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.

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.