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Although his work on the application of invariance concepts to the sequential testing of composite hypotheses is better known, Cox (1963) has also outlined a large sample approach to the same problem. His method is based on Bartlett's (1946) recognition that the sequence of maximum likelihood estimates (MLE) of the parameter of interest, calculated from an increasing number of observations, resembles asymptotically a random walk of normally distributed variables. However, the large sample theory needed to justify this approach rigorously is left largely implicit. At the end of his paper, Cox suggests that these msthods may be extended to yield a sequential comparison of survival curves (Armitage (1959)), a suggestion which has been reiterated as a research problem in the monograph of Wetherill (1966). In this paper we first present a general theoretical framework in which the asymptotic validity of a wide class of large sample sequential tests may be examined, thus making explicit the justification for Cox's approach. The results of this section are fairly straightforward consequences of the increasingly well known theory of convergence in distribution for random variables which take values in separable metric spaces. Next we illustrate the theory by re-examining Cox's results on the comparison of two binomial parameters. Finally, and of greater consequence from the practical point of view, we present a large sample solution to the problem of the sequential comparison of exponential survival

In a previous paper (2), one of us has derived a formula for the counting series for bicoloured graphs.2 These are graphs each of whose points has been coloured with exactly one of two colours in such a way that every two adjacent points have different colours.In this paper we first enumerate bicoloured graphs without isolated points and connected bicoloured graphs. This leads us to corresponding problems for bicolourable graphs. Such a graph has the property that its points can be coloured with two colours so as to obtain a bicoloured graph.

In this paper, we consider linear spaces and algebras with real scalars. It is well known that if X is a Banach space and is the set of all bounded linear operators which map X into itself, then is a Banach algebra. In this paper we shall show that can be partially ordered so that it becomes a partially ordered algebra in which norm convergence is equivalent to order convergence. This motivates a study of Banach algebras of operators in which one uses the order structure to obtain various results. In addition, it encourages a study of partially ordered algebras in general, since our result shows that among such algebras one finds all real Banach algebras of operators. Of course, there are many other real algebras which are naturally partially ordered and which have been studied from that point of view.

This paper was originally planned as an exposition of finite geometry in a form in which it could be used to illustrate all the work covered in the normal school geometry course. The beginning of it, however, was anticipated by an article for the Gazette by Mr. H. Martyn Cundy (1), which he has kindly allowed me to read before publication. This present paper shows how a coordinate system can be constructed for the geometry described in his article by using the field of residue classes (mod. 5); that is, the Galois field GF(5). A definition of distance is introduced which extends that adopted by Cundy to a form suited to the discussion of a wider range of theorems.

1. Introduction. In one of his papers on limits for the characteristic roots of a Matrix, Brauer (1) has stated a theorem, which connects the roots of a given square matrix A, with those of a matrix A* derived from A by a certain process. The proof of this theorem involves a continuity argument and in a recent paper on the construction of stochastic matrices Hazel Perfect (5) has given a proof which avoids considerations of continuity.

The theorem that every properly primitive binary quadratic form is capable of representing infinitely many prime numbers was first proved completely by H. Weber (5). The purpose of this paper is to give an elementary proof of the case where the form is ax2 + 2bxy + cy2, with a > 0, (a, 2b, c) = 1, and D = b2 — ac not a square. The cases where the form is ax2 + bxy + cy2 with b odd, and the case where the form is ax2+ 2bxy + cy2 with D a square, can be settled very simply once the first case is taken care of, and this is done in a page and a half in the Weber paper.

T. Evans in [4] introduced a concept generalizing the associative law for groups. Certain very restricted cases of this generalization had occurred, somewhat as side results, in the considerations of A. K. Suškevič [7], D. C. Murdoch [5], [6], and R. H. Bruck [2] associated with quasigroups which obey theorems generalizing certain results in group theory.In this paper we consider quasigroups obeying an instance of Evans' generalization. We show two means of constructing such quasigroups, and we demonstrate these constructions with several examples. The principal result of this paper is a theorem which reveals a peculiar trait of quasigroups obeying an instance of the generalized associative law we are considering. Such a quasigroup obeys a law expressed with permutations in which the nontrivial permutations involved either are both automorphisms of the quasigroup or else both fail to distribute over any product from the quasigroup.

In their paper (1), Beurling and Livingston established a generalization of the Riesz-Fischer theorem for Fourier series in Lp using a theorem on duality mappings of a Banach space B into its conjugate space B*. It is our purpose in the present paper to give another proof of this theorem by deriving it from a more general result concerning monotone mappings related to recent results on non-linear functional equations in Banach spaces obtained by the writer (2, 3, 4, 5) and G. J. Minty (6).

The purpose of this paper is to record some progress on the problem of determining which (bounded, linear) operators A on a separable Hilbert space H are commutators, in the sense that there exist bounded operators B and C on H satisfying A = BC — CB. It is thus natural to consider this paper as a continuation of the sequence (2; 3; 5). In §2 we show that many infinite diagonal matrices (with scalar entries) are commutators and that every weighted unilateral and bilateral shift is a commutator.

In another paper [l] measures determined from base functions, that have finite right and left limits everywhere and are of generalized bounded variation in the restricted sense, are studied and used to define non absolutely convergent integrals of Denjoy type. In this paper base functions of bounded variation and the corresponding measures are studied as a background for that paper. The results supplement parts of [2].

The paper represents a generalization of one of the main theoretical results of my Ph.D. thesis. The work is an outgrowth of work first begun by E. J. Hannan and a correct 'conjecture' by P. Whittle. The main theorem of this paper proves the existence of, and gives an explicit formula for, the asymptotic best linear predictor of a certain type of non-stationary multivariate time series from noise distorted observations. The non-stationarity arises from the fact that the signal satisfies a difference equation, which when considered as a polynomial, has only elementary divisors. The proof is accomplished by showing, through Hilbert space and harmonic analysis methods, that the generating function is a limit of the generating functions of the stationary analogue; that is, where the difference function has elementary divisors. Finally, it is shown that this asymptotic generating function exactly predicts null solutions to the difference equation. The proof is direct and due to E. J. Hannan.

In an earlier paper [8], one of the authors has established some Vandermonde type convolution identities involving multinomial coefficients with several summations which evidently are regeneralizations of identities in [1] with one summation. In this paper similar identities are derived for coefficients (defined below) of a general type, in the line of the results in [2] and [3], Furthermore, in a series of papers [4], [5], [6], Gould has obtained results on inversion of series and on classical polynomials by an extensive use of these identities with one summation.

Let A = R/, where R is a regular local ring of arbitrary dimension and is an ideal of R. If A is a Gorenstein ring and if height = 2, it is easily proved that A is a complete intersection, i.e., is generated by two elements (Serre [5], Proposition 3). Hence Gorenstein rings which are not complete intersections are of embedding codimension at least three. An example of these rings is found in Bass’ paper [1] (p. 29). This is obtained as a quotient of a three dimensional regular local ring by an ideal which is generated by five elements, i.e., generated by a regular sequence plus two more elements. In this paper, suggested by this example, we prove that if A is a Gorenstein ring and if height = 3, then is minimally generated by an odd number of elements. If A has a greater codimension, presumably there is no such restriction on the minimal number of generators for , as will be conceived from the proof.

This paper is primarily a brief elaboration on the axioms for a bicategory introduced in (3). From this point of view, the main aim is the development of the structure of certain systems of topological and uniform spaces, and the present paper merely points out some very general properties which follow from axioms so weak that they are satisfied by any system likely to be considered.

The object of this paper is the extension to linear partial differential equations of order m in N independent variables, of the existence theorems for mixed initial and boundary value problems which have been established for systems of first order equations in (3). In such mixed problems an initial surface S and a boundary surface T are the carriers of the two types of data, and the number of datum functions to be assigned on T depends on the configuration of the characteristic surfaces relative to S and T.For the first part of the paper (§§ 1-5) the coefficients in the differential equation, the initial and boundary surfaces, and the data prescribed are all taken to be real analytic in the variables x1 … xN. In this “analytic” case an existence theorem is established for boundary conditions of considerable generality. We assume that the differential equation is regularly hyperbolic with respect to 5 and T, a notion which is stated precisely in § 1, and is weaker than the usual regular hyperbolic condition.

Throughout this paper all rings are assumed to have unit elements. A ring Λ is said to be semi-primary if its Jacobson radical N is nilpotent and Г = Λ/N satisfies the minimum condition. The main objective of this paper isTHEOREM I. Let A be a semi-primary algebra over a field K. Let N be the radical of Λ and Г = Λ/N. IfThendim Λ = gl.dim Λ.

Summary In this paper a stochastic calculus, based on limits in quadratic mean and in probability, is introduced and applied to some fundamental theorems of natural selection. The paper is divided into five principal sections. In Section 2 some elements of stochastic calculus are given, and in Section 3 the stochastic calculus is applied to the haploid case. Section 4 is devoted to the development of an analysis of variance structure applicable to a haploid population, and in Section 5 the results of Sections 3 and 4 are generalized to cover the diploid case. Finally, Section 6 is devoted to two specific examples in which the results of previous sections apply.

The relation between the notions of measurability and continuity of a function has received a great deal of attention. The best-known result in this connection is the Vitali-Lusin theorem. Various versions of it can be found in (2; 3; 5; 8; 9). We prove one in this paper (Theorem 3.5) under very weak assumptions and state the classical one in Corollary 3.6. Functions satisfying the property stated in the theorem are frequently called quasicontinuous.Here we are interested in the notion of approximate continuity, which we call μ-continuity for short and which is closely connected to differentiation. It was first introduced by Lebesgue (4) and it has so far required knowledge of density theorems in order to prove its relation to measurability. In this paper, making use of a certain type of Vitali property, we prove first the relation between μ-measurable functions and those μ-continuous almost everywhere (Theorems 3.8 and 3.9). Then, in §4, with Theorem 3.8 as the main tool, we derive several theorems about density and differentiation which extend results found in (1; 3; 7; 8).

on the interval I. However, it has also long been known that for n functions which are only (n 1)-times differentiable (so that their Wronskian is defined) the sufficiency part of the above statement no longer holds. Peano [12] seems to have been the first to point this out, and Bocher [3] has given an example which shows that even if the functions involved are infinitely differentiable on I, the identical vanishing of the Wronskian is still not sufficient to imply their linear dependence on I. It was then recognized that the functions involved must satisfy other conditions, supplementary to the vanishing of their Wronskian, in order to guarantee their linear dependence. Peano [13] and Bocher [5] have given such conditions (for example, Lemma 3 of Sec. 2 of this paper) and Bocher has shown that his conditions include those of Peano. In this paper we look at this problem from a slightly different point of view. Namely, instead of placing linear dependence in the forefront and looking for a condition to supplement the identical vanishing of the Wronskian, rather, we shall put the vanishing of the Wronskian in the forefront and ask for a kind of generalized dependence (necessarily weaker than linear dependence) which is equivalent to it. This point of view has led the author to define a new type of dependencef relation for functions of a real variable which he has called "local linear dependence." It is shown that local linear dependence and the identical

Our aim in this paper is to generalize certain ideas and results of Bary (1) on biorthogonal systems in separable Hilbert spaces to their counterparts in separable lp-spaces, 1 < p.The main result of Bary is to characterize a natural generalization of the concept of orthonormal basis for a Hilbert space. That of this paper is to characterize the concept of a Bary basis which is a generalization of the idea of standard basis of an lp-space. The result is interesting for lp-spaces because of the paucity of standard bases in these spaces.Before summarizing our results, we shall introduce some notation and recall a few pertinent definitions and facts. The symbols and denote mutually conjugate lp-spaces, where is the space lt and the space lswith 1 < r <2 and 2 < s = r/(r – 1).

In this paper, we continue the study of integral extensions begun in [7]. Whereas in the previous paper, we dealt exclusively with the extension A[x]/(α(x)), α(x) a monic polynomial over A, we now deal with arbitrary integral extensions. Applications of the results presented herein will be made in subsequent papers.To simplify our presentation, we make the following conventions. By an algebra, we will always mean a commutative complex algebra with an identity element, usually denoted by e

A “correlated queue” is defined to be a queueing model in which the arrival pattern influences the service pattern or vice versa. A particular model of this nature is considered in this paper. It is such that the service time of a customer is directly proportional to the interval between his own arrival and that of his predecessor. The initial busy period, state and output processes are analyzed in detail. For completeness, a sketch is also given of the analysis of the waiting time process which forms the subject of another paper. The results are used in the analysis of the state and output processes.

Indices of normal spaces with countable basis for equivariant mappings have been investigated by Bourgin [4; 6] and by Wu [11; 12] in the case where the transformation groups are of prime order p. One of us has extended the concept to the case where the transformation group is a cyclic group of order pt and discussed its applications to the Kakutani Theorem (see [10]). In this paper we will define the Jp-index of a normal space with countable basis in the case where the transformation group is a cyclic group of order n, where n is divisible by p. We will decide, by means of the spectral sequence technique of Borel [1; 2], the Jp-index of SO(n) where n is an odd integer divisible by p. The method used in this paper can be applied to find the Jp-index of a classical group G whose cohomology ring over Jp has a system of universally transgressive generators of odd degrees.

Recently the writer has obtained some results concerning meromorphic or algebroidal functions with the set of essential singularities of capacity zero, with an aid of a theorem of Evans. In the present paper, suggested from recent interesting papers of Sario and Pfluger, the writer will extend his results to single-valued analytic functions defined on open abstract Riemann surfaces with null boundary in the sense of Nevanlinna, using a lemma instead of Evans’ theorem.

Properties and characterizations for prime and semiprime rings have been provided by A. W. Goldie (2, 3). In a previous paper (1), the authors used the results of (2) to characterize prime and uniform prime modules. It is the aim of the present paper to generalize Goldie's work on semi-prime rings (3) to modules. In this setting certain new properties will appear.Notationally, in the work to follow, the symbol R always denotes a ring and all R-modules will be right R-modules.In the theory of rings an ideal C is said to be prime if and only if whenever AB ⊆ C for ideals A and B, then either A ⊆ C or B ⊆ C. A ring is prime if the zero ideal is prime.

In a recent paper we studied systems of equations of the form(1) (2) where as usual [a,b] = ab — ba and ϕ(λ) is a polynomial. Equations of this type have arisen in quantum mechanics. In our paper we gave a method of determining the matrix solutions of such equations. The starting point of our discussion was the observation that if the elements xi satisfy (1) then the elements xi, [xj,xk] satisfy the multiplication table of a certain basis of the Lie algebra of skew symmetric (n + 1) ⨯ (n + 1) matrices. We proved that if (2) is imposed as an added condition, then the algebra generated by the has a finite basis, and we obtained the structure of the most general associative algebra that is generated in this way.

Throughout this paper R and Rn will denote v-rings, that is, complete discrete rank-one valuation rings of characteristic zero, having a common residue field k of characteristic p. R is assumed unramified and Rn has ramification index n. Let π be a prime element in Rn. Then Rn = R[π], where π is a root of an Eisenstein polynomial ƒ = xn + pƒn-1 xn-1 + … + pƒ0 with coefficients in R and ƒ0 a unit. Thus Rn is inertially isomorphic to R[[x]]/ƒR[(x)], that is, the rings are isomorphic by a mapping which induces the identity mapping on the common residue field. R[[x]] represents the power series ring in the indeterminate x over R. In this paper we identify Rn with R[[x]]/ƒR[[x]], R with its natural embedding in Rn and π with x + ƒR[[x]].

A service unit I, with Poisson input and general service times is in series with a unit II, with negative-exponential service times. The intermediate waitingroom can accomodate at most k persons and a customer cannot leave unit I when the waitingroom is full. The paper shows that this system of queues can be studied in terms of an imbedded semi-Markov process. Equations for the time dependent distributions are given, but the main emphasis of the paper is on the equilibrium conditions and on asymptotic results. 1. Description of the model The system of queues, discussed in this paper, consists of two units. Customers arrive at a first unit (I) according to a homogeneous Poisson process of rate 2. Their service times in unit I are independent, identically distributed random variables with common distribution function H(.). We assume that H(.) has a positive, finite mean a and we will denote the Laplace-Stieltjes transform of H( ) by h(s), Re s 0. Upon completion of service in unit I, all customers go on to a second unit (II) via a finite waitingroom. We assume that there can be not more than k customers in unit II and in the waitingroom at any time. If upon completion of service in unit I a customer finds the waitingroom full, then the unit I blocks until a service in unit II is completed. At that time he is allowed to enter the waitingroom. We assume that the service times in unit II are independent, identically distributed random variables with a negative-exponential distribution with mean 1/a. The service times in unit II are also stochastically independent of those in unit I and of the arrival process. The case k = 1, i.e., when no customers can wait between the two units, was studied by B. Avi-Itzhak and M. Yadin[1], T. Suzuki [11] and N. U. Prabhu [7]. These authors allow the service times in the second unit to have a general distribution. In the case of general k, we impose the requirement that these service Received 3 July 1967. Research supported in part by Office of Naval Research Contract NONR 1100(26). 123 This content downloaded from 157.55.39.217 on Mon, 18 Apr 2016 07:16:32 UTC All use subject to http://about.jstor.org/terms

The treatment of semigroups given in a previous paper (3) is based upon representations of a semigroup by means of transformations of a set (cf. also 12). In this paper we try to remove the assumption of the existence of a zero element proposed in (3). In accordance with our general programme explained at the beginning of (3) we utilize certain minimum conditions in order to gain more information on the structure of semigroups.Our main results are structure theorems on primitive semigroups which have irreducible right ideals generated by idempotents (§§15-17). As we have shown in (5), these theorems permit the explicit construction of primitive semigroups.

As a generalization of the notion of Frobenius algebras over a field Kasch [103 introduced that of Frobenius extensions of a ring. The present writers [13] recently freed one of Kasch’s main theorems from its rather strong S-ring assumption of the ground ring. However, even with the removal of the S-ring assumption of the ground ring the notion does not seem general enough, and we wish, in the present paper and its sequel, to develope the theory upon the basis of a more general notion of Frobenius extensions. Thus, we replace the free module property of the extension by the projective module property (according to a general tendency in algebra), which has been done in fact in case of Frobenius algebras over a commutative ring in a previous work by Eilenberg and one of the writers [4], and, further, take automorphisms of the ground ring into the definition of Frobenius extensions (which seems quite natural particularly in case of non-commutative rings). To such generalized notion of Frobenius extensions we may extend many of Kasch’s theorems, including those which are immediate extensions of classical theorems for Frobenius algebras and those which are essentially new, as the above alluded endomorphism ring theorem. Also homological properties of Frobenius extensions, as were developed in Hirata’s [6] recent paper in succession to Eilenberg-Nakayama [4], can be extended to our present generalized case; we shall also exceed [4], [6] somewhat in considering injective and weak dimensions.

In a recent interesting paper (3) H. Schwerdtfeger answered a question of W. R. Utz (4) on the structure of the real solutions A of A* = B, where A is skew-symmetric. (Utz and Schwerdtfeger call A* the "adjugate" of A ; A* is the n-square matrix whose (i, j) entry is (—1)i+j times the determinant of the (n — 1)-square matrix obtained by deleting row i and column j of A. The word "adjugate," however, is more usually applied to the matrix (AT)*, where AT denotes the transposed matrix of A ; cf. (1, 2).)The object of the present paper is to find all real n-square skew-symmetric solutions A to the equation

Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires.Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdös and Turán deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turán who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn, what kind of properties hold for almost all subgroups of the class?

A considerable volume of research on the theory of regular maps is now in existence. Systematic enumerations of regular maps on the surfaces of genus 1 and 2 were begun by Brahana (1; 2) and completed by Coxeter (6; 7, p. 141). In addition Coxeter enumerated the regular maps on the simplest non-orientable surfaces (7, pp. 116, 139), and constructed tables of some interesting families of regular maps (3; 7, p. 140).Most of the regular maps on a surface of genus 3 have appeared in these papers, but no systematic enumeration of them seems to have been attempted. The ultimate goal of this paper is a complete list of these regular maps. However, the families of maps {j.p,q}and {j.p,j.q} which are defined in § 4 and listed in Tables I and II are of considerable interest in themselves. Also of some importance is the complete list of regular maps of type {p,3} with six or fewer faces (§ 5 and Table III).

This paper deals with the discrete groups of rigid motions of the hyperbolic plane. It is known (12) that the finitely generated, orientation-preserving groups have the following presentations:Generators: .Defining relations: where km = ambmam-1bm-1. We shall denote this group by F(p; n1, … , nd; r).In particular, the finitely generated free groups are contained among these. Indeed, one purpose of this paper is to indicate some geometrical methods for investigating free groups.

In [0], the writer proved some theorems of Hasse type for two orthogonal groups which operate on the same vector space. In this paper, we shall further generalize those results in two directions. One is to consider the propositions of that type for two orthogonal groups which operate respectively on two vector spaces whose dimensions are different from each other, and the other is to deal with some conspicuous subgroups of an orthogonal group simultaneously which play important roles in the structure theory for orthogonal groups. For this reason, the present paper consists of three steps §1, §2 and §3 which give the generalizations in the above sense of the results in the corresponding sections of [0].

Letbe an ordinary differential operator of order h whose coefficients are (η, η) matrices defined on the interval 0 ≤ x < ∞, hη = n = 2v. Let the operator L0 be formally self adjoint and let v boundary conditions be given at x = 0 such that the eigenvalue problem(1.1)has no non-trivial square integrable solution. This paper deals with the perturbed operator L∈ = L0 + ∈q where ∈ is a real parameter and q(x) is a bounded positive (η, η) matrix operator with piecewise continuous elements 0 ≤ x < ∞. Sufficient conditions involving L0, q are given such that L∈ determines a selfadjoint operator H∈ and such that the spectral measure E∈(Δ′) corresponding to H∈ is an analytic function of ∈, where Δ′ is a subset of a fixed bounded interval Δ = [α, β]. The results include and improve results obtained for scalar differential operators in an earlier paper (3).

This paper applies the invariance principle to the problem of testing a distribution on a compact homogeneous space for uniformity. The notion of using a reduction by invariance in such a situation is due to Ajne[1], who considers tests invariant under rotation on a circle. In his paper, he derives the distribution of the maximal invariant and gives the general form of the most powerful invariant test for uniformity on the circle.

The concept of homotopy for homomorphisms of modules, suggested by analogy with fundamental properties of the topological homotopy, was developed by Eckmann and Hilton (3; 9; 10). In the present paper, this concept of homotopy is generalized to additive categories with an additional structure, and the theory of homotopy, including in particular various exact homotopy sequences, is established in full detail. It may be helpful to the reader to realize that most of our theorems, concepts, and constructions have their analogues in the homotopy theory of topological spaces. Many of these analogues may be found in a recent paper by Eckmann and Hilton (4). Explicit references will be given at various places in the paper.

Summary In a previous paper [4] the author studied the stochastic process {wn , n = 1,2, …}, recursively defined by with K a positive constant, τ1, τ2, … σ1, σ2, …, independent, nonnegative stochastic variables. τ1,τ2…, are identically distributed, and σ1,σ2,…, are also identically distributed variables. For this process the generating function of the Laplace-Stieltjes transforms of the joint distribution of Wn , σ2 + … + σ n and τ1 + … + τ n−1 has been obtained. Closely related to the process {wn , n = 1, 2,…} is the process {un , n = 1, 2,…} with {un = K + [wn + τ n − K]−, n = 1,2,…; these are dual processes. In the present paper we study the stationary distributions of the processes {wn , n= 1,2, …} and {un , n = 1,2, …}, and the distributions ot the entrance times and return times of the events “wn , n = 0” and “un = K” for some n, for discrete as well as for continuous time. For these events various taboo probabilities are also investigated. The mathematical descri ption of the processes {wn , n = 1,2, …} and {un , n= 1,2, …} gives all the necessary information about the time-dependent behaviour for the general dam model with finite capacity K, since the process {wn , n= 1,2, …} is the basic process for such dam models. In Sections 5, 6 and 7 the general theory is applied to the models M/G/1 and G/M/1. Complete explicit solutions are obtained for these models. The present theory also leads to new and important results for the queueing system or dam model G/G/1 with infinite capacity. For instance the joint distribution of the busy period (or wet period) and of the supremum of the dam content dunng this period is obtained.

Recently H.-E. Richert [10] introduced a new method of summability, for which he completely solved the “summability problem” for Dirichlet series, and which led also to an extension of our knowledge of the relations between the abscissae of ordinary and absolute Rieszian summability. This non-linear method, which may best be characterized by the notion “strong Rieszian summability” †, depends on three parameters, on the order k;, the type λ, and the index p;. While Richert's paper deals almost exclusively with the application of that method of summability in a specialized form (namely the case p = 2, λn=log n) to Dirichlet series, it is the object of the present paper, to consider the general theory of strong Rieszian summability.

The main theorem of this paper names seven ring classifications which coincide with the class of rings named in the title. Three of them are (1) rings over which every module is rationally complete, (2) left and right perfect rings over which every module is corationally complete, and (3) right perfect rings over which every right module is rationally complete. Corational completeness (introduced in this paper) and rational completeness are generalizations of projectivity and injectivity, respectively. One recalls that a proper subclass of the rings investigated in this paper, the artinian rings with zero radical, is known to have one-sided characterizations in terms of the injectivity (projectivity) of modules. An example proves, however, that the class of rings over which every right module is rationally complete properly contains the rings of the title.

In [2], J. Gross provides an infinite collection of topologically distinct irreducible homology 3-spheres. In this paper, we construct for any finitely generated abelian group A, an infinite collection {Mi} of topologically distinct irreducible closed 3-manifolds such that H1(Mi) = A for each i.The proof consists of first constructing a closed irreducible 3-manifold MA with H(MA) = A, and then providing a method for producing more such manifolds with the same first homology group.All maps and spaces in this paper are assumed to be in the piecewise linear category, and all subspaces are assumed to be piecewise linear subspaces.A 3-manifold M is irreducible if each 2-sphere in M bounds a 3-cell in M. A compact 2-manifold (or surface) F in a compact 3-manifold M is properly embedded in M if F ∩ bdM = bdF.

This paper is a continuation of the earlier papers (1, 5) in which the author studied matrices with entries from the algebra C() of all continuous, complex-valued functions on an extremely disconnected, compact Hausdorff space . (Such spaces are sometimes called Stonian, after M. H. Stone, who first considered them in (8). They arise naturally as maximal ideal spaces of abelian W*-algebras.) In this note, three theorems are proved.

This paper continues the study of continuous selections begun in (13; 14; 15) and the expository paper (12). The purpose of these papers, which is described in detail in the introduction to (13), can be summarized here as follows. If X and Y are topological spaces, and ϕ a function (called a carrier) from X to the space 2Y of non-empty subsets of F, then a selection for ϕ is a continuous f: X → Y such that f(x) ∈ ϕ(x) for every x ∈ X. For reasons which are explained in (13), we restrict our attention to carriers which are lower semi-continuous (l.s.c), in the sense that, whenever U is open in Y, then is open in X. Our purpose in these papers is to find conditions for the existence and extendability of selections.The principal purpose of this paper is to generalize the following result, which is half of the principal theorem (Theorem 3.2˝) of (13) (and is repeated as Theorem I of (12)).

This paper contains part of the author’s Ph.D. dissertation directed by Frederick Bagemihl at Wayne State University. The research was supported by a grant from the Michigan Institute of Science and Technology.Alice Roth has made an extensive study of entire meromorphic functions with prescribed behavior along half rays emanating from the origin (6). The question arose whether analogous results could be found for functions meromorphic in the unit disk with the same behavior prescribed along an exhaustive class of spirals emanating from the origin. In this paper, I present a class of spirals which satisfactorily fills this role. However, I make no claim to the effect that only this class will suffice.

It is well known that an idealAin a. Dedekind domain has a prime radical if and only ifAis a power of a prime ideal. The purpose of this paper is to determine necessary and sufficient conditions in order that a commutative ring with unit element have this property and to study the ideal theory in such rings. Domains with unit element having the above property possess many of the characteristics of Dedekind domains (however, they need not be Noetherian) and will be referred to in this paper as "almost Dedekind domains"—these domains are considered in Section 1.

We consider a countable number of independent random uniform lines in the hyperbolic plane (in the sense of the theory of geometrical probability) which divide the plane into an infinite number of convex polygonal regions. The main purpose of the paper is to compute the mean number of sides, the mean perimeter, the mean area and the second order moments of these quantities of such polygonal regions. For the Euclidean plane the problem has been considered by several authors, mainly Miles [4]–[9] who has taken it as the starting point of a series of papers which are the basis of the so-called stochastic geometry.

The well-known results of Krull concerning the minimal prime divisors and the radical of an ideal in a commutative ring have been extended to the noncommutative case in a recent paper [5] by N. H. McCoy. In that paper systematic use was made of the concept of an m-system, a set M of elements of the ring such that if a ∈ M and b ∈ M then axb ∈ M for some element x of the ring.

This paper originated mainly from results presented in a paper by J. Ohm (13), and, to a lesser degree, from results of Gilmer in (3). Ohm's paper is concerned with the validity of the equation (x, y)n = (xn, yn) for each pair of elements x, y of an integral domain D with identity. If D is a Prüfer domain, the above equation is valid for all x, y ϵ D (7, p. 244). Butts and Smith have shown (2) that if (x, y)2 = (x2, y2) for all x, y of the integrally closed domain D, then D is a Prüfer domain.

A few years ago, in a short paper (4) Ryser introduced an interesting topic in number theory, viz. the connection between integer matrices (i.e., matrices having only integers as their elements) satisfying certain conditions and 0-1 matrices (i.e., matrices that have no element different from 0 and 1). In this series of papers we shall pursue this topic further.To make the statements of our theorems short we introduce some terminology. We need the definitions of certain 0-1 matrices related to a few well-known combinatorial configurations. By an incidence matrix of a balanced incomplete block (b.i.b. for conciseness) design we mean a 0-1 matrix with v rows and b columns, such that the sum of the elements in each column of A is k, k < v, and the scalar product of any two row vectors of A is λ, λ ≠ 0.