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The topology of abstract 2-metric (area-metric) spaces has been the object of study in recent papers of Gähler (1) and Froda (2). The geometric properties of such spaces, however, have remained largely untouched since the initial work of Menger (3). As in ordinary metric spaces, a notion of 2-betweenness, or interiorness, can be easily defined in 2-metric spaces. In abstract metric spaces the betweenness relation is characterized among all relations defined on each triple of points of every metric space by six natural properties (4, pp. 33-40; 5). The purpose of this paper is to prove a similar theorem characterizing the relation of 2-betweenness in 2-metric spaces.

Let N be a finite set of n elements. A collection ﹛S1, S2, … , Sm﹜ of subsets of N is called a determining collection if an arbitrary subset T of N is uniquely determined by the cardinalities of the intersections Si ⋂ T, 1 ≤ i ≤ m. The purpose of this paper is to study the minimum value D(n) of m for which a determining collection of m subsets exists.This problem can be expressed as a coin-weighing problem (1; 7).In a recent paper Cantor (1) showed that D(n) = O(n/log log n), thus proving a conjecture of N. J. Fine (3) that D(n) = o(n). More recently Erdös and Rényi (2), Söderberg and Shapiro (7), Berlekamp, Mills, and Leo Moser have independently found proofs that D(n) = O(n/log n).

Throughout this paper the word “ring” will mean an associative ring which need not have an identity element. There are Artinian rings which are not Noetherian, for example C(p∞) with zero multiplication. These are the only such rings in that an Artinian ring R is Noetherian if and only if R contains no subgroups of type C(p∞) [1, p. 285]. However, a certain class of Artinian rings is Noetherian. A famous theorem of C. Hopkins states that an Artinian ring with an identity element is Noetherian [3, p. 69]. The proofs of these theorems involve the method of “factoring through the nilpotent Jacobson radical of the ring”. In this paper we state necessary and sufficient conditions for an Artinian ring (and an Artinian module) to be Noetherian. Our proof avoids the concept of the Jacobson radical and depends primarily upon the concept of the length of a composition series. As a corollary we obtain the result of Hopkins.

The existence of certain formulae analogous to Poisson's summation formula (9, pp. 60-64),where αβ = 2π, α > 0, and Fc(x) is the Fourier cosine transform of f(x), but involving number-theoretic functions as coefficients, was first demonstrated by Voronoï (10) in 1904. He proved thatwhere r(n) is an arithmetic function,/(x) is continuous in (a, b) and a(x) and i?(x) are analytic functions dependent on τ(n). Later, numerous papers were published by various authors giving formulae of this type involving d(n), the number of divisors of n (3), and rp(n), the number of ways of expressing n as the sum of p squares of integers (8).In 1937, Ferrar (4) developed a general theory of summation formulae, using complex analysis. Around that time, Guinand (5) also published papers where he developed the general theory from a different point of view. He applied the theory of mean convergence for the transforms of class L2(0, ∞ ). Later in 1950, Bochner (1) gave a general summation formula.

In two previous papers (see4;5) O. T. O'Meara and I investigated the problem of generating the integral orthogonal group of a quadratic form by symmetries in the case where the underlying ring of integers was the integers of a dyadic local field of characteristic not 2. In this paper, the investigation is concerned with a local field of characteristic 2. As in (5), only the unimodular case is treated. As in (4) and (5), groupsS(L), Xh(L), andO(L) are introduced for a unimodular latticeLand the relationship betweenS(L) andO(L) studied. As in the previously cited papers, generation by symmetries means thatS(L) =O(L). The following result is obtained.

In an earlier paper [5], the author defined a zeta function on the real sphere , whereas in the present paper it is proposed to define one on the unitary sphere where xi's are complex numbers and their complex conjugates. Following E. Cartan, harmonics on the unitary sphere are defined and then a zeta function formed just as in the case of a real sphere. The unitary sphere is seen to behave like an even-dimensional closed manifold, since results similar to the ones proved by the author and A. Pleijel [6] for closed manifolds (of even dimensions) are observed here also.

In a recent paper (3), Grünbaum has found a general and unifying setting for a number of properties of some special lines associated with a planar convex body. Besides various interesting results, two conjectures are stated and two kinds of convexity and polygonal connectedness are introduced.In the present paper, we shall prove one of Grünbaum's conjectures (§ 3, Theorem 1); we consider the other in § 4 and establish some related results in §§ 5 and 6. Six-partite problems are studied in this general setting (§ 7) and a question raised by Ceder (2) is answered. We give a generalization of the notion of a continuous family of curves in § 8, and discuss some new kinds of connectedness in § 9.

All groups considered in this paper are abelian. A subgroup N of a group G is said to be a quasi-essential subgroup of G if G = 〈H, K〉 whenever H is an N-high subgroup of G and K is a pure subgroup of G containing N. We started the study of such subgroups in [5]; in particular, we characterized subsocles of a primary group which were both quasi-essential and centres of purity. In this paper we show that quasi-essential subsocles of a primary group are necessarily centres of purity answering thus in the affirmative a question raised in [5].We obtain the following theorem: A subsocle S of a p-group G is quasi-essential if and only if either S ⊂ G1or (pnG)[p] ⊃ S ⊃ (pn+1G)[p] for some non-negative integer n. The notation is that of [1]. If G is a group, thenwhere p is a prime integer.

1. In this paper, X will always denote a locally compact Hausdorff space, C0(X) the algebra of all complex-valued continuous functions vanishing at infinity on X and B(X) the algebra of all bounded continuous complex-valued functions defined on X. If X is compact, C0(X) is identical to B (X) and all the results of this paper are obvious. Therefore, we will assume at the outset that X is not compact. If A represents an algebra of functions, AR will denote the algebra of all real-valued functions in A.

Error-correcting codes are used in several constructions for packings of equal spheres in ^-dimensional Euclidean spaces En. These include a systematic derivation of many of the best sphere packings known, and construction of new packings in dimensions 9-15, 36, 40, 48, 60, and 2m for m g 6. Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density A of the packings in En for n = 2m satisfies log A ~ — \n log log n as n —* oo. 1.1. Introduction. In this paper we make systematic use of error-correct ing codes to obtain sphere packings in En, including several of the densest packings known and several new packings. By use of cross-section s we then obtain packings in spaces of lower dimension, and by building up packings by layers we obtain packings in spaces of higher dimension. Collectively, these include all of the densest packings known, and further new packings are also con­ structed. Part 1 of the paper is devoted to groundwork for the constructions. § 1.2 introduces sphere packings, and §§ 1.3-1.8 survey the error-correcting code theory used in the later Parts. Part 2 describes and exploits Construction A, which is of main value in up to 15 dimensions. Part 3 describes Construction B% of main value in 16-24 dimensions. Part 4 digresses to deal with packings built up from layers, while Part 5 gives some special constructions for dimen­ sions 36, 40, 48 and 60. Part 6 deals with Construction C, applicable to dimensions n = 2m and giving new denser packings for m ^ 6. We conclude with tables summarizing the results. Table I, for all n S 24, supersedes the tables of [18; 19], and Table II gives results for selected n > 24. The tables may be used as an index giving references to the sections of the paper in which the packings are discussed. Partial summaries of this work have appeared in [22; 23]. General references for sphere packing are [18; 19; 31] and for coding theory [4; 25].

In this paper various concepts intrinsically defined by the differential equation1.1are interpreted geometrically by concepts analogous to those in the Minkowski plane. This is carried out in § 2. The point of such a development is that one may apply the techniques or transfer known results in the theory of curves (in particular, convex curves) to (1.1), thereby gaining an additional tool in the investigation of this equation. For an application of a result obtained in this way, namely (3.12), see (4).Throughout this paper,R(t)is a real-valued, continuous function ofton the real line (— ∞ < t < + ∞) and only the real solutions of (1.1) are considered.

The projective planes discovered in 1957 by Hughes [3] were originally described by means of a nearfield F satisfying the following conditions:(a) F is finite,(b) the centre and kernel of F coincide,(c) F is of rank 2 over its kernel.(The definitions of these terms will be given in § 2; the terminology used throughout the paper is that of [1].)Rosati [5] showed in 1960 that condition (a) is not necessary, thus constructing the first “infinite Hughes planes”. Condition (b), however, plays an essential part also in Rosati's work.The aim in this paper is to show that condition (b) is superfluous as well. For the finite case, this has been remarked by Ostrom [4] without proof; here we shall show that a “generalized Hughes plane” can be constructed over any nearfield satisfying condition (c) only.

In this paper we apply various results and methods of previous papers on the kernel to four-person games.Section 2 contains the basic definitions needed. In §3 we prove that the kernel of the general-sum four-person game consists of a line segment (which may shrink to a point). A method for classifying games according to their kernels is suggested in §4 and is used there to characterize all four-person games whose kernel consists of a non-degenerate interval. In the last section, §5, we offer a bargaining procedure, based on principles established in (1), which leads to the kernel in the case of a non-degenerate interval.

I t is straightforward, but tedious, to write down the integers whose representations in a given base do not have particular digits in certain positions. In the first section of this paper we give a computational scheme that enables us to carry out such operations in a rapid and simple fashion.In the second section of the paper we derive a general identity involving the digits of integers in arbitrary Cantor systems of notation.In the third section we apply this identity and deduce a number of results concerned with the splitting of integers into classes with equal power sums. The computational scheme of the first section leads us to an algorithm for the determination of such splittings.

A ring A has finite norm properly, abbreviated FNP, if each proper homomorphic image of A is finite. In [3], Chew and Lawn described some of the structural properties of FNP rings with identity, which they called residually finite rings. The twofold aim of this paper is to extend the results of [3] to arbitrary rings with FNP and to give characterizations of FNP rings independent of the results of [3].If A is a ring, let A+ denote A regarded as an abelian group. In the first section of this paper, we explore the effects of FNP upon the structure of A+. The following theorem is typical of the results in this section.

This paper is based on part of the thesis of one of the authors (5), submitted at the University of Toronto in 1963. In the first part of the paper a result on induced representations (2, 4, 9) is generalized slightly and a number of corollaries are derived. In the rest of the paper a special case of this result is applied to put the representation theory of the alternating group on a par with that of the symmetric group. A knowledge of the representation theory of Sn (7) on the part of the reader is assumed.

LetRbe the discrete group ring of the groupGover the ringA. In this paper we attempt to find necessary and sufficient conditions onGandAso thatRwill have some standard ring-theoretic property ; among the properties considered are those of being artinian, regular, self-injective, and semi-prime.The contents of this paper form essentially the author's doctoral thesis. The author would like to thank his supervisor Dr. J. Lambek for his generous encouragement and continued interest.

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

In a recent paper (7) Yood developed the beginnings of a theory of modular annihilator algebras. In this paper we extend his work on these algebras.The definition of modular annihilator algebra is algebraic in nature (see §4) ; in fact the algebra need not be assumed even topological. However, a significant number of important normed algebras are modular annihilator algebras. A list of examples is given in §8.The theory of modular annihilator algebras is related to the theory of certain important topological algebras. In §5 we consider the relationships between dual and annihilator algebras and modular annihilator algebras, and in §7, the relationship between completely continuous normed algebras and modular annihilator algebras.

Let Mm be a closed PL manifold of dimension m. Then a concordance between two PL-homeomorphisms h0, h1:M → M is a PL-homeomorphismH: M × I → M × I such that H|M × 0 = h0 and H|M × 1 = h. Concordance is an equivalence relation and in his paper [2], M. Kato classifies PL-homeomorphisms of Sp × Sq up to concordance. To do this he treats first the problem of classifying those homeomorphisms that induce the identity in homology, and then describes the automorphisms of the cohomology ring that can arise from homeomorphisms of Sp × Sq. In this paper we show that for sufficiently connected PL-manifolds that embed in codimension 1, one can extend Kato's classification of the homeomorphisms that induce the identity in homology.

For a formal definition of “map” the reader is referred to (7, §2). The maps in this paper are rooted by specifying an orientation for one of the edges. This also specifies a root vertex, the negative end of the root, and a root face, the face on the left of the root edge. Counting is, as usual, defined on isomorphism classes.Regular maps of even valence have been enumerated in a recent paper by Tutte. In this paper we determine the average number of trees in such maps, and include similar results for regular tri valent maps, that is, maps with three edges incident on every vertex. In the development for the latter, a formula for the number of trivalent maps with 2t vertices is produced.

In a paper published in 1956, Rund (4) developed the differential geometry of a hypersurface of n— 1 dimensions imbedded in a Finsler space of n dimensions, considered as locally Minkowskian.The purpose of the present paper is to provide an extension of the results of (4) and thus develop a theory for the case of m-dimensional subspaces imbedded in a generalized (Finsler) metric space.We consider an n-dimensional differentiable manifold Xn and we restrict our attention to a suitably chosen co-ordinate neighbourhood of Xn in which a co-ordinate system xi (i= 1, 2, … , n), is defined. A system of equations of the type xi = xi(t) defines a curve C of Xn the tangent vector dxi/dt of which is denoted by xi.

In an important paper (7), long overlooked, J. H. Redfield dealt with several aspects of enumerative combinatorial analysis. In a previous paper (1) I showed the relation between a certain repeated scalar product of a set of permutation characters of a symmetric group and Redfield's composition of his group reduction functions. Here I consider, from a group representational point of view, Redfield's idea of a range-correspondence and its application to enumeration of linear graphs. The details of the application of these ideas to more general enumerations are also given.

SummaryIn an earlier paper [2] we denned the chromatic sums g, q, l and h. We determined the derivatives of these sums with respect to the colour-number λ at the special values λ = 1 and λ = 2. In the present paper we find parametric equations for h, l and q in the case λ = τ + 1, where τ is the golden ratio. We obtain h, l and the basic indeterminate z explicitly in terms of the parameter u, but for q we exhibit only a cubic equation with coefficients depending on u. We obtain an exact formula for the coefficients in h by applying Lagrange's theorem to the parametric equations.

The definition and some of the properties of what may be called a Perron second integral (P2-integral) were given in a previous paper [4]. This integral starts with a function f(x) defined in an interval (a, c) and goes directly to a second primitive F(x) with the property that the generalized second derivative D2F is equal to f(x) for almost all x in (a, c). In the present paper the definition is changed slightly and further properties are deduced.

HNN groups have appeared in several papers, e.g., [3; 4; 5; 6; 8]. In this paper we use the results in [6] to obtain a structure theorem for the subgroups of an HNN group and give several applications.We shall use the terminology and notation of [6]. In particular, if K is a group and {φi} is a collection of isomorphisms of subgroups {Li} into K, then we call the group1the HNN group with base K, associated subgroups { Li,φi(Li)} and free part the group generated by t1, t2, …. (We usually denote φi(Li) by Mi or L–i.) The notion of a tree product as defined in [6] will also be needed.

Throughout this paper, Z is the ring of integers, ƒ*(t) (ƒ(t)) is an integer monic (co-monic) polynomial in the indeterminate t (i.e., each coefficient of ƒ* (ƒ) is in Z and its highest (lowest) coefficient is 1 (5, p. 121, Definition) and M* (M) is the multiplicative semigroup of all integer monic (co-monic) polynomials ƒ* (ƒ) having no constant term. In (3, Theorem 2), Herstein proved that if R is a division ring with centre C such that1then R = C. In this paper we seek a generalization of Herstein's result to semi-simple rings. We also study the following condition:(1)*Our results are quite complete for a semi-simple ring R in which there exists a bound for the codegree ofƒ (ƒ*) (i.e., the degree of the lowest monomial of ƒ(ƒ*)) appearing in the left-hand side of (1) ((1)*).

Soon after the publication of Tutte's paper [5] on m-cages, H. S. M. Coxeter asked in a letter to the author whether one-regular graphs of degree 3 exist. The purpose of the following paper is to show by an example that the answer is in the affirmative.

In a previous paper (1) the authors considered the problem of finding an asymptotic formula for numbers or functions Bn,m whose generating function is of the form(1.1),where Pm(x) is a polynomial of degree m in x given by(1.2), am≠0.The above-mentioned paper contained the restriction that ak ≥ 0.

Throughout this paper, the symbol P = [Pij] will represent the transition probability matrix of an irreducible, null-recurrent Markov process in discrete time. Explanation of this terminology and basic facts about such chains may be found in (6, ch. 15). It is known (3) that for each such matrix P there is a unique (except for a positive scalar multiple) positive vector Q = {qi} such that QP = Q, or1this vector is often called the "invariant measure" of the Markov chain.The first problem to be considered in this paper is that of determining for which vectors U(0) = {μi(0)} the vectors U(n) converge, or are summable, to the invariant measure Q, where U(n) = U(0)Pn has components2In § 2, this problem is attacked for general P. The main result is a negative one, and shows how to form U(0) for which U(n) will not be (termwise) Abel summable.

The Szegö polynomials are defined on T, the real numbers modulo 1. In this paper and in its sequel we give a generalization of Szegö polynomials in which T is replaced by an arbitrary locally compact abelian group θ on whose dual there has been distinguished a measurable order relation compatible with the group structure. The present paper is devoted to the case where θ is compact and therefore discrete. The general case will be taken up in the sequel mentioned above. It is desirable to proceed in this way because the case θ compact is much simpler and much more like the classical situation than is the general case, in which various measure-theoretic difficulties obtrude. Moreover, as it happens, it is possible to develop the theory in this way with relatively little repetition.

In a previous paper (2) I defined the fundamental group σ(X, x0, G) of a group Gof homeomorphisms of a space X, and showed that if the transformation group admits a family of preferred paths, then σ(X, x0, G) can be represented as a group extension of π1(X, x0) by G. In this paper the homotopy groups of a transformation group are defined. The nth absolute homotopy group of a transformation group which admits a family of preferred paths is shown to be representable as a split extension of the nth absolute torus homotopy group τn(X, x0) by G.In § 6 it is shown that the action of G on X induces a homomorphism of Ginto a quotient group of a subgroup of the group of automorphisms of τn(X, x0). This homomorphism is used to obtain a necessary condition for the embedding of one transformation group in another, in particular, for the embedding of a discrete flow in a continuous flow with the same phase space.

For various classesQof metric spaces, there are several well-known results characterizing the localn-connectivity of a metric space in terms ofn-ANR(Q)'s. Specifically, we have in mind the results of Kuratowski (13, p. 265) and Kodama (10, p. 79). The main purpose of this paper will be to obtain similar results along these lines for non-metric classesQ.In the last part of the paper we specifyQto be the class of totally normal spaces and characterize the localn-connectivity of ann-dimensional separable metric space in terms of ANR(Q)'s.

The problem of the field equations and the equations of motion in general relativity theory is now sufficiently clarified. The equations of motion can be deduced from pure field equations by treating matter as singularities, [2; 3], or from field equations with the energy momentum tensor [4]. Recently two papers appeared in which the problem of the coordinate system was considered [5; 8]. The two papers are in general agreement as far as the role of the coordinate system is concerned. Yet there are some differences which require clarification.

Measures may be obtained from suitable non-negative valued functions in a number of ways. It is the purpose of this paper to present an abstract formulation of certain principles which may be used to construct measures, and to show that the various methods most frequently encountered in the literature are in fact all special applications of these principles.The basic requirements A1-A4 are set forth in § 2, and it is there shown that a measure can be denned when these are fulfilled. These requirements are satisfied in every case known to the writer. Usually, however, conditions stronger than A1-A4 hold, and it is these extra restrictions which yield information on the class of measurable sets and other matters. In §§ 3, 4, and 5 certain abstracts form of such restrictions are considered, and results are derived thereforom. The paper concludes with an analysis of how a number of measures occur as special cases of the theory.

The aim of this paper is to present a unified treatment of certain theorems in Combinatorial Analysis (particularly in enumerative graph theory), and their relations to various results concerning symmetric functions and the characters of the symmetric groups. In particular, it treats of the simplification that is achieved by working with S-functions in preference to other symmetric functions when dealing with combinatorial problems. In this way it helps to draw closer together the two subjects of Combinatorial Analysis and the theory of Finite Groups. The paper is mainly expository; it contains little that is really new, though it displays several old results in a new setting.

This paper deals with a particular class of distributive properties which appear to be important for Analysis and which I call frequencies. They can be defined for any kind of sets but it essential for proper application that a condition L (the statement of Lindelöf's lemma) is satisfied. From this condition follows Theorem 1, which is characteristic for frequencies but does not hold for other distributive properties. For every frequency F of a space Σ, one can build up an Analysis mod F of Σ the classical case is the Analysis mod F0. It is convenient to introduce the words “nearly every” with such a meaning that “every” and “almost every” are the special cases which, when we use the notation of this paper, correspond to F = F0 and F = FC.

In 1955 Taylor [6] constructed an AP-integral sufficiently strong to integrate Abel summable series with coefficients o(n). He showed that the AP-integral includes the special Denjoy integral and further that, when applied to trigonometric series, the AP-integral is more powerful than the SCP-integral of Burkill [1] and the P2-integral of James [3]. The present paper shows that the AP-integral includes the SCP-integral, and, under natural assumptions, the P2-integral.After completing this manuscript I was advised by Skvorcov that he had shown [5] under more general conditions that the P2-integral is included in the AP-integral. The proof in the present paper seems to have some value in its own right and is considerably shorter.Since the definition of the AP-integral is essentially for a function defined in (0, 2π] and elsewhere by 2π-periodicity, we shall consider SCP-integrable and P2-integrable functions defined similarly.

In two earlier papers * the writer discussed involutions of periods five and seven on certain cubic surfaces in S3. In this paper, a quartic surface containing a cyclic involution of period eleven is considered. The surface

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.

R. E. Johnson (10), Utumi (18), and Findlayand Lambek (7) have defined for each ring R a unique maximal "ring of right quotients" Q. When R is a commutative integral domain (in this paper an integral domain need not be commutative) or an Ore domain, then Q is the usual division ring of quotients of R. Moreover, it is well known that in these special cases, if R is totally ordered, then so is Q.The main purpose of this paper is to study the ring of quotients Q, and in particular its order properties, for certain lattice-ordered rings R.

1. 1. This paper is the second in a series of papers discussing linear groups of prime degree, the first being (8). In this paper we discuss only linear groups of degree 7. Thus, G is a finite group with a faithful irreducible complex representation Xof degree 7 which is unimodular and primitive. The character of Xis x- The notation of (8) is used except here p= 7. Thus Pis a 7-Sylow group of G.In §§ 2 and 3 some general theorems about the 3-Sylow group and 5-Sylow group are given. In § 4 the statement of the results when Ghas a non-abelian 7-Sylow group is given. This corresponds to the case |P| =73 or |P|= 74. The proof is given in §§ 5 and 6. In a subsequent paper the results when Pis abelian will be given.

This paper continues the study, begun in [7], of the spectral theory of non-self-ad joint second order ordinary differential operators on a half-line. The case of a ‘Very small” potential was studied in [4; 5; 6]. The case considered in [7], and in the present paper, is that where the potential is not so small.

In a series of papers which will follow this paper the authors will present a theory of functionals which are bilinear over a product A × B of two normed vector spaces A and B. This theory will include a representation theory, a variational theory, and a spectral theory. The associated characteristic equations will include as special cases the Jacobi equations of the classical variational theory when n = 1, and self-adjoint integrodifferential equations of very general type. The bilinear theory is oriented by the needs of non-linear and non-bilinear analysis in the large.

This paper is a partial solution of problem 24 in (2) which suggests that the finiteness of the partially ordered semigroups generated by various combinations of operators on classes of universal algebras be investigated. The main result is that the semigroups generated by the following sets of operators (for definitions see §2) are finite: {H, S, P, Ps}, {C, H, S, P, PF} {C, H, S, PU, PF}.This paper is part of the author's Master's thesis written in the Department of Mathematics at McMaster University. The author is indebted to the referee for his helpful suggestions.

Eigenvalue problems of the form Af = λBf, where λ is a complex parameter and A and B are operators on a Hilbert Space, have been considered by a number of authors (e.g., [1; 3; 5; 7; 10]). In this paper, we shall be concerned with the existence and nature of eigenfunction expansions associated with such problems, with no assumptions of self-adjointness. The form of the theorems to be given here is: if the system (A, B) is spectral and complete (definitions below), and F and G are operators satisfying certain “smallness” conditions, then (A + F, B + G) is also spectral and complete. The hypotheses for these theorems are chosen with an eye to applying the results to boundary-value problems on a compact interval. Such applications, together with an examination of circumstances under which the system (Dn, Dm) (D denoting differentiation) is spectral and complete under a broad class of boundary conditions, will be made in a later paper.

In this paper I use techniques developed by Mirsky and Perfect (5) to generalize the extremely close relationship between transversal theory and the theory of matroids or independence structures. I extend in two directions a fundamental theorem of Rado (8) and use the techniques of Mirsky and Perfect to obtain easy proofs of known and unknown results about systems of representatives with repetition.2. Basic concepts. In this section I review the results used subsequently. Throughout the paper, S will denote a finite set and A will denote the collection of subsets of S, {Ai i ∈ I}, where I is a finite index set. |K| will denote the cardinality of a set K and I use the notation

By a Reidemeister plane we mean a projective plane having the property that every ternary ring coordinatizing it has associative addition. Finite Reidemeister planes have been investigated by Gleason (2), Liineburg (6), and Kegel and Luneburg (4). In the first paper, Gleason proved that if the order of the plane is a prime power, then it is Desarguesian. Luneburg showed that this is still true if the order is not 60. In the third paper, this last restriction is removed. For infinite planes, the only result is the following theorem due to Pickert (7, p. 301).

In this paper we study the problem of representing groups as groups of automorphisms on an orthomodular lattice or poset. This problem not only has intrinsic mathematical interest but, as we shall see, also has applications to other fields of mathematics and also physics. For example, in the “quantum logic” approach to an axiomatic quantum mechanics, important parts of the theory can not be developed any further until a fairly complete study of the representations of physical symmetry groups on orthomodular lattices is accomplished [1].We will consider two main topics in this paper. The first is the analogue of Schur's lemma and its corollaries in this general setting and the second is a study of induced representations and systems of imprimitivity.

These notes are to supplement my paper (4), and should be read in conjunction with it. Both are divided into three parts, and in these notes the section numbers have a further digit added; thus §1.41 here supplements §1.4 of (4). References by section numbers are always to (4) or to the present notes, but references to other papers are numbered independently.The principal results of these notes are the following. New sphere packings are given in [2m], m ⩾ 6, and in [24], which are twice as dense as those of §§1.6, 2.3. Others are given in [2m], m ⩾ 5, with the same density as those of §1.6, but in which each sphere touches fewer other spheres than in the earlier packings.