In this paper priority queues with K classes of customers with a preemptive repeat and a preemptive resume policy are considered. Customers arrive in independent Poisson processes, are served, within classes, in order of arrival, and have general requirements for service. Transforms of stationary waiting time and queue size distributions and busy period distributions are obtained for individual classes and for the system; the moments of the distributions are considered.

In his Topics in Number Theory, vol. 2, chapter 2 (Reading, Mass., 1956) W. J. LeVeque proved an important generalisation of Roth's theorem (K. F. Roth, Mathematika 2,1955, 1—20).

Let ξ be a fixed algebraic number, σ a positive constant, and K an algebraic number field of degree n. For κ∈K denote by κ(1), …, κ(n) the conjugates of κ relative to K, by h(κ) the smallest positive integer such that the polynomial has rational integral coefficients, and by q(κ) the quantity

Certain integers have the property that they can be partitioned into distinct positive integers whose reciprocals sum to 1, e.g., and In this paper we prove that all integers exceeding 77 possess this property. This result can then be used to establish the more general theorem that for any positive rational numbers α and β there exists an integer r(α, β) such that any integer exceeding r(α, β) can be partitioned into distinct positive integers exceeding β whose reciprocals sum to α.

Certain optimization problems involving inequality constraints, known as optimal control problems have been extensively studied during recent years especially in relation to the calculation of optimal rocket thrusts and trajectories. A summary of these works is given by Berkovitz [1] who also establishes necessary conditions for the existence of solutions for a wide class of such problems.

Lagrange's theorem on the reversion of power series may be stated in the following form (e.g. Whittker and Watson [3]).

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.