In this paper we shall consider a generalization of a very old problem in differential geometry; namely, given a second-order covariant tensor field aij(x) on an n-dimensional manifold, when does there exist a connection such that the covariant derivative, defined by

vanishes?

The earliest question of this type arose in the case when is symmetric and positive definite. A solution connection of the problem is then given by the Christoffel symbols

Throughout this paper X denotes a fixed Hausdorff locally compact group with left Haar measure dx. Various spaces of functions and measures on X will recur in the discussion, so we name and describe them forthwith. All functions and measures on X will be scalarvalued, though it matters little whether the scalars are real or complex.

C = C(X) is the space of all continuous functions on X, Cc = Cc(X) its subspace formed of functions with compact supports. M = M(X) denotes the space of all (Radon) measures on X, Mc = MC(X) the subspace formed of those measures with compact supports. In general we denote the support of a function or a measure ξ by [ξ].

In 1961, S. T. Hu published a paper (1) in which he discussed the desirability of discovering those topological properties which are preserved under homotopy and isotopy equivalences. In that paper he gave general tests in terms of weakly hereditary and hereditary topological properties for homotopy and isotopy properties.

In this paper, general tests for homotopy and isotopy properties in terms of weakly hereditary properties and of a class of properties which the author calls open properties are given. In the last sections, we shall show the strong role played by the notions of dimension and separating subsets in forming isotopy properties.

In general, given the fact that every element in a ring is nilpotent, one cannot conclude that the ring itself is nilpotent. However, there are theorems which do assert that, in the presence of certain side conditions, nil implies nilpotent. We shall prove some theorems of this nature here; among them they contain or subsume many of the earlier known theorems of this sort.

Let n be a positive integer and put N = {1, 2, . . . , n}. A collection {S1, S2, . . . , St} of subsets of N is called determining if, for any T ⊂ N, the cardinalities of the t intersections T ∩ Sj determine T uniquely. Let €1, €2, . . . , €n be n variables with range {0, 1}. It is clear that a determining collection {Sj) has the property that the sums

The purpose of this note is to point out some connexions between generalized Hadamard matrices (4, 5) and various tactical configurations such as group divisible designs (3), affine resolvable balanced incomplete block designs (1), and orthogonal arrays of strength two (2). Some constructions for these arrays are also indicated.

A balanced incomplete block design (BIBD) with parameters v, b, r, k, λ is an arrangement of v symbols called treatments into b subsets called blocks of k < v distinct treatments such that each treatment occurs in r blocks and any pair of treatments occurs in λ blocks.

In this paper a ring will always mean a commutative ring with identity element. Furthermore, a ring R is called a multiplication ring if, whenever A and B are ideals of R and A is contained in B, there is an ideal C such that A = BC. Noetherian multiplication rings have been studied by Asano (1), Krull (4, 5), and Mori (6, 7). Krull also studied non-Noetherian multiplication rings (3). In (8, 9), Mori studied non-Noetherian multiplication rings which did not necessarily contain an identity element.

It has been shown elsewhere (1, 4) that the number of rooted non-separable planar maps with n edges is

In the present paper we improve upon this result by finding the number fi,j of rooted non-separable planar maps with i + 1 vertices and j + 1 faces. We use the definitions of (1).

Among the non-separable planar maps only the loop-map and the link-map have i = 0 or j = 0. We therefore confine our attention to the case in which i and j are both positive.

We present here an investigation of the theory of one-sided ideals in a topological ring R. One of our aims is to discuss the question of "left" properties versus "right" properties. A problem of this sort is to decide if (a) all the modular maximal right ideals of R are closed if and only if all the modular maximal left ideals of R are closed. It is shown that this is the case if R is a quasi-Q-ring, that is, if R is bicontinuously isomorphic to a dense subring of a Q-ring (for the notion of a Q-ring see (6) or §2). All normed algebras are quasi-Q-rings. Also (a) holds if R is a semisimple ring with dense socle.

The intent of this paper is to apply the following theorem in several particular instances:

Theorem 1. For any finite ring of q elements, let {} be a collection of s subsets of , each containing hi (i = 1, 2, . . . , s) members, and let denote the set of all differences d′ — d″ with d′ and d″ from including d′ = d″. Furthermore, suppose that

This paper deals with the problem of finding the solutions of the integral equation

where the constant a and the function giipc) are both given and both assumed to be real, while the function f1(x) is to be determined.

On writing

equation (1) simplifies to

All the matrices considered in this paper have their elements in the field of residues mod 2.

Two non-singular matrices are equivalent if each row of either matrix is a linear combination of rows of the other. The matrices then have equal numbers of rows and equal numbers of columns.

A nodal matrix is a non-singular matrix in which no column has more than two l's. A graphic matrix is a non-singular matrix equivalent to a nodal matrix.

This paper discusses projective planes from the viewpoint of their classification into singly-generated and non-singly-generated planes. (Singly-generatedness, a property explicit in Hall (6) and Wagner (19), and implicit in Hughes (10), is defined in Section 5.) The elements (points and lines) of a singly-generated plane are expressible in four basic points called a quadrangle. A t'completion procedure" enables us to obtain expressions for the elements of a plane.

In the article "Moulton Planes" (10), I studied F. R. Moulton's construction over any field containing a multiplicative subgroup of index 2. In "Collineations of Affine Moulton Planes" (11), I determined the collineations between two arbitrary affine Moulton planes.

The purpose now is to describe the collineations between two projective Moulton planes. Since the affine collineations are known from (11), we are concerned with collineations mapping ideal lines onto ordinary lines. Notations and conventions of (10) and (11) are retained.

The purpose of this paper is to clarify and sharpen the argument in the last two chapters of the author's Representation theory of the symmetric group(3). When these chapters were written the peculiar properties of the case p = 2 were not fully appreciated. No difficulty arises in the definition of the block in terms of the p-core, or in the application of the general modular theory based on the formula

In (3), Chevalley constructed, for every field K and every semi-simple Lie algebra g over the complex number field, a group GK(g) by using a system of root vectors Xr which satisfies a certain condition (more precisely, (1.2) in Section 1). The main point of Chevalley's construction lies in the fact that the above-mentioned root vectors furnish a basis of g such that, for every root r, exp(t ad Xr) is represented by a matrix (Aij(t)) whose entries Aij(t) are polynomials in t with integral coefficients.

This paper is concerned with the packing of equal spheres in Euclidean spaces [n] of n > 8 dimensions. To be precise, a packing is a distribution of spheres any two of which have at most a point of contact in common. If the centres of the spheres form a lattice, the packing is said to be a lattice packing. The densest lattice packings are known for spaces of up to eight dimensions (1, 2), but not for any space of more than eight dimensions. Further, although non-lattice packings are known in [3] and [5] which have the same density as the densest lattice packings, none is known which has greater density than the densest lattice packings in any space of up to eight dimensions, neither, for any space of more than two dimensions, has it been shown that they do not exist.

Metric spaces in which the distances are not real numbers have been studied by several people (2, 3, 4, 7, 9). Any ring R together with a mapping, X —> ϕ(X), of R into a lattice A with 0 and 1 satisfying

is called a "lattice-valued ring," where the operations union, ∪, and intersection, ∩, are the usual lattice operations. The mapping ϕ is called a "valuation" and A is a 'Valuation lattice." If R is a lattice-valued ring and a mapping d is defined by

If A = (aij) is a matrix whose elements are 0's and l's, more briefly a (0, l)-matrix, the trace of A, defined as the sum of the diagonal elements aii and denoted by Tr A, is the number of l's on the main diagonal. Matrices which can be obtained from A by permutation of its rows or of its columns, or both, may be expressed as PA, AQ, or PAQ, respectively, where P and Q are permutation matrices.