An interesting set of geometric figures is composed of the convex polyhedra in Euclidean 3-space whose faces are regular polygons (not necessarily all of the same kind). A polyhedron with regular faces is uniform if it has symmetry operations taking a given vertex into each of the other vertices in turn (5, p. 402). If in addition all the faces are alike, the polyhedron is regular.

That there are just five convex regular polyhedra—the so-called Platonic solids—was proved by Euclid in the thirteenth book of the Elements (10, pp. 467-509). Archimedes is supposed to have described thirteen other uniform, “semi-regular” polyhedra, but his work on the subject has been lost.

Knowing the isotopy of cones is a crucial first step in knowing the isotopy of finitely triangulable spaces, for the cones are exactly the stars of vertices. Furthermore, they are the simplest examples of contractible spaces, and the non-triviality of the contractible spaces is one of the distinguishing characteristics of isotopy theory as contrasted with homotopy theory.

The present paper is concerned with the cones over 1-dimensional finitely triangulable spaces. It is clear that homeomorphic spaces have homeomorphic cones, hence cones of the same isotopy type. The surprising result of §2 is that there are very few exceptions to the converse statement. The exceptional isotopy classes of cones all contain cones over spaces that are themselves cones.

In 1938, Frucht (2) proved that for any given finite group G there exists a finite symmetric graph X such that G(X) is abstractly isomorphic to G. Since G(X) is a permutation group, it is natural to ask the following related question : If P is a given finite permutation group, does there exist a symmetric (and more generally a directed) graph X such that G(X) and P are isomorphic (see Convention below) as permutation groups? The answer for the symmetric case is negative as seen in (3) and more recently in (1). It is the purpose of this paper to deal with this problem further, especially in the directed case. In §3, we supplement Kagno's results (3, pp. 516-520) for symmetric graphs by giving the corresponding results for directed graphs.

In two recent papers by V. F. Cowling and C. L. Miracle (1; 2), the regularity of generalized Lototsky transformations, as well as their application to the geometric series, has been investigated. The main interest of the papers centres around (1, Theorems 3.1 and 4.1) and (2, Theorem A). In (1, Theorem 3.1) and in (2, Theorem A), the authors stated and proved a set of sufficient conditions for the regularity of Lototsky-type transformations. In (1, Theorem 4.1), they proved under certain additional conditions that these transformations sum the geometric series ∑zn to (1 — z)–l if Re z < 1.

Block design games have been developed by Richardson (12) and by Hoffman and Richardson (8), who proved a number of theorems concerning such games by studying the number of elements in a blocking coalition. Hoffman and Richardson listed as unsolved (except for PG(2, 3)) the following problem: What is the minimum number of elements in a blocking coalition of a block design game?

This note considers blocking coalitions in those games that are dual to block designs having λ = 1 and r — k > 0. For such games certain blocking coalitions are shown to be related to sets of mutually disjoint blocks in the design to which the game is dual. In particular, for Steiner triple systems the largest odd-numbered set of mutually disjoint triples is shown to yield a minimum blocking coalition in the dual.

In this paper we prove:

Theorem. Let be an ordered pseudo-symmetric connected graph with l lines and v vertices. Let there be aij lines directed from vertex i to vertex j, i, j = 1, 2, … , v. Let gcd(aij) = d, and define

The purpose of this paper is to extend various invariant subspace theorems for the circle group to multiply connected domains. Such attempts are not new. Actually, Sarason (4) studied the invariant subspaces of annulus operators acting on L2 and showed certain parallelisms between the unit disk case and the annulus case. Voichick (8) observed analytic functions on a finite Riemann surface and generalized the Beurling theorem on the closed invariant subspaces of H2 as well as the Beurling–Rudin theorem on the closed ideals of the disk algebra. Here we shall consider LP(Γ) and C(Γ) defined on the boundary Γ of a finite orientable Riemann surface R.

Let D and C denote respectively the open unit disk and the unit circle in the complex plane. Further, γ = z(t), 0 ⩽ t ⩽ 1, will denote a simple continuous arc lying in D except for Ƭ = z(l) ∈ C, and we shall say that γ is a boundary arc at Ƭ.

We use extensively the notions of non-Euclidean hyperbolic geometry in D and employ the usual metric

where a and b are elements of D. For a ∈ D and r > 0 let

For details we refer the reader to (4).

In (2; 3; 4), A. L. Foster denned Boolean extensions of framal algebras and bounded Boolean extensions of framal-in-the-small algebras. Foster proved that the class of Boolean (of bounded Boolean) extensions of a framal (a framal-in-the-small) algebra A is coextensive up to isomorphism with a certain class of subdirect powers of A, namely, the class of normal (of bounded normal) subdirect powers of A. His proofs apply, however, to considerably more general situations. Indeed, as remarked in (2), the construction of Boolean extensions may be carried out for an arbitrary universal algebra with finitary operations; this is done, in fact, in (4).

In this paper we discuss a generalization of the split exceptional Jordan algebra M38() of the 3 X 3 hermitian matrices with elements in the split Cayley-Dickson algebra (1). The generalization consists of replacing by the non-commutative Jordan algebra ≡ (A, ƒ, s, t) discussed in (2; 3) and forming the set of 3 X 3 hermitian matrices M3m() ≡ M with elements in the m-dimensional algebra . With the usual definition of multiplication X · Y = ½(XY + YX), M becomes a commutative algebra and we have the following theorem, which shows how the structure of M is reflected by that of .

We shall consider a generalization of the sieve process introduced by W. E. Briggs (1) in 1963. Let A(1) be the sequence {ak(1)}, where ak(1) = k + 1, so that A(1) = {2, 3, 4, … }. Suppose inductively that A(1), A(2), … , A(n) has been defined. 4(n+1) will be defined from A(n) = {a1(n), a2(n), a3(n), …} in the following manner: For each integer t ⩾ 0, choose an arbitrary element αt(n) from the set where an = an(n), and delete the elements αt(n) from A(n) to form A(n+1). The sequence A is defined to be the sequence {an}. It is also the set-theoretic intersection of all the sequences A(n), n — 1, 2, … . Let be the class of all sequences that can be generated by this sieve process.

We prove here a result closely related both in spirit and technique to those obtained in our paper (1). We also take this opportunity to present a counter example, due to E. Sąsiada, to the conjecture we made there. Finally, we extend a result of Wedderburn (3) from algebras over fields to algebras over commutative rings; the result follows easily from the paper by Posner (2) although it does not appear there explicitly.

The author (2) has shown that corresponding to each positive square matrix A (i.e. every aij > 0) is a unique doubly stochastic matrix of the form D1AD2, where the Di are diagonal matrices with positive diagonals. This doubly stochastic matrix can be obtained as the limit of the iteration defined by alternately normalizing the rows and columns of A.

In this paper, it is shown that with a sacrifice of one diagonal D it is still possible to obtain a stochastic matrix. Of course, it is necessary to modify the iteration somewhat. More precisely, it is shown that corresponding to each positive square matrix A is a unique stochastic matrix of the form DAD where D is a diagonal matrix with a positive diagonal. It is shown further how this stochastic matrix can be obtained as a limit to an iteration on A.

This paper extends the results of A. L. Foster (1) on elementary factorization in Boolean-like rings to commutative π-regular rings. After proving some preliminary lemmas we proceed to the partition of the set of non-units of a π-regular ring into irreducible and composite elements. Finally, we prove a number of theorems concerning factorization rings, weakly unique factorization rings, principal ideal rings, etc. The principal result is that a π-regular ring is a weakly unique factorization ring if and only if it is a principal ideal ring.

Suppose S = [a, b] is a number interval and F is a function from S X S to a normed algebraic ring N with multiplicative identity I. We consider the problem of finding, for appropriate conditions on F, a function M from S X S to N such that for all t and x,

where the integral is a Cauchy-left integral.

We introduce certain n X n matrices with integral elements that constitute a free semigroup with identity and generate the n-dimensional unimodular group. In terms of these matrices we define a certain sequence of n-dimensional vectors, which we show is the natural generalization to n dimensions of the Fibonacci sequence. Connections between the generalized Fibonacci sequences and certain Jacobi polynomials are found. The various basic identities concerning the Fibonacci numbers are shown to have natural extensions to n dimensions, and in some cases the proofs are rendered quite brief by the use of known theorems on matrices.

In (6) (see also 7), Sobolev introduced a class of function spaces Wm,p(Ω) (m a non-negative integer, 1 < p < ∞) defined on open subsets Ω of Euclidean space En, which have important applications in partial differential equations. They are defined as follows. For each n-tuple α = (α1, … αn) of non-negative integers let

In this paper we develop a theory of Laplace transforms for generalized functions. Some fundamental results in this field were given by Schwartz in (3) for the n-dimensional bilateral case from the point of view of topological vector spaces, and in (4) in a form amenable to operational use. Our presentation characterizes a one-dimensional theory of Laplace transforms with a half-plane of convergence (indicating an extension of the usual classical transform) and with the property that Laplace transforms are analytic functions satisfying the fundamental convolution-multiplication theorem. Section 1 is devoted to defining the Laplace transform of generalized functions and also to showing how the property of a half-plane of convergence is intrinsic to this definition.

The question posed by Sylvester (6) concerning the collinearity of a finite set of points in E2 having the property that each two together with some third be collinear has been the inspiration for numerous investigations. The original question was answered by the following theorem.

A generalized sequential machine (abbreviated gsm) is a 6-tuple (K, ∑, Δ, δ, λ, p1), where K, ∑, Δ are finite non-empty sets (of “states,” “inputs,” and “outputs” respectively), δ (the “next state” function) is a mapping of K X ∑ into K, λ (the “output” function) is a mapping of K X ∑ into Δ*, and p1 (the “start“ state) is a distinguished element of K.