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.

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.

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.

In [1] we considered various aspects of the quotient semigroup H. · H2 where H is an ℋ-class of a semigroup S. In particular, the action of the Schützenberger group of H upon SH was studied to obtain various results on the existence of subcontinua. Crucial in [1] was the notion of the (right handed) core of an ℋ-class which may be considered as a generalization of the notion of the core of a homogroup, [2].

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.

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.

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.

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.

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.

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.

We consider the case of a genetic population for which the selective advantages of the various genotypes are not constant but for each generation depend linearly on the gene frequencies in the population in the previous generation. For such populations, the effect of competition between similar genotypes may be allowed for by suitable choice of the frequency-dependent selective advantages, or, by a reversal of sign, the case where genotypes are favoured by the presence of similar genotypes may also be considered. favoured by the presence of similar genotypes may also be considered. All populations are finite and of constant size so that eventually only one type of gene will survive. The probabilities of survival for each gene are found and compared with the case where there are no frequency-dependent factors. If a small amount of mutation is allowed, gene fixation will not occur and a steady-state distribution of gene frequency will appear. The form of this distribution may be derived simply from the survival probabilities in the corresponding cases where there is no mutation. The main result is that in some cases, frequency-dependent factors have a marked effect on survival probabilities, while in other cases they can be completely ignored. The latter wifi only occur in certain cases where there exists competition between similar genotypes.

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.