Mathematics learning and tools from theoretical, historical and practical points of view: the productive notion of mathematics laboratories

In our research work, we have looked at the way in which artefacts become, for teachers as well as for students, instruments of their mathematical activity. The issues related to the use of tools and technologies in mathematical education are now widely considered. A look to history highlights the different ways in which the same questions have been studied at different times and in different places. This suggests that the contribution of artefacts to mathematics learning should be considered in terms of various contexts. Our “visits” to these contexts will be guided by the coordination of two main theoretical frameworks, the instrumental approach and the semiotic mediation approach from the perspective of mathematics laboratory. This journey through history and schooling represents a good occasion to address some questions: Are there “good” contexts in which to develop mathematical instruments? Are there “good” teaching practices which assist students’ instrumental geneses and construct mathematical meanings? How is it possible to promote such teaching practices? Some study cases are discussed.     

A Simple Product Formula for Certain Kazhdan-Lusztig R-Polynomials

We give a new characterization of the analytic spread of ideals in local rings with infinite residue fields. This new characterization answers a question of Huneke and can simplify Huneke's constructions in [1]     

Investigation of free and conjugated seleno‐amino acids in wheat bran by hydrophilic interaction liquid chromatography with tandem mass spectrometry

An analytical method for determining seleno‐methionine, methyl‐seleno‐cysteine, and seleno‐cystine in wheat bran was developed and validated. Four different extraction procedures were evaluated to simultaneously extract endogenous free and conjugated seleno‐amino acids in wheat bran in order to select the best extraction protocol in terms of seleno amino acid quantitation. The extracted samples were subjected to a clean‐up by a reversed phase/strong cation exchange solid‐phase extraction and analyzed by chiral hydrophilic interaction liquid chromatography‐tandem mass spectrometry. The optimized extraction protocol was employed to validate the methodology. Process efficiency ranged from 58 to 112% and trueness from 73 to 98%. Limit of detection and limit of quantification were lower than 1 ng/g. Four wheat bran samples were analyzed for both total Se and single seleno‐amino acids determination. The results showed that Se‐ seleno‐methyl‐l selenocysteine was the major seleno‐amino acid in wheat bran while seleno‐methionine and seleno‐cysteine were both minor species.     

(−)-Quinic acid: a versatile precursor for the synthesis of analogues of 2-crotonyloxymethyl-(4R,5R,6R)-4,5,6-trihydroxycyclohex-2-enone (COTC) which possess anti-tumour properties

Syntheses of three novel analogues of the Streptomyces metabolite COTC are described, using the versatile chiral pool starting material, (−)-quinic acid. The results of bioassays of the target compounds against two lung cancer cell lines, A549 and H460, are presented.     

Heegner divisors in the moduli space of genus three curves

S. Kondo used periods of surfaces to prove that the moduli space of genus three curves is birational to an arithmetic quotient of a complex 6-ball. In this paper we study Heegner divisors in the ball quotient, given by arithmetically defined hyperplane sections of the ball. We show that the corresponding loci of genus three curves are given by hyperelliptic curves, singular plane quartics and plane quartics admitting certain rational ``splitting curves'.

     

Graphic Calculators and Micro-Straightness: Analysis of a Didactic Engineering

This paper concerns the analysis of a didactic engineering, the aim of which is to introduce Calculus, at secondary-school level, through the relationship between global and local points of view. It was designed for a graphic–symbolic calculator environment and structured in accordance with a learning trajectory from identifying the graphical phenomenon of local linearity to its mathematical formulation. This learning trajectory involves the reconstruction of the relationship with the tangent line to a curve at a chosen point. The analysis shows the use of different semiotic systems in order to grasp this phenomenon and construct its mathematical meaning.

Accurate long-range distance measurements in a doubly spin-labeled protein by a four-pulse, double electron-electron resonance method

Distance determination in disordered systems by a four-pulse double electron-electron resonance method (DEER or PELDOR) is becoming increasingly popular because long distances (several nanometers) and their distributions can be measured. From the distance distributions eventual heterogeneities and dynamics can be deduced. To make full use of the method, typical distance distributions for structurally well-defined systems are needed. Here, the structurally well-characterized protein azurin is investigated by attaching two (1-oxyl-2,2,5,5-tetramethylpyrroline-3-methyl) methanethiosulfonate spin labels (MTSL) by site-directed mutagenesis. Mutations at the surface sites of the protein Q12, K27, and N42 are combined in the double mutants Q12C/K27C and K27C/N42C. A distance of 4.3 nm is found for Q12C/K27C and 4.6 nm for K27C/N42C. For Q12C/K27C the width of the distribution (0.24 nm) is smaller than for the K27C/N42C mutant (0.36 nm). The shapes of the distributions are close to Gaussian. These distance distributions agree well with those derived from a model to determine the maximally accessible conformational space of the spin-label linker. Additionally, the expected distribution for the shorter distance variant Q12C/N42C was modeled. The width is larger than the calculated one for Q12C/K27C by 21%, revealing the effect of the different orientation and shorter distance. The widths and the shapes of the distributions are suited as a reference for two unperturbed MTSL labels at structurally well-defined sites.     

Modeling the resting state of oxalate oxidase and oxalate decarboxylase enzymes

In view of the biological and commercial interest in models for Oxalate Decarboxylases (OxDC) and Oxalate Oxidases (OxOx), we have synthesized and characterized three new Mn (II) complexes ( 1- 3) employing N3O-donor amino-carboxylate ligands (TCMA, 1,4,7-triazacyclononane- N-acetic acid; K (i) Pr 2TCMA, potassium 1,4-diisopropyl-1,4,7-triazacyclononane- N-acetate; and KBPZG, potassium N,N-bis(3,5-dimethylpyrazolyl methyl)glycinate). These complexes were characterized by several techniques including X-ray crystallographic analysis, X-band electron paramagnetic resonance (EPR), electrospray ionization mass spectrometry (ESI-MS), and cyclic voltammetry. The crystal structures of 1 and 3 revealed that both form infinite polymeric chains of Mn (II) complexes linked by the pendant carboxylate arms of the TCMA (-) and the BPZG (-) ligands in a syn-antipattern. Complex 2 crystallizes as a mononuclear Mn (II) cation, six-coordinate in a distorted octahedral geometry. Although complexes 1 and 3 crystallize as polymeric chains, all compounds present the same N3O-donor set atoms around the metal center as observed in the crystallographically characterized OxDC and OxOx. Moreover, complex 2 also contains two water molecules coordinated to the Mn center as observed in the active site of OxDC and OxOx. ESI-MS spectrometry, combined with EPR, were useful techniques to establish that complexes 1- 3 are present as mononuclear Mn (II) species in solution. Finally, complexes 1- 3 are able to model the resting state active sites, with special attention focused on complex 2 which provides the first exact first coordination sphere ligand structural model for the resting states of both OxDC and OxOx.