Let V be an n–dimensional inner product space over , let H be a subgroup of the symmetric group on {l,…, m}, and let x: H → be an irreducible character. Denote by (H) the symmetry class of tensors over V associated with H and x. Let K (T) ∈ End((H)) be the operator induced by T ∈ End(V), and let DK(T) be the derivation operator of T. The decomposable numerical range W*(DK(T)) of DK(T) is a subset of the classical numerical range W(DK(T)) of DK(T). It is shown that there is a closed star-shaped subset of complex numbers such that

⊆ W*(DK(T)) ⊆ W(DK(T)) = con

where conv denotes the convex hull of . In many cases, the set is convex, and thus the set inclusions are actually equalities. Some consequences of the results and related topics are discussed.

Novel bile acid sequestrants based on a polyammonium backbone were synthesized using molecular imprinting technique. These imprinted polymer networks were prepared by crosslinking different polymeric amines with crosslinking agents in the presence of sodium cholate as the template. The template molecules were completely removed from the polymer matrices by repeated washings. The bile acid sequestration properties of these polymeric resins were evaluated under both in vitro and in vivo conditions. Adsorption isotherms performed in physiologically relevant media revealed that molecular imprinting led to improvement in bile acid sequestration with about a twofold increase in the Ka (association constant). More importantly, hamsters fed with imprinted polymers in their diet excreted more bile acids than the non-imprinted control polymer. These results suggest that molecular imprinting may be potentially an interesting approach to prepare novel polymer therapeutics.

This paper discusses a sequence of extensions of NFU, Jensen's improvement of Quine's set theory “New Foundations” (NF) of [16].

The original theory NF of Quine continues to present difficulties. After 60 years of intermittent investigation, it is still not known to be consistent relative to any set theory in which we have confidence. Specker showed in [20] that NF disproves Choice (and so proves Infinity). Even if one assumes the consistency of NF, one is hampered by the lack of powerful methods for proofs of consistency and independence such as are available for use with ZFC; very clever work has been done with permutation methods, starting with [18] and [5], and exemplified more recently by [14], but permutation methods can only be applied to show the consistency or independence of unstratified sentences (see the definition of NFU below for a definition of stratification). For example, there is no method available to determine whether the assertion “the continuum can be well-ordered” is consistent with or independent of NF. There is one substantial independence result for an assertion with nontrivial stratified consequences, using metamathematical methods: this is Orey's proof of the independence of the Axiom of Counting from NF (see below for a statement of this axiom).

We mention these difficulties only to reassure the reader of their irrelevance to the present work. Jensen's modification of “New Foundations” (in [13]), which was to restrict extensionality to sets, allowing many non-sets (urelements) with no elements, has almost magical effects.

For a real vector space V acted on by a group K and fixed x and y in V, we consider the problem of finding the minimum (respectively, maximum) distance, relative to a K-invariant convex function on V, between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V. Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component ρ of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.

Cholestyramine, the first bile acid sequestrant to be marketed, has been in use for over 20 years. Despite its low potency, requiring 16-24 g of polymer to achieve 20% LDL cholesterol reduction in hypercholesterolemic individuals, only one other sequestrant, colestipol, has come to market in the ensuing period. GelTex Pharmaceuticals has been involved for over six years in the discovery and development of new, more potent polymeric sequestrants. Two binding mechanisms are presented — one that operates via an aggregate binding structure and one that is effective via a defined site binding structure. These two binding mechanisms are compared and contrasted through bile acid binding isotherms. The best of these new sequestrants bind bile acids through a combination of hydrophobicity and ion exchange. Optimization and balancing of each of these interactions led us to more potent materials. The first of these, colesevelam hydrochloride is expected to be three to four times more potent than cholestyramine. A third generation product is still in research at GelTex. With another twofold increase in potency possible, single tablet therapy may become a reality.

An ω-model (a model in which all natural numbers are standard) of the predicative fragment of Quine's set theory “New Foundations” (NF) is constructed. Marcel Crabbé has shown that a theory NFI extending predicative NF is consistent, and the model constructed is actually a model of NFI as well. The construction follows the construction of ω-models of NFU (NF with urelements) by R. B. Jensen, and, like the construction of Jensen for NFU, it can be used to construct α-models for any ordinal α. The construction proceeds via a model of a type theory of a peculiar kind; we first discuss such “tangled type theories” in general, exhibiting a “tangled type theory” (and also an extension of Zermelo set theory with Δ0 comprehension) which is equiconsistent with NF (for which the consistency problem seems no easier than the corresponding problem for NF (still open)), and pointing out that “tangled type theory with urelements” has a quite natural interpretation, which seems to provide an explanation for the more natural behaviour of NFU relative to the other set theories of this kind, and can be seen anachronistically as underlying Jensen's consistency proof for NFU.

Using sol-gel technology, thin organic/ceramic (ceramer) coatings have been applied to metal surfaces in order to enhance such surface properties as adhesion promotion and corrosion prevention. Isotropic coatings have been found to be effective in certain applications such as corrosion prevention, but the formation of anisotropic coatings permits greater flexibility over the resulting properties. Isotropic coatings derived from tetraethoxysilane, for example, effectively inhibit corrosion while being only 100 to 1000 Å thick. These coatings do not, however, promote adhesion. Thin coatings made from traditional silane adhesion promoters alone are unable to prevent corrosion of metallic substrates.

Using monomers with appropriate reactivities permits the single-step synthesis of anisotropic coatings that can both promote adhesion and prevent corrosion. These types of anisotropic coatings allow the physical and chemical properties of a coating to be varied as a function of distance from the substrate and confer properties to the substrate that would not be possible from a single isotropic coating. The principle behind the construction of these anisotropic coatings is general enough that it can be used in many applications where microengineering of surface structures is important.