Among the many solutions of the wave equation investigated by the late Professor Bateman there is one type which has so far received little attention.

It is shown that an extended Schlömilch formula for Stirling-type pairs of numbers and the inversion formula of Lagrange are implied by each other. Also proved are some congruence relations modulo a prime number p(>2) associated with generalized Stirling numbers. The third result is concerned with the asymptotic expansions of Stirling-type pairs involving large parameters.

If $T:A_{0}\rightarrow B$ boundedly and $T:A_{1}\rightarrow B$ compactly, then a result of Lions–Peetre shows that $T:A\rightarrow B$ compactly for a certain class of spaces $A$ which are intermediate with respect to $A_{0}$ and $A_{1}$. We investigate to what extent such results can hold for arbitrary intermediate spaces $A$. The ‘dual’ case of an operator $S$ such that $S:X\rightarrow Y_{0}$ boundedly and $S:X\rightarrow Y_{1}$ compactly, is also considered, as well as similar questions for other closed operator ideals.

AMS 2000 Mathematics subject classification: Primary 46B70; 47D50

Let n be a positive integer and D a division algebra of finite dimension m over its centre. We describe in detail the structure of a soluble subgroup G of GL(n,D). (More generally we consider subgroups of GL{n,D) with no free subgroup of rank 2.) Of course G is isomorphic to a linear group of degree mn and hence linear theory describes G, but the object here is to reduce as far as possible the dependence of the description on m. The results are particularly sharp if n=l. They will be used in later papers to study matrix groups over certain types of infinite-dimensional division algebra. This present paper was very much inspired by A. I. Lichtman's work: Free subgroups in linear groups over some skew fields, J. Algebra105 (1987), 1–28.

If T ∈ L(X) is such that T′ is a scalar-type prespectral operator, then Re T′ and Im T′ are both dual operators. It is shown that that the possession of a functional calculus for the continuous functions on the spectrum of T is equivalent to T′ being scalar-type prespectral of class X, thus answering a question of Berkson and Gillespie.

Recently A. D. Sands [7] solved a problem posed in [6], and characterised the semisimple classes of associative rings as classes being regular, coinductive and closed under extensions. It is the purpose of this note to prove the same assertion for alternative rings. This result is perhaps not surprising, nevertheless its proof cannot be considered an easy one, and it requires a technique of dealing with ideals of ideals. In addition, semisimple classes of hereditary radicals and those of supernilpotent radicals will be characterised as easy consequences of our theorem.

The Cartan determinant conjecture for left artinian rings was verified for quasihereditary rings showing detC(R) = detC(R/I), where I is a protective ideal generated by a primitive idempotent. This article identifies classes of rings generalizing the quasihereditary ones, first by relaxing the “projective” condition on heredity ideals. These rings, called left k-hereditary are all of finite global dimension. Next a class of rings is defined which includes left serial rings of finite global dimension, quasihereditary and left 1-hereditary rings, but also rings of infinite global dimension. For such rings, the Cartan determinant conjecture is true, as is its converse. This is shown by matrix reduction. Examples compare and contrast these rings with other known families and a recipe is given for constructing them.

In this paper, we consider a reaction infiltration problem consisting of a parabolic equation for the concentration, an elliptic equation for the pressure, and an ordinary differential equation for the porosity. Existence and uniqueness of a global classical solution is proved for bounded domains Ω⊂RN, under suitable boundary conditions.