The aim of this paper is to establish some new integral inequalities of the Sobolev type involving functions of several independent variables. The analysis used in the proofs is elementary and the results established provide new estimates for this type of inequality.

In [13], Nikaido and Isoda generalised von Neumann's symmetrisation method for matrix games. They showed that N-person noncooperative games can be treated by a minimax method.

We apply this method to N-person differential games. Lukes and Russell [11] first studied N-person nonzero sum linear quadratic games in 1971. Here we have reproduced and strengthened their results. The existence and uniqueness of equilibria are completely determined by the invertibility of the decision operator, and the nonuniqueness of equilibrium strategies is only up to a finite dimensional subspace of the space of all admissible strategies.

In the constrained case, we have established an existence result for games with a much weaker convexity assumption subject to compact convex constraints. We have also derived certain results for games with noncompact constraints. Several examples of quadratic and non-quadratic games are given to illustrate the theorem.

Numerical computations are also possible and are given in the sequel [3].

We consider the bifurcation problem of the Föppl–Kármán equations for a thin elastic rectangular plate near a multiple eigenvalue allowing for a small perturbation parameter related to the aspect ratio of the plate. The first step in the study is to introduce equivalent operator equations in the energy spaces of the problem which explicitly contain the perturbation parameter. By dealing partially with a general formulation, we obtain the main results for the double eigenvalue and Z2 ⊓ Z2 symmetry of bifurcation equations. We are chiefly interested in the degenerate cases of bifurcation equations.

In this paper we give two generator, two relation presentations for the following perfect groups which were not previously known to have deficiency zero: SL(2, 32), SL(2, 64), SL(2, 27), SL(2, 49), Â7, Ŝz(8), SL(2, 5) × SL(2, 5), SL(2, 5) × SL(2, 25). We also give two generator, two relation presentations for three other finite perfect groups, two having SL(2, 7) as an image and one having SL(2, 5) as an image. We also discuss presentations for certain other perfect groups which were known to have deficiency zero and find some neat new presentations for them.

In this paper we show that for the whole class of differential expressions in the limit-point case considered by Kauffman in [2], the perturbation theory yields a limit-point criterion for a much wider class of ordinary differential expressions. More general coefficients are admitted which may be eventually negative provided they are “dominated” by some other positive coefficients. This generalises results in [4], [5] and [6].

For radiating solutions to the time-harmonic Maxwell equations, it is shown that the boundary operator mapping the tangential components of the electric field into the tangential components of the magnetic field is a bounded bijective operator from the space of Holder continuous tangential fields with Hölder continuous surface divergence onto itself.

We consider the equations for the isothermal motion of a one-dimensional unbounded body composed of a material with viscosity and capillarity. Using a technique derived from the theory of compensated compactness, we find conditions which guarantee that, as viscosity and capillarity approach zero, the solutions to these equations converge to a solution to the corresponding equations in elasticity.

Self-adjoint operators in L2(0, 1) associated with a formally symmetric differential operator regular in [0, 1] can be determined by boundary conditions or as extensions of the minimal operator. These extensions are determined by extensions of the Cayley transform of the minimal operator. This paper establishes an explicit expression for the extension of the Cayley transform in terms of the boundary conditions and vice versa.

We consider classes of self-adjoint operators for which the nonpositive part of the spectrum consists of eigenvalues ρ0(λ)≦ ρ1(») ≦ … repeated according to multiplicity. The sets of λ where ρi(λ) is negative and zero are labelled Ni and Zi respectively, and Pi = ℝk\(Ni ∪ Zi). We study conditions on the Vj sufficient to ensure nonemptiness of at least one of Ni, Zi and Pi for all T or for all positive definite T, as well as conditions which are necessary in the sense that failure permits emptiness for at least one T.

As an example of our results, we show in the Sturm–Liouville case

with L∞ coefficients and separated end conditions, that nonemptiness of Zi for all T (i.e. for all p > 0, all q and all boundary data) is equivalent to the i-independent condition that the ftj do not vanish simultaneously on a set of positive measure.

We construct firstly a single tactical configuration which has the structure of the dual of the affine plane of order 4, and show how to obtain a further set of 3 such dual planes which, together with , satisfy a certain set of intersection properties. This set of 4 dual planes is used to extend the 20 points of to the Steiner system = S(5, 8, 24). The construction leads to the production of involutions of the type which fix the points of an octad. It is shown that 3 involutions each of this type suffice to generate M24, each of the simple Mathieu groups inside M24, the Todd group, and all the intransitive maximal subgroups of M24.

This paper deals with the existence of bounded plane wave fronts of reaction-diffusion systems. The main result ensures the partial invariance of a certain region, under kinetic conditions commonly used in the literature. This allows us to construct bounded plane wave fronts taking their values in that domain. We also give an estimate of the minimum permissible value of the propagation velocity of those plane wave fronts. Some examples are given.