In a recent paper (Frazer, Jones, and Skan, 1937) some methods are discussed for the approximate representation of functions by means of polynomials. The coefficients in the polynomials are determined by the equation

where c is the column of coefficients, h is a column of known constants, and M is a matrix which depends on the method of representation adopted. The present paper shows how the reciprocal matrix M-1 can be computed rapidly and simply in the two cases where M is a moment matrix or an alternant matrix.

The physicist, in his study of natural phenomena, has two methods of making progress: (1) the method of experiment and observation, and (2) the method of mathematical reasoning. The former is just the collection of selected data; the latter enables one to infer results about experiments that have not been performed. There is no logical reason why the second method should be possible at all, but one has found in practice that it does work and meets with remarkable success. This must be ascribed to some mathematical quality in Nature, a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important rôle in Nature's scheme.

We prove uniqueness of solutions to scalar conservation laws with space discontinuous fluxes. To do so, we introduce a partial adaptation of Kružkov's entropies which naturally takes into account the space dependency of the flux. The advantage of this approach is that the proof turns out to be a simple variant of the original method of Kružkov. In particular, we do not need traces, interface conditions, bounded variation assumptions (neither on the solution nor on the flux), or convex fluxes. However, we use a special ‘local uniform invertibility’ structure of the flux, which applies to cases where different interface conditions are known to yield different solutions

We consider a homogenization problem for highly anisotropic conducting fibres embedded into an isotropic matrix. For a ‘double porosity’-type scaling in the expression of high contrast between the conductivity along the fibres and the conductivities in the transverse directions, we prove the homogenization theorem and derive two-scale homogenized equations using a version of the method of two-scale convergence, supplemented in the case when the spectral parameter λ = 0 by a newly derived variant of high-contrast Poincaré-type inequality. Further elimination of the 'rapid' component from the two-scale limit equations results in a non-local (convolution-type integro-differential) equation for the slowly varying part in the matrix, with the non-local kernel explicitly related to the Green function on the fibre. The regularity of the solution to the non-local homogenized equation is proved.

Hitherto the Lamé functions of u, of real periods 2K and 4K, and the Lamé polynomials in particular, have been expressed as series of integral powers of sn u (multiplied, in appropriate cases, by cn u, dn u, or cn u dn u). The necessity to use these, or the corresponding Weierstrassian forms, is a serious handicap to research, because the terms of the expression are not mutually orthogonal. In particular, results that can be deduced from the orthogonal properties of the Fourier series in which periodic Mathieu functions are developed, find no analogues in the Lamé functions as expressed in the customary forms. For instance, the integral equations that generate the Mathieu functions have proved to be valuable sources of information, but those for the Lamé functions cannot be much more than scientific curiosities until the development of Lamé functions as series of orthogonal functions enables them to be put to use.

The paper is concerned with a non-local time-delayed reaction–diffusion equation. We prove the (time) asymptotic stability of a travelling wavefront without a smallness assumption on its wavelength, i.e. the so-called strong wavefront, by means of the (technical) weighted energy method, when the initial perturbation around the wave is small. The exponential convergent rate is also given. Selection of a suitable weight plays a key role in the proof.

In this paper we show that in O(2) symmetric systems, structurally stable, asymptoticallystable, heteroclinic cycles can be found which connect periodic solutions with steady states and periodic solutions with periodic solutions. These cycles are found in the third-order truncated normal forms of specific codimension two steady-state/Hopf and Hopf/Hopf mode interactions.

We find these cycles using group-theoretic techniques; in particular, we look for certainpatterns in the lattice of isotropy subgroups. Once the pattern has been identified, the heteroclinic cycle can be constructed by decomposing the vector field on fixed-point subspaces into phase/amplitude equations (it is here that we use the assumption of normal form). The final proof of existence (and stability) relies on explicit calculations showing that certain eigenvalue restrictions can be satisfied.

Existence and uniqueness results are proved for positive solutions of a class of quasilinear elliptic equations in a domain Ω⊂ℝN via a generalisation of Serrin's sweeping principle. In the case when Ω is an annulus, it is shown that the solution is radially symmetric.

In this paper, we study boundary problems with dynamic boundary conditions, that is, with boundary operators containing time derivatives. The equations under consideration are transformed into abstract Cauchy problems x – Cx = f and x(0) = x0. Abstract theoretical results concerning the operators C are obtained by the study of a naturally arising pseudodifferential operator. For existence and uniqueness theorems concerning solutions of parabolic and hyperbolic equations, we then apply the theory of semigroups in Banach spaces. Some examples of semilinear and quasilinear problems, to which our results apply, are given.

The subsemigroup Singn of singular elements of the full transformation semigroup on a finite set is generated by n(n − l)/2 idempotents of defect one. In this paper we extend this result to the subsemigroup K(n, r) consisting of all elements of rank r or less. We prove that the idempotent rank, defined as the cardinality of a minimal generating set of idempotents, of K(n, r) is S(n, r), the Stirling number of the second kind.

We give a survey of the current state of knowledge on the Arens second dual of a Banach algebra, including some simplified proofs of known results, some new results, some open problems and a full bibliography of the subject.

We investigate a non-autonomous ratio-dependent predator–prey system, whose autonomous versions have been analysed by several authors. For the general non-autonomous case, we address such properties as positive invariance, permanence, non-persistence and the globally asymptotic stability for the system. For the periodic and almost-periodic cases, we obtain conditions for existence, uniqueness and stability of a positive periodic solution, and a positive almost-periodic solution, respectively.

We consider the following problem

where for all ≦f(x,u)≦c1up-1 + c2u for all x ∈ℝN,u≧0 with c1>0,c2∈(0, 1), 2<p<(2N/(N – 2)) if N ≧ 3, 2 ≧ + ∝ if N = 2. We prove that (*) has at least two positive solutions if

and h≩0 in ℝN, where S is the best Sobolev constant and

Degond and Markowich discussed the existence and uniqueness of a steady-state solution in the subsonic case for the one-dimensional hydrodynamic model of semiconductors. In the present paper, we reconsider the existence and uniqueness of a globally smooth subsonic steady-state solution, and prove its stability for small perturbation. The proof method we adopt in this paper is based on elementary energy estimates.

Based on Fenichel's geometric idea, invariant manifold theory is applied to singular perturbation problems. This approach clarifies the nature of outer and inner solutions. A specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds. A method to approximate the connections is also presented.

We study the multiplicity and concentration behaviour of positive solutions for a quasi-linear Choquard equation

where Δp is the p-Laplacian operator, 1 < p < N, V is a continuous real function on ℝN, 0 < μ < N, F(s) is the primitive function of f(s), ε is a positive parameter and * represents the convolution between two functions. The question of the existence of semiclassical solutions for the semilinear case p = 2 has recently been posed by Ambrosetti and Malchiodi. We suppose that the potential satisfies the condition introduced by del Pino and Felmer, i.e.V has a local minimum. We prove the existence, multiplicity and concentration of solutions for the equation by the penalization method and Lyusternik–Schnirelmann theory and even show novel results for the semilinear case p = 2.

We study the existence of T-periodic positive solutions of the equation

where f(t, .) has a singularity of repulsive type near the origin. Under the assumption that f(t, x) lies between two lines of positive slope for large and positive x, we find a non-resonance condition which predicts the existence of one T-periodic solution.

Our main result gives a Fredholm alternative-like result for the existence of T-periodic positive solutions for

We study a semilinear boundary value problem with the feature that the vanishing boundary value makes the equation singular. We prove that the positive solution is in general Hölder-continuous up to the boundary and has even better regularity in some special cases.

We consider the existence and smoothness of global centre unstable manifolds for finite and infinite dimensional flows or maps. We show that every global centre unstable manifold can be expressed as a graph of a Ck map, provided that the nonlinearities are Ck smooth. The proofs are based on a lemma by D. Henry on a necessary and sufficient condition for a Lipschitz map to be continuously differentiable.

In this paper we study the diffusive Nicholson's blowflies equation. Generalizing previous works, we model the generation delay by using an integral convolution over all past times and results are obtained for general delay kernels as far as possible. The linearized stability of the non-zero uniform steady state is studied in detail, mainly by using the principle of the argument. Global stability both of this state and of the zero state are studied by using energy methods and by employing a comparison principle for delay equations. Finally, we consider what bifurcations are possible from the non-zero uniform state in the case when it is unstable.