We present a geometric approach for systems of ordinary differential equations which generate an order preserving flow. One of our main goals is to describe qualitatively the asymptotic behaviour of trajectories of dynamical systems enjoying a uniformly monotone principle.

A one-dimensional scalar neural network with two stable steady states is analysed. It is shown that there exists a unique monotone travelling wave front which joins the two stable states. Some additional properties of the wave such as the direction of its velocity are discussed.

The Dirichlet, Neumann and mixed boundary-value problems for the two-dimensional Helmholtz equation in the interior or exterior of a quadrant are considered in a Sobolev space setting. It is shown that the potential operators arising in the interior problems can be used to derive systems of boundary integral equations to the exterior problems, which can be solved explicitly.

Let B(ℤ)* be the Banach dual of the space of all bounded complex-valued functions on ℤ. For each n ε ℤ, let Ln be the translation operator on B(ℤ) and Tn be its adjoint operator on B(ℤ)*. This paper concerns itself with equations of the form

where (an)nεℤ is a sequence of complex numbers.

We construct the Conley index for maps. We do not assume any compactness of map or space. We prove the Ważewski property, additivity property and continuation property.

A study is made of the minimum number of generators of the n-th direct power of certain 2-generator groups.

The problem

is considered, where X is a normed space, F: X →] –∞, + ∞] is a (possibly non-convex) functional and L ∈ X'. We look for the values of γy for which the infimum above is attained. Applications to nonconvex functionals denned on measures and on the BV space are given.

This paper is a sequel to [4]. Its purpose is to show that the concept of isometric foldings of Riemannian manifolds can be extended to a much wider class of manifolds without losing the main structure theorem. We present here what we believe to be a definitive form of the folding concept for smooth manifolds.

The theory discussed here is based on the idea of a 1-spread [2], where the role of geodesies on a Riemannian manifold is assumed by smooth, unoriented and unparametrised curves on a smooth manifold. The absence of metrical structure forces a fresh approach to the basic definitions. A crucial feature of the Riemannian theory does survive, however, in this general setting: a 1-spread on a sufficiently smooth manifold M induces a 1-spread on sufficiently small spheres surrounding any point of M. With the help of this fact, we are able to construct an inductive definition of “star folding” f:M → N between smooth manifolds M and N, and to retain the theorem that the manifold M is stratified by the “folds”, each of which has the character of a “totally geodesic” submanifold with respect to the above-mentioned curves.

In this paper we obtain new results which relate the number of conjugacy classes of л-elements of a finite group and an arbitrary subgroup, which are analogous to some results about normal subgroups. We also prove some new results which show the relationship between class numbers and splitting theorems. Our proofs only involve elementary techniques.

Solution branches of a semilinear elliptic problem with Neumann boundary conditions are studied at its corank-2 bifurcation points. It is shown that generally there are exactly four different nontrivial solution branches passing through a corank-2 bifurcation point. The bifurcating solution branches are parametrised via a nonsingular enlarged problem. Branch switching at bifurcation points is incorporated with a continuation method.

In the first part of the paper a variational characterisation of the periodic eigenvalues (the so-called Fučik spectrum) of a semilinear, positive homogeneous Sturm–Liouville equation is given. The proof relies on the S1-invariance of the equation.

In the second part a nonlinear Sturm–Liouville equation with, typically, an exponential nonlinearity is considered. It is proved that under certain conditions this equation is solvable for arbitrary forcing terms. The proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values in the Fučik spectrum.

In this paper we prove an extension of inequality (1.1) due to A. Meir.

In this paper we characterise left orders in strongly regular rings, both for classical left orders and left orders in the sense of Fountain and Gould [2] where the ring of quotients need not have an identity.

Let Cf, Pf, Qf and Rf be respectively the convex, polyconvex, quasi-convex and rank-one-convex envelopes of a given function f. If fp: RNxM→R and fq(ξ) behaves as |ξ|q at infinity q ∈ (1, ∞), we show that . This is the case for (Pfp)p provided that q ≠1,…, min (N, M), otherwise . In the last part of this work, we show that f(ξ) = g(|ξ|) does not imply in general Pf = Qf.

A fundamental comparison theorem is established for general semilinear parabolic systems via the notions of sectorial operators, analytic semigroups and the application of the Tychonoff Fixed Point Theorem. Based on this result, we establish a maximum principle for systems of general parabolic operators and general comparison theorems for parabolic systems with quasimonotone or mixed quasimonotone nonlinearities. These results cover and extend most currently used forms of maximum principles and comparison theorems. A global existence theorem for parabolic systems is derived as an application which, in particular, gives rise to some global existence results for Fujita type systems and certain generalisations.

We treat the problem of determining a crack inside a conductor when two pairs of current and voltage boundary measurements are given. We prove a theorem of continuous dependence from the data.

It is known that the parametric equation u'∊+ a∊u∊ = f, u∊ (0)= 0,with α ≦ a∊ ≦ β, for all ∊ > 0 and almost everywhere in a bounded domain Ω of ℝN, and f in L∞((0, T) × Ω), shows, at the limit, a memory effect. In this work the associated minimisation problem is considered and we describe how the memory effect appears in the Γ-limit, for the weak topology H1:(0, T; L2(Ω)) of the corresponding functional. The sequence a∊ has no dependence in time.

We study a semilinear boundary value problem with the feature that the vanishing boundary value makes the equation singular. We prove that the positive solution is in general Hölder-continuous up to the boundary and has even better regularity in some special cases.

Hyperasymptotic expansions were recently introduced by Berry and Howls, and yield refined information by expanding remainders in asymptotic expansions. In a recent paper of Olde Daalhuis, a method was given for obtaining hyperasymptotic expansions of integrals that represent the confluent hypergeometric U-function. This paper gives an extension of that method to neighbourhoods of the so-called Stokes lines. At each level, the remainder is exponentially small compared with the previous remainders. Two numerical illustrations confirm these exponential improvements.

Prime Malcev superalgebras over fields of characteristic not two and three have been studied by Shestakov [8]. He obtains the remarkable result that if these superalgebras have a nonzero odd part then they are Lie superalgebras. The main purpose of this note is to extend this result to fields of characteristic three. To this aim, it is enough to use adequately a result of Filippov [3]. Commutative and anticommutative superalgebras will be considered too, showing that they are prime, semiprime or simple as superalgebras if and only if they are as algebras. Finally, some conclusions for finite-dimensional semisimple Malcev superalgebras will be deduced. Any such superalgebra is the direct sum of a semisimple Lie superalgebra and a direct sum of simple non-Lie algebras.