Mathematical modelling and the learning trajectory: tools to support the teaching of linear algebra

In this article we present a didactic proposal for teaching linear algebra based on two compatible theoretical models: emergent models and mathematical modelling. This proposal begins with a problematic situation related to the creation and use of secure passwords, which leads students toward the construction of the concepts of spanning set and span. The objective is to evaluate this didactic proposal by determining the level of match between the hypothetical learning trajectory (HLT) designed in this study with the actual learning trajectory in the second experimental cycle of an investigation design-based research more extensive. The results show a high level of match between the trajectories in more than half of the conjectures, which gives evidence that the HLT has supported, in many cases, the achievement of the learning objective, and that additionally mathematical modelling contributes to the construction of these linear algebra concepts.     

In‐line preconcentration capillary zone electrophoresis for the analysis of haloacetic acids in water

Two in‐line enrichment procedures (large volume sample stacking (LVSS) and field amplified sample injection (FASI)) have been evaluated for the CZE analysis of haloacetic acids (HAAs) in drinking water. For LVSS, separation on normal polarity using 20 mM acetic acid–ammonium acetate (pH 5.5) containing 20% ACN as BGE was required. For FASI, the optimum conditions were 25 s hydrodynamic injection (3.5 kPa) of a water plug followed by 25 s electrokinetic injection (?10 kV) of the sample, and 200 mM formic acid–ammonium formate buffer at pH 3.0 as BGE. For both FASI and LVSS methods, linear calibration curves (r²>0.992), limit of detection on standards prepared in Milli‐Q water (49.1–200 μg/L for LVSS and 4.2–48 μg/L for FASI), and both run‐to‐run and day‐to‐day precisions (RSD values up to 15.8% for concentration) were established. Due to the higher sensitive enhancement (up to 310‐fold) achieved with FASI‐CZE, this method was selected for the analysis of HAAs in drinking water. However, for an optimal FASI application sample salinity was removed by SPE using Oasis WAX cartridges. With SPE‐FASI‐CZE, method detection limits in the range 0.05–0.8 μg/L were obtained, with recoveries, in general, higher than 90% (around 65% for monochloroacetic and monobromoacetic acids). The applicability of the SPE‐FASI‐CZE method was evaluated by analyzing drinking tap water from Barcelona where seven HAAs were found at concentration levels between 3 and 13 μg/L.     

Protonation of water clusters induced by hydroperoxyl radical surface adsorption

We have investigated the HO(2) adsorption and acid dissociation process on the surface of (H(2)O)(20) and (H(2)O)(21) clusters by using quantum-chemistry calculations. Our results show that the radical forms a stable hydrogen-bond complex on the cluster. The HO(2) acid dissociation is more favorable in the case of the (H(2)O)(21) cluster, for which the inner water molecule plays a crucial role. In fact, acid dissociation of HO(2) is found to occur in two steps. The first step involves H(2) O autoionization in the cluster, and the second one involves the proton transfer from the HO(2) radical to the hydroxide anion. The presence of the HO(2) radicals on the surface of the cluster facilitates water autoionization in the cluster.     

Reactivity of metallic nitride endohedral metallofullerene anions: electrochemical synthesis of a Lu3N@Ih-C80 derivative

Electrochemically generated Lu(3)N@I(h)-C(80) dianions react with the electrophile, PhCHBr(2), to produce a methano derivative of Lu(3)N@I(h)-C(80)(CHC(6)H(5)) (1) with high regioselectivity. The compound was characterized by MALDI-TOF, NMR, and UV-vis-NIR absorption spectroscopy. Electrochemical characterization of this Lu(3)N@I(h)-C(80)(CHC(6)H(5)) derivative showed the typical irreversible reductive behavior of the pristine Lu(3)N@I(h)-C(80), similar to those observed for Bingel adducts of Lu(3)N@I(h)-C(80). Using the same conditions, the reaction between Sc(3)N@I(h)-C(80) dianions and PhCHBr(2) was conducted for comparison. Unexpectedly, no nucleophilic reaction was observed, indicating that Sc(3)N@I(h)-C(80) dianions are not reactive toward the electrophile. Theoretical studies for both dianionic Lu(3)N@I(h)-C(80) and Sc(3)N@I(h)-C(80) showed that the HOMO is more highly localized on the fullerene cage for [Lu(3)N@I(h)-C(80)](2-) and more localized on the inside cluster for [Sc(3)N@I(h)-C(80)](2-), providing an explanation for the drastically different reactivities observed.     

Development and validation of micellar liquid chromatographic methods for the determination of antibiotics in different matrixes

Antibiotics are the most important bioactive and chemotherapeutic compounds to be produced by microbiological synthesis, and they have proved their worth in a variety of fields, such as medicinal chemistry, agriculture, and the food industry. Interest in antibiotics has grown in parallel with an increasingly high degree of productivity in the field of analytical applications. Therefore, it is necessary to develop chromatographic procedures capable of determining various drugs simultaneously in the shortest possible time. Micellar liquid chromatography (MLC) is an RP-HPLC technique that offers advantages over conventional HPLC as far as sample preparation, selectivity, and versatility are concerned. Its main advantage is that samples can be injected directly into the chromatographic system with no previous preparation step. This paper mainly focuses on the results of the authors' own recent research and reports the chromatographic conditions for determination of various antibiotics (penicillins, quinolones, and sulfonamides) in different matrixes (pharmaceuticals, biological fluids, and food). The work of other authors on MLC-based antibiotic determination has been included.     

Translocation versus cyclisation in radicals derived from N-3-alkenyl trichloroacetamides

Under radical reaction conditions, two different and competitive reaction pathways were observed for N-(α-methylbenzyl)trichloroacetamides with a N-3-cyclohexenyl substituent: 1,4-hydrogen translocation and radical addition to a double bond. However, for radicals with an acyclic alkenyl side chain, the direct cyclisation process was exclusively observed. The dichotomy between translocation and direct radical cyclisation in these substrates has been theoretically studied using density functional theory (DFT) methods at the B3LYP/6-31G** computational level.     

Exploring the accuracy level of new potential energy surfaces for the F + HD reactions: from exact quantum rate constants to the state-to-state reaction dynamics

Exact quantum reactive scattering calculations in the collision energy range 1-250 meV have been carried out for both the isotopic product channels of the title system. The dynamical studies compares an ab initio potential energy surface (PES) recently appeared in the literature (J. Chem. Phys., 2008, 129, 011103) with other phenomenological PESs. Vibrational branching ratios, cross sections and rate constants are presented and compared with molecular beam scattering experiments as well as with chemical kinetics data. In particular, the agreement of the vibrational branching ratios with experimental measurements is improved with respect to previous studies on other PESs, mainly because of the presence of a broad peak in the HF(v' = 3) integral cross section completely absent in the previous simulations. This feature, observed by molecular beam experiments, is the fingerprint of a new reaction mechanism operative in the dynamics described by the new PES. A conjecture for its origin, able to explain many of its characteristic aspects, is analyzed and discussed.     

Controlling the length and location of in situ formed nanowires by means of microfluidic tools

Progress in microelectronics, sensors and optics is strongly dependent on the miniaturization of components, and the integration of nanoscale structures into applicable systems. In this regard, conventional top-down technologies such as lithography have limits concerning the dimensions and the choice of material. Therefore, several bottom-up approaches have been investigated to satisfy the need for structures with large aspect ratios in the nanometre regime. For further implementation, however, it is crucial to find methods to define position, orientation and length of the nanowires. In this study, we present a microchip to trap in situ formed bundles of nanowires in microsized cages and clamps, thereby enabling immobilisation, positioning and cutting-out of desired lengths. The microchip consists of two layers, one of which enables the formation of metal-organic nanowires at the interface of two co-flowing laminar streams. The other layer, separated by a thin and deflectable PDMS membrane, serves as the pneumatic control layer to impress microsized features ("donuts") onto the nanowires. In this way, a piece of the nanowire bundle with a prescribed length is immobilised inside the donut. Furthermore, partly open ring-shaped structures enabled trapping of hybrid wires and subsequent functionalisation with fluorescent beads. We believe that the method is a versatile approach to form and modify nanoscale structures via microscale tools, thereby enabling the construction of fully functional nanowire-based systems.     

A comprehensive dynamic model for class-1 tensegrity systems based on quaternions

In this paper we propose a new dynamic model, based on quaternions, for tensegrity systems of class-1. Quaternions are used to represent orientations of a rigid body in the 3-dimensional space eliminating the problem of singularities. Moreover, the equations based on quaternions allow to perform more precise calculations and simulations because they do not use trigonometric functions for the representation of angles. We present a thorough introduction of tensegrities and the current state of research. We also introduce the quaternions and provide in the appendix some important details and useful properties. Applying the Euler–Lagrange approach we derive a comprehensive dynamic model, first for a simple rigid bar in the space and, at last, for a class-1 tensegrity system. We present two model forms: a matrix and a vectorial form. The first more compact and easier to write, the latter more suitable to apply the tools and the theory based on vector fields.