In [9] J. L. Krivine and B. Maurey introduced the class of stable Banach spaces: a separable Banach space is called stable if for every pair of bounded sequences (xn)n, (yn)n and for every pair of ultrafilters on the natural numbers we have

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.

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.

Asymptotic behaviour of solutions of polyharmonic equations

on exterior domains in Rn is considered under suitable conditions on f. It is shown that finite energysolutions u have the asymptotic property

This partially extends results of Egnell [6] for m = 1.

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.

The paper is concerned with equations of the form x' = A(t)x +f(t, x), where A is a continuous matrix function defined on ℝ, f is a continuous vector-valued function of (t, x) with f(t, 0) = 0. It is proved that if x' = A(t)x has an exponential trichotomy, A is bounded and f satisfies the Lipschitz condition with coefficient sufficiently small, then these equations are topologically equivalent to the systems of equations of the form , where B, g satisfy the same conditions as A, f.

We give several results which extend our recent proof of the Payne-Pólya–Weinberger conjecture to ratios of higher eigenvalues. In particular, we show that for a bounded domain Ω⊂ℝn the eigenvalues of its Dirichlet Laplacian obey where λm denotes the mth eigenvalue and jp,k denotes the kth positive zero of the Bessel function Jp(x). Certain extensions of this result are given, the most general being the bound where k≧2 and l(m) denotes the number of nodal domains of an mth eigenfunction. Our results imply certain further conjectures of Payne, Pólya, and Weinberger concerning λ3/λ2 and λ4/λ3. In addition, we find a resonably good bound on λ4/λ1. We also briefly discuss extensions to Schrödinger operators and other elliptic eigenvalue problems.

Normal-convex embeddings are introduced for inverse semigroups, generalizing the group-theoretic concept, due to Papakyriakopoulos [4]. It is shown that every E-unitary inverse semigroup admits a normal-convex embedding into a semidirect product of a semilattice by a group, a stronger version of a result by O'Carroll [3]. A general embedding result for inverse semigroups is also obtained.

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.

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.

The behaviour of a microsensor thermistor is described by a system of nonlinear coupled elliptic equations subject to mixed Dirichlet-Neumann boundary conditions, to be solved on different domains. We employ the Implicit Function Theorem in Banach space to show that the system has a solution for small applied bias. It does not appear that earlier approaches for similar thermistor problems can be employed in this physically important situation. The fact that the problem is cast in a subset of R3 is significant in our presentation.

The asymptotic behaviour has been determined for several natural geometric or topological quantities related to (degrees of) compactness of bounded linear operators on Banach spaces; see for instance [24], [25] and [17]. This paper complements these results by studying the spectral properties of some quantities related to weak compactness.

All Archimedean commutative semigroups S are described such that every S-homogeneous hereditary radical is S-normal. It is shown that this result is in a sense unimprovable.

We consider a class of measure-valued Markov processes constructed by taking a superprocess over some underlying Markov process and conditioning it to stay alive forever. We obtain two representations of such a process. The first representation is in terms of an “immortal particle” that moves around according to the underlying Markov process and throws off pieces of mass, which then proceed to evolve in the same way that mass evolves for the unconditioned superprocess. As a consequence of this representation, we show that the tail σ-field of the conditioned superprocess is trivial if the tail σ-field of the underlying process is trivial. The second representation is analogous to one obtained by LeGall in the unconditioned case. It represents the conditioned superprocess in terms of a certain process taking values in the path space of the underlying process. This representation is useful for studying the “transience” and “recurrence” properties of the closed support process.

Piecewise-circular (PC) curves are made up of circular arcs and segments of straight lines, joined so that the (undirected) tangent line turns continuously. PC curves have arisen in various applications where they are used to approximate smooth curves. In a previous paper, the authors introduced some of their geometrical properties. In this paper they investigate the ‘symmetry sets’ of PC curves and one-parameter families of such curves. The symmetry set has also arisen in applications (this time to shape recognition) and its mathematical properties for smooth curves have been investigated by Bruce, Giblin and Gibson. It turns out that the symmetry sets of general one-parameter families of plane curves are mirrored remarkably faithfully by the symmetry sets arising from the much simpler class of PC curves.

Let S be a cancellative semigroup. This paper is motivated by the problem of finding a description of semigroup rings K[S] over a field K that are semiprime or prime. Results of this type are well-known in the case of a group ring K[G], cf. [8]. The description, as well as the proofs, involve the FC-centre of G defined as the subset of all elements with finitely many conjugates in G. In [4], [5] Krempa extended the FC-centre techniques to the case of an arbitrary cancellative semigroup S. He defined a subsemigroup Δ(S) of S which coincides with the FC-centre in the case of groups, and can be used to describe the centre and to study special elements of K[S]. His results were strengthened by the author in [7], where Δ(S) was also applied in the context of prime and semiprime algebras K[S]. However, Δ(S) itself is not sufficient to characterize semigroup rings of this type. We note that in [2], [3] Dauns developed a similar idea for a study of the centre of semigroup rings and certain of their generalizations.