The Riemann–Hilbert–Poincaré problem with general coefficient for the inhomogeneous Cauchy–Riemann equation on the unit disc is studied using Fourier analysis. It is shown that the problem is well posed only if the coeffcient is holomorphic. If the coefficient has a pole, then the problem is transformed into a system of linear equations and a finite number of boundary conditions are imposed in order to find a unique and explicit solution. In the case when the coefficient has an essential singularity, it is shown that the problem is well posed only for the Robin boundary condition.

This paper studies the overall evolution of fronts propagating with a normal velocity that depends on position, υn = f(x), where f is rapidly oscillating and periodic. A level-set formulation is used to rewrite this problem as the periodic homogenization of a Hamilton–Jacobi equation. The paper presents a series of variational characterization (formulae) of the effective Hamiltonian or effective normal velocity. It also examines the situation when f changes sign.

We study the following polyharmonic Hénon equation:

where (m)* = 2N/(N – 2m) is the critical exponent, B1(0) is the unit ball in ℝN, N ⩾ 2m + 2 and K(|y|) is a bounded function. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.

In this paper we obtain the existence of a radial solution for some elliptic non-local problem with constraints. The problem arises from some mean field equation which models, among other things, a system of self-gravitating particles when one looks for its stationary solutions. We include the cases of Maxwell—Boltzmann, Fermi—Dirac and polytropic statistics.

Let A be a closed linear transformation from a real Hilbert space ℋ, with symmetric inner product 〈, 〉, into itself; and let f ∈ ℋ be given such that the problem Aø = f has a solution ø ∈ D(A), the domain of A. Then bivariational upper and lower bounds on 〈g, ø〉 for any g ∈ ℋ are exhibited when there exists a positive constant a such that 〈AΦ, AΦ⊖ ≧ a2〈Φ, Φ〉 for all Φ ∈ D(A). The applicability of the theory both to Fredholm integral equations and also to time-dependent diffusion equations is demonstrated.

In this paper, we study the zero dissipation limit of the initial boundary-value problem of the multi-dimensional Boussinesq equations with viscosity and heat conductivity. Such equations are used as models for the motion of multi-dimensional incompressible fluids in atmospheric and oceanographic turbulence. In particular, they describe the thermal convection of an incompressible flow, and constitute the relations between the velocity field, the pressure and the local temperature. Under the Navier slip boundary condition in the velocity field and the thermal isolation boundary condition for the temperature, we prove the existence of weak amplitude characteristic boundary layers. Then, by a standard energy method, we prove the L2 convergence of the solutions when both the viscosity and the heat conductivity coefficients tend to 0.

This paper considers the Cauchy problem for hyperbolic conservation laws arising in chromatography:

with bounded measurable initial data, where the relaxation term g(δ, u, v) converges to zero as the parameter δ > 0 tends to zero. We show that a solution of the equilibrium equation

is given by the limit of the solutions of the viscous approximation

of the original system as the dissipation ε and the relaxation δ go to zero related by δ = O(ε). The proof of convergence is obtained by a simplified method of compensated compactness [2], avoiding Young measures by using the weak continuity theorem (3.3) of two by two determinants.

We reconcile two classical models of edge dislocations in solids. The first, from the early 1900s, models isolated edge dislocations as line singularities in locally Euclidean manifolds. The second, from the 1950s, models continuously distributed edge dislocations as smooth manifolds endowed with non-symmetric affine connections (equivalently, endowed with torsion fields). In both models, the solid is modelled as a Weitzenböck manifold. We prove, using a weak notion of convergence, that the second model can be obtained rigorously as a homogenization limit of the first model as the density of singular edge dislocation tends to infinity.

We clear the decks for an attack on the problem of the breadth and class of a finite p-group by exhibiting some examples with properties relevant to breadth. These were constructed by a mixture of hand computation, machine search and serendipity.

We study positive solutions to the porous media equation of degenerate logistic type −Δu = a(x)u1/m – b(x)f(u), m > 1, which blow up at the inner boundary and satisfy a homogeneous boundary condition at the outer boundary of a multiply connected domain. In particular, we investigate uniqueness and blow-up rate near the inner boundary for such solutions. We also look at the limiting behaviour as m ↘ 1 for the special case where b(x) > 0 and f(u) = up/m with p > m.

The semilinear elliptic differential operator L[u] = Δu + c(x, u) is studied and sufficient conditions are derived for all solutions of uL[u] ≦ 0 with suitable boundary conditions to be oscillatory in unbounded domains of Rn. Here, unbounded domains to be considered are cones, strips and cylinders in Rn. The results are based on the conditions for the non-existence of positive solutions of ordinary differential inequalities.

In 1994 Boyd derived a resurgence representation for the gamma function, exploiting the 1991 reformulation of the method of steepest descents by Berry and Howls. Using this representation, he was able to derive a number of properties of the asymptotic expansion for the gamma function, including explicit and realistic error bounds, the smooth transition of the Stokes discontinuities and asymptotics for the late coefficients. The main aim of this paper is to modify Boyd’s resurgence formula, making it suitable for deriving better error estimates for the asymptotic expansions of the gamma function and its reciprocal. We also prove the exponentially improved versions of these expansions complete with error terms. Finally, we provide new (formal) asymptotic expansions for the coefficients appearing in the asymptotic series and compare their numerical efficacy with the results of earlier authors.

In this paper some special entropy–entropy flux pairs of Lax type are constructed for nonlinear hyperbolic systems of types (1.1) and (1.2), in which the progression terms are functions of a single variable. The necessary estimates for the major terms are obtained by the use of singular perturbation theory. The special entropies provide a convergence theorem in the strong topology for the artificial viscosity method when applied to the Cauchy problems (1.1), (1.3) and (1.2), (1.3) and used together with the theory of compensated compactness.

Non-negative post-blow-up solutions of the quasilinear degenerate parabolic equation in RN (or a bounded domain with Dirichlet boundary condition) are studied. Various sufficient conditions for complete blow-up of solutions are given.

We introduce a second numerical index for real Banach spaces with non-trivial Lie algebra, as the best constant of equivalence between the numerical radius and the quotient of the operator norm modulo the Lie algebra. We present a number of examples and results concerning absolute sums, duality, vector-valued function spaces…which show that, in many cases, the behaviour of this second numerical index differs from the one of the classical numerical index. As main results, we prove that Hilbert spaces have second numerical index one and that they are the only spaces with this property among the class of Banach spaces with one-unconditional basis and non-trivial Lie algebra. Besides, an application to the Bishop-Phelps-Bollobás property for the numerical radius is given.

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.

1. The adsorption of a gas has been investigated thermodynamically with special reference to the heat effects accompanying adsorption.

2. Expressions are developed for three isothermal heats of adsorption of a gas and for the heat of immersion of a powder in a liquid.

3. The effect of the variation of the surface of an adsorbent when adsorbing is examined, and it is shown from Titoff's observations (assuming, of course, the adsorption to be reversible) that the divergence between calculated and observed values of the heat of adsorption can be explained on the assumption of a change of surface area. The fractional change of surface per c.cm. adsorbed can be calculated, and also the surface energy per grm. adsorbent in vacuo.

Descended from marine ancestors, Ligia oceanica, like other members of the sub-order (Oniscoidea or “slaters”) to which it belongs, has adopted a terrestrial mode of life. The other Oniscoidea have spread inland, invading regions far from the sea. Ligia, known as the sea-slater, is confined to the coastal fringe, and may possibly form a link between marine isopods and the more truly terrestrial forms.

The elaborate system of molecules, atoms, ions, and electrons modern chemistry and physics have constructed to meet their needs invites the biologist to inquire into the problem of material particles as bearers of life. Leaving aside all questions of finality in the organism touching the vexed problem of purpose in our world, which science has also not been able to solve in lifeless matter (1), we may try to get glimpses of the construction of living material by different methods of attack.

The Dipnoi form an archaic group of Fish with so many unusual features that it is of importance to work out the development of all their organs in detail, and this investigation was therefore undertaken. The early stages were studied in the collection of slides in the Zoology Department of Glasgow University, and the adult organs were obtained from the material brought back from the Gran Chaco by Dr. G. S. Carter in 1927. The work was done in Glasgow under Professor J. Graham Kerr, F.R.S., to whom I am deeply indebted. The larval stages are numbered according to his system in Keibel's Normentafeln, Heft x (3).