On the equipartition of plane convex bodies and convex polygons

In this paper we consider the problem of partitioning a plane compact convex body into equal-area parts, i.e., an equipartition, by means of chords. We prove two basic results that hold with some specific exceptions: (a) When chords are pairwise non-crossing, the dual tree of the partition has to be a path, (b) A convex n-gon admits no equipartition produced by more than n chords having a common interior point.     

Carbosilane dendrons as stabilizing agents for the formation of gold nanoparticles

The synthesis and characterization of two new carbosilane dendrons functionalized in the focal point with a single thiol group is described. These molecules were used as stabilizing agents for the formation of gold nanoparticles. These materials were obtained by the reduction of hydrogen tetrachloroaurate phase-transferred into toluene in the presence of the carbosilane dendrons. The gold nanoparticle-cored dendrons (NCDs) were characterized by TEM, XPS, AFM, IR, UV, and NMR techniques. Average particle diameter ranged from 2.5 to 2.6 nm with narrow size distribution and remarkable stability; in particular, the elimination of the protecting shells to give naked gold, after NCDs deposition on a support, is more difficult using carbosilane dendrons than in the case of conventional dodecanethiol-protected gold nanoparticles.     

Non-crossing matchings of points with geometric objects

Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a non-crossing matching between point–object pairs. In this paper, we address the algorithmic problem of determining whether a non-crossing matching exists between a given point–object pair. We show that when the objects we match the points to are finite point sets, the problem is NP-complete in general, and polynomial when the objects are on a line or when their size is at most 2. When the objects are line segments, we show that the problem is NP-complete in general, and polynomial when the segments form a convex polygon or are all on a line. Finally, for objects that are straight lines, we show that the problem of finding a min-max non-crossing matching is NP-complete.     

An experimental limit on the decayK +→μ+ v e

We have searched for the decay modeK ⁺→μ⁺ v _e by looking for thev _e interactions in a neon/hydrogen bubble chamber. The observed events with electrons are consistent with being produced byv _e from the decaysK ⁺→π⁰ e ⁺ v _e . We set a 90% confidence level upper limit of 0.4% for the μ⁺ v _e decay mode of positive kaons.     

X-ray diffraction structures of two N-salicylidene tryptophananato diaquocopper(II) complexes: erythro and threo isomers

X-ray crystal structures are reported for two novel N-salicylidene tryptophanato diaquocopper(II) isomers, [Cu(Sal-Trp)(h₂O)₂, erythro (1) and hreo (2). The coordination geometry about the copper in both complexes is approximately square-pyramidal with the tridentate Sal-Trp Schiff base ligand and the oxygen atom O(W1) of one water molecule occupying the corners of a square. The coordination sphere about the copper is completed by an axial OW(2) water molecule. In 1, O(W2) [Cu---O(W2) distance 2.25(1) Å] is at the same side of the indole ring (syn disposition—“erythro” isomer), whereas in 2 the O(W2) and the α-amino acid chain are disposed anti (“threo” isomer), with a longer Cu---O(W2) distance, 2.485 Å. IR, ESR, electronic spectral data and magnetic properties are discussed and related to the copper binding mode derived from the crystal structure determination.     

Chlorotrimethylsilane, a reagent for the direct quantitative analysis of fats and oils present in vegetable and meat samples

Acylglycerides present in oil seeds and meat can be transformed into volatile fatty esters using chlorotrimethylsilane (CTMS) and 1-pentanol as reagents. The volatile esters can then be analysed by GC. The method is quantitative and involves only minor sample manipulation. It often permits major recoveries of the total saponifiable lipids present in solid samples. A 40 min reaction time is enough to ensure the total conversion of saponifiable lipids to the corresponding FAPEs.