The deleted product space X* of a space X is X × X — △. In (4), I computed the homology groups of the deleted product of a polyhedron in a subcollection (see §2 of this paper for the definition of ) of the finite, contractible, 2-dimensional polyhedra. In the present paper, I show that there is an infinite subcollection of such that the deleted product of each member of has the homotopy type of the 2-sphere.

In (2) Sinkhorn showed that corresponding to each positive n × n matrix A (i.e., every aij > 0) is a unique doubly stochastic matrix of the form D1AD2, where each Dk is a diagonal matrix with a positive main diagonal. The Dk themselves are unique up to a scalar multiple. In (3) the result was extended to show that D1AD2 could be made to have arbitrarypositive row and column sums (with the reservation, of course, that the sum of the row sums equal the sum of the column sums) where A need no longer be square.

In analogy with Hamiltonian groups, an associative ring in which every subring is a two-sided ideal is called a Hamiltonian ring, or, more concisely, an H-ring. Several attempts at classification of H-rings have been made. H-rings generated by a single element have been studied by M. Šperling (5), L. Rédei (4), and A. Jones and J. J. Schäffer (2). H-rings enjoying additional properties have been characterized by F. Szász (e.g., 6), and by S.-X. Liu (3). A class of closely related rings has been studied by P. A. Freĭdman (1). In the present paper and its sequel all H-rings are classified and completely described in terms of their generators and relations.

Słomiński (9), as well as the author (8), gave a descriptive, i.e., noncategory-theoretic, definition of the direct sum of partial algebras, i.e., the co-product in the category of partial algebras (A,ƒ), where ƒ = (ƒi)i∈I, ƒi: dom ƒi ⟶ A, dom ƒi ⊂ AKi, of fixed type A = (Ki)i∈I.

The coarseness, c(G), of a graph G is the maximum number of edge-disjoint, non-planar subgraphs of G. We consider only the complete graph, Kp, on p vertices here. For p = 3r, Erdös conjectured that the coarseness was , but it has been shown (1) that

1

where square brackets denote integer part.

Y. Utumi (14 and 15) obtained some interesting results on self-injective rings. He showed that, if R is right self-injective, then so is R/J, where J is the Jacobson radical of R. Also, if R is right self-injective and regular, then R is left self-injective for any set of orthogonal idempotents is an essential extension of . This note extends these results to endomorphism rings of quasi-injective modules.

Let be a Hilbert space with inner product (•, •) and let E(•) and E0(•) be spectral measures in corresponding to self-adjoint operators and . In this paper we consider the set function ƒ(I × J) = E(I)E0(J) defined on the semiring of bounded rectangles, and obtain an integral representation for this set function for disjoint I, J under the hypotheses that H — H0 is a type of Carleman operator.

Let R be a ring satisfying the identity

1

for all x, y, z ϵ R, where (x,y,z) = (xy)z — x(yz). If R also satisfies the identity (x, x, x) = 0 for all x ϵ R, then R is alternative. It is known that if R satisfies (1), it need not be an alternative (see 6). Thus, the class of rings satisfying (1) is a non-trivial extension of the class of alternative rings. P. Jordan remarked that (x, x, x)2 = 0 is an identity in R (see 9).

Our aim in this paper is to obtain some theorems concerning spaces of analytic functions on a finite open Riemann surface R which extend known results for the disc △ = {|z| < 1}. Suppose that R has a smooth boundary bR consisting of t closed curves, and that the interior genus of R is s. The first Betti number of R is then r = 2s + t — 1.

In this paper some methods for the construction of balanced incomplete block (b.i.b. for conciseness) designs are given. In the last section it is established that the existence of an affine resolvable b.i.b. design implies the existence of two other b.i.b. designs; §§ 6 and 7 are independent of §§ 3, 4, and 5.

In their classic treatment (5) Veblen and Young build n-dimensional projective geometry from points and lines. Naturally, each line becomes identified with the set of points with which it is incident, and many treatments build from points alone, postulating the existence of certain distinguished subsets of the set of points. From either point of view, some labour is required, even in the two-dimensional case, to establish duality; hence a considerable interest attaches to self-dual systems of axioms; cf. (2; 3).

Let X be a set, a σ-ring of subsets of X, and let μ be a measure on . Following (1), we define μ to be semifinite if

We show (Theorem 1) that every measure can be reduced to a semifinite measure for many practical purposes. In many cases, this reduction can be made even more significantly (Theorems 2 and 3). Finally, necessary and sufficient conditions that a semifinite measure be c-finite are given as a corollary to Theorem 3.

Let the sequence {λi} (i ≧ 0) satisfy the following conditions.

1.

2.

3.

We shall deal with the following moment problems: what are the conditions, necessary and sufficient, on a sequence {μn} (n ≧ O) in order that it should possess the representation

1.1

Suppose that a sequence of spheres is packed in order of decreasing diameters into the unit cube In of Rn . In recent work (2), I have shown that for n = 2, there exist positive constants K2, s ( = 0.97) such that the area of has an asymptotic lower bound K2(d(Sm))s . Although the methods used were complicated and possibly only viable in two dimensions, it is intuitively clear that such a result should also be true in higher dimensions.

This paper originated mainly from results presented in a paper by J. Ohm (13), and, to a lesser degree, from results of Gilmer in (3). Ohm's paper is concerned with the validity of the equation (x, y)n = (xn, yn) for each pair of elements x, y of an integral domain D with identity. If D is a Prüfer domain, the above equation is valid for all x, y ϵ D (7, p. 244). Butts and Smith have shown (2) that if (x, y)2 = (x2, y2) for all x, y of the integrally closed domain D, then D is a Priifer domain.

It is our aim in this note to introduce methods from homological algebra in the study of some problems in valuation theory. In particular, we will use such methods to give a new, and, in some respect, simpler proof of a well-known theorem of Krull and Ribenboim; see (2). We shall also show that the same methods can be used to prove the Riemann-Roch theorem for algebraic curves and the Weierstrass product theorem.

The subsemigroups of the projective group on the line that are described in this paper are those that can be generated by a pair of infinitesimal transformations. One denotes by G the connected component of the identity of this group; Theorem 1 gives a necessary and sufficient condition for a pair of infinitesimal transformations to generate a subsemigroup which is equal to G (and hence is actually a group). This condition is reformulated in a geometric manner in Theorem 1*.

It is our aim in this paper to give an elementary solution to the problem of simultaneous unitary equivalence of two finite sets of matrices, i.e., given two ordered sets {Aj} and {Bj} of n × n matrices, j = 1, 2, … , m, we wish to determine whether there exists a unitary matrix U such that Bj = U*AjU for all j. A special case of this problem is that of unitary equivalence of two arbitrary matrices.

In recent work of E. Arthurs and L. A. Shepp on a problem of H. Dym concerning the existence of an ergodic stationary stochastic process with zero entropy (cf. 1), the function dθ(n) was introduced as follows:

For an irrational number θ, let

be the sequence of points {lθ}, 1 ≦ l ≦ n, (where {x} denotes x — [x], the fractional part of x) and define*

dθ(n) = max(ai — ai-1), 1 ≦ in + 1.

The literature on Finsler geometry contains more than one definition for the normal curvature vector of a hypersurface and for coefficients of the second fundamental form; see Berwald (1), Davies (3), and Rund (5). In the first case this situation has arisen from the basically different approach to the subject adopted by the authors; Davies, following the locally Euclidean school and Rund the locally Minkowskian theory. In both cases, a comparison of the definitions shows that they are linked by expressions in a vector Mαwhich was introduced in the paper by Rund (7).