The purpose of the present paper is to give a general theory of the quantum mechanical representation of particle populations.

The first part of the paper, Sections 1 to 5, is devoted to a review of mathematical principles of quantum theory, with particular emphasis on the role played by probability concepts, using an approach adapted to the subsequent development of the theory of particle populations. This approach, which goes back in its essentials to von Neumann [20], leans heavily on the subsequent work of Wigner, Mackey, Jauch, Segal, Wightman and many others (see e.g., Mackey [15], Jauch [11], Streater and Wightman [26]). Sections 6 to 9 deal with the representation of finite particle populations: i.e., quantum systems where the total number of particles is an observable. In Section 10 a brief sketch is given of the generalization of the theory to infinite populations where the total number of particles is not an observable, as e.g., in the statistical theory of an infinitely extended gas (see Ruelle [22]). Finally, Section 11 treats some simple examples.

This paper discusses two-terminal series-parallel networks occurring in Applied Probability and other contexts where the mathematical theory of reliability is developed. Eighty years ago Mac Mahon successfully discussed the number of different structures built of n identical components [1]; later Knödel [3] and Carlitz and Riordan [4] in the 1950's investigated the number of different structures built of n different components. However, in many new applications, engineers and statisticians have to study structures having n components with a specification defining various types of components and stating how many identical components of each kind should be used. This general problem is investigated in Section 7 whilst in Sections 1–6 results obtained earlier are presented and it is shown there how modern approaches to combinatorics (particularly Pólya's Enumerative Combinatorial Analysis) can simplify some reasoning of previous authors.

In the homogeneous process X(t) defined on a finite irreducible Markov chain R(t) was studied. The process was characterized by an overall transition rate v per unit time, a matrix β of transition probabilities for the chain and a linear growth for X(t) dependent on the chain; viz. dX(t)/dt|R(t) = i = vi. The central limit behavior of the process was exhibited in. For the case when the chain had only two states, the ruin and ergodic problems were considered in for the bounded process. The object of this paper is to investigate the ruin and ergodic problems for the process of in the presence of boundaries.

The paper consists of three parts A, B and C. In part A, “A contribution to the methodology of biomathematics”, general considerations on mathematical analysis and automaton theory are given in order to show how and why new approaches for biomathematics are required from the standpoint of information science. In fact three requirements are mentioned: (1°) general system approach; (2°) historical approach; (3°) three indispensable constituent fields of biological science. Regarding (1°), the author refers to his work on the logic of information science explained in 1968 at the CAS Seminar in Dubrovnik and published in 1969 in a Japanese monograph. Regarding (2°), reference is made to his paper “A contribution to the methodology of futurology” published in International Conference on Futurology, April 1970, Kyoto. The three indispensable constitutent fields are given by (a) theoretical, (b) experimental and (c) engineering biologies. In view of these three requirements, characteristic features of biomathematics in connection with an information science approach are (1°) discrete, (2°) combinatorial, (3°) dynamical, (4°) evolutionary, and (5°) design mathematics.

This Symposium was organized “to explore existing mathematical theories of disease and the possibilities offered to the medical sciences by further research in this field”. Because of the extreme breadth of these terms of reference, the discussion was limited to problems of communicable disease. The intention was to bring academic mathematical research into closer contact with applied problems in public health. It was thought that in this way, WHO would have greater access to theoretical development, while mathematical work in Universities and research institutes would achieve closer contact with practical problems.

We consider a single GI/G/1 queueing system in which customer number 0 arrives at time t0 = 0, finds a free server, and experiences a service time v0. The nth customer arrives at time tn and experiences a service time vn. Let the interarrival times tn - tn-1 = un, n ≧ 1, and define the random vectors Xn = (vn-1, un), n ≧ 1. We assume the sequence of random vectors {Xn : n ≧ 1} is independent and identically distributed (i.i.d.). Let E{un} = λ-1 and E{vn} = μ-1, where 0 < λ, μ < ∞. In addition, we shall always assume that E{v02} < ∞ and that the deterministic system in which both vn and un are degenerate is excluded. The natural measure of congestion for this system is the traffic intensity ρ = λ/μ. In this paper we shall restrict our attention to systems in which ρ < 1. Under this condition, which we shall refer to as light traffic, our system is of course stable.

When a basal cell divides, both daughter cells remain in the basal layer of the epithelium, with one of the neighbouring cells being pushed out to make room. This fact opens the possibility that a cell with a heritable advantage over the normal cells may gradually produce a clone covering more and more of the basal layer. The advantage in question may consist in a faster rate of division than normal, or a more tenacious hold on the basement membrane; we shall limit consideration to the former situation.

There exists already a considerable literature about a mathematical approach to the description and control of malaria (see Ross (1911); Martini (1921); Lotka (1923); Moshkovskii (1950); Macdonald (1957), (1968)). This note gives an outline of the main assumptions of a simulation model which has been developed in close co-operation with epidemiologists, entomologists and immunologists in connection with a field project in Northern Nigeria where the necessary data for testing the model are being collected.

This report is based mainly on the research conducted in 1969–70 in the Division of Research in Epidemiology and Communications Science of the World Health Organization, and gives a short résumé of methods elaborated for concise descriptions of what has been observed for prognosis of subsequent development of disease, and for the planning of new studies.

The purpose of this paper is to bring to the attention of epidemiologists the possibility of setting up and utilizing structural models, more precisely, mathematical models based on graph theory. We shall present the first general results in this investigation using some statements and definitions.

The aim of this study is an investigation of the joint limiting distribution of the sequence of partial sums of the positive parts and negative parts of a sequence of independent identically distributed random variables. In particular, let {Xn} be a sequence of independent identically distributed random variables with common distribution function F, assume F is in the domain of attraction of a stable distribution with characteristic exponent α, 0 < α ≦ 2, and let {Bn} be normalizing coefficients for F. Let us denote Xn+ = XnI[Xn > 0] and Xn− = − XnI[Xn<0], so that Xn = Xn+ - Xn−. Let F+ and F− denote the distribution functions of X1+ and X1− respectively, and let Sn(+) = X1+ + · · · + Xn+, Sn(-) = X1− + · · · + Xn−. The problem considered here is to find under what conditions there exist sequences of real numbers {an} and {bn} such that the joint distribution of (Bn-1Sn(+) + an, Bn-1Sn(-) + bn) converges to that of two independent random variables (U, V). As might be expected, different results are obtained when α < 2 and when α = 2. When α < 2, there always exist such sequences so that the above is true, and in this case both U and V are stable with characteristic exponent a, or one has such a stable distribution and the other is constant. When α = 2, and if 0 < ∫ x2dF(x) < ∞, then there always exist such sequences such that the above convergence takes place; both U and V are normal (possibly one is a constant), and if neither is a constant, then U and V are not independent. If α = 2 and ∫ x2dF(x) = ∞, then at least one of F+, F− is in the domain of partial attraction of the normal distribution, and a modified statement on the independence of U and V holds. Various specialized results are obtained for α = 2.

Influenza epidemics at a city or country level can be observed only through a daily officially registered morbidity, or briefly DORM. The influenza model for any country is defined as some set of equations containing the DORM for cities in that country as unknown functions of time, these being designed to approximate the cities' real DORM.

Let n and m be the initial numbers of susceptibles and infectives respectively in the well-known model of Bailey. The successive states of a population will be described by (i,j), where i is the number infected since the beginning of the epidemic, and j is the number removed (dead, isolated or recovered and immune) during the same period. The transition probabilities obviously are the following

It has been shown that for many epidemic models, the stochastic theory leads to essentially the same results as the deterministic theory provided that one identifies mean values with the functions calculated from the deterministic differential equations (cf. [1]). If one considers a generalization of Bailey's simple epidemic for a fixed population of size N, represented schematically by where I refers to an infected, S refers to a susceptible, and α and β are appropriate rate constants, then it is evident that at time t = ∞, the expected number of infected individuals must be zero provided that β > 0. If x(t) denotes the number of infected at time t, then the deterministic model is summarized by

Recent research at Sheffield, which is briefly described below, has been concerned with chain-binomial models, and stochastic path integrals under infective curves.

Bailey (1957) has considered the general stochastic epidemic with relative removal rate ρ, one initial infective and n susceptibles. His Equation (5.51) may be re-cast as where the fk satisfy the recurrence relationship

Let X = {1, …, a} and Y = {1, …, a} be the input and output alphabets, respectively. In the (unsynchronized) channels studied in this paper, when an element of X is sent over the channel, the receiver receives either nothing or a sequence of k letters, each a member of Y, where k, determined by chance, can be 1, or 2, or … or L, a given integer. The channel is called unsynchronized because the sequence received (for each letter sent) is not separated from the previous sequence or the following sequence, so that the receiver does not know which letters received correspond to which letter transmitted.

In Sections 1 and 2 we give the necessary definitions and auxiliary results. In Section 3 we extend the results of Dobrushin [2] by proving a strong converse to the coding theorem1 and making it possible to compute the capacity to within any desired accuracy.

In Section 4 we study the same channel with feedback, prove a coding theorem and strong converse, and give an algorithm for computing the capacity.

In Section 5 we study the unsynchronized channel where the transmission of each word is governed by an arbitrary element of a set of channel probability functions. Again we obtain the capacity of the channel, prove a coding theorem and strong converse, and give an algorithm for computing the capacity.

In Section 6 we apply results of Shannon [4] and supplement Dobrushin's results on continuous transmission with a fidelity criterion.