Determinant of a matrix that commutes with a Jordan matrix

Given a Jordan matrix J, we obtain an explicit formula for the determinant of any matrix T that commutes with it.     

Self‐Assembled Tetragonal Prismatic Molecular Cage Highly Selective for Anionic π Guests

The metal‐directed supramolecular synthetic approach has paved the way for the development of functional nanosized molecules. In this work, we report the preparation of the new nanocapsule 3? (CF₃SO₃)₈ with a A₄B₂ tetragonal prismatic geometry, where A corresponds to the dipalladium hexaazamacrocyclic complex Pd‐1 , and B corresponds to the tetraanionic form of palladium 5,10,15,20‐tetrakis(4‐carboxyphenyl)porphyrin ( 2 ). The large void space of the inner cavity and the supramolecular affinity for guest molecules towards porphyrin‐based hosts converts this nanoscale molecular 3D structure into a good candidate for host–guest chemistry. The interaction between this nanocage and different guest molecules has been studied by means of NMR, UV/Vis, ESI‐MS, and DOSY experiments, from which highly selective molecular recognition has been found for anionic, planar‐shaped π guests with association constants (K_a) higher than 10⁹ M ^?1, in front of non‐interacting aromatic neutral or cationic substrates. DFT theoretical calculations provided insights to further understand this strong interaction. Nanocage 3? (CF₃SO₃)₈ can not only strongly host one single molecule of M(dithiolene)₂ complexes (M=Au, Pt, Pd, and Ni), but also can finely tune their optical and redox properties. The very simple synthesis of both the supramolecular cage and the building blocks represents a step forward for the development of polyfunctional supramolecular nanovessels, which offer multiple applications as sensors or nanoreactors.     

Determination of ageing time of spirits in oak barrels using a headspace–mass spectrometry (HS-MS) electronic nose system and multivariate calibration

The aromatic composition of sugar cane spirits and, in general, of alcoholic beverages, is mainly influenced by the ageing process in wood barrels. There are several factors that affect the quality of the final aged product, but the time of the storage in the barrel is perhaps the most important one. Ageing time must therefore be controlled in order to detect counterfeits; however, this parameter is very difficult to control and, at present, there is no analytical method available to determine it. We propose a quantitative method for determining the ageing time of sugar cane spirits in oak barrels by using an electronic nose based on coupling directly a headspace sampler to a mass spectrometer (HS-MS), and multivariate calibration. The method developed is simple and provides, in 5 min, the ageing time of spirits with an accuracy of about 1 month.     

Bingel-Hirsch reactions on non-IPR Gd3N@C2n (2n = 82 and 84)

The Bingel-Hirsch reactions on non-isolated pentagon rule (non-IPR) Gd(3)N@C(2n) (2n = 82, 84) are studied. Computational results show that the two metallofullerenes display similar reactivity according to their related topologies. Long C-C bonds with large pyramidalization angles lead to the most stable adducts, the [5,6] bonds in the adjacent pentagon pair being especially favored. The lesser regioselectivity observed for Gd(3)N@C(82) is probably due to the activation of some C-C bonds by means of the metal cluster.     

A networking method to compare theories: metacognition in problem solving reformulated within the Anthropological Theory of the Didactic

An important role of theory in research is to provide new ways of conceptualizing practical questions, essentially by transforming them into scientific problems that can be more easily delimited, typified and approached. In mathematics education, theoretical developments around ‘metacognition’ initially appeared in the research domain of Problem Solving closely related to the practical question of how to learn (and teach) to solve non-routine problems. This paper presents a networking method to approach a notion as ‘metacognition’ within a different theoretical perspective, as the one provided by the Anthropological Theory of the Didactic. Instead of trying to directly ‘translate’ this notion from one perspective to another, the strategy used consists in going back to the practical question that is at the origin of ‘metacognition’ and show how the new perspective relates this initial question to a very different kind of phenomena. The analysis is supported by an empirical study focused on a teaching proposal in grade 10 concerning the problem of comparing mobile phone tariffs.     

Development of an analytical methodology to quantify melamine in milk using micellar liquid chromatography and validation according to EU Regulation 2002/654/EC

Melamine is a toxic triazine, illegally used as an additive in milk to apparently increase the amount of protein. A chromatographic procedure using a C18 column and a micellar mobile phase of sodium dodecyl sulphate (0.05 M) and propanol (7.5%), buffered at pH 3, and a detection set by absorbance at 210 nm, was reported for the resolution and quantification of melamine in liquid and powdered milk samples. In this work, samples were diluted with a SDS solution and were directly injected, thus avoiding long extraction and experimental procedures. Melamine was eluted in nearly 9.3 min without overlapping the protein band or other endogeneous compounds. The optimal mobile phase composition was taken using a chemometrical approach that considers the retention factor, efficiency and peak shape. Validation was performed following the European Commission's indications (European Decision 2002/657/EC), and the main analytical parameters studied were: linearity (0.02-100 ppm; r² = 0.999), limit of detection (5 ppb), intra- and inter-day precision (R.S.D. <7.6% and <9.7%, respectively) and robustness (R.S.D. <7.4% for retention time and <5.0% for area). Sensitivity was adequate to detect melamine under the safety limits proposed by the US FDA. Finally, recoveries for several milk samples were found in the 85-109% range.