For α≥0 and β≥0 we denote by K (α, β) the Kaplan classes of functions f analytic and non-zero in the open unit disk U = {z: |z| < 1} such that f ∈ K(α, β), if, and only if, for θ1 < θ2 < θ1 + 2π and 0 < r < 1,

Where

Consider an impermeable container Ω in ℝ3 filled with a porous material saturated with a Boussinesq fluid. The boundary ∂Ω of Ω consists of two parts Γ1 and Γ2, i. e., ∂Ω = Γ1 ⋃ Γ2. Γ1 is the intersection of the horizontal planes at z = 0 and z = 1 with a vertical cylinder of arbitrary cross section G (a bounded smooth domain in ℝ2), i.e., Γ1 = G × {0, 1}. Γ2 is the sidewall of the vertical cylinder between the planes z = 0 and z = l, i.e., Γ2Γ = ∂G × [0, 1].

In [S1] we introduced and in [S2, S3, S4] developed a class of topological spaces that is useful in the study of the classification of Banach spaces and Gateaux differentiation of functions defined in Banach spaces. The class C may be most succinctly defined in the following way: a Hausdorff space T is in C if any upper semicontinuous compact valued map (usco) that is minimal and defined on a Baire space B with values in T must be point valued on a dense Gδ subset of B. This definition conceals many interesting properties of the family C. See [S2] for a discussion of the various definitions. Our main result here is that if X is a Banach space such that the dual space X* in the weak* topology is in C and K is any weak* compact subset of X* then the extreme points of K contain a dense, necessarily Gδ, subset homeomorphic to a complete metric space. In [S4] we studied the class K of κ-analytic spaces in C. Here we shall show that many elements of K contain dense subsets homeomorphic to complete metric spaces. It is easy to see that C contains all metric spaces and it is proved in [S4] that analytic spaces are in K. We obtain a number of topological results that may be of independent interest. We close with a discussion of various examples that show the interaction of these ideas between functional analysis and topology

We construct a universal function φ on the real line such that, for every continuously differentiable function f the range of f – φ has measure zero. We then apply this to obtain results on curve packing that generalize the Besicovitch set. In particular, we show that given a continuously differentiable family of measurable curves, there exists a plane set of measure zero containing a translate of each curve in the family. Examples are given to show that the differentiability hypothesis cannot be weakened to a Lipschitz condition of order α for any 0<α<1.

Sandwich semigroups were introduced in [4], [5] and [6]. Green's relations (for regular elements) were characterized for these semigroups in [11] and [13]. Sandwich semigroups of continuous functions first made their appearance in [5]. In this paper, we consider only sandwich semigroups of continuous functions and we refer to them simply as sandwich semigroups. We now recall the definition. Let X and Y be topological spaces and fix a continuous function α from Y into X. Let S(X, Y, α) denote the semigroup of all continuous functions from X into Y where the product fg of f, g ε S(X, Y, α) is defined by fg = f ∘ α ∘ g. We refer to S(X, Y, α) as a sandwich semigroup with sandwich function α. If X = Y and α is the identity map then S(X, Y, α) is, of course, just S(X), the semigroup of all continuous selfmaps of X.

The Minkowski-Hlawka bound implies that there exist lattice packings of n-dimensional “superballs” |x1|σ + … + |xn|σ ≤ 1 (σ = 1,2,…) having density Δ satisfying log2 Δ ≥ −n(l + o(l)) as n → ∞. For each n = pσ (p an odd prime) we exhibit a finite set of lattices, constructed from codes over GF(p), that contain packings of superballs having log2 Δ ≥ −cn(l + o(l)), where for σ = 2 (the classical sphere packing problem), worse than but surprisingly close to the Minkowski-Hlawka bound, and c = 0·8226 … for σ = 3, c = 0·6742 … for σ = 4, etc., improving on that bound.