The paper deals with genetic algebras for populations in which given fractions of the population practise specified types of inbreeding. It is shown that the technique of linearising the quadratic transformation of genetic type frequencies, which has been studied for triangular algebras, can be extended to a much wider class, including those for inbreeding systems.

We prove that a finite semigroup which is decomposable into a direct product of cyclic semigroups which are not groups is uniquely so decomposable, and show how the non-group cyclic direct factors of a finite semigroup, if they exist, can be found.

Linear semi-group theory can be used to prove the existence of solutions to the equations of linear elasticity when the elasticity tensor is positive definite. Here, it is shown that this condition is also necessary for the existence of a semi-group. The method is also applied to linear dissipative equations.

The Poincaré-Bendixson theorem, concerning the existence of periodic orbits of plane autonomous systems, is extended to higher order systems under certain conditions. Under similar conditions, a complementary theorem on the existence of recurrent orbits is also proved. For the feedback control equation, these conditions are reduced to a form which can be easily verified in practice.

We investigate the existence, uniqueness and regularity of solutions to the linear differential equation Lu = f under nonlinear mixed boundary conditions on domains with singular boundary points.

We introduce a class of essentially self-adjoint Schrödinger operators, where essential self-adjointness is stable under positive potential perturbations. We show that this class is “stable” under certain perturbations and contains operators discussed by Simon and Kato. Finally, an extended essentially self-adjointness criterion is given.

Asymptotic approximations are derived for the Whittaker functions Wκ,μ (z), Mκ, μ (z), Wικ, ιμ (iz) and Mικ, ιμ(iZ) for large positive values of the parameter μ that are uniform with respect to unrestricted values of the argument z in the open interval (0, ∞), and bounded real values of the ratio κ/μ. The approximations are in terms of parabolic cylinder functions, and in most instances are accompanied by strict error bounds.

The results are derived by application of a recently-developed asymptotic theory of second-order differential equations having coalescing turning points, and an extension of the general theory of equations of this kind is also included.

The nonlinear operator equation (N), , describes both semilinear elliptic boundary value problems (P) and their natural discretisations (Ph). (Here is a positive compact linear operator and [t]+ ≡ (t + |t|)/2 for all t ∊ ℝ.) It is proved that, for q ≧ 0 (q ≢ 0), in a Banach lattice E the equation (N) has an unbounded continuum ℱ of nontrivial solutions (λ, ψ) ∊ ℝ × E bifurcating from infinity at (λ1, ∞) (here is the first (positive) eigenvalue of ). All nontrivial solutions (λ, ψ) have λ ≧ λ1, and if maps a smaller cone Ks, into itself then there is a ℱs ⊂ℝ×Ks with similar properties to ℱ. The existence theory for (N) is applied to problems (P) and (Ph) which are defined on the simply-connected domain Ω. It is shown that the projection of ℱ on the λ-axis is either unbounded if the continuous function q > 0 except on a set of measure zero in Ω, or is bounded if q ≡ 0 on a subdomain of Ω. If is the second eigenvalue for in (Ph) then there is at most one nontrivial solution for each λ satisfying λ1 < λ < λ2; in the corresponding uniqueness result for (P) q is restricted to being strictly positive somewhere on ∂Ω Additional properties for the solutions of (P) and (Ph) are also proved.

The analysis developed by the author in a previous paper is used to discuss two magnetohydrodynamic duct flows for which the boundary conditions are not of Dirichlet type. To the accuracy stated, the results obtained confirm those obtained by previous authors who used different approaches to the problems. A feature of the present analysis is that it yields the magnitude of the error entailed by the use of approximate forms for flow quantities. This has been lacking in previous analyses.

Bivariational principles are constructed that yield upper and lower bounds to the quantity 〈g0, f〉, where f is the solution of the equation f0−Tf = 0, g0 is a given function and T is a non-self-adjoint linear operator from a Hilbert space into itself. The theory is illustrated by an integral equation of Fredholm type.

Let N be a zero-symmetric near-ring with identity and let G be an N-group. We consider in this paper nilpotent ideals of N and N-series of G and we seek to link these two ideas by defining characterizing series for nilpotent ideals. These often exist and in most cases a minimal characterizing series exists. Another special N-series is a radical series, that is a shortest N-series with a maximal annihilator. These are linked to appropriate characterizing series. We apply these ideas to obtain characterizing series for the radical of a tame near-ring N, and to show that these exist if either G has both chain conditions on N-ideals or N has the descending chain condition on right ideals. In the latter case this provides a new proof of the nilpotency of the radical of a tame near-ring with DCCR, and an internal method for constructing minimal and maximal characterizing series for the radical.

A new method is developed for obtaining the asymptotic form of solutions of the fourth-order differential equation

where m, n are integers and 1 ≦ m, n ≦ 2. The method gives new, shorter proofs of the well-known results of Walker in deficiency index theory and covers the cases not considered by Walker.

Wirtinger-type inequalities of order n are inequalities between quadratic forms involving derivatives of order k ≦ n of admissible functions in an interval (a, b). Several methods for establishing these inequalities are investigated, leading to improvements of classical results as well as systematic generation of new ones. A Wirtinger inequality for Hamiltonian systems is obtained in which standard regularity hypotheses are weakened and singular intervals are permitted, and this is employed to generalize standard inequalities for linear differential operators of even order. In particular second order inequalities of Beesack's type are developed, in which the admissible functions satisfy only the null boundary conditions at the endpoints of [a, b] and b does not exceed the first systems conjugate point (a) of a. Another approach is presented involving the standard minimization theory of quadratic forms and the theory of “natural boundary conditions”. Finally, inequalities of order n + k are described in terms of (n, n)-disconjugacy of associated 2nth order differential operators.

In the following paper, the horizontal line method (the Rothe method) is applied to Maxwell's initial boundary value problem. By means of results from abstract perturbation theory, convergence results and error estimates are established.

In this paper, the formally J-symmetric Sturm—Liouville operator with complex-valued coefficients is considered, and as a preliminary two criteria are established for the J-selfadjointness of certain of its extensions. The main purpose of the work is to give results on the location of the point spectra of such operators. Some of the results are extensions of results known for real-valued coefficients, whilst others are new.

By using a Hilbert space decomposition theorem for two polar cones it is shown that the method of the hypercircle can be extended to determine solutions to best approximation problems involving unilateral constraints. The method is applied to abstract boundary value problems for linear operators involving such constraints.

When K is a convex body in d-dimensional euclidean space E and 0 ≤ s ≤ d, the s-skeleton of K, denoted skel(s)K, consists of those points of K which are not centres of (s + l)-dimensional balls contained in K. The s-skeleton is thus the union of the extreme faces of K having dimension at most s. The s-skeleton is a -set [see 6] and it is therefore measurable with respect to the s-dimensional Hausdorff measure ℋ(s) [see 7]; here we normalize ℋ(s) so that it assigns unit measure to the s-dimensional unit cube.