A graph G consists of a finite set of p points and q lines joining pairs of these points. Each line joins two distinct points and no pair of points is joined by more than one line. A subgraph of G is a graph whose points and lines are also in G. If every pair of points of a graph is joined by a line, the graph is called complete and is denoted by Kp. A planar graph can be embedded in the plane, that is, drawn in the plane in such a way that none of its lines intersect.

An interesting identity is obtained for ideals A and B in a Noetherian ring :

A = (A + Bn) ∩ (A : Bn)

for sufficiently large n. This identity is applied to obtain Fuchs' quasi-primary decomposition of A in an improved form, and to obtain Krull's theorem on the intersection of the powers of A, both developments making no use of the Noetherian primary decomposition of A. Finally, the identity is used to obtain the primary decomposition without reference to irreducible ideals, in a largely constructive manner which yields the decomposition in an illuminating, automatically normal form, and which, subject to certain simple conditions, is unique.

In the first part of this paper we define a partial ordering on the set of all homogeneous identities and find necessary and sufficient conditions that an identity does not imply any identity lower than it in the partial ordering (we call such an identity irreducible). Perhaps the most interesting property established for irreducible identities is that they are skew-symmetric in any two variables of the same odd degree and symmetric in any two variables of the same even degree. The results of the first section are applied to commutative algebras in the remainder of the paper.

Many authors have made interesting and important contributions to the study of vector fields or infinitesimal transformations on compact orientable Riemannian manifolds and Hermitian manifolds without boundary. Recently, Hsiung (6, 7, 8) has extended some of these results to compact orientable Riemannian manifolds with boundary. The purpose of this paper is to continue Hsiung's work by studying vector fields and infinitesimal transformations on almost-Hermitian manifolds with boundary.

It will be shown in this paper that if two chords of a closed plane convex curve θ divide θ into four arcs of equal length and intersect inside the domain bounded by θ, then the sum of the lengths of the two chords is at least equal to (√5 — 2)½ times the length of θ. We shall show firstly that we need only consider the case when θ is a convex (possibly degenerate) quadrilateral and then prove the result in this case.

This result is related to a conjecture of P. Ungar in which the chords are assumed to be perpendicular and the factor (√5 — 2)½ is replaced by §. But Ungar's conjecture is neither proved nor disproved by this result.

A semigroup S is a λ- [ρ-, σ-] semigroup if and only if each subsemigroup of S is a left [right, two-sided] ideal of S. Since the concept of ρ-semigroup is the dual of that of λ-semigroup, the results for ρ-semigroups are generally not stated explicitly, σ-semigroups are treated as a special case of λ-semigroups ; in fact, a semigroup S is a σ-semigroup if and only if it is a λ-semigroup and a ρ-semigroup. The purpose of this paper is to determine the structure of λ- [ρ-, σ-] semigroups.

With every connected graph G there is associated a metric space M(G) whose points are the vertices of the graph with the distance between two vertices a and b defined as zero if a = b or as the length of any shortest arc joining a and b if a ≠ b. A metric space M is called a graph metric space if there exists a graph G such that M = M (G), i.e., if there exists a graph G whose vertex set can be put in one-to-one correspondence with the points of M in such a way that the distance between every two points of M is equal to the distance between the corresponding vertices of G.

Let R be a simply connected closed region in the Euclidean plane E2 whose boundary is a simple closed curve C. A triangular map, or simply "map," is a representation of R as the union of a finite number of disjoint point sets called cells, where the cells are of three kinds, vertices, edges, and faces (said to be of dimension 0, 1, and 2, respectively), where each vertex is a single point, each edge is an open arc whose ends are distinct vertices, and each face is a simply connected open region whose boundary consists of the closure of the union of three edges. Two cells of different dimension are incident if one is contained in the boundary of the other.

If A is an algebra over a commutative ring with unity, Φ, then the Jacobson radical of the algebra A is equal to the Jacobson radical of A, thought of as a ring (1, p. 18, Theorem 1). The present note extends this result to all radical properties (in the sense of Kurosh 2) and allows ϕ to be any set of operators on A.

If A is a ring and Φ is an arbitrary set, we say that Φ is a set of operators for A if for any α in Φ and any x in A, the composition ax is defined and is an element of A, and if this composition satisfies the following two conditions:

Let L be the total length of the edges of a convex polyhedron containing a unit sphere. If the number of edges is small, the edges must be, on the average, comparatively long. If, on the other hand, the edges are short, their number must be great. So the problem arises to find a polyhedron with a possibly small value of L.

By a simple argument the author (5; 6) proved that L > 20 and announced the conjecture that L ≥ 24 with equality only for the cube (of in-radius 1). The same argument shows that for trigonal-faced polyhedra L > 28. This supports the conjecture that for such polyhedra L ≥ 12√6 = 29.4 . . . , with equality only for the tetrahedron and octahedron.

A matrix ω having p rows and 2p columns of complex number elements is called a Riemann matrix of genus p if there exists a rational 2p-rowed skew matrix C such that

(1)

is positive definite Hermitian. The matrix C is then called a principal matrix of ω. If ω and ω0 are two Riemann matrices of the same genus, we say that ω and ω0 are isomorphic if there exists a non-singular p-rowed complex matrix α, and a non-singular 2p-rowed rational matrix A, such that

(2)

If (L, ≥) is a lattice or partial order we may think of its Hesse diagram as a directed graph, G, containing the single edge E(c, d) if and only if c covers d in (L, ≥). This graph we shall call the graph of (L, ≥). Strictly speaking it is the basis graph of (L, ≥) with the loops at each vertex removed; see (3, p. 170).

We shall say that an undirected graph Gu can be realized as the graph of a (modular) (distributive) lattice if and only if there is some (modular) (distributive) lattice whose graph has Gu as its associated undirected graph.

1.1. Throughout this note, will denote an associative ring but we shall not require to possess a unit.

If A and B are subsets of , then A + B will denote the set {x + y| x ∊ A, y ∊ B}. Aτ will denote the set {u ∊ | au = 0 for all a ∊ A} .

Elements a and b will be said to be orthogonal if ab = ba = 0.

A sequence is said to be well distributed (mod 1) (3, 4, 5) if the limit

(1.1)

exists, uniformly in p ≥ 0, for all intervals I in [0, 1], with length |I|, characteristic function where (x) is the fractional part of x. If (1.1) is true for p = 0 and all I in [0, 1] we say that is uniformly distributed (mod 1).

Let E be a completely regular space, let C* denote the Banach algebra of continuous bounded real-valued functions on E, and let A* denote a Banach subalgebra which contains the constant function 1. Let

The purpose of this note is to discuss some properties of the subsets L(A*) of the ring C of continuous real-valued functions on E. They arose out of an unsuccessful attempt to distinguish C* from its Banach subalgebras.

In (1), M. Davis and M. Maschler define the kernel K of a characteristic-function game; they also prove, among other theorems, that K is a subset of the bargaining set M1(i) and that it is never void, i.e. that for each coalition structure b there exists a payoff vector x such that the payoff configuration (x, b) belongs to K. The main advantage of the kernel, as it seems to us, is that it is easier to compute in many cases than the bargaining set M1(i).

Eigenvalue problems for the biharmonic operator (iterated Laplacian) L = ΔΔ will be studied on bounded plane domains. Our purpose is to obtain asymptotic variational formulae for eigenvalues and eigenfunctions under the deformation of removing an ∊-disk and adjoining additional boundary conditions on the new boundary component thereby introduced, valid on a positive interval 0 < ∊ < ∊0. Eigenvalue problems can be considered in connection with each of the following sets of homogeneous boundary conditions.

In recent years a number of algebraic characterizations of the groups of Moebius transformations over finite fields have been given in the literature; see (1, 3, 6). H. W. E. Schwerdtfeger has noticed (4) that the group G of Moebius transformations over the real, complex, and certain other fields has the property:

Every abstract open Riemann surface can be made "concrete" (in the terminology of (1)) by considering it as a covering surface (in general branched) of the complex plane by means of a suitable projection map p. Since this covering map is not unique, it seems natural to single out some such maps by an extremal property. The use of Riemannian metrics compatible with the conformai structure on the given surface for the study of $1 is well known ; from the point of view of differential geometry it suggests an investigation of the distortion caused by p between such a metric ds^ and the Euclidean metric of .

In (3) Ore poses two problems concerning the decomposition of graphs into edge-disjoint paths. The first is to find the conditions on a graph so that it can be decomposed into a finite number k of edge-disjoint, two-way infinite paths and no fewer. In (2) Nash-Williams solves this problem. The results of (2) are used here to solve the second problem, to find conditions on a graph so that it can be decomposed into a finite number k of edge-disjoint paths (finite, one-way infinite, and two-way infinite) and no fewer.