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.

It is well known that any set of four anticommuting involutions (see §2) in a four-dimensional vector space can be represented by the Dirac matrices

(1)

where the B1,r are the Pauli matrices

(2)

(See (1) for a general exposition with applications to Quantum Mechanics.) One formulation, which we shall call the Dirac-Pauli theorem (2; 3; 1), is

Theorem 1. If M1,M2, M3, M4 are 4 X 4 matrices satisfying

then there is a matrix T such that

and T is unique apart from an arbitrary numerical multiplier.

Shapiro and Forman have presented in (4) an abstract formulation of prime number theorems which includes the various prime number theorems; for primes in arithmetic progressions, for prime ideals in ideal classes etc. The methods of proofs are “elementary” and follow closely Shapiro's proof for the primes in arithmetic progression (for reference see bibliography in (4)).

The author has followed in (1) some ideas of Yamamoto (5) on arithmetic linear transformations to introduce a symbolic calculus in dealing with arithmetic functions. This calculus proved to be very useful in unifying many of the “elementary” proofs in the behaviour of arithmetic functions. In (6) Yamamoto has extended his theory to ideals in algebraic number fields, and with this extension the symbolic calculus of (1) can be extended to cover the abstract case of prime number theorem in countable free abelian groups as discussed in (4).

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.

The only known examples of nondegenerate homogeneous plane continua are the simple closed curve, the circle of pseudo-arcs (6), and the pseudo-arc (1; 13). Another example, called the pseudo-circle, has been suggested by Bing (2), but it has not been proved to be homogeneous. (Definitions of some of these terms and a history of results on homogeneous plane continua can be found in (6).) Of the three known examples, the pseudo-arc is both linearly chainable and circularly chainable, and the simple closed curve and the circle of pseudo-arcs are circularly chainable but not linearly chainable. It is not known whether every homogeneous plane continuum is either linearly chainable or circularly chainable. Bing has shown that a homogeneous continuum is a pseudo-arc provided it is linearly chainable (4).

Let L be a Lie algebra over a field of characteristic 0 and let D(L) be the derivation algebra of L, that is, the Lie algebra of all derivations of L. Then it is natural to ask the following questions: What is the structure of D(L)? What are the relations of the structures of D(L) and L? It is the main purpose of this paper to present some results on D(L) as the answers to these questions in simple cases.

Concerning the questions above, we give an example showing that there exist non-isomorphic Lie algebras whose derivation algebras are isomorphic (Example 3 in § 5). Therefore the structure of a Lie algebra L is not completely determined by the structure of D(L). However, there is still some intimate connection between the structure of D(L) and that of L.

In this paper, we define and begin the study of an extensive family of simple n-person games based in a natural way on block designs, and hitherto for the most part unexplored except for the finite projective games (13). They should serve at least as a proving ground for conjectures about simple games. It is shown that many of these games are not strong and that many do not possess main simple solutions. In other cases, it is shown that they have no equitable main simple solution, that is, one in which the main simple vector has equal components. On the other hand, the even-dimensional finite projective games PG(2s, pn) with s > 1 possess equitable main simple solutions, although they are not strong either. These results are obtained by means of the study of the possible blocking coalitions. Interpretations in terms of graph theory, network flows, and linear programming are discusssed, as well as k-stability, automorphism groups, and some unsolved problems.

In this paper, A denotes a ring with an identity element 1, and B a subring of A containing 1 such that B satisfies the left and right minimum conditions, and A is a finitely generated left and right B-module. The identity element 1 is required to act as the identity operator on all modules which we shall consider. For any left B-module V, there is a standard construction of a left A -module which is, roughly speaking, the smallest A -module containing V.

A graph on n labelled nodes is a set of n objects called “nodes”, distinguishable from each other, and a set (possibly empty) of “edges,” that is, pairs of nodes. Each edge is said to join its pair of nodes, at most one edge joins any two nodes and no edge joins a node to itself. By a k-colouring of such a graph we mean a mapping of the nodes of the graph onto a set of k distinct colours, such that no two nodes joined by an edge are mapped onto the same colour. We take k > 1.

The equation Px = y in Banach spaces has aroused considerable interest, particularly in view of the various situations in applied analysis which it encompasses, and consequently it has been the topic of numerous investigations (2; 9; 10; 12). Detailed references may be found in (10). The equation is of special interest because of its interpretation as an integral equation; and in turn, many problems related to differential equations can be reformulated as integral equations (5; 7; 13).

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 following result has been conjectured by Dr. Birch. Let z1 z2, . . . , zn be any n complex numbers such that

(1)

Then

(2)

attains its greatest value when the z are at the vertices of a regular n-sided polygon inscribed in the circle |z| =1.

It seems to be difficult to prove this but Dr. Birch informs me that some work by Mullholland shows that the result is false for large n. I can, however, prove that the result is true for n = 3, and then Δ ≤ 27. The suggested general result would be Δ ≤ nn.

We study representations of o-orders, that is, of o-regular -algebras, in the case that o is a Dedekind domain. Our main concern is with those -modules, called -representation modules, which are regular as o-modules. For any -module M we denote by D(M) the ideal consisting of the elements x ∈ o such that x.Ext1(M, N) = 0 for all -modules N, where Ext = Ext(,0) is the relative functor of Hochschild (5). To compute D(M) we need the small amount of homological algebra presented in § 1. In § 2 we show that the -representation modules with rational hulls isomorphic to direct sums of right ideal components of the rational hull A of , called principal-modules, are characterized by the property that D(M) ≠ 0. The (, o)-projective -modules are those with D(M) = 0. We observe that D(M) divides the ideal I() of (2) for every M , and give another proof of the fact that I() ≠ 0 if and only if A is separable. Up to this point, o can be taken to be an arbitrary integral domain.

In (3) R. Lashof and S. Smale proved among other things the following theorem. If the compact oriented manifold M is immersed into the oriented manifold M', with dim M' ≥ dim M + 2, then the normal degree of the immersion is equal to the Euler-Poincaré characteristic x of M reduced module the characteristic x’ of M'. If M’ is not compact, x' is replaced by 0. “Manifold” always means C∞-manifold. An immersion is a differentiable (that is, C∞) map f whose differential df is non-singular throughout. The normal degree is defined in a certain fashion using the normal bundle of M in M', derived from f, and injecting it into the tangent bundle of M'

It is our purpose to give an elementary proof, using vector fields, of this theorem, and at the same time to identify the homology class that represents the normal degree (Theorem I), and to give a proof, using the theory of Morse, for the special case M’ = Euclidean space (Theorem II).

The object of this paper is to establish a simple connection between Thorn's theory of cobounding manifolds and the theory of modifications. The former theory is given in detail in (8) and sketched in (3), while the latter is worked out in (1). In particular in (1) it is shown that the only modifications which can transform one differentiable manifold into another are what I call below spherical modifications, which consist in taking out a sphere from the given manifold and replacing it by another. The main result is that manifolds cobound if and only if each is obtainable from the other by a finite sequence of spherical modifications.

The technique consists in approximating the manifolds by pieces of algebraic varieties. Thus if M1 and M2 form the boundary of M, the last is taken to be part of an algebraic variety such that M1 and M2 are two members of a pencil of hyperplane sections.

The problem of spectral analysis of non-self-adjoint (and non-normal) operators has received considerable attention recently. Livšic (5), and more recently Brodskii and Livšic (1) have considered operators on Hilbert space with completely continuous imaginary parts. Dunford (3) has generalized the notion of spectral measure and defined a class of spectral operators on Hilbert and Banach space. Schwartz (8) and Rota (7) have investigated conditions under which a differential operator will be spectral. The work of Naimark (6) and the author (4) on non-self-adjoint differential operators leads to an expansion theorem which implicitly defines a type of spectral measure. However the projections involved in this will not in general be bounded, much less uniformly bounded.

The problem of arranging v elements into v sets in such a way that every set contains exactly k distinct elements and that every pair of sets has exactly λ = k(k — l)/(v — 1) elements in common, where 0 < » < k < v, is equivalent to finding a normal integral v by v matrix A such that AT A = B, where B is the v by v matrix having k in every position on the main diagonal and λ in all other positions (10). Utilizing the fact that for the existence of a λ, k, v design it is necessary that I (the v by v identity matrix) represent B rationally, (2) and (3) have proved the non-existence of certain λ, k, v designs. Neither of the proofs utilize the fact that it is necessary that A be normal. However, Albert (1) for the projective plane case and Hall and Ryser (5) for the general design proved that if there exists a rational A such that ATA = B then there exists a normal rational matrix satisfying the same equation. Thus the requirement of normality does not exclude any λ, k, v which were not previously excluded.

Recently I (1) gave some new basic hypergeometric identities of the Cayley and Orr type with the help of a certain basic differential operator. The present paper deals with some bilateral generalizations of those identities together with certain new identities of the same type. In § 4 is indicated how the generalizations of Orr's identities given recently by Shukla (8, Theorems I, II) may be connected with each other. Later in § 5 certain general expansions of hypergeometric functions are deduced.

Let L0 be a differential operator of even order n = 2v on the half open interval 0 ≤ t < ∞ which is formally self adjoint and satisfies the conditions of Kodaira (5, p. 503). We consider a perturbed operator of the form L∈ = Lo + ∈q where q(t) is a real-valued bounded function and ∈ is a real parameter. The object of this paper is to set up conditions on the operator L0 and the function q(t) such that L∈ determines a self-adjoint operator H∈ and such that the spectral resolution operator E∈(Δ) corresponding to H∈ is analytic in a neighbourhood of ∈ = 0, where Δ is a closed bounded interval.

Our conditions are a natural generalization of conditions considered by Moser for the case n = 2(6). Moser has given a number of examples showing that when his conditions do not hold E∈(Δ) need not be analytic.

Combinatorial configurations may generally be phrased in terms of arrangements of objects into sets subject to certain conditions. In view of this, the question arises as to whether given a set S and its power-set Us (the class of all subsets of S), it might be possible to structure Us in a combinatorially significant manner. This paper proposes and investigates one such structuring achieved by defining a distance function over US.

Given A, B in Us, define their distance by

where N(E) denotes the number of elements in E, + ∞ being an admissible value.