Search

The main problem of the present paper is the existence of the Burkill integral of an interval function ƒ(I) which is not supposed to be continuous. Little is known about this case, though otherwise the theory of the integral can be considered as complete: we may refer to Ringenberg's comprehensive paper (2) in which further references are given.

By an Euler polyhedron of valence three or a trivalent convex polyhedron in Euclidean 3-space (4) we mean in the present paper an Euler polyhedron in the sense of Steinitz (8, p. 113), such that each vertex is incident with exactly three edges.In the present paper we establish a theorem concerning the colouring of trivalent polyhedra. A specialization of this theorem solves the following problem implicit in Eberhard (1, p. 84): Does there exist a trivalent Euler polyhedron with an odd number of faces such that the number of edges incident with any face is divisible by three?

In (1) Baer studied the following problem: If a torsion-free abelian group G is a direct sum of groups of rank one, is every direct summand of G also a direct sum of groups of rank one? For groups satisfying a certain chain condition, Baer gave a solution. Kulikov, in (3), supplied an affirmative answer, assuming only that G is countable. In a recent paper (2), Kaplansky settles the issue by reducing the general case to the countable case where Kulikov's solution is applicable. As usual, the result extends to modules over a principal ideal ring R (commutative with unit, no divisors of zero, every ideal principal).The object of this paper is to carry out a similar investigation for pure submodules, a somewhat larger class of submodules than the class of direct summands. We ask: if the torsion-free i?-module M is a direct sum of modules of rank one, is every pure submodule N of M also a direct sum of modules of rank one? Unlike the situation for direct summands, here the answer depends heavily on the ring R.

If G is a finite group which has a faithful complex representation of degree nit is said to be a linear group of degree n. It is convenient to consider only unimodular irreducible representations. For n ≦ 4 these groups have been known for a long time. An account may be found in Blichfeldt's book (1). For n= 5 they were determined by Brauer in (4). In (4), many properties of linear groups of prime degree pwere determined for pa prime greater than or equal to 5.In a forthcoming series of papers these results will be extended and the linear groups of degree 7 determined. In the first paper, some general results on linear groups of degree p, p≧ 7, will be given. These results will later be applied to the prime p = 7.

We say a (plane) set A contains all sets of some type if, for each B of type , there is a subset of A that is congruent to B. Recently, Besicovitch and Rado [3] and independently, Kinney [5] have constructed sets of plane measure zero containing all circles. In these papers it is pointed out that the set of all similar rectangles, some sets of confocal conies and other such classes of sets can be contained in sets of plane measure zero, but all these generalizations rely in some way on the symmetry, or similarity of the sets within the given type.In this paper we construct a set of plane measure zero containing all finite polygonal arcs (i.e., the one-dimensional boundaries of all polygons with a finite number of sides) with slightly stronger results if we restrict our attention to k-gons for some fixed k.

In this paper we give local conditions for a rectilinear embedding of a non-bounded combinatorial manifold,Mn, in Euclidean space, which are sufficient to prove thatMnhas a transverse field (see 1.1 and 1.2, definitions).In a sequel to this paper (6), we will show how with this transverse field we can construct a normal microbundle for the embedded manifoldMn.Our object in this research was only to obtain an existence theorem for normal microbundles. However, the method of proof via the construction of a transverse field yields as corollaries by Cairns (1), Whitehead (9), or Tao (8), results on smoothing. Earlier smoothing results achieved by the construction of transverse fields in the special case of (global) codimension 1 were obtained by Noguchi (5), and Tao (7; 8).After the research for this paper was completed, a paper of Davis (2) came to our attention.

In this paper we deal with the number of zeros of a solution of the nth order linear differential equation1.1where the functions pj(z) (j = 0, 1, …, n – 2) are assumed to be regular in a given domain D of the complex plane. The differential equation (1.1) is called disconjugate in D, if no (non-trivial) solution of (1.1) has more than (n – 1) zeros in D. (The zeros are counted by their multiplicity.)The ideas of this paper are related to those of Nehari (7; 9) on second order differential equations. In (7), he pointed out the following basic relationship. The function1.2where y1(z) and y2(z) are two linearly independent solutions of1.3is univalent in D, if and only if no solution of equation(1.3) has more than one zero in D, i.e., if and only if(1.3) is disconjugate in D.

In the preceding paper Bose, Shrikhande, and Parker give their important discovery of the disproof of Euler's conjecture on Latin squares. In this paper we show that their results can be strengthened to imply that N(n), the maximal number of pairwise orthogonal Latin squares of order n, tends to infinity with n. In fact there exists a positive constant c, such that N(n) > nc for all sufficiently large n.Our proof involves no new combinatorial insights, but is based entirely on a number-theoretical investigation of the following inequality due to Bose and Shrikhande.

In the first part of this paper we define a partial ordering on the set of all homogeneous identities and find necessary and sufficient conditions that an identity does not imply any identity lower than it in the partial ordering (we call such an identity irreducible). Perhaps the most interesting property established for irreducible identities is that they are skew-symmetric in any two variables of the same odd degree and symmetric in any two variables of the same even degree. The results of the first section are applied to commutative algebras in the remainder of the paper.

In previous papers (7; 8), we have investigated certain properties of general function algebras which may be regarded as generalizations or analogues of familiar results in the theory of analytic functions of several complex variables. This investigation is continued and expanded in the present paper. The main results concern a notion of holomorphic convexity for the general situation. We also extend somewhat several of the results obtained in the earlier papers.

In a recent paper by Douglas Underwood [8] several definitions of “associated prime divisors” were discussed and shown to be unique. In this note we produce a fifth type, which is due to W. Krull, and is found in his classical paper [2] and further discussed by B. Banaschewski in[1].Historically this characterization considerably predates the other four definitions.Throughout this note,Rdenotes a commutative ring with unity, and all ideals and elements are assumed to be in such a ring. We shall let upper case letters, most frequently the beginning of the alphabet, denote ideals and lower case letters, elements ofR.On the whole, our terminology will be that of [9]. We do, however, take exception with [9] in two instances, viz.

1.1 This paper gives a lattice theoretic investigation of “finiteness“ and “continuity of the lattice operations” in a complemented modular lattice. Although we usually assume that the lattice is-complete for some infinite,3we do not require completeness and continuity, as von Neumann does in his classical memoir on continuous geometry (3); nor do we assume orthocomplementation as Kaplansky does in his remarkable paper (1).1.2. Our exposition is elementary in the sense that it can be read without reference to the literature. Our brief preliminary § 2 should enable the reader to read this paper independently.1.3. Von Neumann's theory of independence (3, Part I, Chapter II) leans heavily on the assumption that the lattice is continuous, or at least upper continuous.

In this paper we shall derive a concise formula for the number of Euler graphs on n labelled nodes and k edges. An Euler graph is a connected graph in which every node has even valency, where by the valency of a node is meant the number of edges which are incident with that node. Throughout most of the paper we shall be dealing with graphs whose nodes have even valencies but which may or may not be connected. For convenience we shall refer to these graphs as Euler graphs, although the usage is not, strictly speaking, correct. We shall impose the condition of connectedness in § 4.

In this paper we consider the value distribution of a meromorphic function whose behaviour is prescribed along a spiral. The existence of extremely wild holomorphic functions is established. Indeed a very weak form of one of our results would be that there are holomorphic functions (in the unit disc or the plane) for which every curve “tending to the boundary” is a Julia curve.The theorems in this paper generalize results of Gavrilov [7], Lange [9], and Seidel [11].I wish to express my thanks to Professor W. Seidel for his guidance and encouragement.2. Preliminaries. For the most part we will be dealing with the metric space (D, ρ) where D is the unit disc, |z| < 1, and ρ is the non-Euclidean hyperbolic metric on D. The chordal metric on the Riemann sphere will be denoted by x.

The theorems of the present paper deal with conditions which are necessary and sufficient in order that a solvable or nilpotent infinite group should have a completely reducible matrix representation over a given algebraically closed field.It is known (17) that a locally nilpotent group of matrices is always solvable. Thus the first theorem of the present paper is a partial generalization of Theorem 1 of (16), which states:If G is a locally nilpotent subgroup of the full linear group GL(n, P) over a perfect field P, then G is completely reducible if and only if each matrix of G is diagonizable (by a similarity transformation over some extension field of P).

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.

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.