The form described in the present paper was found amongst the contents of the stomach of a Weddell seal (Leptonychotes weddelli), taken by the Scottish National Antarctic Expedition in Scotia Bay, South Orkneys. In all about sixty specimens were collected. They occurred unattached amongst material in a semi-fluid condition, so that it is not possible to state definitely whether the seal or some animal upon which it fed is the normal host of this parasite. This question, however, in the case of the Echinorhynchi, appears from the work of de Marval to be one of minor importance. He has shown that in the Echinorhynchi of birds there is a most marked absence of specialisation of hosts, and quotes from other authors various instances of the same parasite occurring in different vertebrate classes. The worms in the present case showed no indication of having been affected by the digestive juices of the seal. As will appear from the description given below, their structural peculiarities suggest relationships with certain forms known to inhabit aquatic birds.

We consider a method for determining the asymptotic solution to a sufficiently wide class of ordinary linear homogeneous differential equations in a sector of a complex plane or of a Riemann surface for large values of the independent variable z. The main restriction of the method is the condition that the coefficients in the equation should be analytic and single-valued functions in the sector for | z | ≫ 1 possessing the power order of growth for |z| → ∞. In particular, the coefficients can be any powerlogarithmic functions. The equation

can be taken as a model equation. Here ai are complex numbers, aij are real numbers (i = 1,2,…, n; j = 0, 1, …, m) and ln1 Z≡ln z, lnsz= lnlnS−1z = S = 2, … It has been shown that the calculation of asymptotic representations for solution to any equation in the class considered may be reduced to the solution of some algebraic equations with constant coefficients by means of a simple and regular procedure. This method of asymptotic integration may be considered as an extension (to equations with variable coefficients) of the well known integration method for linear differential equations with constant coefficients. In this paper, we consider the main case when the set of all roots of the characteristic polynomial possesses the property of asymptotic separability.

One considers the linear differential systems where is a (not necessarily diagonal) matrix and one relates the computation of a general multiplicity defined from this system to the corresponding multiplicity of some eigenvalues of . Then applying these conclusions, one gives simple conditions ensuring the existence of odd or even periodic solutions for systems having the form .

We consider a variety of integral representations, single and multiple, old and new, for solutions of the hyper-Bessel equation u(n) – zmu =0. In particular, we show how a very early multiple Laplace integral solution of Molins (1876) may be related to recent Mellin–Barnes integral representations given by the present authors by way of multiple integral solutions given by Saxton and the second author for an associated equation. Although both these multiple integral solutions may be found by elementary methods, it is not easy to find their asymptotic expansions for large z, and we show how these may conveniently be obtained from our earlier results [10].

Let X be a Banach space and let ℬ be a σ-complete Boolean algebra of projections on X with a cyclic vector. It is shown that there exists a normed Köthe space Lρ, the norm of which has the Fatou property, such that X is linearly homeomorphic to the subspace of Lρ consisting of those functions of absolutely continuous norm and such that, under this homeomorphism, the projections ℬ correspond to operators consisting of multiplication by characteristic functions. This representation theorem for X is used to show that certain operator algebras associated with ℬ are reflexive. As an immediate corollary of the reflexivity result, it is shown that, if T is a scalar type spectral operator whose resolution of the identity has a cyclic vector, then T is reflexive.

We present a new result on the existence of periodic solutions for the equation:

for all positive parameters λ sufficiently large. Our fundamental assumption is the following monotonicity property: if ø ≧ ψ (ø and ψ are data) then x(ø) ≧ x(ψ). The proof consists in applying the global Hopf bifurcation theorem. The main steps are: (i) a classical estimation of the periods; (ii) an a priori estimate for the solutions along a connected branch; (iii) a transformation acting on periodic solutions.

In this paper, appropriate definitions are given for the spectra of multiparameter problems in Hilbert space when the operators involved are not necessarily self-adjoint. Notions of adjoint multiparameter problems are introduced, and some properties of the spectrum of a single linear operator are generalised to multiparameter settings.

The results which are achieved are compared with those in the literature which have beendeveloped for multiparameter problems involving self-adjoint operators.

For the reaction diffusion equation

with homogeneous Neumann boundary conditions, we give results on the generic hyperbolicity of equilibria with respect to a for fixed f and with respect to f for fixed a.

It is shown how to associate eigenvectors with a meromorphic mapping defined on a Riemann surface with values in the algebra of bounded operators on a Banach space. This generalises the case of classical spectral theory of a single operator. The consequences of the definition of the eigenvectors are examined in detail. A theorem is obtained which asserts the completeness of the eigenvectors whenever the Riemann surface is compact. Two technical tools are discussed in detail: Cauchy-kernels and Runge's Approximation Theorem for vector-valued functions.

Presentations of Coxeter type are defined for semigroups. Minimal right ideals of a semigroup defined by such a presentation are proved to be isomorphic to the group with the same presentation. A necessary and sufficient condition for these semigroups to be finite is found. The structure of semigroups defined by Coxeter-type presentations for the symmetric and alternating groups is examined in detail.

For a certain class of first order systems of differential equations several theorems are derived which give sufficient conditions for an appropriate sesquilinear form to be identically zero on suitable spaces of solutions of the system. As a consequence for second order systems limit-point criteria are obtained which include rather general criteria in the case of second order equations. The method used involves sequences of auxiliary functions and is most expedient for the proof of interval limit-point criteria. The theory is also applicable to second order equations with complex coefficients yielding sufficient conditions for the existence of solutions which are not of integrable square.

We use the modern tools of the duality principles and the calculus of variations to formulate, analyse and solve a class of plasticity problems involving second order partial derivatives. The Serrin-type integrals can most appropriately facilitate the existence statements for the extrema from either side of the duality relation in a larger class of BV functions, and interpret the solutions with possible discontinuities on sets of measure zero. The exact solutions of a beam and numerical solutions of a circular plate are presented to demonstrate the theoretical conclusions.

This paper deals with conditions which guarantee that a meromorphic function on the plane cannot satisfy any algebraic differential equation having coefficients in a given field of meromorphic functions. Some of the conditions are of growth type, while others depend on a representation for the function.

This paper deals mainly with showing that meromorphic solutions of Poincaré functional equations cannot satisfy algebraic differential equations. Results of this type for entire solutions have previously been obtained by H. Wittich.

It is shown that every weakly l1-stable linear and bounded operator (which represents a linear discrete-time system) on a Hilbert space is power stable. It solves (at least partially) a discrete-time version of a problem posed by A. J. Pritchard and J. Zabczyk for strongly continuous semigroups of bounded linear operators.

In this paper we consider a one-dimensional Stefan problem with a source term. Under the assumption that the initial profile is monotone, we obtain continuity of the free boundaries between the solid, the mushy and the liquid region.

We consider the generalisation from central series to marginal series in groups and set up firstly various basic results. The main section of the paper is concerned with the study of which group theoretical properties may be transferred from a marginal factor in a group to the corresponding verbal subgroup and which properties may be transferred from one factor of a lower marginal series to successive factors of the series.

We study the bifurcation diagram and uniqueness of solutions of

By using a rescaling technique and the Implicit Function Theorem, we establish the global bifurcation diagram. Uniqueness is proved by a separation argument to complete the bifurcation picture of the problem. Our study suggests that the bifurcation diagrams have different behaviour at λ = 0, depending on whether g(∞) > 0 or g(∞) < 0 in L∞ norm, but quite similar behaviour in Lp or W2,1.

Each singular element α of the full transformation semigroup on a finite set is generated by the idempotents of defect one. The length of the shortest expression of α as a product of such idempotents is given by the gravity function g(α).We use certain consequences of a result by Tatsuhiko Saito to explore connections between the defect and the gravity of α, and then determine the number of elements that have maximum gravity. Finally, we obtain formulae for the number of elements of small gravity. Such elements must have defect 1, and we determine their number within each ℋ-class. Many of the results obtained were suggested, and all have been verified, by programs written in PROLOG, a logic programming language very well suited for algebraic calculations.

Given an elliptic operator L on a bounded domain Ω ⊆ Rn, and a positive Radon measure μ on Ω, not charging polar sets, we discuss an explicit approximation procedure which leads to a sequence of domains Ωh ⊇ Ω with the following property: for every f ∈ H−1(Ω) the sequence uh of the solutions of the Dirichlet problems Luh = f in Ωh, uh = 0 on ∂Ωh, extended to 0 in Ω\Ωh, converges to the solution of the “relaxed Dirichlet problem” Lu + μu = f in Ω, u = 0 on ∂Ω.