For an H-space with a generating subspace, we construct a space whose K-cohomology is a direct sum of a truncated polynomial algebra and an ideal, which enables technical restrictions to be removed from several known results in the homotopy theory of H-spaces.

Kuiper has proved that there is no tight topological immersion of RP2 into E3. Thus any continuous tight map RP2 → E3 has to have singular points. In this note we consider tight C∞-stable maps of RP2 and the torus into E3; i.e. maps with the simplest possible singularities (Whitney pinchpoints) and transversal crossings. We classify the forms that the outer part of such maps can take; we prove also some facts about the inner part. The result about the outer part of C℞-stable tight maps RP2 → E3 establishes the first half of Banchoff's conjectured classification of such mappings.

We present several general methods of constructing tight hypersurfaces f: Mn → En+1, n ≧ 2. We prove a smoothing lemma, which allows us to approximate tight continuous hypersurfaces by C∞ tight ones. We show that given a tight immersion or C∞-stable map f: M2 → E3 of a compact surface M2 other than S2, there is a tight C∞-stable map g: M2 # RP2:→ E3. We prove that given C∞ tight immersions f: Mn → En+1 and g: Nn → En+1 of compact n-manifolds Mn and Nn into En+1, there is a tight C∞ immersion of Mn # Nn into En+1. Two other methods involve hypersurfaces of rotation and sets in En+1 at a fixed distance from a tightly embedded n-manifold with boundary in En. One consequence of these methods is that the outer part of C∞ tight hypersurfacesf: Mn → En+1, n ≧ 3, is far more complicated than in the case of tight surfaces in En. For example, given any C∞n-manifold M with boundary tightly embedded in En, there is a tight immersion f:Nn → En+1 of a closed n-manifold having M as a topset. Kuiper's theorem describing tight immersions of surfaces into E3 does not generalise to the case of hypersurfaces f: Mn → En+1, n ≧ 3, without substantial restrictions on Mn and/or f.

In this paper we give a proof on existence of non-negative solutions of weakly coupled systems of non-linear elliptic PDE's which model isothermal chemical reactions in a bounded volume Ω. The boundaries of this “diffusion reactor” will be allowed to be differently permeable to different species, giving rise to homogeneous Neumann, or inhomogeneous mixed, boundary conditions. We assume a “conservation of mass” condition and a second condition, the choice of which is motivated in an example.

In the second section we prove that the PDE system that describes the reaction:

in the diffusion reactor admits a unique stationary solution.

For this proof we must show invertibility for a class of non inverse-positive linear elliptic operators.

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.

This paper describes a proof that the solutions of Neumann problems for the exterior of spheres {x ∊ ℝ3∣ converge to the solutions of the exterior problem for the halfspace {x ∊ ℝ3 } x1 ≧ 0} as b → ∞ provided that the boundary data converge in a certain sense. The method requires that there be some dissipation which can be arbitrarily small. Fredholm integral equations are set up for the boundary data, and these are solved by means of Neumann series for large b. Estimates on the terms of the series (which involve singular integrals), in terms of b, allow the convergence proof to be carried through.

The procedure of expressing problems with infinite boundaries in terms of problems with finite boundaries allows the implementation of an effective numerical procedure for determining, for example, the entire near-field of a baffled piston, a problem whose solution has proved elusive for many years.

Theorems of Gersgorin-type are established for a diagonally dominant, unbounded, infinite matrix operator A acting on lp for some l ≦p≦∞. The results are established using an approximating sequence of infinite matrices An that converges to A in the generalised sense as n → ∞. This constructive approach admits approximation of the spectral properties of A by those of An.

Any non-ruled quartic surface with a double line in space of three dimensions has plane sections of genus 2 and so is a projection of a non-singular duodecimic surface in space of eleven dimensions. This was accepted as an established fact by 1890, but there seems not to be any account of such a projection in action. The following pages are submitted to aid the removal of this century-old anomaly.

In this note, we consider the Hardy-Littlewood maximal function on R for arbitrary measures, as was done by Peter Sjögren in a previous paper. We determine the best constant for the weak type inequality.

Necessary and sufficient conditions on weights u and v are given for the inequality

to hold for all g ∊ .

In this note we show that certain perturbations, involving rank-one stabilising operators, which correspond to small delays in some feedback control problems, shift a part of the point spectrum of the unperturbed operator into the right-half plane.

We consider a weakly coupled set of two partial differential equations where the coupling matrix has variable elements and the principal part of each equation is the same uniformly elliptic operator. Weobtain necessary conditions that the system of equations can be decoupled. By decoupling the system and using a positivity lemma due to Hess and Kato, we determine the algebraic sign of the solution components. This work extends recent results of de Figueiredo and Mitidieri. Further, one can use these results to determine the sign of the solution to certain fourth order elliptic boundary value problems.

The method of viscosity solutions for nonlinear partial differential equations (PDEs) justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions. We describe in this paper a “perturbed test function” device, which entails various modifications of the test functions by lower order correctors. Applications include homogenisation for quasilinear elliptic PDEs and approximation of quasilinear parabolic PDEs by systems of Hamilton-Jacobi equations.