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In the paper of the title [1], a number of problems are posed. Ne­ gative solutions of two of them (Problems 2 and 3) are derived in a straightforward way from a paper of L. Gillman and the present author [2]. Motivation will not be supplied since it is given amply in [1], but enough definitions are given to keep the presentation reasonably self­ contained. 1. A Hausdorff space X is said to satisfy (Qm), where m is an in finite cardinal, if, whenever U and V are disjoint open subsets of X such that each is a union of the closures of less than m open subsets of X, then U and V have disjoint closures. In particular, a normal (Hausdorff) space X satisfies (Q*,) if and only if disjoint open F.-SUbsets of X have disjoint closures. (For, an open set that is the union of less than ~, closed sets is a fortiori an F.. Conversely if U is the union of countably many closed subsets Fo> then since X is normal, for each n there is an open set Un containing Fn whose closure is contained in U. Thus U is the union of the closures of the open sets Un.) In Prob­ lem 3 of [1], it is asked if every compact (HaUSdorff) space satisfying (Qm) for some m>~o is necessarily totally disconnected, and it is re­ marked that this is the case if the first axiom of countability is also as­ sumed. If X is a completely regular space, let C(X) denote the ring of all continuous real-valued functions on X, and let Z(f)= {x EX: f(x)=O}, let P(f)={xEX:f(x»O}, and let N(f)=P(-f). As usual, let (IX denote the Stone-Cech compactification of X. If every finitely generated ideal of C(X) is a principal ideal, then X is called an F-space. The fol­ lowing are equivalent. ( i) X is an F-space. ( ii) If f E C(X), then P(f) and NU) are completely separated [2, Theorem 2.3]. (iii) If f E C(X), then every bounded g E C(X-ZU» has an ex­ tension g E C(X) [2, Theorem 2.6]. A good supply of compact F-spaces is provided by the fact that if X is locally compact and ,,-compact, then {lX-X is an F-space [2, Theo­ rem 2.7].

In a beautiful paper which appeared in 1939 ([4]), F. Gantmacher made a thorough study of automorphisms of semi-simple Lie algebras over the field of complex numbers. Among other things, he defined the index n(G i ) of a connected component G i of the automorphism group G = G(8) as the minimum multiplicity of the characteristic root 1 for elements of G i . The main purpose of this note is the determination of these indices. It is somewhat surprising that this does not appear in Gantmacher’s paper since all the methods for deriving the formula for index G i are available in his paper. The secondary purpose of this note is to extend Gantmacher’s theory to the case of Lie algebras over arbitrary algebraically closed base fields of characteristic 0. This can be done by using algebraic group concepts and techiques which are by now well known. Nevertheless, it seems worthwhile to carry out the program in detail since Gantmacher’s results give a real insight into the action of an automorphism in a semi-simple Lie algebra. For example, as we indicate, they can be used to give a new derivation and sharpening of theorems on fixed points which are due to Borel and Mostow ([1]).

An Artίnίan module can be characterized in terms of certain properties of its factor modules. A module Mis Artinian if and only if the following two conditions hold for M: (I) Every nonzero factor module of M contains a minimal submodule. (A) The socle of every factor module ofM is finitely generated. The dual to the factor module is the submodule. We state the dual of (I): (Π) Every nonzero submodule ofM contains a maximal submodule. We call a module with property (Π) a Max module and one with property (I) a Min module. Every Noetherian module is a Max module but not conversely. This paper investigates these generalizations of the Artinian and Noetherian conditions and the relationships among them. Throughout this paper M denotes a right module over an arbitrary

The oscillatory or Apollonian packing in two dimensions is well known and is described for example in [13]. Recently we investigated the three dimensional osculatory packing of a sphere [4] However, the results of that paper indicate that, for N > 3, ΛΓ-dimensional osculatory packings are irregular and not invariant under inversion as is the case for N = 2 and 3. In this paper we introduce a class of packings which we call discrete packings, and produce some examples. This class is analogous to the class of lattice packings which appear in the theory of packings of equal spheres. We shall use the systems of polyspherical coordinates developed in [4]. Section 2 contains a description of these as well as the proofs of some additional results needed here. The idea of the separation Δ{X, Y) between two spheres X and Y will again play an important role. In § 3 we consider inversively generated configurations obviously generalizing the construction used in [4]. That is, we begin with a 'cluster' of (N + 2) disjoint spheres and by successive inversions replace the spheres one at a time with new spheres in such a way that the separations between the spheres in the new cluster are the same as for the initial cluster. In terms of polyspherical coordinates the necessary inversions are represented by matrices which preserve a certain indefinite quadratic form. Repetition of the process leads to a configuration of spheres in EN which may or may not be a packing, depending on the initial cluster. In § 4 we give sufficient conditions under which an inversively generated configuration is a packing. The conditions force the separations between the spheres in the configuration to lie in a discrete subset of the rational numbers, hence the name 'discrete packing'. In addition to the two and three dimensional osculatory packings, we give examples of discrete packings for dimensions 2, 3, 4, 5, and 9. We do not know yet whether such packings exist in all dimensions. The examples we have found are given in § 6. The packings described in § 4 are not in general osculatory; that is, the largest possible sphere is not generated at each step. However

In this paper it is shown that every strongly locally homogeneous Hausdorff continuum is locally connected. It is known that every connected representable space is homogeneous and that every locally connected representable space is strongly locally homogeneous; this paper investigates the problem of whether or not every connected representable space is locally connected. The discovery that homogeneity did not characterize the circle among planar continua motivated other classifications of continua in terms of the actions of their homeomorphism groups, and the pseudoarc remains the most tortuous proving grounds for testing such topological properties. Those homogeneity-like properties that are not possessed by the pseudo-arc appear to be closely related to local connectedness. Such properties include strong local homogeneity, representability, (strong) 2-homogeneity and countable dense homogeneity. In this paper it is shown that every strongly locally homogeneous Hausdorff continuum is locally connected. It is known that every connected representable space is homogeneous and that every locally connected representable space is strongly locally homogeneous; this paper investigates the problem of whether or not every connected representable space is locally connected. It follows from results of Ben Fitzpatrick, Jr. and Ralph Bennett that every connected locally compact separable metric representable space is locally connected. Arguments of Fitzpatrick and de Groot are used to show that if (X, τ) is a representable connected complete metric space such that τ does not have a base of totally disconnected sets, then (X, τ) is locally connected (and hence arcwise connected and strongly locally homogeneous). The authors do not know of even a homogeneous connected complete metric space with a base of totally disconnected sets; however, there exists a separable connected metric topology that has such a base. Let (X, τ) be a topological space. We let H(X) denote the group of all homeomorphisms from the space (X, τ) onto itself and let i denote the identity of H(X). If A c X, then A' = {he H(X): h\A = i I A) and if G is a subgroup of H(X), then G' = {x e X: g(x) = x for each geG}. If GaH{X) and AcX, then G(A) = {g(a):geG and aeA}. We write G(x) in room of G({x}). Throughout this paper all spaces are assumed to be Hausdorff.

Sario's capacity function of a closed subset γ of the ideal boundary is known to be unique if γ is of positive capacity. The present paper will determine the number of capacity functions of γ in terms of the Heins harmonic dimension when γ has zero capacity, under the assumption that γ is isolated. This includes the special case where γ is the ideal boundary. 1* Capacity functions* Denote by β the ideal boundary of an open Riemann surface R in the sense of Kerekjartό-Stoϊlow. We consider a fixed nonempty closed subset 7 c β which is isolated from S — β — 7. Throughout this paper D will denote a fixed parametric disk about a fixed point ζ e R with a fixed local parameter z and the uniqueness is always referred to this fixed triple (ζ, D, z). Here we do not exclude the case where 7 = β. For a regular region Ω Z) D we denote by yΩ the part of dΩ which is "homologous" to 7. The remainder δΩ = dΩ — yΩ consists of a finite number of analytic Jordan curves δΩj. For a regular exhaustion {RfΛn^o with Ro => D and nonempty τΛo, set yn = ΎE% and δni = dRnJ. Then there exists a unique function py% e H(Rn — ζ) satisfying (a) pΐn I D = log I z - ζ I + hn(z) with hn e H(D) and Λw(ζ) = 0, (b) pϊn I 7n = fcnCr) (const.) and pϊn \ δnj = d ni (const.) so that \ %dpr% = 0, which is called a capacity function of jn (Sario [6]). It is known that kn(y) increases with n and the limit k(y) is independent of the choice of {i2J*= 0. We call β~fc(r) the capacity of 7 and denote it by cap 7. When cap 7 > 0, pTn converges to a functions pr, which

Introduction^ This paper originated in trying to show that the one point compactification of an orientable generalized ^-manifold (n-grn) with cohomology isomorphic to Euclidean n-space was an orientable n-gm. Heretofore, in papers on transformation groups where this was relevant, it was stated as an extra assumption. The solution to this problem is given as a corollary to the main theorems which characterize the orientable (or locally orientable) generalized manifolds whose Freudenthal end point compactification is again an orientable (or locally orientable) generalized ^-manifold (see 4.5 and 4.13). In the first section we give a new characterization of the Freudenthal end point compactification in terms of inverse limits. We show as in Specker [10] that a certain O-dimensional cohomology group measures the extent of this compactification. Higher dimensional analogues of this cohomology group have been used by Conner [4] in proving, e.g., that a simply connected locally Euclidean ^-manifold whose 1 point compactification is a locally Euclidean w-manifold cannot be fibered by a non-trivial compact fiber. He has called these groups the cohomology of the ideal boundary. These groups are further studied and the homology analogue is derived (see § 2). The main Lemma (2.16) is an exact sequence which relates the homology of the ideal boundary with the homology of a given compactification and the local homology groups at infinity. This exact sequence together with our characterization of the Freudenthal compactification gives the main theorems. Applications are given in § 3 to Poincare duality, in § 5 to open 2manifolds, and in § 6 special mappings of manifolds. Throughout this paper X will denote a locally campct, locally connected, connected Hausdorff space. If S is a locally compact Hausdorff space, A(S) will denote the collection of open subsets of S whose closure is compact. When the generic space is not necessarily locally connected, as is usually the case in § 2 and part of § 4, it will be denoted by the letter S.

This paper studies the nilpotent ring analogues of several well-known results on finite p-groups. We first prove an analogue for finite nilpotent p-rings [a ring is called a p-ring if its additive group is a p-group] of the Burnside Basis Theorem, and use this to obtain some information on the automorphism groups of these rings. Next we obtain Anzahl results, showing that the number of subrings, right ideals, and two-sided ideals of a given order in a finite nilpotent p-ring is congruent to 1 mod p. Finally, we characterize the class of nilpotent p-rings which have a unique subring of a given order. The analogy between nilpotent groups and nilpotent rings which motivates the results of this paper is the replacement of group commutation by ring product. A nilpotent ring, of course, is itself a group under the circle composition x o y = x + y + xy but the structure of this group implies little about the invariants to be studied here, as shown by the examples in the last section of the paper. All rings considered here are associative. The reader may verify, however, that all results of §§ 1-3 hold without the assumption of associativity, with the exception of (3.3). The unqualified word "ideal" means two-sided ideal. The letter p always denotes a prime number. If 31 is a ring, we denote the additive group of 9t by 3ϊ +. The order of a ring 3ΐ, denoted |9t|, is the order of the group 9ϊ+; the index of a subring @ in a ring 3ΐ, denoted [9t: @], is the index of Θ + in 3ϊ +. A ring is called null if all products are 0. A ring 9t is called nilpotent of exponent e if all products of e elements from 9ΐ are 0, but not all products of e — 1 elements are 0. The characteristic of a finite ring is the maximum of the additive orders of its elements. The smallest ideal containing ideals @ and % is denoted @ + %. We shall need the following elementary results

The algebra of operators commuting with a Boolean algebra of projections of finite uniform multiplicity (in the sense of Bade [3]) has been studied by Foguel [10] and the author [12]. The aim of the present paper is to show that most properties described in these two papers can be extended (sometimes, under additional conditions) to the algebra 6 of all the operators which commute with a complete countably decomposable Boolean algebra of projections containing no projections of infinite uniform multiplicity. It should be mentioned that one cannot get interesting general results in the case in which there are projections of infinite uniform multiplicity in the Boolean algebra of projections, since every operator on a Banach space commutes with the Boolean algebra of projections composed from the identities 0 and I. We shall start by proving that for every operator A E 6 there is a sequence of projections belonging to the above mentioned Boolean algebra of projections which increases to the identity and A multiplied by any element of this sequence is a spectral operator. Relying on this result we study the spectrum of operators of E and give a necessary and sufficient condition for such an operator to be spectral. In the following section we generalize Theorem 8 of [12] showing that in 6 a strong limit of spectral operators on a Hilbert space is spectral provided that they are of the same finite type and their resolutions of the identity are uniformly bounded. Adequate examples elucidate why we require the boundedness of the type of the spectral operators in most theorems.

Let X be a topological space and 2X the space of all closed subsets of X with the finite topology. Assuming the continuum hypothesis it is shown that 2X is normal if and only if X is compact. It is not known if the continuum hypothesis is a necessary assumption, but it is shown that for X a A -space, 2X normal implies X compact. A theorem about the compactification of the n-t\v symmetric product of a space X is first proved which then plays an important part in the proof of the above results. Throughout this paper we will assume that X is any completely regular 2\ space. By 2X we will mean the space of all closed subsets of X with the finite topology [13, Definition 1.7, p. 153] except that we include the empty set as an isolated point as in [12]. The finite topology is also known as the exponential or Vietoris topology. Let ^l(X) be the subspace of 2X consisting of all nonempty subsets of X with n points or less. This space is known as the n-th symmetric product of X. In this paper the normality of 2X is studied. If X is compact, it is known that 2X is compact Hausdorff [13, Th. 4.2, p. 161] and thus normal. The main result of this paper is that if we assume the continuum hypothesis (CH), then 2X is normal if and only if X is compact. The first result in this direction was obtained by Ivanova [9] who proved that if X is a well ordered space with the order topology, then 2X normal implies X compact. In [10] it is shown that 22 is normal if and only if X is compact. These results were obtained without the use of CH. The paper is divided into three sections. In the first section our main result is that JΓSβX) = β^Z(X) if and only if J^(X) is pseudocompact. This result is related to the work of Glicksberg in [7] and the proof makes use of his work. In the second section of the paper we investigate the normality of 2X without the aid of CH using the results of the first section. One significant result in this section is that if 2X is normal with X noncompact, then X is normal and countably compact, but Xn is not pseudocompact for some n. As a corollary one obtains that if X is first countable or locally compact, then 2X normal implies that X is compact. Also 2X x βN is normal only when X is compact. In the last section of the paper it is shown that if CH is assumed

In the first part of this paper the structure of w-aposyndetic continua is studied. In particular, those continua which are w-aposyndetic but fail to be in + l)-aposyndetic are investigated. Unicoherence is shown to be a sufficient condition for an %-aposyndetic continuum to be in + l)-aposyndetic. In the final portion of the paper a stronger form of unicoherence is defined. As a point-wise property, aposyndesis and connected im kleinen are shown to be equivalent in continua with this property. Throughout this paper a continuum is a compact connected metric space and M will denote a continuum. If N is a subcontinuum of M, the interior of N in M will be denoted by int N. Suppose pe M and F is a closed subset of M such that p ί F. M is aposyndetic at p with respect to F if there is a subcontinuum N of M such that p e int N c N c M — F. Let n be a positive integer. If M is aposyndetic at p with respect to each subset of M consisting of n points, then M is n-aposyndetic at p. M is n-aposyndetic if it is ^-aposyndetic at each point. By convention if M is 1-aposyndetic then M is said to be aposyndetic. For other terms not defined herein, see [3], [4] and [6]. LEMMA 1. Suppose M is n-aposyndetic, peM,F is a subset of M — {p} consisting of n + 1 points, and M is not aposyndetic at p with respect to F. If F 1 and F2 are disjoint nonempty subsets of F such that F = Fι U F21 there exist subcontinua H and K such that Fιa H — K, F2 c K- H,pemt(HΓ\K), and M= H\JK.

Let Z = {O, 1, 2} and define two binary operations /\ and V on Z as follows: 0/\ 1 = 0,0 V 1 = 1, 1 /\ 2 = 1,1 V 2 = 2,2 A 0 = 2, 2 V 0 = 2, both operations are idempotent and commutative. This paper deals with the equational class Z generated by the algebra . The class Z contains the class of all distributive lattices and Z is a subclass of the class of weakly associative lattices (trellis, T·lattice) in the sense of E. Fried and H. Skala. The purpose of this paper is to prove that Z shares the most important properties of the class of distributive lattices.

In this paper a proof of the Jordan curve theorem will be presented. Some familiarity with the basic notions of nonstandard analysis is assumed. The rest of the paper is selfcontained except for some standard theorems about polygons. The theorem will be proved in what ought to be a natural way: by approximation by polygons. This method is not usually found in the standard proofs since the approximating sequence of polygons is often unwieldly. But by using nonstandard analysis, one can approximate a Jordan curve by a single polygon that is infinitesimal ly close to the curve. This allows types of reasoning which are extremely difficult and unnatural on sequences of polygons. Preliminaries. The basic concepts of nonstandard analysis and some acquaintance with polygons are assumed. Some basic definitions and theorems of point set topology are also assumed. Throughout this paper the following notations and conventions will be used: (1) All discussion, unless otherwise stated, is assumed to be about a nonstandard model of the Euclidean plane. 'Otherwise stated" will often mean that the notion or concept will be prefaced by the word "standard". (2) A standard concept and its extension will be denoted by the same symbol. If it is necessary to distinguish between them, reference to the model in which they are to be interpreted will be made. (3) If A and B are sets of points and x is a point, then \x, A\ will denote the distance from x to A and \B, A\ = mίxeB\x, A\. (Thus if A Π B Φ 0 then \A, B\ = 0.) \x,y\ will denote the distance from the point x to the point y. (4) / will denote a fixed continuous function on [0,1] into the Euclidean plane with the property that x < y and f(x) — fyy) if and only if x — 0 and y = 1. C will denote the range of /. ( 5 ) x ~ y will mean that the distance from x to y is infinitesimal. If x is near-standard then °χ will denote the standard y such that x ~ y. (6) \i x and y are points then xy will denote the ordered, closed line segment that begins at x and ends at y. (7) If x and y are points then intv (x, y) is the set of all points z of xy such that z Φ x and z Φ y.

It will be shown that (i) the absolute value of every locally normal linear functional is again locally normal; (ii) two locally normal representations πi and π2 of S/ generate isomorphic von Neumann algebras ^(πϊ) and ^^(π 2) if and only if there exists an automorphism a of S>/ such that πx o β and π2 are quasi-equivalent, provided that either ^(πϊ) or ^^(τr 2) is afinite. This paper is motivated by a recent work [6] of R. Haag. R. V. Kadison and D. Kastler. As they mentioned, the recent progress in mathematical physics has made a precise analysis of representatio ns of a C*-algebra furnished with a net of von Neumann algebras a growing necessity. In the first half of this paper, we shall show that the space of all locally normal linear functionals of a C*-algebra with a net of von Neumann algebras is a closed invariant subspace of the conjugate space in the sense of [14], which will imply that the absolute value of a locally normal linear functional is locally normal too. The last half of this paper will be devoted to extending a result of Powers [11] for UHF algebra to a C*-algebra sf with a proper sequential type 1^ funnel. Namely it will be shown that two locally normal representations πλ and π2 of the C*-algebra s%? generate isomorphic von Neumann algebras if and only if they are connected by an automorphism of s^. This is proven under the assumption that one of the generated von Neumann algebras is σ-finite. 2* The locally normal conjugate space of a C*-algebra with a net of von Neumann algebras* Let Szf be a C*-algebra. Suppose a system % = (jQ (ϋ) \}a^fa is dense in s^ with respect to the norm topology. The system % = {s/a} is called a net (in Ssf) of von Neumann algebras and each j^ς is called local subalgebra of DEFINITION 1. A continuous linear functional φ (resp. representa

1* Introduction* Let X be a real or complex Orlicz space of functions on an atomic measure space; an additional (not very restrictive) condition will be imposed on X which implies in particular that X Φ L°°. If X is a Hubert space, there are numerous strongly continuous one parameter groups of isometries on X, according to a classical theorem of M. H. Stone; namely, each skew-adjoint operator on X generates such a group. We shall show that this property characterizes the Hubert spaces among the Orlicz spaces under consideration on an atomic measure space. Our main result is, roughly, if {Tt: teR ~ (—°°, °°)} is a strongly continuous (or (Co)) group of linear isometries on X and if X is not a Hubert space, then for each real t, Tt has the following form: (Ttf)(w) = exp {i tg(w)}f(w) for fe X and w e Ω if X is complex, where g is a real-valued function on Ω; or Tt = I (= the identity operator on X) if X is real. Section 2 contains some preliminaries, including a discussion of duality maps for Orlicz spaces. The main result is stated and proved in § 3. Section 4 contains some complements and examples, including a proof of the main theorem for finite dimensional L°° spaces. The present paper has several points of contact with Lumer's paper [9], which we became aware of shortly after the present paper was submitted for publication.

Recent investigations concerning the application of variational principles to nonlocal field theories [7, 8, 30, 34-36, 461 have brought to light the fact that, in order to construct a variational theory for nonlocal phenomena, one has to introduce integrals which depend on several hypersurfaces, rather than on just one, as do the multiple integrals usually considered in the calculus of variations. It therefore appears that a systematic study of such integrals, using the direct methods of the calculus of variations, is worthwhile and should be initiated.l It is our aim to devote a series of papers to the variational theory of integrals depending on several hypersurfaces. In this paper, which is the first of this series, we shall establish geometrical conditions, i.e. conditions concerning the figurative, which are both necessary and sufficient for the continuity of these integrals. The necessity of these conditions is proved by reducing the problem to one that has been solved previously by Cinquini [l]. Their sufficiency is proved by applying, recursively, a proof given by Tonelli [48] for multiple integrals of the usual kind. It will be assumed throughout that all the surfaces considered are of the same dimension, +z, since surfaces of dimensions less than n may be represented as n-dimensional.

A group G is called an R-group if for every pair of elements x and y, and every natural number nf it follows from x n = y that x = y. In other words, G is an ϋϊ-group if G has not more than one solution for every equation x = a. If G is an iϋ-group, G is locally infinite. The converse, however, need not be true even if G is restricted to the group of a knot in S. For example, let G be the group of K(m, ri), the torus knot of type (m, n). G has a presentation G = (α, b: a = b). Then the equation x = a has infinitely many distinct solutions, x = a, (ba)a(ba)"ιf (ba) a(ba)~, This observation gives immediately a negative answer to Problem N in [3]. Neuwirth asks if a knot group can be ordered. In fact, the group of K{m, n)(\ m |, | n | ^ 2) cannot be ordered, since an ordered group is always an i?-group. Therefore, Problem N now leads slightly weaker problems: Can a knot group other than torus knot groups be ordered? Or, is a knot group other than torus knot groups an iZ-group? The purpose of this paper is to give a sufficient condition for the group of a fibred knot to be an iϋ-group. (See Theorem 2.) Using this condition, we can prove, for example, that the group of the figure eight knot is an i2-group. (See Proposition 3 or Proposition 5.)

If K is a Choquet simplex and X is a metrizable compact Hausdorff space, we let dsK denote the set of extreme points of K with the facial topology and let S(L(C(X), A{K))) denote the set of continuous operators from C(X) into A(K) with norm not greater than 1. Our main purpose in this paper is to characterize the extreme points of S(L(C(X), A(K))). We show that T is an extreme point of S(L(C(X), A(K))) if and only if its adjoint T* sends extreme points of Kivdo XU - XQ C(X)*> also, the set of extreme points of S(L{C(X), A{K))) equals C(daK,X\J -X).

In this paper we investigate the relationship between total positivity and reproducing kernels. We extend the notion of total positivity to domains in the complex plane. In doing so, we also give a geometrical interpretation to certain Wronskians of reproducing kernels. These geometrical quantities are connected to Gaussian curvatures of Kahler metrics induced by these kernels. For simply-connected domains these curvatures are negative constants, thereby showing that the kernels are totally positive and moreover yielding an efficient method for computing the relevant determinants. In general, the reproducing kernels of multiplyconnected domains are not totally positive. The motivation for this work stems from the work of Karlin [7] which deals with "optimal" quadrature formulas. Let H be a Hubert space of functions analytic in a plane domain D and possessing a reproducing kernel K(z, t), z,teD. Let LeH*, where iϊ* is the dual of H. A subset & of H* is specified and a member Q e & is called a quadrature formula. To each Q e & is associated a remainder functional RQ = L — Q. An optimal quadrature formula, if it exists, is any member Q* e ^ satisfying

commutes. In this paper, a necessary and sufficient condition for the existence of a stabilizing homomorphism (hence isomorphism) between any two crossed product extensions is obtained. The result is applied to obtain a necessary and sufficient condition for the existence of an automorphism φ: E -> E making the diagram 0 >G >E >Π >1

The two which have received the greatest attention are theclassical (Ore) quotient ring and the maximal (Utumi) quotient ring.The classical quotient ring has a relatively straightforward description,but it is only defined for rings which satisfy the so-called Ore con-dition. In contrast the maximal quotient ring is less easy to describebut is defined for all rings. In both cases there are distinct notionsof left and right quotient rings and we will always consider leftquotient rings.For group rings the classical quotient ring has been studied byHerstein and Small [2], Passman [5, 6], M. Smith [7], and P. F. Smith[81, and the maximal quotient ring has been studied by Burgess [1].This paper investigates the relationship of the maximal quotientrings of group rings, subgroup rings, and the centers of group rings.The object is to obtain for the maximal quotient ring analogues oftheorems of Passman and M. Smith on the classical quotient ring.Their techniques are used for the group ring arguments while thequotient ring arguments reflect the formalism of the maximal quotientring.

The basic results in this paper show that each collectively compact set of linear operators can be viewed as an equicontinuous collection followed by a single compact operator. This observation not only gives insight into the character of collectively compact sets of linear operators, but also yields easier proofs of many of the results obtained by earlier workers in the field. I* Factorizations of collectively compact operators* A fairly complete treatment, with applications, of collectively compact sets of linear operators is given in the recent book [1] by Anselone. Collectively compact sets of linear operators on normed linear spaces were originally studied by Anselone and Moore [2] in connection with approximate solutions of integral and operator equations. The general properties of such sets of operators, again in normed linear spaces, were studied by Anselone and Palmer in [3] and [4]. Collectively compact sets of linear operators were studied in the more general setting of linear topological spaces by DePree and Higgins [5]. In the current work new characterizations are given for collectively compact sets of operators on a linear topological space.

It is proved in the present paper that if A is a normal Hubert space operator, and if the operator B leaves invariant every invariant subspace of A, then B belongs to the weakly closed algebra generated by A and the identity. This may be regarded as a refinement of the von Neumann double commutant theorem. A generalization is given in which the single operator A is replaced by a commuting family of normal operators. Also the same result is proved for the case where A is an analytic Toeplitz operator.

In a previous paper on this subject, the author gave a new proof of the theorem of "analytic independence of systems of parameters". The methods used can be applied to prove a converse result. Hence: An ideal of a Noetherian ring is of the principal class if, and only if, it is generated by an analytically independent set.

In a fundamental paper Marshall Hall defines a free plane to be a projective plane which either is degenerate or is generated as follows from a 'basis configuration/ ττ0, consisting of at least two points on a line together with two isolated points. For each pair of points not already joined in π0 create a distinct line that joins them and add it to 7Γ0. In the resulting configuration, πlf consider pairs of lines that do not intersect, and for each create a distinct intersection point and add it to πu thus forming π2. Continuing, construct 7Γ3, τr4, πδ, π6, etc. adding alternately lines and points as indicated above. Then π = U*^* (with the obvious incidence relation) is a projective plane. It is bydefinition a free plane. Hall proved that a free plane contains no confined configuration, that is, no finite configuration that, like the Desargues configuration, has ^ 3 points on each line and ^ 3 lines through each point. Further, using a complicated argument, he showed that, if a finitely generated plane contains no confined configuration, it is free. It follows that any finitely generated subplane of a free plane is free. L. I. Kopejkina [2] proved, shortly after, that an arbitrary subplane of a free plane is free. (Of interest is the analogy with free groups.) An exposition of Kopejkina's theorem appears in [3]. Because it suggests a more symmetrical definition of free plane that leads to very direct proofs of the above theorems, we introduce the notion of an extension process.

Contraction semigroups of linear operators on a Banach space appear in a wide variety of physical and mathematical situations, and for this reason there has been much interest attached to studying the properties of such semigroups and their infinitesimal generators. In this paper we examine in some detail the question of when one can (left or right) multiply the infinitesimal generator A by another (bounded or unbounded) operator B and still preserve the generator property. Also, we state when polynomials in a generator retain the generator property. Our treatment is not exhaustive but is meant to be representative of the type of results which may be obtained.

In this paper we examine the relation between i?-sets, which are a purely set-theoretic concept, and various concepts associated with planar maps, for instance, four-colorings, five-colorings, Ήamiltonian circuits, and Petersen's theorem. Moreover, the introduction of the notion of a i?-set into graph theory enables us to ask questions which may be more tractable than the four-color conjecture and shed light on it. 1* Introduction and definitions* Let F be a family of sets. A set that meets every member of F and yet contains none of the members of F is called a i?-set for F. Observe that if B is a Bset for F, then so is its complement, (\JEeFE) — B. In fact, F has a £-set if and only if \JEeF E can be partitioned into two sets, A and B, such that neither A nor B contains a member of F. Observe that if F has a J5-set, and if G S F, then G has a 5-set. Also, if G C F, and every member of F contains some member of G, and if G has a i?-set, then F also has a B-set. (The notion of j?-set goes back to Bernstein, who used it in 1908 to deal with a topological question.) We shall be concerned with maps covering the surface of the sphere, S2. For the most part, we will assume that these maps are 3-regular, that is, each vertex has degree three. Each region of the map will be a topological cell. Two regions are adjacent if they share at least one edge. A sequence of distinct regions Ru R2, , Rn+1, n^>l such that Ri is adjacent to Ri+1, 1 ^ i ^ n is a path of regions. If we have Rn+1 = Rι, and Ru R2, •••!?» are still distinct, we call the path a region-cycle of length n (or n-r eg ion-cycle). A region-cycle consisting simply of the regions around a vertex we call a basic cycle. In a 3-regular map the basic cycles have length three. If the union of any two regions in a map is simply connected, then the regions bordering any given region form a region-cycle, which we call a face cycle. Its length is just the number of edges of the surrounded region. We shall not be interested in region-cycles of length two, unless their union is not simply connected. A region-cycle is odd if its length is odd; otherwise, it is even.

Let T be the set of minimal primes of a Krull domain A. If S is a subset of T9 we form B = n AP for PeS and study the relation of the class group of B to that of A. We find that the class group of B is always a homomorphic image of that of A. We use this type of construction to obtain a Krull domain with specified class group and then alter such a Krull domain to obtain a Dedekind domain with the same class group. Let A be a Krull domain with quotient field K. Thus A is an intersection of rank 1 discrete valuation rings; and if x e K, x is a unit in all but a finite number of these valuation rings. If P is a minimal prime ideal of A, then AP is a rank 1 discrete valuation ring and must occur in any intersection displaying A as a Krull domain. In fact, if T denotes the set of minimal prime ideals of A, then A = OPΘΓAP displays A as a Krull domain. Choose a subset S of T (S Φ 0) and form the domain B — Γ\pesAp. It is immediate that B is also a Krull domain which contains A and has quotient field K. If one of the AP were eliminable from the intersection representing B, it would also be eliminable from that representing A. Thus the AP for PeS are exactly the rings of the type BQ, where Q is a minimal prime ideal of B. If Q is minimal prime ideal of B, then Q Π A = P for the Pe S such that BQ = AP. Let A and B be generic labels throughout this paper for a Krull domain A and a Krull domain B formed from A as above. We recall that the valuation rings AP are called the essential valuation rings, and we will denote by VP the valuation of A going with AP. We summarize and add a complement to the above. PROPOSITION 1. With A and B as above, B is a Krull domain containing A, and the AP for PeS are the essential valuation rings of JB. Every ring B is of the form AM for some multiplicative set M if and only if the class group of A is torsion.

J. P. Jans has shown that if a ring R is right perfect, then a certain torsion in the category Mod R of left ϋί-modules is closed under taking direct products. Extending his method, J. S. Alin and E. P. Armendariz showed later that this is true for every (hereditary) torsion in Mod R. Here, we offer a very simple proof of this result. However, the main purpose of this paper is to present a characterization of perfect rings along these lines: A ring R is right perfect if and only if every (hereditary) torsion in Mod R is fundamental (i.e., derived from "prime" torsions) and closed under taking direct products; in fact, then there is a finite number of torsions, namely 2 for a natural number n. Finally, examples of rings illustrating that the above characterization cannot be strengthened are provided. Thus, an example of a ring R± is given which is not perfect, although there are only fundamental torsions in Mod Ri, and only 4 = 2 of these. Furthermore, an example of a ring R2* is given which is not perfect and which, at the same time, has the property that there is only a finite number (namely, 3) of (hereditary) torsions in Mod i?2* all of which are closed under taking direct products. Moreover, the ideals of R2* form a chain (under inclusion) and Rad R2* is a nil idempotent ideal; all the other proper ideals are nilpotent and R2* can be chosen to have a (unique) minimal ideal.

~ (λt + [h)t + o(t) if j = i where Xt > 0 for i > 0, μt > 0 for i > 1 and μ0 > 0. We further assume that Pu(ί) satisfies the foward and backward equation in the usual form. In this paper we restrict attention to the case μ0 — 0 so that when the particle enters the state zero it remains there a random length of time according to an exponential distribution with parameter λ0 and then moves into state one etc. In order to avoid inessential difficulties we assume henceforth that the infinitesimal birth and death rates Xt and μi uniquely determine the process. This is equivalent to the condition Σ~-o (πn + l/λn7Γn) = °° where

In 1963, Gilmer characterized all finite commutative rings with a cyclic group of units and, in 1967, Eldridge and Fischer generalized these results to rings with minimum condition. In the present paper these results are extended to semiperf ect rings and generalizations of the three theorems are obtained. It is shown that a semiperfect ring with cyclic group of units is finite and is either commutative or is the direct sum of a commutative ring and the 2 x 2 upper triangular matrix ring over the field of two elements. Let R be semiperfect with an abelian group of units. It is shown that R is finite if either the group of units is finite or the group of units is finitely generated and the Jacobson radical is nil. The proofs of all these results depend on our main theorem: The structure of a semiperfect ring R with an abelian group of units is described completely up to the structure of commutative local rings. (That is commutative rings with a unique maximal ideal.) The groups of units of these local rings are shown to be direct factors of the group of units of R.

The notions of nonexpansive, contractive, iteratively contractive and strictly contractive mappings have been generalized to a Hausdorff topological space whose topology is generated by a family of pseudometries . A fixed point theorem for strictly contractive mappings is obtained which generalizes the Banach's contractive mapping principle. Several examples and an implicit function theorem are given as well as some applications in solving functional equations in topological vector spaces. For iteratively contractive mappings, some results obtained by D. D. Ang and E. D. Daykin, S. C. Chu and J. B. Diaz, by M. Edelstein, by K. W. Ng and by E. Rakotch respectively are generalized. 1* Definitions and Notations* Throughout this paper X is a

This paper is concerned with the condition for the convergence to a doubly stochastic limit of a sequence of matrices obtained from a nonnegative matrix A by alternately scaling the rows and columns of A and with the condition for the existence of diagonal matrices A and D2 with positive main diagonals such that ΏγAΏ2 is doubly stochastic. The result is the following. The sequence of matrices converges to a doubly stochastic limit if and only if the matrix A contains at least one positive diagonal. A necessary and sufficient condition that there exist diagonal matrices A and D2 with positive main diagonals such that D1AD2 is both doubly stochastic and the limit of the iteration is that AφO and each positive entry of A is contained in a positive diagonal. The form DιAD2 is unique, and A and D2 are unique up to a positive scalar multiple if and only if A is fully indecomposable.

Let G be a topological group and let G* denote the space of all Lebesgue measurable functions from the unit interval [0, 1] into G with the topology of convergence in measure. With this topology and with pointwise multiplication as the group operation, G* is a topological group. If G is separable and has a complete metric and has more than one point, then Bessaga and Peiczyήski have shown that G* is homeomorphic to h, separable infinite-dimensional Hubert space. This fact is used in this paper to show the existence of separable Frechet manifolds which are topological groups and which have certain algebraic and topological properties.

This paper deals with the asymptotic behavior at infinity of the solutions to /(y) = λwy on [a, oo) where / is an nth order ordinary linear differential operator, λ is a nonzero complex number and w is a suitably chosen positive valued continuous functions. As an application the deficiency indices of certain symmetric differential operators in Hubert space are computed. 1* Preliminaries* Throughout the first three sections / will denote an operator of the form, (1.1) S(v) = y + Σ PkV' on [α, oo) , where each of p2, , pn is a continuous complex valued function on [α, oo). In view of the transformation indicated on p. 309 of [2] it results in no great loss of generality to take the coefficient of y~~ to be zero, and in order to simplify the exposition we shall do this. We shall be concerned with the behavior at infinity of the solutions to (1.2) s{y) = Xwy on [α, oo) where λ is a nonzero complex number and w is an appropriate weight (i.e., positive valued continuous function). For a given / we shall consider the weights w indicated by the following definition. £f(a, oo) denotes the Banach space of all complex valued measurable functions which are absolutely Lebesgue integrable on [α, oo). DEFINITION. If / is as in 1.1 the statement that w is an /-admissible weight means that (1) w is differentiate, strictly increasing, and unbounded on (2) each of [w'/w]' and [(w'/wy(l/w)] is continuous on [α, oo) and is in £f{a, oo); and (3) pόlw G £f{a, oo) for j = 2, 3, . , n. For example if s(y)(t) = y"{t) ± ty{t) for t ^ 1 and w(t) = P then w will be an /-admissible weight if and only if β > 0 and β > 2(α + 1). We shall demonstrate that when w is an /-admissible weight the solutions of 1.2 have a particularly simple asymptotic behavior and

Let T be a unitary operator on a Hubert space H. Then in particular, (i) T is a contraction, i.e. || T\\ 1; and (ii) The spectrum of T is a subset of the unit circle, i.e. Sp (T)cC, where C denotes the set of complex numbers of absolute value one. Call an arbitrary operator T a unimodular contraction if it satisfies conditions (i) and (ii) above. Then several questions immediately come to mind. Do there exist nonunitary unimodular contractions? If so, what is the nature of their spectra, e.g. what subsets of the unit circle arise as spectra of nonunitary unimodular contractions; when does the spectrum contain point, residual, or continuous spectrum? Under what conditions is a unimodular contraction unitary? What is the nature of operator algebras containing nonunitary unimodular contractions? In this paper examples are given of nonunitary unimodular contractions. It is shown (Theorem 2) that such exist with arbitrarily prescribed spectrum, which however can contain no residual spectrum. It is also shown (Theorem 1) that nonunitary unimodular contractions exist only in infinite von Neumann algebras. This result is applied to a mapping problem of operator algebras.

The question whether a system of n — 1 real algebraic numbers (n = 2, 3, •) chosen from an algebraic field of degree not higher than n9 yields periodicity by Jacobi's Algorithm is still as open and challenging as hundred years ago. The present paper gives an affirmative answer to this problem in the following case: let K(w) be an algebraic number field generated by w = (Dn — d: m)ί/n, where m, n, d, D are natural numbers satisfying the conditions m^ 1, n έ 3, d | D, 1 ^ d ^ D/2(n — 1). Then n — 1 numbers can be chosen from K(w), so that their Jacobi Algorithm becomes purely periodic. The length of the period equals n2 (or n, if d = m = 1). This is the longest period of a periodic Jacobi Algorithm ever known. In three corollaries the following special cases are investigated

The purpose of this paper is to study inclusion theorems for some of the more familiar sequence spaces. Necessary and sufficient conditions are given for an FϋΓ-space to contain each of the spaces bv0, bv, S?, 0 < p < oo, and c0. It is also shown that the Hardy space H, 0 < p < 1, is a barrelled subspace of its containing Banach space B. This leads to new results concerning multipliers of H and to new estimates on the growth of the Taylor coefficients of B functions.

In this paper, the zeta functions on the real general linear group, formed relative to arbitrary positive definite zonal spherical functions, are defined and studied. The main result asserts that such functions satisfy a functional equation and can be meromorphical ly continued to the whole complex plane. 1* Introduction* Let R denote the field of real numbers. Let M = M(n, R) denote the w2-dimensional vector space over R comprised of all n x n matrices with real coefficients, G — GL(n, R) the subset of M consisting of those elements with nonzero determinant. As is known, G is a locally compact unimodular group whose Haar measure dG(x) (xeG) is given, up to a constant factor, by dσ(x) = \x\~ ndM(x) (\x\ = |det(α?)|), dM(x) being normalized Lebesgue measure on M. Let K= {xeGix-'x = 1}

This paper treats a class of stochastic processes called biadditive processes and gives a proof of a decomposition of their sample functions. Informally, a biadditive proces X(s, t) is a process indexed by two time parameters whose " increments " over disjoint rectangles are independent. The increments of such a process are the second differences X(s2, U) - X(8U U) - X(s2, U) + X(8U ti) where s± < s2 and U < t2. The decomposition theorem states that every centered biadditive process is the sum of four independent biadditive processes: one with jumps in both variables, two with jumps in one variable and continuous in probability in the other, and a fourth process which is jointly continuous in probability. This decomposition is similar to one for processes with independent increments and in the proofs of both results a major role is played by the theory of centralized sums of independent random variables. More formally, let P1 = {slf s2, , sn} and P2 = {tl9 t2, , tm) be two partitions of [0, sn] and [0, tm] respectively. Define Px x P2 to be the corresponding partition of [0, sn] x [0, tm] into rectangles whose vertices are the (si9 ί, )'s. Let ΔiS denote the increment

In [5] Tsai defined the Brown-McCoy prime radical for Jordan rings in terms of the quadratic operation and proved basic results for the radical. In this paper we give a definition of the prime radical for arbitrary nonassociative rings in terms of a ^-operation defined on the family of ideals and of a function / of the ring into the family of ideals in the ring. The prime radical for Jordan or standard rings is obtained by a particular choice of the ^-operation and the function /. We also extend the results for the Jordan case to weakly W admissible rings which include the generalized standard rings and therefore alternative and standard rings as well as Jordan rings.

In this paper we prove that under certain conditions the sum and the product of two commuting spectral operators on a locally convex space are again spectral operators. We also obtain expressions for the spectral measures of such sum and product. The technique used is, in principle, similar to one employed by Pedersen [9]. Also, if X is a locally compact and σ-compact space and μ is a positive Radon measure on X, we consider the space ΩP(X, μ)(l

(unbounded spectrum, domain Φ X) in finite-dimens ional spaces which linear operators exhibit only in infinite-dime nsional spaces. We present an outline of the paper. In § 2 we define p(T) where p is a polynomial with coefficients in the field Φ involved in X. We prove that (pq)(T) = p(T)q(T), (poq)(T) = p{q{T)), and point out that sometimes (p + q){T) Φ p(T) + q(T), etc. In § 3 we turn to relations in dual pairs. In this situation, adjoints can be defined. We build an automorphism λ —> λ of Φ into the theory of dual pairs, so as not to exclude the Hubert space situation, which dual pairs are intended to imitate. (Thus the transpose is a special kind of adjoint.) Closedness is defined algebraically, but in a way compatible with the topological concept. Closure of T 7* and other algebraic properties of * are established. Finally, it is shown that if T is closed and its resolvent is not void then p(T) is also closed. Section 4 considers the self-dual case. We give a simple condition (4.3) always true in Hubert space, that T*T be self-adjoint, T being closed. In § 5 we give the spectral analysis of self-ad joint linear relations in Hubert space. In a 1:1 manner these correspond to the unitary operators, via the Cay ley transform. However, it can be shown directly that X is the direct sum of orthogonal subspaces Y, Z which reduce T (= T7*) giving in Za self-ad joint operator and in Fthe inverse of the zero-operator. 2 Linear relations* A relation T between members of a set X and members of a set Y is merely a subset of X x Y. For x e X, T(x) = {y (x, y) e T}. The domain of T consists of those x such that T(x) is not void. T is called single-valued if T(x) never contains more than one element. The range of T is the union of all T(x).

J. Leray defined a local fixed point index for functions defined in what he called convexoid spaces. From the standpoint of analysis, the most important example of a convexoid space is a compact subset C c X, X a locally convex topological vector space, such that C = U ?=1 d, where d are compact, convex subsets of X or a homeomorphic image of such a C. In this paper a simple geometric approach is given (see Lemma 2 below) by means of which a fixed point index can be defined for functions with domain in a class of spaces ^ which contains the spaces C mentioned above and also the compact metric ANR's. The usual properties of the fixed point index are established, and it is shown that they axiomatically determine the index for the class of spaces J^~.

for all i, j . Thus in the (two-person, zero-sum) game with matrix Λf, player I has a strategy insuring an expected gain of at least v, and player II has a strategy insuring an expected loss of at most v. An alternative statement, which follows from the von Neumann theorem and an appropriate law of large numbers is that, for any e>0, I can, in a long series of plays of the game with matrix M, guarantee, with probability approaching 1 as the number of plays becomes infinite, that his average actual gain per play exceeds v — e and that II can similarly restrict his average actual loss to v-he. These facts are assertions about the extent to which each player can control the center of gravity of the actual payoffs in a long series of plays. In this paper we investigate the extent to which this center of gravity can be controlled by the players for the case of matrices M whose elements m(i9 j) are points of ΛΓ-space. Roughly, we seek to answer the following question. Given a matrix M and a set S in iV-space, can I guarantee that the center of gravity of the payoffs in a long series of plays is in or arbitrarily near St with probability approaching 1 as the number of plays becomes infinite ? The question is formulated more precisely below, and a complete solution is given in two cases: the case JV=1 and the case of convex S. Let

An axiomatic setting for the theory of convexity is provided by taking an arbitrary set X and constructing a family ^ of subsets of X which is closed under intersections. The pair consisting of any ordered vector space and its family of convex subsets thus become the prototype for all such pairs (X, ^). In this connection, Levi proved that a Radon number r for ^ implies a Helly number h ^ r — 1; it is shown in this paper that exactly one additional relationship among the Carathέodory, Helly, and Radon numbers is true, namely, that if ^ has Carathέodory number c and Helly number h then ^ has Radon number r ^ ch+1. Further, characterizations of (finite) Caratheodory, Helly, and Radon numbers are obtained in terms of separation properties, from which emerges a new proof of Levi's theorem, and finally, axiomatic foundations for convexity in euclidean space are discussed, resulting in a theorem of the type proved by Dvoretzky.

p o h = f \ L. A map g: K-> E such that p ° g — f and g I L = h we call a lifting of f rel h. In this paper single obstruction Γ(f) e Hf (K, L, f if) is defined, ί? is a so-called Z?-spectrum, and H* ( if) is cohomology in that spectrum. If a lifting of / rel h exists, Γ{f) = 0; this condition is also sufficient if the fiber of p is /^-connected and dim (K/L) ^ 2k + 1. If g0 and #i are liftings of / rel h, a single obstruction