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.

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].

Let C denote Cayley's algebra defined over the field of rational numbers. This paper contains a simple characterization of arithmetics of C in terms of a given basis i0 = 1, i1, i2, …, i7. We deduce that certain of the arithmetics of C are isomorphic. The result that the maximal arithmetics are isomorphic is also contained in the work of van der Blij and Springer [2].

Dr V. O. Ennola has pointed out that there is a mistake in this paper. The inequality for dox at the foot of page 76 does not imply the stated inequality between the second and third lines on page 77. This error vitiates the entire argument, and I have not so far been able to put it right.

Let A be an associative ring. Given a ∊ A, an element b ∊ A is called a left identity for a if

Given a subset S of A, an element b ∊ A is, called a left identity for S if (1) is satisfied for all a ∊ S. An element of A need not have a left identity; for example, if A is nilpotent then no non-zero element of A has a left identity. If a does have a left identity, the latter need not be unique; if every element of a subset S of A has a left identity, then it is not necessarily true that S has a left identity.

Let T be a bounded symmetric operator in a Hilbert space H. According to the spectral theorem, T can be expressed as an integral in terms of its spectral family {Eλ}, each Eλ being a certain projection which is known to be the strong limit of some sequence of polynomials in T. It is a natural question to ask for an explicit sequence of polynomials in T that converges strongly to Eλ. So far as the author knows, no complete solution of this problem has been given even when H has finite dimension, i.e. when T is a finite symmetric matrix. Since a knowledge of the spectral family {Eλ} of a finite symmetric matrix carries with it a knowledge of the eigenvalues and eigenvectors, a solution of the problem may have some practical value.

By the group G(2 cos π/q) we mean the group of linear fractional transformations of the complex plane onto itself, generated by V(z)= — 1/z and U(z) = z+λq, where λq = 2 cos (π/q), qbeing a positive integer greater than 2. In this paper we shall be concerned only with the group given by q = 5, and we shall therefore omit the subscript 5 on the λ. We note that λ = λ5 satisfies the equation

x2–x–l=0;(1)

henceλ = (l + 5½)/2.

It is proposed to establish the two following integrals.

where n is a positive integer, x is real and positive, μi and ν are complex, and Δ (n; a) represents the set of parameters

where n is a positive integer and x is real and positive.

In this note I show that

where Jdenotes the Bessel function of the first kind of the orders and arguments indicated, n = 0, 1, 2, 3, … and the real parts of both μand v exceed — 1. This is a generalization of Sonine's first finite integral [1, p. 373] to which it reduces in the special case n = 0.

The integral function

is known as Airy's Integral since, when z is real, it is equal to the integral

which first arose in Airy's researches on optics. It is readily seen that w= Ai(z) satisfies the differential equation d2w/dz2 = zw, an equation which also has solutions Ai(ωz), Ai(ω2z), where ω is the complex cube root of unity, exp 2/3πi. The three solutions are connected by the relation.

In this note a theorem, giving a relation between the Hankel transform of f(x) and Meijer's Bessel function transform of f(x)g(x), is proved. Some corollaries, obtained by specializing the function g(x), are stated as theorems. These theorems are further illustrated by certain suitable examples in which certain integrals involving products of Bessel functions or of Gauss's hypergeometric function and Appell's hypergeometric function are evaluated. Throughout this note we use the following notations:

Our purpose is to develop a new and simple procedure for embedding graphs into orientable surfaces. This will involve the identification of the oriented edges of two oriented polygons, subject to certain rules.

A steady motion of incompressible viscous liquid caused by the slow rotation of a rigid sphere of radius a is considered. The medium is bounded by an infinite rigid plane and the axis of rotation is parallel to, and at a distance d from, this plane. To complete the analysis the solution by successive approximation of an infinite set of linear equations is required. Satisfactory solutions have been found numerically for four values of d/a, of which 1·13 is the smallest; we gratefully acknowledge valuable help from Miss S. M. Burrough in this part of the work.

Denote by |E| the cardinal of a set E. The purpose of the present paper is to prove the following result, constituting the solution of an unpublished problem of Erdös, Hajnal and Milner.

Recent papers [1, 2, 3[ have considered dual series equations in Legendre and associated Legendre functions and have given applications of these series to various potential and diffraction problems. This note gives a further application to the problem of the axisymmetric Stokes flow of a viscous fluid past a spherical cap. The stream-function of the flow is found by solving two pairs of dual series equations in associated Legendre functions, these equations being of a form considered previously [1, henceforth referred to as DSE]. As an example a uniform flow past the cap is considered and the drag of the cap calculated. This flow has previously been investigated by Payne and Pell [4], who by a suitable limiting process derive the stream-function for the flow past the cap from the stream-function for the flow past a lens-shaped body. Their method, however, involves the use of peripolar coordinates, besides much complicated algebra, and results are given only for a cap whose semi-angle is π/2. Further, their value for the drag of this cap is incorrect.

Let be a set of points on a sphere, centre O, radius R, in (n+1)– dimensional space. Suppose a spherical cap of angular radius α≤½π is centred at each point of . Let k be a positive integer and suppose that no point of the sphere is an inner point of more than k caps. We say that provides a k–fold packing for caps of radius α.

It is shown that in a certain sense, inversion transforms biharmonic functions into biharmonic functions. The first boundary value problem of elastostatics is also largely unchanged by this transformation, and known solutions can be used to obtain new results for inverse regions. As an example, the problem of a stress free dumb-bell shaped hole in an infinite plate is solved.