We consider analytic maps $f_j:D\to D$ of a domain D into itself and ask when does the sequence $f_1\circ\dotsb\circ f_n$ converge locally uniformly on D to a constant. In the case of one complex variable, we are able to show that this is so if there is a sequence $\{w_1,w_2,\dotsc\}$ in D whose values are not taken by any fj in D, and which is homogeneous in the sense that it comes within a fixed hyperbolic distance of any point of D. The situation for several complex variables is also discussed.

The following question of V. Stakhovskii was passed to us by N. Dolbilin [4]. Take the barycentric subdivision of a triangle to obtain six triangles, then take the barycentric subdivision of each of these six triangles and so on; is it true that the resulting collection of triangles is dense (up to similarities) in the space of all triangles? We shall show that it is, but that, nevertheless, the process leads almost surely to a flat triangle (that is, a triangle whose vertices are collinear).

Roughness parameters of sample surface and buried interfaces in a series of thin layers of Si0.4 GeO.6 grown on Si(100) by molecular beam epitaxy (MBE) were measured by using the technique of grazing-incidence x-ray scattering (GIXS). The strain in the layer and the critical thickness of the film were determined from x-ray diffraction of the Si(004) peak. The roughness parameters can be described by a scaling-law with an exponent β = 0.71 for both the surface and interfacial roughness. Establishment of a scaling law thus allows a possibility of predicting the interfacial roughness as a function of the epilayer thickness.

The algebraic tensor product A1⊗A2 of two Banach algebras is an algebra in a natural way. There are certain norms α on this tensor product for which the multiplication is continuous so that the completion, A1αA2, is a Banach algebra. The representation theory of such tensor products is the subject of this paper. It will be shown that, under certain simple conditions, the tensor product of two semi-simple Banach algebras is semi-simple although, without these conditions, the result fails.

An H′-algebra which is not an operator algebra is constructed. This is used to prove that the class of operator algebras is not equal to the class of α-algebras for any tensor norm α.

When A1 and A2 are Banach algebras, their algebraic tensor product A1 ⊗ A2 has a natural multiplication. In this paper we investigate when the condition that A1 and A2 are ℒp-spaces constrains this multiplication to extend to the injective tensor product A1A2, making it a Banach algebra.