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.     

A fractional step method for 2D parabolic convection-diffusion singularly perturbed problems: uniform convergence and order reduction

In this work, we are concerned with the efficient resolution of two dimensional parabolic singularly perturbed problems of convection-diffusion type. The numerical method combines the fractional implicit Euler method to discretize in time on a uniform mesh and the classical upwind finite difference scheme, defined on a Shishkin mesh, to discretize in space. We consider general time-dependent Dirichlet boundary conditions, and we show that classical evaluations of the boundary conditions cause an order reduction in the consistency of the time integrator. An appropriate correction for the evaluations of the boundary data permits to remove such order reduction. Using this correction, we prove that the fully discrete scheme is uniformly convergent of first order in time and of almost first order in space. Some numerical experiments, which corroborate in practice the robustness and the efficiency of the proposed numerical algorithm, are shown; from them, we bring to light the influence in practice of the two options for the boundary data considered here, which is in agreement with the theoretical results.     

On the mean value parametrization of natural exponential families — a revisited review

It is well known that any natural exponential family (NEF) is characterized by its variance function on its mean domain, often much simpler than the corresponding generating probability measures. The mean value parametrization appeared to be crucial in some statistical theory, like in generalized linear models, exponential dispersion models and Bayesian framework. The main aim of the paper is to expose the mean value parametrization for possible statistical applications. The paper presents an overview of the mean value parametrization and of the characterization property of the variance function for NEF’s. In particular it introduces the relationships existing between the NEF’s generating measure, Laplace transform and variance function as well as some supplemental results concerning the mean value representation. Some classes of polynomial variance functions are revisited for illustration. The corresponding NEF’s of such classes are generated by counting probabilities on the nonnegative integers and provide Poisson-overdispersed competitors to the homogeneous Poisson distribution.     

Computer-Assisted Analysis of the Anderson–Hájek Ontological Controversy

A universal reasoning approach based on shallow semantical embeddings of higher-order modal logics into classical higher-order logic is exemplarily employed to analyze several modern variants of the ontological argument on the computer. Several novel findings are reported which contribute to the clarification of a long-standing dispute between Anderson and Hájek. The technology employed in this work, which to some degree realizes Leibniz’s dream of a characteristica universalis and a calculus ratiocinator for solving philosophical controversies, is ready to be fruitfully adopted in larger scale by philosophers.     

Antimicrobial Activity of Pyrazinamide Coordination Frameworks Synthesized by Mechanochemistry

The urge for the development of a more efficient antibiotic crystalline forms led us to the disclosure of new antibiotic coordination frameworks of pyrazinamide, a well-known drug used for the treatment of tuberculosis, with some of the novel compounds unravelling improved antimycobacterial activity. Mechanochemistry was the preferred synthetic technique to yield novel compounds, allowing the reproduction of a 1D zinc framework, the synthesis of a novel hydrogen bonding manganese framework, and three new compounds with silver. The structural characterization of the novel forms is presented along with stability studies. The increased antimicrobial activity of the new silver-based frameworks against Escherichia coli, Staphylococcus aureus, and Mycobacterium smegmatis is particularly relevant.     

Nanocomposite Polymer Electrolytes of Sodium Alginate and Montmorillonite Clay

Nanocomposite polymer electrolytes (NPEs) were synthesized using sodium alginate (Alg) and either sodium (SCa-3-Na⁺)- or lithium (SCa-3-Li⁺)-modified montmorillonite clays. The samples were characterized by structural, optical, and electrical properties. SCa-3-Na⁺ and SCa-3-Li⁺ clays’ X-ray structural analyses revealed peaks at 2θ = 7.2° and 6.7° that corresponded to the interlamellar distances of 12.3 and 12.8 Å, respectively. Alg-based NPEs X-ray diffractograms showed exfoliated structures for samples with low clay percentages. The increase of clay content promoted the formation of intercalated structures. Electrochemical Impedance Spectroscopy revealed that Alg-based NPEs with 5 wt% of SCa-3-Na⁺ clay presented the highest conductivity of 1.96 × 10⁻² S/cm², and Alg with 10 wt% of SCa-3-Li⁺ showed conductivity of 1.30 × 10⁻² S/cm², both measured at 70 °C. From UV-Vis spectroscopy, it was possible to infer that increasing concentration of clay promoted a decrease of the samples’ transmittance and, consequently, an increase of their reflectance.     

Enantioselective Total Synthesis of (+)-Nordasycarpidone, (+)-Dasycarpidone,and (+)-Uleine

The structure of uleine type alkaloids is characterized by the presence of a bridged tetracyclic hexahydro-1H-1,5-methanoazocino[4,3-b]indole ring system 1 . Various strategies have been developed to access this polycyclic structural motif. We report herein a one-step conversion of appropriately functionalized 1,3,4-trisubstituted cyclopent-1-ene to 1 by way of an integrated oxidation/reduction/cyclization (iORC) process. This domino sequence, initiated by oxidative cleavage of cyclopentene ring, generated subsequently a cyclohexenone, an indole and a 1,3-bridged piperidine ring through formation of one C−C and two C−N bonds. Compound 1 is subsequently converted to nordasycarpidone, dasycarpidone and uleine. The chirality of the molecule was introduced by enzymatic desymmetrization of commercially available meso cis-3,5-diacetoxy-1-cyclopentene.     

Secondary Metabolites of Clinacanthus nutans

Chemistry of Natural Compounds -