It is known that various cases of the steady isentropic irrotational motion of a compressible fluid are expressible as variational principle [1], [5]. in particular, the aerofoil problem i.e. the case of plane flow in which a uniform stream is locally deflected, without circulation, by a bounded obstacle, can be expressed in such a form. Thus we make stationary where the region R is that bounded internally by the obstacle (C0) and externally by a circle (CR) of radius R. In this expression φ∞ is the velocity potential for a uniform stream, and φ0 is the velocity potential for the corresponding incompressible flow.

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.

Let G be an n-generator group and let g = (g1, g2, …, gn) be an ordered set of n elements which generate G, then g is called a generating n-vector of G. Let denote the set of all generating n-vectors of G.

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.

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.

If G is a finite linear group of degree n, that is, a finite group of automorphisms of an n-dimensional complex vector space (or, equivalently, a finite group of non-singular matrices of order n with complex coefficients), I shall say that G is a quasi-permutation group if the trace of every element of G is a non-negative rational integer. The reason for this terminology is that, if G is a permutation group of degree n, its elements, considered as acting on the elements of a basis of an n-dimensional complex vector space V, induce automorphisms of V forming a group isomorphic to G. The trace of the automorphism corresponding to an element x of G is equal to the number of letters left fixed by x, and so is a non-negative integer. Thus, a permutation group of degree n has a representation as a quasi-permutation group of degree n.

The random processes discussed here may be specified in the following way. A fixed population of N members is spilt into two distinct classes. Individuals move about randomly between the classes, and we are interested in the size of each class at any time, rather than in the behaviour of particular individuals. Let i(t) and N —i(t) be the numbers present in the repective classes at the time t. It is assumed that the process {i(t), t ≧ 0} is Markovian, and that transitions between the states j = 0, 1, … N, occur according to the conditional probabilities; and.

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.

When n objects are to be compared in pairs, a complete experiment requires N= comparisons. There are frequent occasions when it si desirable to make only a fraction F of the possible comparison, either because N is large or because even an individual comparison is laborious. The problem of what constitutes a satisfactory subset of the comparisons has been considered by Kendall [5[ who lays down the following two minimum requirements: (a) every object should appear equally often; (b) the design should be ‘connected‘ so that it is impossible to split the objects into two sets with no comparison made between objects in one set and objects in the other.

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.

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.

We consider a set of η first order simultaneous differential equations in the dependent variables y1, y2, …, yn and the independent variable x ⋮ No loss of gernerality results from taking the functions f1, f2, …, fn to be independent of x, for if this were not so an additional dependent variable yn+1, anc be introduced which always equals x and thus satisfies the differential equation

Huta [1], [2] has given two processes for solving a first order differential equation to sixth order accuracy. His methods are each eight stage Runge-Kutta processes and differ mainly in that the later process has simpler coefficients occurring in it.

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.

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.

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.

A mathematical theory is developed which enables the wind-tunnel corrections to the lift and moment forces acting on an aerofoil in subsonic two-dimensional flow to be calculated. The usual “averaged” boundary condition for slotted walls is assumed and the corrections obtained by successive approximation from the open-jet tunnel.