On coverings of plane convex sets by translates of strips

In the euclidean planeE ² letS ₁,S ₂, ... be a sequence of strips of widthsw ₁,w ₂, .... It is shown thatE ² can be covered by translates of the stripsS _i if w ₁ ^3/2 = . Further results concern conditions in order that a compact convex domain inE ² can be covered by translates ofS ₁,S ₂, ....This research was supported by National Science Foundation Research Grant MCS 76-06111.     

Sur la forme canonique des formes extérieures

Sans résumé

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The article is the text of a talk given at the Symposium on Differential Geometry in Debrecen, Hungary, on August 28–September 3, 1975.     

Molecular modeling of cytochrome P450 3A4

The three-dimensional structure of human cytochrome P450 3A4 was modeled based on crystallographic coordinates of four bacterial P450s: P450 BM-3, P450cam, P450terp, and P450eryF. The P450 3A4 sequence was aligned to those of the known proteins using a structure-based alignment of P450 BM-3, P450cam, P450terp, and P450eryF. The coordinates of the model were then calculated using a consensus strategy, and the final structure was optimized in the presence of water. The P450 3A4 model resembles P450 BM-3 the most, but the B helix is similar to that of P450eryF, which leads to an enlarged active site when compared with P450 BM-3, P450cam, and P450terp. The 3A4 residues equivalent to known substrate contact residues of the bacterial proteins and key residues of rat P450 2B1 are located in the active site or the substrate access channel. Docking of progesterone into the P450 3A4 model demonstrated that the substrate bound in a 6-orientation can interact with a number of active site residues, such as 114, 119, 301, 304, 305, 309, 370, 373, and 479, through hydrophobic interactions. The active site of the enzyme can also accommodate erythromycin, which, in addition to the residues listed for progesterone, also contacts residues 101, 104, 105, 214, 215, 217, 218, 374, and 478. The majority of 3A4 residues which interact with progesterone and/or erythromycin possess their equivalents in key residues of P450 2B enzymes, except for residues 297, 480 and 482, which do not contact either substrate in P450 3A4. The results from docking of progesterone and erythromycin into the enzyme model make it possible to pinpoint residues which may be important for 3A4 function and to target them for site-directed mutagenesis.     

The structure of 1-formyl-3-phenyl-Δ-pyrazoline in the gas phase (DFT calculations), in solution (NMR) and in the solid state (X-ray crystallography)

The molecular structure of 1-formyl-3-phenyl-Δ²-pyrazoline was determined by X-ray crystallography (triclinic, P-1). The geometry thus obtained was compared with that obtained by DFT calculations. The GIAO method was used to calculate absolute shieldings, which agree conveniently with those measured by ¹³C and ¹⁵N NMR. The title compound appears to be an essentially planar molecule.     

Ion-selective electrode for gallium determination in nickel alloy, fly-ash and biological samples

A poly(vinyl chloride)-based membrane of 2,9-dimethyl-4,11-diphenyl-1,5,8,12-tetraazacyclotetradeca-1,4,8,11-tetraene (DDTCT) with sodium tetraphenyl borate (STB) as an anion excluder and dibutyl phthalate (DBP), dibutyl butylphosphonate (DBBP), tris(2-ethylhexyl) phosphate (TEP) and tributyl phosphate (TBP) as plasticizing solvent mediators was prepared and investigated as a Ga(III)-selective electrode. The best performance was observed with the membrane having the ligand-PVC-DBP-STB composition 1:4:1:1, which worked well over a wide concentration range (1.45 × 10⁻⁶ to 0.1 mol L⁻¹) with a Nernstian slope of 28.7 mV per decade of activity between pH 4.0 and 10.0. This electrode showed a fast response time of 12 s and was used over a period of 100 days with good reproducibility (s = 0.3 mV). The selectivity coefficients for monovalent, divalent and trivalent cations indicate excellent selectivity for Ga(III) ions over a large number of cations. Anions such as Cl⁻ and SO₄²⁻ do not interfere and the electrode also works satisfactorily in partially water-alcohol medium. The practical utility of the membrane sensor has also been observed in solutions contaminated with detergents, i.e., cetyltrimethylammonium bromide and sodium dodecyl sulfate and used for the determination of gallium in nickel alloy, fly-ash and biological samples.     

Determination of ochratoxin A in maize bread samples by LC with fluorescence detection

Ochratoxin A (OTA) is a secondary fungal metabolite produced by several moulds, mainly by Aspergillus ochraceus, A. carbonarius, A. niger and by Penicillium verrucosum. The present work shows the results of comparative studies using different procedures for the analysis of OTA in maize bread samples. The studied analytical methods involved extraction with different volumes of PBS/methanol, different extraction apparatus, and clean-up through immunoaffinity columns. The separation and identification were carried out by high-performance liquid chromatography with fluorescence detection. The optimized method for analysis of OTA in maize bread involved extraction with PBS:methanol (50:50), and clean-up with IAC column. The limit of quantification was 0.033 ng g⁻¹. Recoveries ranged from 87% to 102% for fortifications at 2.000 and 0.500 ng g⁻¹, respectively, within-day R.S.D. of 1.4% and 4.7%. The proposed method was applied to 15 samples and the presence of OTA was found in nine samples at concentrations ranging from nd to 2.650 ng g⁻¹.     

Multiplicative posets

A functionf from the posetP to the posetQ is a strict morphism if for allx, y P withx we havef(x). If there is such a strict morphism fromP toQ we writeP Q, otherwise we writeP Q. We say a posetM is multiplicative if for any posetsP, Q withP M andQ M we haveP ×Q M. (Here (p ₁,q ₁)<(p ₂,q ₂) if and only ifp ₁<p ₂ andq ₁<q ₂.) This paper proves that well-founded trees with height are multiplicative posets.This research was supported in part by NSERC Grant #69-1325.     

Orthmodular lattices whose MacNeille completions are not orthomodular

The only known example of an orthomodular lattice (abbreviated: OML) whose MacNeille completion is not an OML has been noted independently by several authors, see Adams [1], and is based on a theorem of Ameniya and Araki [2]. This theorem states that for an inner product space V, if we consider the ortholattice ?(V,⊥) = {A \( \subseteq \) V: A = A ^⊥⊥} where A ^⊥ is the set of elements orthogonal to A, then ?(V,⊥) is an OML if and only if V is complete. Taking the orthomodular lattice L of finite or confinite dimensional subspaces of an incomplete inner product space V, the ortholattice ?(V,⊥) is a MacNeille completion of L which is not orthomodular. This does not answer the longstanding question Can every OML be embedded into a complete OML? as L can be embedded into the complete OML ?(V,⊥), where V is the completion of the inner product space V. Although the power of the Ameniya-Araki theorem makes the preceding example elegant to present, the ability to picture the situation is lost. In this paper, I present a simpler method to construct OMLs whose Macneille completions are not orthomodular. No use is made of the Ameniya-Araki theorem. Instead, this method is based on a construction introduced by Kalmbach [7] in which the Boolean algebras generated by the chains of a lattice are glued together to form an OML. A simple method to complete these OMLs is also given. The final section of this paper briefly covers some elementary properties of the Kalmbach construction. I have included this section because I feel that this construction may be quite useful for many purposes and virtually no literature has been written on it.     

On the support of midconvex operators

Summary LetX be a real vector space,D a convex subset ofX and (Y, K) an order complete ordered vector space. The following sandwich theorem holds: Iff: D Y is midconvex,g: D Y {– } is midconcave andg f onD, then there exists a Jensen mappingh: D Y {– } such thatg h f onD. Using this theorem we show that a mappingf: D Y is midconvex if and only if it has Jensen support at every point ofD. Moreover, ifX is a Baire topological vector space and (Y, K) is an ordered topological vector space satisfying some additional conditions, then a mappingf: D Y is continuous whenever it has continuous Jensen support at every point ofD. As an application of these results we obtain the equality of some set-classes connected with additive and midconvex operators.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth