In [1] I showed that for any θ belonging to (0,1) there exists a constant C(θ) such that for at least θp of the non-principal characters χ modulo the prime p we have both

and

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.

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.

This note is concerned with the decay of temperature in a finite, inhomogeneous, and one-dimensional, rigid body, which occupies an interval, [0,1] say, of the real line.

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].

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

We prove that, if (fn)n∈ω is a sequence of continuous functions on some recursively presentable Polish space, such that any pointwise cluster point of (fn)n∈ω is a Borel function, then there exists a -subsequence of (fn)n∈ω which is pointwise convergent. This is an effective version of a well known result of Bourgain, Fremlin and Talagrand.

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.

Let M be a smooth surface in Euclidean space E3 and L the Weingarten map. The fundamental forms I1, I2, I3,… on M are defined in terms of L and the usual inner product 〈, 〉 of E3 as follows. If X and Y are in the tangent space TPM of M (Pε M), then I1(X, Y) = 〈X, Y), I2(X, Y) = 〈LX, Y〉, I3(X, Y) = 〈L2X, Y), etc. Moreover, if M is convex, i.e., the Gaussian curvature K = k1k2, where ki, (i = l,2) are the principal curvatures of M, is everywhere positive, then one can also define on M the forms I0(X, Y) = 〈L−1X, Y), I−1,(X, Y) = 〈L−2X, Y), I−2(X, Y) = 〈 L−3X, Y) etc., where L−1 is the inverse of L. Since L is self-adjoint, the forms Im are, for any integer m, symmetric bilinear functions on TPM × TPM. Furthermore Im are C∞ in the sense that if X and Y are vector fields with domain A ⊂ M, then 〈 LmX, Y〉P = 〉LmXP, YP) is a C∞ real function on A. If the convex surface M is appropriately oriented, then the forms Im define metrics on M, which we also denote by 〈, 〉m (〈, 〉1)≡ 〈, 〉).

Recall that a Poincaré Duality group G is said to be smoothly realisable when there exists a smooth closed manifold XG of homotopy type K(G, 1). In this note we prove

Theorem 1. Let

be an exact sequence of groups in which each Si is a Surface group, withfor i ≠ j, Ф is finite and G is torsion free. Then the Poincaré Duality group G is smoothly realisable.

It is shown first that internal or external boundary-layer flow over obstacles or other surface distortions is susceptible to a novel kind of viscous-inviscid instability, involving growth rates much larger than those of traditional Tollmien-Schlichting and Gortler modes for instance. The same instabilities arise in liquid-layer flow at sub-critical Froude number, and they are associated with an interacting boundary-layer problem where the normalized pressure is equal to the normalized displacement decrement. Second, certain limiting linear and nonlinear disturbances are studied to shed more light on the overall instability process and each form of disturbance leads to a finite-time collapse, although different in each case. Thirdly, and in consequence, the work finds the significant feature that the whole interacting boundary layer can break down nonlinearly within a finite scaled time.

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