A sequence of transformations of the type Y = (I + o(1))Z is developed for the system Y′(x) = {Λ(x) + R(x)}Y(x), where Λ is diagonal. The transformations bring in the derivatives of R in succession until the Levinson form is obtained when a given derivative is reached. This theory covers rapidly varying coefficients and it extends results which are known for constant Λ.

In this paper we study a problem in multilinear algebra which consists of finding small values of a certain quotientμ/α. Here μ is the minimal eigenvalue of a positive definite operator determinant Δ of the type introduced by F. V. Atkinson, and α is the minimum of the quadratic form corresponding to Δ with respect to all decomposable tensors of unit norm. Our results are connected with earlier results of P. Binding.

We demonstrate local Lipschitz regularity for minimisers of certain functionals which are appropriately convex and quadratic near infinity. The proof employs a blow-up argument entailing a linearisation of the Euler—Lagrange equations “at infinity”. As a application, we prove that minimisers for the relaxed optimal design problem derived by Kohn and Strang [3] are locally Lipschitz.

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 study the perturbation of zeros of maps of Banach spaces where the maps are invariant under continuous groups of symmetries. In some cases, we allow the perturbed maps partially to break the symmetries. Our results improve earlier results of the author by removing smoothness conditions on the group action. The key new idea is a regularity theorem for the zeros of invariant Fredholm maps.

Assume f ∈ C(RM, SN−1) (M ≧ 1, N ≧ 1) is periodic with period one in each variable and satisfies:

Then, every probability measure on [0, 1)M such that

is a Dirac measure at some point of [0, 1)M.

In this paper, we prove the existence of at least one solution to the problem

where ∆k is an eigenvalue of the linear part, h is orthogonal to the eigenspace corresponding to ∆R and g is a nonlinear perturbation which can be, for instance, a continuous periodic real function with mean value zero. We employ the techniques used by the second author in a previous paper in which the same result was obtained in the case in which ∆R is assumed to be simple. The final result is obtained by using variational methods and in particular a suitable version of the saddle point theorem of P. Rabinowitz.

Some second order ordinary differential equations of the form ξ2ϕ″ + ξ2(N − 1)″′/r + ½(ϕ − ϕ3) + ½k = 0 are studied. Properties such as existence and monotonicity of solutions are considered for N ≧ 1, ξ > 0 and two sets of boundary conditions. For N = 1, some explicit results are obtained for small ξ. These ODE's arise from a phase field approach to free boundary problems involving a phase transition.

We derive an asymptotic expansion for the Titchmarsh–Weyl m-function associated with the second order linear differential equation

in the case where the only restriction on the real-valued function q is

The correspondence between radicals of associative rings and A-radicals is studied. It is shown that corresponding to each A-radical there is an interval of radicals and that each radical belongs to exactly one such interval. The question of the nature of the radical of a one-sided ideal is considered. It is shown that the radicals such that the radical of a one-sided ideal is always a one-sided ideal are those which contain their associated A-radicals. Radicals such that the radical of a one-sided ideal always equals the intersection of a left ideal and a right ideal are described, as are those A-radicals such that every associated radical has this property.

It is proved that if a ring R has the extension property in containing ringSi, then the amalgam [R; Si,] is strongly embeddable. Using a result of P. M. Cohn, a necessary and sufficient condition for a ring to be an amalgamation base is then given. It is also shown that R is a level subring of a ring S if and only if S/R is flat. From this, a classical result of P. M. Cohn on flat amalgams is proved.

By using asymptotic estimates for the eigenvalues and eigenfunctions or irregular boundary value problems, we state necessary conditions for the pointwise convergence and for the divergence of the corresponding eigenfunction expansions.

Initial-boundary value problems for nonlinear first order partial differential equations ∂tu + H(x, t, u, Dxu) = 0 and corresponding boundary value problems H(x, u, Dxu) = 0 are studied in bounded sets, using Crandal's and Lions' notion of viscosity solutions. We give pointwise conditions on the boundary data that guarantee the existence of such solutions and estimate their moduli of continuity in terms of continuity properties of the data. The results are applied to properties of the value function for certain differential games.

We give conditions on pairs of non-negative weight functions u and v which are sufficient that, for 1<p, q <∞,

where T is the Hankel-or the K-transformation.

The proofs are based on a weighted Marcinkiewicz interpolation theorem for linear operators. In the case that T is the Hankel transformation and 1<p≦q <∞, the result is similar to a weighted estimate of Heywood and Rooney [9], but with different weight conditions.