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Erdös [1] has shown that, if the square-free numbers in ascending order be denoted by s1, s2, … , sn, … , then for 0 ≦ γ ≦ 2as x → ∞. In this paper we shall extend this result by proving that the asymptotic formula in fact holds for the wider range 0 ≦ γ ≦ 3.Similar results have been obtained previously by the author in respect of both the sequence of numbers expressible as the sum of two squares and also sequences of numbers relatively prime to given large integers, although the method used here differs from that of the earlier papers [2; 3].

A detailed probabilistic treatment is given of a birth-and-death process proposed by Williams (1969) in which the elements of the process bear up to s inanimate marks. Equations for the second-order moments of the process, and approximate marginal univariate solutions, are derived. The exact bivariate solution is given for the case s = 1. For general s the variance of the mark population is also derived. MARKED BIRTH-AND-DEATH PROCESS; MULTIVARIATE LAGRANGE DIFFERENTIAL EQUATIONS; MULTITYPE BRANCHING PROCESS Introduction In a recent paper Williams (1969) has considered the following process. We have a population of elements developing as a linear birth-and-death process and each element bears a number of inanimate marks. When a marked element dies then it and its marks are removed from the population; when a marked element divides, its marks segregate independently and without loss, each with probability I into one of the two daughter elements. It is assumed that marking an element does not affect its birth and death rates (cf. Clifford and Sudbury (1972)). Williams' process was proposed as a model of an experiment conducted on bacteria (the elements) infected by phage (the inanimate marks), by Meynell (1959). (The bacteria are lysogenic with respect to the phage, which means that the phage do not interfere with the behaviour of the bacteria.) His object was to use the marking process in order to separate the birth and death rates of the bacteria; if we denote birth and death rates by A and p respectively, then the mean-value measures on the bacterial population (assumed to develop as a linear birth-and-death process) will estimate the difference (A p). As the marks are inanimate, however, they will only be affected by p and so mean-value measures on the mark population enable us to estimate p (see Williams (1969)). The papers of Meynell and Williams taken together provide a most interesting example of a virological experiment and the probability model of that experiment. Received in revised form 10 August 1973 Research sponsored by a Science Research Council grant and a World Health Organisation research traineeship. 423 This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 05:01:40 UTC All use subject to http://about.jstor.org/terms 424 BYRON J. T. MORGAN Williams considered the mean-value solution of his model; this was adequate for Meynell's experiment in which population sizes were large. Here we shall give a fuller probabilistic statement of the model and make some contributions to its solution. Two differences from Williams' work lie in the adoption of an upper bound, s, to the number of marks any element can have, and the assumption (except where stated) of birth and death rates independent of time. We 'shall assume general initial conditions which, in Meynell's experiment, correspond to the distribution of phage over bacteria once these have been initially exposed to a phage population. Williams took this distribution to be Poisson; the distribution of phage over bacteria has, however, been the subject of much probabilistic study (e.g., see Gani (1965) and Morgan (1971)). Finally, let us note that the value s = I, on which we shall particularly concentrate, probably gives the most realistic description of Meynell's experiment. 2. The probability model The model is a continuous-time Markov process with discrete states. Let N,(t) be a random variable denoting the number of elements with i marks at time t, i = 0, 1,2, ..., s. Let n = (n0,n1, l , n.., ), Iu = (Uo, u1, *,us), N(t) = (No(t), N(t), --,N,(t)), and suppose N,(O) = ai, i = 0, 1,2, --, s. Let P(n; t) = P(N(t) = n), then corresponding to death we have the transitions, (No, ..., Ni, ---, N,) (No, --, Ni 1, , N,) at probability rate pNi, i = 0, 1,2, ,s, while corresponding to birth we have the following transitions (No, N,) ,(No + 1, .., N,) at probability rate, 2 =1 N 2-' +,ANo, (No, , Ni ,N2i,..., X (No, Ni + 2, , N2iI, at probability rate, This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 05:01:40 UTC All use subject to http://about.jstor.org/terms On the distribution of inanimate marks over a linear birth-and-death process 425 where we use [is] to denote the integral part of is, and (No, " , Ni, * * *, Nk i Nk, "', Ns) -*(No, ..., Nj + 1, ..., Nk-i + 1, ..., Nj 1, ..., Nl) at probability rate 2N 2-k' i = 1,2,...,C[(k 1)], k = 3,4,...,s (smaller k give transitions already considered above). It is readily verified that the sum of the transition rates is (A + yp) X = oN,, as required in the overall birth-and-death process. Introducing the operator Di, which increases the argument ni of a function of n by 1, we then deduce that the Kolmogorov forward equations (for all possible n) are, d S S (1) +(n; t) P(n; t)(A+ t)1 (n2 + D,P(n;t)(n +1) dt ==0 1=0

A dissectible polyhedron is a natural extension of a concept whose history dates back to at least 1758 and Euler [7]—the concept of a dissection of a polygon. An interesting historical survey of dissections of a polygon is given by Brown [4]. Some approaches to the classical problem have been given by Moon and Moser [9] and by Guy [8] ; the latter provides an approach which is the basis of the work in this paper. A summary of enumeration results on dissections of polygons and polyhedra by automorphism groups has been given by the authors [2].Recent extensions of the problem have been investigated in a series of papers by Brown and Tutte [3; 5; 14; 15] and b y Takeo [10; 11; 12; 13].

In this paper R is a commutative noetherian local ring with unit element 1 and M is its maximal ideal. Let K be the residue field R/M and let {t1,t2,…, tn) be a minimal system of generators for M. By a complex R1. . ., Tp> we mean an R-algebra* obtained by the adjunction of the variables T1. . ., Tp of degree 1 which kill t1,…, tp. The main purpose of this paper is, among other things, to construct an R-algebra resolution of the field K, so that we can investigate the relationship between the homology algebra H (R < T1,…, Tn>) and the homological invariants of R such as the algebra TorR(K, K) and the Betti numbers Bp = dimk TorR(K, K) of the local ring R. The relationship was initially studied by Serre [5].

Introduction. Until recently none of the numerous papers on the distribution of quadratic and higher power residues was concerned with questions of the following sort: Let k and m be positive integers. According to a theorem of Brauer (1), for every sufficiently large prime p there exist m consecutive positive integers r, r + l , . . . , r + m — 1, each of which is a &th power residue of p. Let r(k, m, p) denote the least such r. What can be said about the behaviour of r(k, m, p) as p varies? In particular, for what values of k and m is r(k, m, p) bounded, and for these values what is its maximum? For a fixed k and m we call a prime exceptional and denote it by p* if there do not exist m consecutive integers each of which is a &th power residue of p*. Set A(k,m) = Max r (k, m, p), where the maximum is taken over all nonexceptional primes p. In a previous paper (4) Lehmer and Lehmer showed that

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

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

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

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

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

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

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

This paper applies the theory of semi-Markov processes to the construction of a stochastic model for interpreting data obtained from clinical trials. The model characterizes the patient as being in one of a finite number of states at any given time with an arbitrary probability distribution to describe the length of stay in a state. Transitions between states are assumed to be chosen according to a stationary finite Markov chain.Other attempts have been made to develop stochastic models of clinical trials. However, these have all been essentially Markovian with constant transition probabilities which implies that the distribution of time spent during a visit to a state is exponential (or geometric for discrete Markov chains). Markov models need also to assume that the transitions in the state of a patient depend only on absolute time whereas the semi-Markov model assumes that transitions depend on time relative to a patient. Thus the models are applicable to degenerative diseases (cancer, acute leukemia), while Markov models with time dependent transition probabilities are applicable to colds and epidemic diseases. In this paper the Laplace transforms are obtained for (i) probability of being in a state at timet, (ii) probability distribution to reach absorption state and (iii) the probability distribution of the first passage times to go from initial states to transient or absorbing states, transient to transient, and transient to absorbing. The model is applied to a clinical study of acute leukemia in which patients have been treated with methotrexate and 6-mercaptopurine. The agreement between the data and the model is very good.

This paper applies the constant population approximation to the study of epidemics which involve more than a single type of infective. An example of this would be a situation in which both clinically infected individuals and subclinically infected individuals or carriers are present. We derive equations for the expected numbers of clinically infected individuals and carriers at any time t for the model with zero latent period and infectious periods having negative exponential distributions. From these equations we derive conditions under which a unimodal incidence curve can result, and expressions for the expected total epidemic size. The equations describing the course of an epidemic are analytically intractable except in certain simple cases, [1]. When one examines the reason for the mathematical difficulties that arise, one finds that the principal stumbling block is the assumption of a finite population of susceptibles. It can be argued, however, that in modern societies epidemics very rarely menace an entire population, and that observed epidemic sizes are usually much smaller than the total susceptible population. This is due to a variety of reasons, among which are public health measures and good communication facilities available to a modern society. In consequence it is reasonable to try to bypass some of the mathematical difficulties inherent in the theory of epidemics by assuming a susceptible population whose size does not change through the course of the epidemic. Bailey [2], Morgan [3], and Williams [4] have discussed in some detail the theory of epidemics using the constant population approximation, although such an approximation was used by several authors earlier, [5], [6], [7]. The theory developed so far has dealt mainly with epidemic processes in which there is only a single type of infective and a single type of susceptible. Recently Gart [8] has considered a model for epidemics involving more than one type of susceptible. It is the purpose of this paper to analyze the development in time of epidemics which involve more than a single type of infective individual. This Received in revised form 3 January 1967. 257 This content downloaded from 157.55.39.78 on Mon, 20 Jun 2016 07:20:57 UTC All use subject to http://about.jstor.org/terms 258 HUGH M. PETTIGREW AND GEORGE H. WEISS problem, in which there may be several types of infectives, is suggested by diseases in which carriers are important, [9]. In the present paper we derive equations for the expected numbers of carriers and clinically infected individuals at any time t, in the case of a zero latent period and negative exponential distributions of infectious periods. From these equations we will give the conditions under which an epidemic arises from the introduction of a bearer of the disease, i.e., the conditions under which the reporting curve increases initially. These conditions are of some theoretical interest since Bailey has shown that in the infinite population approximation to an epidemic with only one class of susceptibles and one class of infected individuals, no initial increase in the reporting curve can occur. Finally, we derive expressions; for the expected total epidemic size. Let us consider a homogeneously mixing population which, in the present approximation, can be characterized by four parameters, yi(t), y,(t), zi(t), zc(t). These are, respectively, the number of clinically infected individuals, the number of carriers, the cumulative number of removals of infectives, and the cumulative number of removals of carriers, all evaluated at time t. The probability that a single susceptible individual will become clinically infected in (t, t + dt) is assumed to be fli(y, + yc)dt, and the probability that a susceptible will become a carrier is assumed to be Ic(yi + ye)dt, where both l's are assumed to be constant and it is assumed for simplicity that carriers and infectives are equally infectious. The constancy of these p's is the principal assumption of the present theory; in the more detailed stochastic theory discussed by Bailey [1] the f's are proportional to the number of susceptibles as well. In the simplest model, we assume that the latent period is zero, i.e., that newly infected individuals are immediately capable of infecting others, and that the infectious period is a random variable with a negative exponential distribution. The rate parameter appearing in the distribution for clinically infected will be denoted by vi and that appearing in the distribution for carriers will be denoted by v,. In order to describe the stochastic process we introduce a set of probabilities p(r,,r2, r3, r4, t) defined by p(r, t) = Pr{y,(t) = r, yc(t) = r2, z(t) = r3, zc(t) = r4}. By our assumptions the p(r, t) satisfy ap = fl (r1 + r2 1)p(r, 1,r2,r3, r4, t) + fl(r, + r2 -1)p(r1,r2 1,r3, r4, t) (1) + vA(r1 + 1)p(r + 1,r2, r 1,r4)+ vc(r2 + 1)p(r, r2 + 1, r3,r4 1) [(fl + ic + vi)rl + (fl + flPc + ve)r2] p(r, t). This content downloaded from 157.55.39.78 on Mon, 20 Jun 2016 07:20:57 UTC All use subject to http://about.jstor.org/terms Epidemics with carriers: the large population approximation 259 The moment generating function (2) M(01, 02, 03 , 0,t) = M(O, t) = E{eo'Y'+2yYc+03zi+042C} therefore satisfies aM M = [f#,(ee I1) + flc(e02 1) + vi(ee, +03 1)] (3) aM + [fl(ee' 1) + flc(e02 1) + vc(e-02+4 1)] The interesting features of the development of the epidemic can be determined by analyzing the properties of the mean values Lpi(t) = E{yi(t)}, pc(t) = E{yc(t)} (4) ow(t) = E{z,(t)}, wc(t) = E{zc(t)}. The equations for these mean values are easily determined by equating coefficients of the O's on both sides of Equation (3). They are Pi = (/01v-i)it?+fliltc (5) Pc = flcpi+ (fc v)Pc

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

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

After the publication of the important paper by Rota [9] on Möbius functions a large number of papers have appeared in which the ideas are applied or generalized in various directions, the papers by Crapo [3], Smith [10] and Tainiter [11] are some of them. The theory of Möbius functions is now recognized as a valuable tool in combinatorial and arithmetical research.

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

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

This paper arose out of an effort to present a result which simplifies the application of the theory of blocks to what is often called the theory of exceptional characters. In order to obtain the most effective use of this result, it is necessary to reformulate part of this theory. Thus we present a development which explains an application of R. Brauer’s main theorem on generalized decomposition numbers. In particular, we improve a result of D. Gorenstein and the author [8; Proposition 25]. These results are needed in a forthcoming paper and will simplify somewhat the use of this theory in existing papers. We are interested principally in determining the values of certain irreducible characters on trivial intersection subsets. The organization of the theory presented here is influenced by an exposition of M. Suzuki [13] and uses concepts introduced by W. Feit and J. G. Thompson [7]. Also it is hoped that this exposition will serve as an introduction to the theory.

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

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

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

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

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

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. GEOMETRICAL PROBABILITY; RANDOM LINES; RANDOM POLYGONS; PLANE OF CONSTANT CURVATURE; HYPERBOLIC PLANE; CONVEX DOMAINS; POISSON LINE PROCESS

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

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

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.

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.

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?

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.

Let X be an n-square matrix with elements in a field F. The permanent of X is defined by1.1where σ runs over the symmetric group of permutations on 1, 2, … , n. This function makes its appearance in certain combinatorial applications (13), and is involved in a conjecture of van der Waerden (6; 9). Certain formal properties of per (X) are known (1), and an old paper of Pólya (12) shows that for n > 2 one cannot multiply the elements of X by constants in any uniform way so as to convert the permanent into the determinant. In a subsequent paper we intend to investigate this problem for more general operations on X.

This paper is concerned with the distribution of "types" of individuals in an infinite population after indefinitely many nonoverlapping generations of random mating. The absence of selection and mutation is assumed. The probabilistic law which governs the production of an offspring may be asymmetrical with respect to the "sexes" of the two parents, but the law is assumed to apply independently of the "sex" of the offspring. The question of the existence of a limit distribution of types, the rate at which a limit distribution is approached, and properties of limit distributions are treated. Such questions have already received much attention in the genetic literature. Most of the limit results which have been obtained previously appear in the series of papers [5] [12] by Geiringer (1944-1949). These papers include results of earlier workers such as those of Jennings (1917), Robbins (1918) and Haldane (1930). With the exception of Geiringer's 1948 paper [8], all papers referred to in this paper assume that differences due to sex are absent. All of Geiringer's papers admit an arbitrary number of alleles at each locus. In [5], Geiringer (1944) obtains a limit theorem for haploid gametes, an arbitrary number of loci and linkage. Additional details in the case of three loci are given by Geiringer (1945) in [6]. A limit theorem is proved in Geiringer's 1947 paper, [7], for an arbitrary degree of autopolyploidy, one locus and chromosomal segregetion. In [8], Geiringer proves a limit theorem for haploid gametes, an arbitrary number of loci, and linkage with different recombination values for males and females. In [9], Geiringer (1949a) obtains the limit distribution and rate of convergence for diploid and triploid gametes, one locus and a general chromatid segregation; tetraploid gametes are considered briefly. In the second part of the two-part paper [10]"-[11], Geiringer (1949b, c) proves a limit theorem for an arbitrary degree of autopolyploidy, an arbitrary number of loci and chromosomal segregation with linkage. Paper [12] is a review paper in which Geiringer (1949d) summarizes the results then available, and indicates what remains to be done.

The main result of this paper is the determination of all nonisomorphic local near-rings with = which are not near-fields. Together with the fundamental paper [6] by Zassenhaus on near-fields and the corollary to Theorem 1 of [2], this 2 paper gives a complete description of all local near-rings of order p2.We recall that a unitary near-ring N is called local if the subset L of elements in N without left inverses is an (N, N)-subgroup and N ≠ J(N). (J(N) denotes the radical of N given in [1].) In [3] it was proved that N ≠ J(N) whenever L is an ideal of N. (For previous result s concerning local near-rings we refer the reader to [3].)

It has been demonstrated by Takaics in a series of papers and in his book [6] that combinatorial methods can be successfully applied to derive certain probability distributions in queueing processes. In this paper, we further illustrate the usefulness of combinatorial techniques and determine the stochastic law of the busy period in two queueing systems particularly involving batches. It may be of interest to note that queues involving batches have been dealt with in [3] and [7]. COMBINATORIAL THEORY; QUEUEING PROCESSES 1. Queueing model I Before describing the model, we give a combinatorial result which is required later. Consider a vector x = (x, -...,x,) with xi's integers and 0? X1 _ x2 ?-_ ...

In this paper we consider undirected graphs, with no edges joining a vertex to itself, but with possibly several edges joining pairs of vertices. The first part of the paper deals with the question of characterizing those sets of non-negative integers d1d2 . . . , dn and {cij}, 1 ≤ i < j ≤ n, such that there exists a graph G with n vertices whose valences (degrees) are the numbers di, and with the additional property that the number of edges joining i and j is at most cij. This problem has been studied extensively, in the general case (1, 2, 9, 11), in the case where the graph is bipartite (3, 5, 7, 10), and in the case where the Cij are all 1 (6).

The term rank p of a matrix is the order of the largest minor which has a non-zero term in the expansion of its determinant. In a recent paper (1), the authors made the following conjecture. If S is the sum of all the entries in a square matrix of non-negative real numbers and if M is the maximum row or column sum, then the term rank p of the matrix is greater than or equal to the least integer which is greater than or equal to S/M. A generalization of this conjecture is proved in § 2.The term doubly stochastic has been used to describe a matrix of nonnegative entries in which the row and column sums are all equal to one. In this paper, by a doubly stochastic matrix, the, authors mean a matrix of non-negative entries in which the row and column sums are all equal to the same real number T.

The manifold in this paper is assumed to be connected differentiable of class C∞. Let Dr(M) and Ӿr(M) be the set of all diffeomorphisms and vector fields of class Cr on a manifold M with Whitney Cr topology, respectively. In [2], the concept of weak stability is defined. The definition is equivalent to the following ((2.1) of this paper); f∈Dr(M) or X ∈ Ӿr(M) is weakly (allowably) stable if and only if there is a neighborhood U of f or X in Dr(M) or Ӿr(M) such that for any (a suitable) g or Y ∈ U the set of all elements topologically equivalent to g or Y is dense in U, respectively. Here, f, g ∈ Dr(M) are said to be topologically equivalent if they are topologically conjugate and X, Y ∈ Ӿr(M) are said to be topologically equivalent if there is a homeomorphism mapping any trajectory of X onto a trajectory of Y preserving the orientations of the trajectories. Similarly, weak Ω-stability is defined for f and X.

has not yet been completely solved. Lehmer ([7], [8]) proved that the least integer n with r(n) =0 must be a prime >214,928,639,999. This limit was further raised by Lehmer and others. In this paper we shall take a different approach to the problem. It was proved by Gupta [5] that (1.2) r(21) 0 O (mod 230), r(20) 0 0 (mod 230+1), which implies that r(20) never vanishes. Now let p be an odd prime and let 0 be determined by p = 20n -1, n odd. In Section 2 we shall prove that, for 0=1,. . ,9, (1.3) r(p)0 (mod 20), r(p) $ 0 (mod 20+1). This suggests the conjecture* that (1.3) holds for all 0 > 1 and this in turn would imply that Tr(p) never vanishes. In Section 3, we shall show that (1.3) for all 0 ? 1 implies (1.4) r(p4m+l) = 0 (mod 20), -r(p4mn+1) 0 0 (mod 20+1), (1.5) r(p40+S) =0 (mod 20), 7r(p4m+S) 0 0 (mod 20+1). In Section 4 we give a new proof of Lehmer's result [7] that if r(p) 5d0 for a prime p, then 'r(pn) 5-?0 for n >2. Furthermore, we shall prove that if (1.3) is valid then, for M odd, (1.6) r(pkM-1) 0 (mod 20+k-1), r(p2kM-1) $ 0 (mod 20+k). It is to be noted that throughout the paper p denotes an odd prime and 0 is determined by p = 2'n -1, n odd. 2. We now proceed to prove (1.3) for 0= 1, * , 9. It is known that [9] (2.1) Tr(M) M2cr7(M) (mod 22). Let M=2n-1 be a prime where n is odd. Then from (2.1) we have r(2n-1) 2n (mod 22) and, since n is odd, Tr(2n-1)-0 (mod 2), r(2n-1) 0 0 (mod 22). Also it is known that ([3], [4]) (2.2) r(M) o-e(M) (mod 23). * Professor Lehmer had suggested the form r(p) oan (p) (mod 2?) for (1.3). He made the conjecture independently but did not publish it and it was communicated privately to the author. 757

The theory of classification of homogeneous bounded domains in the complex number space Cn has been developed mainly in the recent papers [10], [6], [3] and [7]. As a result, the classification is reduced to that of S-algebras due to Takeuchi [7] which correspond to irreducible Siegel domains of type I or type II (For the definition of irreducibility see § 1). On the other hand Pjateckii-Sapiro [5] found large classes of homogeneous Siegel domains obtained from classical self-dual cones. Even in lower-dimensional cases, however, there are still homogeneous Siegel domains which do not appear in his results.

In this paper we derive a generation procedure for the simple (3-valent) 3-polytopes with cyclically 5-connected graphs. (A graph is called cyclically n-connected if it cannot be broken into two components, each containing a cycle, by the removal of fewer than n edges.) We define three new types of face splitting and we show, in Theorems 16 and 17, that the simple 3-polytopes with cyclically 5-connected graphs are exactly the polytopes obtained from the dodecahedron by these face splittings.

Our main purpose is to enlarge upon the studies of McAdam [9; 10] on the property of going down (GD) for prime ideals in extensions of (commutative integral) domains. Unlike the investigations of McAdam and the earlier work of Krull [8] and Cohen-Seidenberg [4] on GD and the related property of going up (GU), this paper is not primarily concerned with integral extensions. Consideration of more general extensions of domains A ⊂ B is facilitated by the following basic definitions. A prime ideal P of A is unibranched in B if there exists exactly one prime ideal Q of B satisfying Q ∩ A = P.

There have been several recent papers on the subject of the P-hull and the SP-hull of an l-group (lattice-ordered group). The most natural formulation of the concepts was given by P. Conrad in [6]. T. Speed studied P-groups extensively in [11]; his work was motivated by earlier work by H. Nakano and I. Amemiya in a vector lattice setting. A. Vecksler [12] produced the SP-hull for f-rings. The ortho-completion of S. Bernau [2] is a related concept.

In [4], a subgroup theorem for HNN groups was established. The theorem was proved by embedding the given HNN group in a free product with amalgamated subgroup and then applying the subgroup theorem of [3]. In this paper we obtain a sharper form of the subgroup theorem of [4] by applying the Reidemeister-Schreier method directly, using an appropriate Schreier system of coset representatives. Specifically, we prove (in Theorem 1) that if H is a subgroup of the HNN group1

This paper is concerned with the internal structure of Ext(Q, T) where Q is the group of rationals and T a reduced p-primary group of unbounded order. In [1] Irwin, Khabbaz, and Rayna define the splitting length of an arbitrary abelian group A, written l(A), to be the least positive integer n, otherwise infinity, such that A ⊗ . . . ⊗ A (n factors) splits. The concept of splitting length has been induced on Ext(Q, T), see [2; 5].

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

It is common to represent taxonomic hierarchies of related objects (such as similar plant or animal species or languages of the same family) by rooted trees with labelled terminal vertices which represent the objects. The multivariate data comparing numerous characteristics of the objects is first reduced to indices of similarity (or more often of dissimilarity) between each pair of objects. These are used to classify the objects into groups which are then depicted on a tree. This paper shows that an unrooted tree with labelled terminal vertices may provide a better representation of the relationships between the objects because the similarity indices are required to conform to fewer restrictions. Also for a given number of terminal vertices, there are fewer unrooted than rooted trees so that studies using probability distributions of trees or seeking the most suitable tree to represent the data are more practicable.

In [5] N. D. Lane and P. Scherk discuss arcs in the conformai (inversive) plane which are met by every circle at not more than three points; i.e., arcs of cyclic order three. This paper is concerned with the analysis of normal arcs of cyclic order four in the conformai plane.

We treat the sum , where α and γ are real with α positive. This sum was treated first by Hardy and Littlewood [4], and after them, by Behnke [1] and [2], Mordell [9], Wilton [11] and others. The reader will find its history in [7] and in the comments of the Collected Papers [4]. Here we show that the sum can be expressed explicitly, together with an error term O(N1/2), using the regular continued fraction expansion of α. As the statements have complications we will divide them into two theorems. In the followings all letters except ϑ, i, σ, ζ, χ and those in 3° are real, N is a positive real, and always k, n, a, A, B, C, D and E denote integers. The author expresses his thanks to Professor Tikao Tatuzawa and Professor Tomio Kubota for their encouragements.