the complex number field has infinitely many automorphisms. Moreover, it seems to be part of the folklore that the family of all automorphisms of the complex field has cardinality 2c, where c = 2ℵo. In this article the following generalization of this fact is proved: If k is any algebraically closed field then the family of all automorphisms of k has cardinality 2card k.

The complex field has infinite transcendency degree over its prime subfield. For fields of this type the proof is accomplished by essentially permuting the elements in a transcendency basis and extending each permutation to an automorphism of the field.

We consider an elliptic partial differential equation with constant coefficients and zero on the right hand side. It is well known [1] that every solution of such an equation can be approximated uniformly on each compact set by a sum of products of polynomials and exponential functions which satisfy the equation. Furthermore, if one assumes that the polynomial operator is homogeneous, then the approximation can be made with polynomials alone. It is our purpose to show, in the latter case, when the number of variables is two, that each solution can be written as an infinite series in certain specific polynomials. Our method is to factor the polynomial and build up the solution in terms of solutions of first degree equations.