Various authors have studied the transient behaviour of single-server queues. Notably, Takacs [13], [14] has analysed a queue with recurrent input and exponential service time distributions, Keilson and Kooharian [9], [10] and Finch [5] have considered a queue with general independent input and service times, Finch [6] has analysed a queue with non-recurrent input and Erlang service, and Jaiswal [8] has considered the bulk-service queue with Poisson input and Erlang service.

H. Simpson [1] has proved a theorem about six points P1,…,P6 (no three in line, not all on a conic) of the real projective plane. He calls P1 an “in-point” or an “out-point”, according as it lies inside or outside the conic (denoted by a Roman figure I) through the other five points; and so on. The theorem is that there are either two, three, or six in-points.

In a recent paper on statistical fluid mechanics Professor J. Kampé de Fériet [1] employed several integrals of which the following is a typical example The function u(x, y, t), which it defines, formally satisfies the following three classical differential equations

A theorem on the canonical form of an antisymmetric matrix, which we attribute to Zumino, is used to show how the derivations of the results of two earlier papers by Blatt on expectation values of operators can be considerably simplified.

Consider a random Markovian process in which the state of the system is defined by a random variable which can take the finite set of values i=0, 1, …, N, and which is such that transition can only occur from any state i to the two nearest states i+1. This restriction brings about an essential simplification of the theory for the basic reason that in order for the system to move from i to state j (i < j say) it must first move to i+1, then i+2 and so on until it reaches j. From this it follows that the first passage distribution from i to j is the convolution of the first passage distributions from i to i+l, i+l to i+2,…, j—l to j each of which is comparatively easy to find.

The problem of storage in an infinite dam with a continuous release has been studied by a number of authors ([5], [3], [2]), who have formulated it in probabilistic terms by supposing the input to be a continuous time stochastic process. These authors have encountered difficulties which they have overcome by regarding the continuous time problem as a limit of discrete time analogues. analogues. The purpose of this paper is to suggest that these difficulties are the result of an unfortunate specification of the problem, and to show that the adoption of a slightly different (and more realistic) formulation avoids the difficulties and allows a treatment which does not have recourse to discrete time analogues.

First consider some familiar results, the inequality of the arithmetic and geometric mean is: Kantorovich's inequality (reference [1]) asserts that if 0 < A ≦f(x) ≦ B then: The Cauchy-Schwarz inequality is: This paper discusses a certain class of inequalities which includes the three above. Three theorems are proved which apply to any inequality of this class; then follow some examples. They are mainly to show how the general theory helps in the finding of inequalities, but the result of Example 1 seems worth reporting for its own sake.

The well-known Banach Contraction Principle asserts that any self-map F of a complete metric space M with the property that, for some number k < 1, for all x, y,∈M, possesses a unique fixed point in M. some extensions and analogues have recently been given by Edelstein [1]. For the reader's convenlience we state here the result of Edelstein which we shall employ. It asserts that if F is a self-map of a metric space M having the property that for any two distinct points x and y of M, and if x0 is a point of M such that the sequence of iterates xn = Fn (x0) contains a subsequence which converges in M, then the limit of this subsequence is the unique fixed point of F.

In 1943, Post conjectured that “monogenic normal systems are universal”, and in 1961 Minsky proved a stronger result “‘tag’ systems are universal” which implied the proof of Post's conjecture. The author had independently obtained a simple direct proof of Post's conjecture. The purpose of this note, then, is to present an exposition of Post's conjecture, and to show the full simplicity of its direct verification.

The work described in this paper grew out of an attempt to generalize some results obtained in an earlier paper [4] on the water entry problem of a thin wedge or cone into an incompressible fluid. The object of the generalization was to include the effect of gravity terms. In most papers on hydrodynamic impact it is considered permissible to neglect this effect since gravity terms might be expected to play a minor role in the initial stages of the motion. However, it seems desirable to investigate the effect of including gravity terms in order both to examine the later stages of the motion and to estimate to what extent their neglect is justified in the early stages. It will be seen that it is possible to develop a fairly complete solution for the normal entry of a thin symmetric body, both for two-dimensional and axially symmetric cases, on the basis of a linearized theory. The restriction to a linearized theory means that the whole field of analysis associated with the theory of surface waves of small amplitude becomes available. Most of the problems considered in this paper are initial value problems in which the whole fluid is at rest at t = 0.

The functions f defined by or by for c rational and less than + 1 map the set of rational numbers between 0 and 1 one-to-one onto itself; and they are the only fractional linear functions with this property. Miss Tekla Taylor recently raised the question * whether these are the only differentiable functions with the stated property. In the present note we show, by two different constructions, that the answer is negative; in each case much freedom remains, which could be used to make the functions in question have various additional properties.

Let R denote the set of real numbers, B the σ-field of all Borel subsets of R. A homogeneous Markov Chain with state space a Borel subset Ω of R is a sequence {an}, n≧ 0, of random variables, taking values in Ω, with one-step transition probabilities P(1) (ξ, A) defined by for each choice of ξ, ξ0, …, ξn−1 in ω and all Borel subsets A of ω The fact that the right-hand side of (1.1) does not depend on the ξi, 0 ≧ i > n, is of course the Markovian property, the non-dependence on n is the homogeneity of the chain.

The present note continues the discussion, begun in the first paper with the above title, of classes of spaces H which are extensions of L2 (0, ∞) and whose elements, which we call ‘sequence-functions’, exhibit some of the properties of distributions. The previous paper defined the spaces, and described how they can be used to extend the domain of definition of Waston transforms. Further related applications to transform theory are described in [2] and [3]. In this paper I pursue the analogy between these sequence- functions and other types of generalized functions further by discussing their local behaviour, the existence of ordinary and convolution-type products, and of derivatives and integrals.

We consider the genetic model introduced by Moran [3] of a haploid population of fixed size M with two genotypes A and a for which the possibility of selection is allowed. In this model an individual is randomly chosen to die and is replaced by a new individual whose probability of being a depends on the selective advantages of the two genotypes and on the number of a individuals before the birth-death event. The probability of eventual elimination of the genotype a, both with and without selection, has been found by Moran [3], while Watterson [4] has found the mean time for absorption and the variance in the case where no selection is allowed. We derive here the mean time and the variance in the case where selection is allowed, thus extending Watterson's result. A diffusion approximation is available for the mean time; it is shown that this gives a very close approximation to the exact value. Comparison is made with the non-overlapping generation model due to Wright [5], and finally some numerical results are exhibited.

A perturbation method is developed, and is used to obtain approximate expressions for the expectation values of one-particle and two particle operators in the quasi-chemical equilibrium (pair correlation) approximation to statistical mechanics, for the case of non-extreme Bose-Einstein condensation of the correlated pairs. To lowest order, the approximate results reproduce the results obtained previously for the case of extreme Bose-Einstein condensation.

The aim of this paper is to dervie two formulae for π(N) that need involve only a few of the smallest primes. Here m is a small integer, the b's are integers that will be found later, and Pij…k denotes the number of products figi… ≧ N, in which f, g,?…,h are unequal integers greater than 1 and prime to the first m primes. The suffixes run through all partitions of all integers.

A crack is assumed to be the union of two smooth plane surfaces of which various parts may be in contact, while the remainder will not. Such a crack in an isotropic elastic solid is an obstacle to the propagation of plane pulses of the scalar and vector velocity potential so that both reflected and diffracted fields will be set up. In spite of the non-linearity which is present because the state of the crack, and hence the conditions to be applied at the surfaces, is a function of the dependent variables, it is possible to separate incident step-function pulses into either those of a tensile or a compressive nature and the associated scattered field may then be calculated. One new feature which arises is that following the arrival of a tensile field which tends to open up the crack there is necessarily a scattered field which causes the crack to close itself with the velocity of free surface waves.

In recent years there has been extensive development in the theory and techniques of mathematical programming in finite spaces. It would be very useful in practice to extend this development to infinite spaces, in order to treat more realistically the problems that arise for example in economic situations involving infinitely divisible processes, and in particular problems involving time as a continuous variable. A more mathematical reason for seeking such generalisation is possibly that of obtaining a unification mathematical programming with other branches of mathematics concerned with extrema, such as the calculus of variations.