In this paper we consider Hopf bifurcation in the presence of O(2) symmetry. The system of reaction diffusion equations ut, = D(µ)uxx + f(µ, u) provided with periodic boundary conditions may serve as a model problem. We prove the bifurcation of a torus of standing waves and two circles of travelling waves and we compute the stability (with asymptotic phase) of these periodic solutions, giving explicit formulae. Finally we demonstrate how a small perturbation which breaks part of the symmetry leads to secondary bifurcation.

In this paper we study a non-homogeneous Neumann-type problem which involves a nonlinearity satisfying a non-standard growth condition. By using a recent variational principle of Ricceri, we establish the existence of at least two non-trivial solutions in an appropriate Orlicz–Sobolev space.

We define and prove the existence of free Banach lattices in the category of Banach lattices and contractive lattice homomorphisms, and establish some of their fundamental properties. We give much more detailed results about their structure in the case when there are only a finite number of generators, and give several Banach lattice characterizations of the number of generators being, respectively, one, finite or countable. We define a Banach lattice P to be projective if, whenever X is a Banach lattice, J is a closed ideal in X, Q : X → X/J is the quotient map, T : P → X/J is a linear lattice homomorphism and ε > 0, there exists a linear lattice homomorphism : P → X such that T = Q º and ∥∥ ≤ (1 + ε)∥T∥. We establish the connection between projective Banach lattices and free Banach lattices, describe several families of Banach lattices that are projective and prove that some are not.

This paper is a continuation of [4], where embeddings of certain logarithmic Bessel-potential spaces (modelled upon generalised Lorentz-Zygmund spaces) in appropriate Orlicz spaces (with Young functions of single and double exponential type) were derived. The aim of this paper is to show that these embedding results are sharp in the sense of [8].

In some recent papers (called -functional calculus) for n-tuples of both bounded and unbounded not-necessarily commuting operators. The -functional calculus is based on the notion of -spectrum, which naturally arises from the definition of the -resolvent operator for n-tuples of operators. The -resolvent operator plays the same role as the usual resolvent operator for the Riesz–Dunford functional calculus, which is associated to a complex operator acting on a Banach space. When one considers commuting operators (bounded or unbounded) there is the possibility of simplifying the computation of the -spectrum. In fact, in this case we can use the F-spectrum, which is easier to compute than the -spectrum. In the case of commuting operators, our functional calculus is based on the -spectrum and will be called -functional calculus. We point out that for a correct definition of the -resolvent operator and of the -resolvent operator in the unbounded case we have to face different extension problems. Another reason for a more detailed study of the -spectrum is that it is related to the -functional calculus which is based on the integral version of the Fueter mapping theorem. This functional calculus is associated to monogenic functions constructed by starting from slice monogenic functions.

In this paper, we study the asymptotic decay properties in both spatial and temporal variables for a class of weak and strong solutions, by constructing the weak and strong solutions in corresponding weighted spaces. It is shown that, for the strong solution, the rate of temporal decay depends on the rate of spatial decay of the initial data. Such rates of decay are optimal.

1. The plumage of the Brown Leghorn fowl is analysed and the extent of its dependence on the activity of the thyroid and gonad is discussed.

2. Hypothyroidism results (a) in a diminution in the amount of melanin and a coincident increase in the red pigment; (b) in an increase in the amount of fringing due to a lack of barbule formation. The female pattern tends to disappear.

3. Hyperthyroidism in the male gives exactly the opposite effect—i.e. the melanin increases in amount while the red pigment and fringing tend to disappear. In the female the effect of hyperthyroidism is slight, and with moderate doses of thyroid there is practically no modification of the female pattern of pencilling and no melanin appears in the breast. The neck hackle, however, becomes darker and shows a decrease in fringing.

4. From these results, together with what is known concerning the effect of gonad on plumage, a hypothesis is advanced, that whereas the plumage typical of the male is developed independently of the gonad and depends for its maintenance on a certain level of thyroid functioning, both gonad and thyroid play a part in regard to that of the female; the former stimulates the latter to a higher level of activity than that present in the male and so indirectly causes a hyperthyroid effect on the feathers. At the same time it modifies this condition by acting directly on the feathers and restricting the deposition of melanin.

5. The fact that in both sexes the plumage in the young chick is similar to that of the female puts this theory in question. There is evidence to show, however, that yolk may have a modifying effect on plumage similar to that of the ovary, and thus the doubt regarding the validity of the hypothesis can be removed by the suggestion that the chick plumage develops under the influence of the yolk in its own yolk sac. This second theory gains confirmation by a study of the atypical plumage which is occasionally met with in growing females, and also by a consideration of the type of plumage developed in juvenile females following the successful implantation of testis.

It is now a year and a half since I communicated to this Society a description of a portable form of apparatus for enabling us to count the number of particles of dust in the atmosphere. The working of that instrument in my hands has been most satisfactory, and though it has occasionally given trouble, yet it has not given more than might have been expected. Though that apparatus has worked quite pleasantly with me, and enabled a beginning to be made of an investigation into the amount, and the effects, of dust in our atmosphere, yet very few have as yet followed up this line of inquiry. This has probably been owing to there being something in the complicated nature of the apparatus which has deterred others from joining in the work. I therefore determined to see if a simpler, and at the same time a reliable, form of the apparatus could not be devised.

We treat the time-harmonic Maxwell equations in an exterior domain with prescribed boundary data [n, E] in the Sobolev space of square integrable tangential fields with square integrable surface divergence. By using boundary integral equation methods, existence and uniqueness results are established. Furthermore, we investigate the completeness of electric and magnetic dipoles distributed on an inner surface in this Sobolev space.

In provided with a J-innerproduct we characterize the J-selfadjoint operators generated by a symmetric ordinary differential expression on an open real interval ι. For a subclass of these operators we prove eigenfunction expansion results using Hilbertspace-techniques.

Let D be a domain in Rn with bounded complement and let n ≠ 2. For the initial-boundary value problemwe prove that there are no non-trivial global (non-negative) solutions if 0 < n (p − 1) ≤ 2 and there exist both global non-trivial and non-global solutions if n (p − 1) > 2.

In this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity

$$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$

We study the class of polynomial vector fields of the form = αx — y + Pn(x, y), = x + αy + Qn(x, y), where Pn and Qn are homogeneous polynomials of degree n. If we define the functions f(x, y) = xPn(x, y) + yQn(x, y) and g(x, y) = xQn(x, y)−yPn(x, y), we characterise the number of limit cycles for this class when the function g(αg − f) does not change sign.

Suppose the linear equation x' = A(t)x has an exponential dichotomy and suppose B(t) is close to A(t) in the following sense: on any interval of length 2T, B(t) is close to some translate A(t + τ) of A(t) (actually the conditions in the paper are slightly weaker than this). Then if T is sufficiently large, the equation y' = B(t)y also has an exponential dichotomy. This generalises the usual roughness theorem for exponential dichotomies.

Some known results for different kinds of boundary value problems for second order ordinary differential equations are generalised. Different approaches are compared with one another, using topological and variational methods and the theory of weighted eigenvalue problems.

We discuss the existence of positive solutions of some singularity perturbed elliptic equations on convex domains with nonlinearity changing sign. In particular, we obtain solutions with both a boundary layer and a sharp interior peak.

The existence of a classical solution to the initial boundary value problem for a semi-infinite extensible string is proved. The result is obtained by using a Galerkin procedure on a semi-infinite interval.

In 1966, J. M. Howie characterised the transformations of an arbitrary set that can be written as a product (under composition) of idempotent transformations of the same set. In 1967, J. A. Erdos considered the analogous problem for linear transformations of a finite-dimensional vector space and in 1983, R. J. Dawlings investigated the corresponding idea for bounded operators on a separable Hilbert space. In this paper we study the case of arbitrary vector spaces.

It is now nearly forty years since the presence of chlorophyll in certain species of Planarians was recognised by Schultze. Later observers recognised that the green colour of certain Infusorians, of the common fresh-water hydra (Hydra viridis), and of the freshwater sponge (Spongilla fluviatilis), was due to the same pigment, but little more attention was paid to the subject until Ray Lankester applied the spectroscope to its investigation.

In this article we examine the qualitative behaviour of non-planar equilibrium states ofnon-linearly elastic rods subject to terminal loads. In our geometrically exact theory, a rod is endowed with enough geometric structure for it to undergo flexure, torsion, axial extension, and shear. The constitutive equations give appropriate stress resultants and couples as non-linear functions of appropriate strains. These constitutive relations must meet minimal conditions ensuring that they be physically reasonable. It turns out that the equilibrium states of such a rod are governed by a boundary value problem for a quasilinear fifteenth-order system of ordinary differential equations.