Teaching linear algebra at university

Linear algebra represents, with calculus, the two main mathematical subjects taught in science universities. However this teaching has always been difficult. In the last two decades, it became an active area for research works in mathematics education in several countries. Our goal is to give a synthetic overview of the main results of these works focusing on the most recent developments. The main issues we will address concern: the epistemological specificity of linear algebra and the interaction with research in history of mathematics; the cognitive flexibility at stake in learning linear algebra; three principles for the teaching of linear algebra as postulated by G. Harel; the relation between geometry and linear algebra; an original teaching design experimented by M. Rogalsk

Effective Theory of Magnetization Plateaux in the Shastry-Sutherland Lattice

We use the non-perturbative Contractor-Renormalization method (CORE) in order to derive an effective model for triplet excitations on the Shastry-Sutherland lattice. For strong enough magnetic fields, various magnetization plateaux are observed, e.g. at 1/8, 1/4, 1/3 of the saturation, as found experimentally in a related compound. Moreover, other stable plateaux are found at 1/9, 1/6 or 2/9. We give a critical review of previous works and try to resolve some apparent inconsistencies between various theoretical approaches.Comment: published version with minor change

Theory of magnetization plateaux in the Shastry-Sutherland model

Using perturbative continuous unitary transformations, we determine the long-range interactions between triplets in the Shastry-Sutherland model, and we show that an unexpected structure develops at low magnetization with plateaux progressively appearing at 2/9, 1/6, 1/9 and 2/15 upon increasing the inter-dimer coupling. A critical comparison with previous approaches is included. Implications for the compound SrCu$_2$(BO$_3$)$_2$ are also discussed: we reproduce the magnetization profile around localized triplets revealed by NMR, we predict the presence of a 1/6 plateau, and we suggest that residual interactions beyond the Shastry-Sutherland model are responsible for the other plateaux below 1/3.Comment: 5 pages, 6 figure

En guise de conclusion

Synthèse de la Biennale de la recherche en action sociale du 28 novembre 2012. Un regard par "le bord" du travail social Quelques hypothèses sur "les débordements de l'intervention sociale" : le débordement n'estil pas un régime normal du travail social ? De quelle frontières parle-t-on ? Regard porté sur les débat de la journée : Pratiques, mise en récits de pratiques et théorisation des pratiques. La problématicité des frontières. Que/qui faut-il activer

En guise de conclusion

Synthèse de la Biennale de la recherche en action sociale du 28 novembre 2012. Un regard par "le bord" du travail social Quelques hypothèses sur "les débordements de l'intervention sociale" : le débordement n'estil pas un régime normal du travail social ? De quelle frontières parle-t-on ? Regard porté sur les débat de la journée : Pratiques, mise en récits de pratiques et théorisation des pratiques. La problématicité des frontières. Que/qui faut-il activer

The role of transcription factories-mediated interchromosomal contacts in the organization of nuclear architecture

Using numerical simulations, we investigate the underlying physical effects responsible for the overall organization of chromosomal territories in interphase nuclei. In particular, we address the following three questions: (i) why are chromosomal territories with relatively high transcriptional activity on average, closer to the centre of cell's nucleus than those with the lower activity? (ii) Why are actively transcribed genes usually located at the periphery of their chromosomal territories? (iii) Why are pair-wise contacts between active and inactive genes less frequent than those involving only active or only inactive genes? We show that transcription factories-mediated contacts between active genes belonging to different chromosomal territories are instrumental for all these features of nuclear organization to emerge spontaneously due to entropic effects arising when chromatin fibres are highly crowde

Modelling of crowded polymers elucidate effects of double-strand breaks in topological domains of bacterial chromosomes

Using numerical simulations of pairs of long polymeric chains confined in microscopic cylinders, we investigate consequences of double-strand DNA breaks occurring in independent topological domains, such as these constituting bacterial chromosomes. Our simulations show a transition between segregated and mixed state upon linearization of one of the modelled topological domains. Our results explain how chromosomal organization into topological domains can fulfil two opposite conditions: (i) effectively repulse various loops from each other thus promoting chromosome separation and (ii) permit local DNA intermingling when one or more loops are broken and need to be repaired in a process that requires homology search between broken ends and their homologous sequences in closely positioned sister chromati