Professor R. A. Dean has proved (1, Theorem 3) that a completely free lattice generated by a countable partially ordered set is isomorphic to a sublattice of a free lattice. In particular, it follows that a free product of countably many countable chains can be isomorphically embedded in a free lattice. Generalizing this we show (2.1) that the class of all lattices that can be isomorphically embedded in free lattices is closed under the operation of forming free lattice-products with arbitrarily many factors. We also prove (2.4) that this class is closed under the operation of forming simply ordered sums with denumerably many summands. Finally we show (2.7) that every finite dimensional sublattice of a free lattice is finite.

The purpose of this note is to characterize those distributive lattices that can be isomorphically embedded in free lattices. If it is known (cf. (2)) that in a free lattice every element is either additively or multiplicatively irreducible, and consequently every sublattice of a free lattice must also have this property. We therefore begin by studying the class of all those distributive lattices in which this condition is satisfied.

The problem we study is a generalization of a problem first solved by Tonolo (6), then generalized successively by Schouten (5), Nijenhuis (4), Haantjes (3), and Nijenhuis-Frölicher (2). The Tonolo- Schouten approach is distinct from that of Nijenhuis-Haantjes-Frölicher in the sense that the former consider the problem on a Riemannian space, while the latter consider it on a manifold without any further structure.

The object of investigation is the integrability of the distribution θ of vector subspaces θP of the tangent space Tp to a manifold M, when θP is intrinsically related to a given field h on M, of linear transformations hp on Tv. The research has so far been restricted to certain types of h.

The unboundedness of the sequence of Lebesgue constants (norms), at a point, of certain transforms implies, as is well known, that there exist (i) a continuous function whose transform fails to converge to the function at the point in question (the du Bois-Reymond singularity), and (ii) another such function whose transform, while converging everywhere to the function, does not do so uniformly in any neighbourhood of the stipulated point (the Lebesgue singularity). The converses also hold in our case.

Many extremal properties of the class of normalized univalent functions are shared by the class of typically real functions each considered in the unit circle. By the class T of typically real functions we mean those functions f(z), regular for |z| < 1 with f(0) = 0, f'(0) = 1, and such that f(z) > 0 for z > 0, (z) < 0 for > 0. This class was first studied by Rogosinski (4) who proved various simple properties for it. Later Robertson (3) took up the study and proved the following important representation result.

In this paper we consider complete orthonormal sequences defined on the interval [0, 1] and satisfying an inequality of the type

(1)

for all n and some sequence {Fn}. Such sequences were first considered by Zygmund and Marcinkiewicz (8). They extended the well-known results of Hausdorff-Young and Paley, originally proved for the case v = ∞, Fn = M for all n (12). We will consider cases of equality in t he Hausdorff-Young theorems and certain limiting cases of the Paley theorems. Application of these results and the results in (8) will be made to functions harmonic in the unit α-sphere.

In the considerable literature on linear operators in L2 or Lp arising from ordinary differential operators it has always been assumed that the coefficient of the highest order derivative appearing does not vanish in the interior of the interval under consideration. If this coefficient vanishes at one or both endpoints of the interval, or if one or both of the endpoints is infinite the differential operator is said to be singular. In this paper we shall allow this leading coefficient to vanish in the interior of the interval, and show that the theory of such operators can sometimes be reduced to a consideration of several operators of the well-known type. We shall also indicate how those which cannot be so reduced should be dealt with.

In this paper we shall be concerned with two problems: (i) the asymptotic behavior of solutions of parabolic inequalities and (ii) the uniqueness of the Cauchy problem for such inequalities when the data are prescribed on a portion of a time-like surface. The unifying feature of these rather separate problems is the employment of integral estimates of the same type in both cases.

We consider parabolic operators in self-adjoint form

(1)

as well as the non-self-adjoint operator

(2)

where the coefficients aij(x, t) = aij (x1, x2... , xn, t) are C1 functions of x and t and the bij= bij(x, t) are C2 functions of x and t.

Define f(k, l) as the least integer so t h a t every graph having f(k, l) vertices contains either a complete graph of order k or a set of l independent vertices (a complete graph of order k is a graph of k vertices every two of which are connected by an edge, a set of I vertices is called independent if no two are connected by an edge).

Throughout this paper c1, c2, … will denote positive absolute constants. It is known (1, 2) that

(1)

and in a previous paper (3) I stated that I can prove that for every ∈ > 0 and l > l(∈), f (3, l) > l2-∈. In the present paper I am going to prove that

(2)

A classical theorem states that if a square matrix B over an algebraically closed field F commutes with all matrices X over F which commute with a matrix A over F, then B must be a polynomial in A with coefficients in F (2). Recently Marcus and Khan (1) generalized this theorem to double commutators. Our purpose is to complete the generalization to commutators of any order.

Let F be an algebraically closed field and let Fn be the ring of all n by n matrices with elements in F. We define ΔYZ — = [Z, Y] = ZY — YZ for all Y, Z in Fn.

Euler (6) in 1782 first studied orthogonal latin squares. He showed the existence of a pair of orthogonal latin squares for all odd n and conjectured their non-existence for n = 2(2k + 1). MacNeish (8) in 1921 gave a construction of n — 1 mutually orthogonal latin squares for n = p with p prime and of n(v) mutually orthogonal squares of order v where

with p1 p2, … , Pr being distinct primes and

A ring R over a commutative ring K, that has a basis of elements g1, g2, … , gn forming a group G under multiplication, is called a group ring of G over K. Since all group rings of a given G over a given K are isomorphic, we may speak of the group ring KG of G over X.

Let π be any partition of G into non-empty sets GA, GB, … . Any subring P of KG that has a basis of elements

is a partition ring of G over K.

If P is a partition ring of G over Z, the ring of integers, then the basis A, B, … for P clearly serves as a basis for a partition ring P’ = Q ⊗ P of G over Q, the field of rationals.

We are mainly interested in the study of prime and semi-prime rings and their rings of quotients. However, our argument proceeds largely in the category of modules (§ 1 to 4) and bimodules (§ 5 to 7).

After a brief description of the generalized rings of quotients introduced recently by Johnson, Utumi, and Findlay and the present author, we study a closure operation on the lattice of submodules of a module. For the lattice of left ideals of a ring, the concept of closed submodules reduces to the If-ideals of Utumi. The lattice of closed submodules of a module is always a complete modular lattice. We are specially interested in the case when it is a complemented lattice. This happens, in particular, when the singular submodule of Johnson and Wong vanishes. We consider the lattice of closed right ideals of a prime ring S and determine the maximal ring of right quotients of S in the case when this lattice has atoms.

Shephard and Todd (5) give generators for the finite primitive irreducible groups generated by two unitary reflections in U2. It is the purpose of the present paper to give generating reflections, and defining relations in terms of these reflections, for the seven such groups requiring three generating reflections, that is, for their nos. 7, 11, 12, 13, 15, 19, 22. The reflections are chosen whenever possible so that their product has the property suggested by Theorem 5.4 of (5). That is, except for no. 15, the period of the product of the three generating reflections is h = m2 + 1, and the characteristic roots of this product are 2πim1/h and 2πim2/h, where m1 and m2 are the “exponents“ (5, p. 282) of the group. The reason for the impossibility of such a choice for no. 15 is given in § 4. In § 5 the homomorphisms between these groups and certain groups of motions in elliptic 3-space are determined.

In 1902, F. R. Moulton (12) gave an early example of a non-Desarguesian plane. Its ‘points” are ordered pairs (x, y) of real numbers. Its “lines” coincide with lines of the real affine plane except that lines of negative slope are “bent” on the x-axis, line {y = b + mx}, for negative m, being replaced by {y = b + mx if y ≤ 0, y = [m/2]. [x + (b/m)] if y > 0}. A certain Desarguesian configuration in the classical plane is shifted just enough to vitiate Desargues’ Theorem for Moulton's geometry. The plane is neither a translation plane (“Veblen-Wedderburn” in the sense of Hall (7), p. 364) nor even the dual of one (Veblen and Wedderburn (17). It is natural to ask if the same construction is feasible when real numbers are replaced by elements from an arbitrary field.

Les propriétés anallagmatiques de groupes de sphères ont été étudiées dans des contextes divers ces dernières années (voir (4), (5)). Dans la note qui suit nous étudions les propriétés anallagmatiques de certains groupes de 4 sphères, nous plaçant à un point de vue élémentaire.

A fin de ne pas alourdir la rédaction nous omettrons de spécifier chaque fois que les points, droites ou sphères considérés sont toujours supposés être dans la position relative la plus générale possible compatible avec les définitions données.

L'index i pourra toujours prendre les valeurs i = 1, 2, 3, 4.

This paper deals with processes of combining convex bodies in Euclidean n-space which are, in a sense, dual to the process of Minkowski addition and some of its generalizations.

All the convex bodies considered will have a common interior point Q. Variables x and y denote vectors drawn from Q; we shall speak of their terminal points as the points x and y. Unit vectors will be denoted by u; ||x|| signifies the length of x. Convex bodies will be symbolized by K with distinguishing marks. ∂K means the boundary of K. λK will mean the image of K under a homothetic transformation in the ratio λ : 1. The centre of the homothety will always be Q.

Let K be a subset of BV(0, 1)—the space of functions of bounded variation on the closed interval [0, 1]. By the Hausdorff moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a given sequence μ = {μn|n = 0, 1, 2, …} there should be a function α ∈ K so that

(1)

For various collections K this problem has been solved—see (3, Chapter III)

By the trigonometric moment problem for K we shall mean the determination of necessary and sufficient conditions that corresponding to a sequence c = {cn|n = 0, ± 1, ± 2, …} there should be a function α ∈ K so that

(2)

For various collections K this problem has also been solved—see, for example (4, Chapter IV, § 4). It is noteworthy that these two problems have been solved for essentially the same collections K.

The problem of expanding an arbitrary function in a series of characteristic solutions of the ordinary differential equation

(1.1)

and the boundary relations

(1.2)

is well known. The various discussions are distinguished by the manner in which a parameter λ appears in the differential system and by the number of points at which the boundary conditions apply. The case in which the boundary conditions apply at intermediate as well as at the end points of a fundamental interval has been considered by Wilder (3).

Autonomous systems. This note concerns the stability of systems of (real) differential equations in the large on Euclidean space En and on certain Riemannian manifolds Mn. The results will be refinements of those of Krasovski (3), (4), (5) and of Markus and Yamabe (8) and will make clear the role of the various assumptions in the type of theorems under consideration.

In this section, the main theorems are stated for autonomous systems

(1)

Their proofs are given in § 2, 3, 4. In § 5, 6, 7, generalizations to non-autonomous systems are made.