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.

Let P(D) be a partial differential operator with constant coefficients. If P(D) has a continuous linear right inverse in the space of tempered distributions, then P is the product of a polynomial without real roots and a real polynomial admitting a right inverse. If the polynomial P is real and irreducible, then P(D) admits a right inverse in the tempered distributions if and only if P(×) has the property of zeros of R. Thorn.

Brouwer topological degree theory, the shooting type method, the disconjugacy theory of Hamiltonian systems and the Liapunov-Razumikhin technique of Volterra integrodifferential equations are employed to establish some solvability results for the 2n-dimensional differential system

subject to one of the following boundary conditions:

(i) x(0) = Qx(l), Qg(l x(1), qy(1)= g(0, x(0), y(0)),

(ii) Blx(O) = B2g(O, x(O), y(O)), C1x(l) = −C2g(l, x(l), y(l)),

where Q, Bi, Ci, i = 1, 2, are n x n real matrices. An application is given to the second order equation xn = h(t, x, x') subject to certain nonlinear boundary conditions.

This paper studies the stability and instability properties of solitary wave solutions φ(x – ct) of a general class of evolution equations of the form Muttf(u)x=0, which support weakly nonlinear dispersive waves. It turns out that, depending on their speed c and the relation between the dispersion (i.e. the order of the pseudodifferential operator) and the nonlinearity, travelling waves maybe stable or unstable. Sharp conditions to that effect are given.

We consider linear and nonlinear elliptic equations in divergence form on Riemannian manifolds with or without boundary. In the former case we impose a homogeneous Neumann boundary condition. By making use of isoperimetric inequalities for manifolds, we obtain a priori sharp estimates for the decreasing rearrangement of the solutions to such equations. These estimates enable us to derive bounds for suitable norms of the solutions and of their gradients.

For regular symmetric ordinary differential expressions we show that (i) the minimal operator is bounded below and (ii) the Friedrichs extension is determined by Dirichlet boundary conditions. Both proofs are based on elementary inequalities.

We construct rank-one convex functions which are not convex and have linear growth at infinity. We show that these functions could be useful in some problems concerning weak convergence of gradients if we were able to prove that they are quasiconvex. This question, however, seems to be open.

The equation

has a solution y(t) which is non-oscillating on the interval (0, ∞) and has the asymptotic expansion

Each term of this expansion is even in t so that formally is zero to all orders of ɛ. The estimate of has been obtained by Byatt-Smith [3] who corrects (2) in the complex plane near t = i where the series ceases to be valid. This requires asolution of the equation

the equation for the first Painlevé transcedent. Here we prove rigorously that this method gives the correct asymptotic estimate

where

The proof involves converting (1) and (3) to integral equations. The existence and uniqueness of these integral equations are established by use of the contraction mapping theorem. We also prove that the appropriate solution to (3) provides a uniformly valid approximation to (2) over a suitably defined region of the complex plane.

We also consider the connection problem for the oscillatory solutions of (1) which have asymptotic expansions

where Ã+, Ã− φ+, and φ− are constants. The connection problem is to determine the asymptotic expansion at +∞ of a solution which has a given asymptotic expansion at −∞. In other words, we wish to find (Ã+,φ+) as a function of Ã−and φ−. We prove that there is a unique solution to the connection problem, provided Ã− is small enough, and obtain bounds on the estimate of Ã+

For a Dirac system on the line, we study the Jost solutions and the existence of half-bound states. A characterisation of this phenomenon is given in terms of the M(λ)-function, which supplements a well-known spectral characterisation for the system. As a corollary, we deduce the spectral density behaviour at the spectral gap endpoints, which is shown to depend on the possible existence of half-bound states.

We prove Cm, μ almost everywhere regularity for minimisers of functional of the form ∫f(x, su, Dmu)dx, where f is uniformly strictly quasiconvex.

It is the aim of this article to consider some problems arising from the non local-compactness of the phase space for functional differential equations. The compact boundedness, that is, the boundedness depending on each compact set involving the initial values, is proved to be implied from the ultimate boundedness for periodic systems of functional differential equations on Cγ: = {φ ∊ C((–∞,0]) Note that it is known that the compactness cannot be dropped in the above. An example is also given to show that the asymptotic stability is not necessarily uniform even for periodic functional differential equations on Co.

The paper presents solutions, for a class of shear relaxation functions, ofa linear problem formulated by Joseph [8] to elucidate the steady, supercritical flow of a viscoelastic fluid past a semi-infinite flat plate. The velocity (U, 0) at infinity is parallel to the plate, and‘supercritical flow ‘means that U is greater than the propagation speed C of shear waves. As a result, the vorticity satisfies a hyperbolic equation and is confined to the region downstream of a shock wave from the leading edge of the plate. The disturbance velocity fieldextends upstream of the shock and is continuous across it. In contrast to the case of a Newtonian fluid, the solutions are unique under the condition that the functions representing the vorticity on the two sides of the platebelong to a certain Banach space.

For a domain Ω in the Euclidean space Rd (d = 2, 3) existence of weak solutions for both interior and exterior Dirichlet boundary value problems of the Ginzburg-Landau equations are established without any restriction on the range of the coupling constant λ, thesize of Ω, or the boundary data. For the critical choice λ = 1, we prove the existence of confined multivortices in a bounded domain by a constructive monotone iteration method.

Weak lower semicontinuity theorems in the sense of Chacon's Biting Lemma are proved for multiple integrals of the calculus of variations. A general weak lower semicontinuity result is deduced for integrands which are acomposition of convex and quasiconvex functions. The “biting”weak limit of the corresponding integrands is characterised via the Young measure, and related to the weak* limit in the sense of measures. Finally, an example is given which shows that the Young measure corresponding to a general sequence of gradients may not have an integral representation of the type valid in the periodic case.