Search

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

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].

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

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.

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

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 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.

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 ?-_ ...

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.

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.

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.

The first part of the paper gives a multitude of essentially different representations of a stationary stochastic process. The second part gives a sufficient condition for the sum of two oscillatory processes to be again oscillatory.

Let {Xk : k ≧ 1} be a sequence of independent, identically distributed random variables with EX 1 = μ < 0. Form the random walk {Sn : n ≧ 0} by setting S 0 = 0, Sn = X 1 + … + Xn, n ≧ 1. Let T denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of X 1) that Sn , conditioned on T > n converges weakly to a limit random variable, S∗, and to find the Laplace transform of the distribution of S∗. We also investigate a collection of random walks with mean μ < 0 and conditional limits S∗ (μ), and show that S∗ (μ), properly normalized, converges to a gamma distribution of second order as μ ↗ 0. These results have applications to the GI/G/1 queue, collective risk theory, and the gambler's ruin problem.

Let k denote the quotient field of a complete discrete rank one valuation ring R. The purpose of this paper is to establish a relationship between the Brauer group of k and the set of maximal orders over R which are equivalent to crossed products over tamely ramified extensions of R.

Let P= {p 1,···, PN } be a finite discrete probability distribution. Then the entropy of the distribution P, introduced by Shannon [12], is defined as Throughout this paper ∑ will stand for and logarithms will be taken to the base D (D > 1).

The purpose of this paper is to give the limit formula of the Kronecker’s type for a non-holomorphic Eisenstein series with respect to a Hubert modular group in the case of an arbitrary algebraic number field. Actually, we shall generalize the following result which is well-known as the first Kronecker’s limit formula. From our view-point, this classical case is corresponding to the case of the rational number field Q.

In this paper is developed a theory of branching processes in which the division probability and rate of growth of cells depend only on their ‘size’ and in which ‘size’ is shared between daughters. Specific results are obtained in a linear case including the calculation of the correlation coefficient for the life spans of sisters.

Let G be a connected semi-simple algebraic group defined over Q and let Γ be a discrete subgroup of GR (the subgroup of G consisting of points rational over R) such that Γ\GR is compact. The main purpose of the present paper is to prove that for a certain type of group G there exists an invariant algebraic differential from ω on G of highest degree defined over Q such that

Let k(≧3) be a positive integer and φ(k) be the Euler function. We denote by % one of the φ(k) characters formed with modulus k, and by % the principal character. Let Lis, %) be the L-series corresponding to % Throughout the paper we use c and c(ε) to denote respectively an absolute positive constant and a positive constant depending on parameter ε( >0) alone, not necessarily the same at their various occurrences. We use the symbol y = O(Z) for positive X when there exists a c satisfying Y ≦ in the full domain of existence of X and Y.

In [1] R. Brauer asked the following question: Letbe a finite group,pa rational prime number, andBa p-block ofwith defectdand defect group. Is it true thatis abelian if and only if every irreducible character inBhas height 0 ? The present results on this problem are quite incomplete. Ifd-0, 1, 2 the conjecture was proved by Brauer and Feit, [4] Theorem 2. They also showed that ifis cyclic, then no characters of positive height appear inB. Ifis normal in, the conjecture was proved by W. Reynolds and M. Suzuki, [12]. In this paper we shall show that for a solvable group, the conjecture is true for the largest prime divisorpof the order of. Actually, one half of this has already been proved in [7]. There it was shown that ifis a p-solvable group, wherepis any prime, and ifis abelian, then the condition on the irreducible characters inBis satisfied.

Let k be an algebraic number field of finite degree, and l a rational prime (including 2); k and l being fixed throughout this paper. For any power ln of l, denote by ζn an arbitrarily fixed primitive ln-th root of unity, and put kn = k(ζn). Let r be the maximal rational integer such that ζr∈k i.e. kr = k and kr+1≠k.

In this paper the theory of genetic algebras introduced by Etherington [3], [5] is developed to cover the case where the population is divided into a number of partially isolated subpopulations, between which a limited amount of migration is permitted. It is known [3], [6], [7], [8], [10], that in the absence of selective factors most genetic segregation systems can be represented by algebras belonging to the class of special train algebras, discussed in detail in [4].

For a closed and bounded set E in the complex plane, let A(E) denote the collection of all functions continuous on E and analytic on E°, its interior; let R(E) denote the collection of all functions which are uniform limits on E of rational functions with poles outside E. Then let A denote the collection of all closed, bounded sets for which A(E) = R(E). The purpose of this paper is to formulate a condition on a set, which is essentially of a geometric nature, in order that the set belong to A. Then using approximation techniques, we shall construct a meromorphic function having a certain boundary behavior on a perfect set; this answers a question raised in [1].

If Z(t) denotes the population size in a Bellman-Harris age-dependent branching process such that a non-denenerate random variable W, then it is known that E(W) = 1 and that ϕ (u) = E(e–uW ) satisfies a well-known integral equation. In this situation Athreya [1] has recently found a NASC for E(W |log W| y ) <∞, for γ > 0. This paper generalizes Athreya's results in two directions. Firstly a more general class of branching processes is considered; secondly conditions are found for E(W 1 + βL(W)) < ∞ for 0 β < 1, where L is one of a class of functions of slow variation.

Our object in this paper is to prove the following result.THEOREM. Let G be a finite group satisfying the following conditions:(*) G has subgroups L1, L2 such that L1 ≃ SL(2, q1), L2 ≃ SL(2, q2), [L1 L2] = 1, , where j is an involution, and |C(j): L1L2| = 2.(**) where n2 = 1

Suppose that G is the projective symplectic groupPSp(4, q), the Dickson groupG2(q)>or the Steinberg “triality-twisted” groupwhereqis an odd prime power. Then G is a finite simple group, andGcontains an involutionjsuch that the centralizerC(j)inGhas a subgroup of index 2 which containsjand which is the central product of two groups isomorphic withSL(2,q1)andSL(2,q2)for suitableqlq2.We wish to show that conversely the only finite simple groups containing an involution with this property are the groupsPSp(4,q), G2(q)9.In this first paper we shall prove the following result.

The present paper is concerned with the classification and corresponding extension theorem of mappings of the (n+-2)-complex Kn+2 (n>2) into the space Y whose homotopy groups πi(Y) vanish for i < n and i = n+1, and the n-th homotopy group πn(Y) of which has a finite number of generators. Our methods followed here are essentially analogous to those of Steenrod [2].

This paper deals with asymptotic properties of various models of age- dependent branching processes. Cell growth is considered in which a cell proceeds in a sequential manner through n independent states, state R with its life distribution FR, R=1,2,...,n. At the end of mitosis, the nth state, the cell divides into similar cells, the number of which is governed by a generating function h, independent of the time and other cells of the process. A variation of the given model treats cell growth in which the cell proceeds from state to state according to a general semi-Markov process until the mitotic state is completed, when division into similar cells in accord with h occurs. The random variable is considered in a simple age dependent branching process for m > 1. Another case is that of two types of cells, of which only one type may divide while the other either accumulates or is eventually absorbed in the medium. An example of this is the production of stem cells and red blood cells from parent stem cells. Two related models are considered, both examples of the 'reducible case'. Increasing cell populations are considered. A binary fission case of each model may be represented schematically to indicate the various combinations of cell births with corresponding probabilities. (Author)

Additive processes on finite Markov chains have been investigated by Miller ([8], [9]), Keilson and Wishart ([2], [3], [4]) and by Fukushima and Hitsuda [1]. These papers study a two-dimensional Markov Process {X(t), R(t)} whose state space is R1 × {1, 2, ···, R} characterized by the following properties: (i)R(t) is an irreducible Markov chain on states 1,2, …,R governed by atransition probability matrix Bo = {brs}.(ii)X(t) is a sum of random increments dependent on the chain, i.e., if the ith transition takes the chain from state r to state s, then the increment has the distribution (iii)Nt, is t in discrete time while in the continuous time case Nt, might be an independent Poisson process.

This paper studies the existence, in a stable GI/M/1 queue, of the limit as t → ∞ of the distribution of the virtual waiting time process at time t conditioned on the event that at no time in the interval [0, t] the queue has become empty. The conditional limit distribution obtained when the traffic intensity is strictly less than one is the weighted sum of an exponential and a gamma distribution. Similar conditional limit distributions are obtained for the queue size process and the waiting time process as defined by Prabhu (1964).

The subject of this paper is the study of the distribution of integrals of the type where {X(t); t ≧ 0} is some appropriately defined continuous-time parameter stochastic process, and f is a suitable non-negative function of its arguments. This subject has also sometimes been labelled as “the occupation time or the sojourn time problem” in literature. These integrals arise in several domains of applications such as in the theory of inventories and storage (see Moran [14], Naddor [15]), in the study of the cost of the flow-stopping incident involved in the automobile traffic jams (see Gaver [8], Daley [3], Daley and Jacobs [4]). The author encountered such integrals while studying certain stochastic models suitable for the study of response time distributions arising in various live situations. In fact in [19], it was shown that such a distribution is equivalent to the study of an integral of the type (1). Again, in the study of response of host to injection of virulent bacteria, Y(t) with f(X(t), t) = bX(t), with b > 0, could be regarded as a measure of the total amount of toxins produced by the bacteria during (0, t), assuming a constant toxin-excretion rate per bacterium. Here X(t) denotes the number of live bacteria at time t, the growth of which is governed by a birth and death process (see Puri [16], [17] and [18]).

In this paper, we investigate several classes of groups, among which the most general is defined as follows:Definition 1.1. A finite group G is a SR-group if it contains a subgroup P1 of order 3 satisfying: (a)A/S3-subgroup P2 of NGP1 is elementary of order 9;(b)NG(P2)/P2 acts semi-regularly by conjugation on the conjugates of P1 contained in P2.

The random packing of geometric objects in one-, two- or three-dimensions may afford useful insights into the structure of crystals, liquids, absorbates on crystals, and in higher dimensions, into problems of pattern recognition. Random packing has accordingly received increasing attention in recent years. Two principal packing procedures have been formulated and each gives rise to different packing ratios. In one case, all possible configurations of a sphere-packed volume are assumed to be equally likely. In the other and most widely reported case, there is random sequential addition of spheres to the volume until it is packed. This is the situation we study in this paper. Most of the work to date has been limited to the theoretical study of the one-dimensional lattice or to continuous cases particularly in the limit for long lines. The higher dimensional cases have resisted theoretical attack but have been studied by computer simulation by Palasti [12] and Solomon [14] and by physical simulation by Bernal and Scott (see [14]).

Summary In the theory of phage reproduction, the mathematical models considered thus far (see Gani [5]) assume that the bacterial burst occurs a fixed time after infection, after a fixed number of generations of phage multiplication, or when the number of mature bacteriophages has reached a fixed threshold. In the present paper, a more realistic assumption is considered: given that until time t the bacterial burst has not taken place, its occurence between tand t + Δt is a random event with probability f(· | t)Δt + o(Δt), where f is a non-negative and non-decreasing function of the number X(t) of vegetative phages and of Z(t), the number of mature bacteriophages at time t. More specifically it is assumed that f = b(t)X(t) + c(t)Z(t) with b(t), c(t) ≦ 0. Here X(t) denotes the survivors in a linear birth and death process and Z(t) the number of deaths until time t. The joint distribution of XT and ZT , the respective numbers of vegetative and mature bacteriophages at the burst time is considered. The distribution of ZT is then fitted to some observed data of Delbrück [2].

Let w = f(z) be a normal meromorphic function defined in the upper half plane U = {Im(z) >0}. We recall that a meromorphic function f(z) is normal in U if the family {f(S(z))}, where z′ = S(z) is an arbitrary one-one conformal mapping of U onto U, is normal in the sense of Montel. It is the purpose of this paper to state some results on the behavior of f(z) on curves which approach a point x0 on the real axis R with a fixed (finite) order of contact q at x0.

The purpose of this paper is to prove the following three theorems which were announced in [2].

Summary In recent papers Yassky (1962) and Gani (1965) have considered respectively deterministic and stochastic models for the attachment of phages to bacteria. Following Brenner's (1965) conjecture they assumed that there is a maximum, S (say), to the number of phages that can be attached to a bacterium. In this note, Gani's (1965) results will be obtained starting from a different set of assumptions. This modification enables us to consider the case when S is a random variable, which probably is a more realistic assumption. Some remarks on the problem of estimation for the latter model are given in Section 3.

For superposition of independent, stationary renewal processes, it is well known that the distribution of waiting time between events for the superimposed process is approximately exponential if the number of processes involved is sufficiently large, (see Khintchine (1960), Ososkov (1956)). We assume that all component processes have the same age t, and we generalize the classical result to show that even for t finite (non-stationary case), the limiting waiting time distribution (as the number of processes increases) is exponential with a scale parameter which depends on t through the average of the individual process renewal densities. For moderate values of t, this scale parameter can differ greatly from its limit due to the generally slow convergence of the renewal density function to its limit. While it has been generally assumed that this waiting time distribution is approximately exponential regardless of t, the explicit formulation of the necessary normalizations and conditions seems new. While a variety of papers on superimposed point processes have appeared recently, all assume stationarity (see 4inlar (1968) for references). Franken (1963) considered the time dependent behavior of superimposed renewal processes under the highly restrictive and generally unrealistic assumption that for each n, the superimposed process is itself a renewal process. Among other things, this requires the individual processes to have distributions which depend on n and which generally do not resemble any standard form. We believe that letting the individual processes have some standard form and imposing no particular form on the superimposed process is more realistic.

Let G be a finite group and let p be a fixed prime number. If D is any p-subgroup of G, then the problem whether there exists a p-block with D as its defect group is reduced to whether NG(D)/D possesses a p-block of defect 0. Some necessary or sufficient conditions for a finite group to possess a p-block of defect 0 have been known (Brauer-Fowler [1], Green [3], Ito [4] [5]). In this paper we shall show that the existences of such blocks depend on the multiplicative structures of the p-elements of G. Namely, let p be a prime divisor of p in an algebraic number field which is a splitting one for G, o the ring of p-integers and k = o/p, the residue class field.