Optimal partitions having disjoint convex and conic hulls

LetA ¹,,A ⁿ be distinctk-dimensional vectors. We consider the problem of partitioning these vectors intom sets so as to maximize an objective which is a quasi-convex function of the sum of vectors in each set. We show that there exists an optimal partition whose sets have (pairwise) disjoint conic hulls. We also show that if the number of vectors in each of the sets is constrained, then a weaker conclusion holds, namely, there exists an optimal partition whose sets have (pairwise) disjoint convex hulls. The results rely on deriving necessary and sufficient conditions for a point to be an extreme point of a corresponding polytope.Research of this author was partially supported by NSF Grant ECS-83-10213 and by a Grant for the Promotion of Research at the Technion.     

Benzothiazines in synthesis. Toward the synthesis of pseudopteroxazole

[reaction: see text] The tricyclic benzothiazine 15 was prepared in a straightforward fashion via a completely stereoselective intramolecular Friedel-Crafts alkylation. This compound represents a potential precursor to the antitubercular agent, pseudopteroxazole. Its synthesis proceeded via a completely selective, intramolecular addition of a sulfoximine-stabilized carbanion to an alpha,beta-unsaturated ester, followed by functional group manipulations.     

The use of inverse gas chromatography to study surface thermal oxidation of polypropylene

Thermo-oxidative effects on the surface energy of polypropylene were measured by inverse gas chromatography as a function of exposure time and temperature. Unaltered polypropylene had a surface energy of 33 mJ/m². Oxidized polypropylene, after exposure to air at temperatures of 100 °C and 110 °C, had a range of maximum surface energies from 38 to 41 mJ/m². Comparisons between FTIR carbonyl peak growth and the surface energy showed that both methods detect oxidation, though the increase in surface energy is detected before the carbonyl peak growth is noticeable. The work of adhesion predicted by the surface free energies obtained in this work between a coated calcium carbonate and polypropylene changes by 10% due to the oxidation of the polymer at 110 °C.     

Filling in the gaps: modelling incomplete CBL data using a graphing calculator

This article discusses the real-world problem-solving lesson that emerged when a high school math teacher used a motion detector with a CBL and graphing calculator to obtain the 'bounce' data of a ping-pong ball. While practising the 'bounce' data collection--a series of diminishing parabolas--the teacher accidentally pulled the motion detector away and then, realizing his mistake, pulled it back. The resulting data showed a series of parabolas, but one was missing. The teacher used this opportunity to create a lesson in which his students collect 'bad data' and then fill in the missing parabola using critical components of parabolas, such as the latus rectum and the vertex, and using matrices. The article provides all the necessary directions, formulas, and names of resources needed to replicate the lesson. The creation of this lesson demonstrates that a serendipitous error can create a genuine and authentic problem-solving activity for math students.     

A moving mesh algorithm for aero‐thermo‐mechanical modelling in turbomachinery

This paper describes the development of a mesh deformation method used for aero‐thermo‐mechanical coupling of turbo‐engine components. The method is based on the nonlinear solution of an elastic medium analogy, solved using finite element discretisation and modified to let the boundary nodes be free to slide over the deflected surfaces. This sliding technique relies on a B‐spline reconstruction of the moving boundary and increases the robustness of the method in situations where the boundary deflection field presents significant gradients or large relative motion between two distinct boundaries. The performance of the method is illustrated with the application to an interstage cavity of a turbine assembly, subjected to the deformations computed by a coupled thermo‐mechanical analysis of the engine component. Copyright © 2011 John Wiley & Sons, Ltd.     

Differentiating Isobaric Steroid Hormone Metabolites Using Multi-Stage Tandem Mass Spectrometry

Steroid hormones and their metabolites are currently undergoing clinical trials as potential therapeutics for traumatic brain injury (TBI). To support this work, it is necessary to develop improved procedures for differentiating isobaric species in this compound class. Equilin sulfate (E-S), estrone sulfate (E1-S), 17α-dihydroequilin sulfate (ADHE-S), and 17β-dihydroequilin sulfate (BDHE-S) are primary constituents in hormone replacement therapies, such as Premarin, which are among pharmaceuticals being investigated for TBI treatment. The latter three compounds are isomers and can be difficult to differentiate in trace analytical determinations. In this work, a systematic study of the fragmentation of ADHE-S, BDHE-S, E1-S, and E-S under different stages of higher order tandem mass spectrometry (MSⁿ) and variation of collision energy, allowed optimization of conditions for distinguishing the isomeric structures. For epimeric variants (e.g., ADHE-S versus BDHE-S; α- versus β-stereoisomerization in the C-17 position), differentiation was achieved at MS⁴ and fragmentation was demonstrated through MS⁵. Computational analysis was performed to further explore differences in the fragmentation pathways due to changes in stereochemistry.