Récréations mathématiques, géométrie de situation... De nouveaux outils pour enseigner les mathématiques à la fin du XIXe siècle

International audienceAt the end of the XIXth century, some mathematicians like Charles-Ange Laisant (1841-1920) or Édouard Lucas (1842-1891) develop new ideas about teaching mathematics using innovative tools or original visualizations. These ideas are linked to their interest for recreational games viewed through a mathematical background and other specific problems including combinatory, number theory, algorithms and the “géométrie de situation”... They are studied especially in L'Arithmétique amusante (Lucas, 1895) or the Initiation mathématique (Laisant, 1906), two books addressed to educators in a large sense. We propose to give some examples of these mathematical representations and these ingenious games, explicitly presented by their authors in a didactical scheme. Thus, we talk in particular about : - “l'éventail mystérieux” proposed in order to introduce binary computation - “le vol des grues” and other drawings proposed in order to give visual proofs of particular sums of integers - a singular representation of decomposition of integers into prime numbers using paths in specific drawings - graph of functions, especially linear ones, where pupils can see solutions of meetings problems - the use of chessboards in combinatory and so on... Moreover, several of these approaches could be presented nowadays to students of secondary schools. We discuss the role of visualizations through these experimentations and point out the connections between diagrams and symbolisms. We study the need of arousing curiosity through striking results in order to present new mathematical notions. We also underline the goals of representing processes in certain moments of mathematical learning and the generalizations enabled by many of these presentations, much more than simple recreational questions.</p

Empreintes d’échanges au sein de la Société mathématique de France dans les pages de son Bulletin : le cas de Charles-Ange Laisant

La Société mathématique de France (SMF) créée en 1872 pour promouvoir les travaux des mathématiciens français accueille rapidement le polytechnicien Charles-Ange Laisant (1841-1920) qui s’investit par ailleurs dans de multiples communautés savantes à la fin du xixe siècle. Son implication durable, ses itinéraires d’administrateur et d’auteur permettent, à partir des renseignements fournis par le Bulletin de la SMF, une approche originale de la circulation de ses travaux au sein de la Société. Une étude des comptes rendus de séances met en lumière tant sa production d’écrits mathématiques proprement dits que ses interventions orales sur des sujets variés lors de ces réunions. L’exemple de celui qui sera président de la SMF en 1888 offre ainsi une perspective particulière sur les mathématiques discutées lors des réunions de la Société, sur les liens qui se tissent avec d’autres cercles savants ou sur les réseaux et collaborations ponctuelles à l’intérieur même de la SMF.The Société mathématique de France (SMF) was created in 1872 to promote the work of French mathematicians. Charles-Ange Laisant (1841-1920) quickly joined the society while also contributing to the work of many other academic communities in the late nineteenth century. His long-term involvement within the SMF as an administrator or an author enables an original approach to the circulation of his work within the Society based on information given in the SMG Bulletin whereas the minutes published therein shed light on both Laisant’s own mathematical production and his spoken contributions on various topics during the society’s meetings. The example of Laisant, who was elected president of the SMF in 1888, offers an original standpoint on the topics discussed during the society’s meetings, on the relationships established with other scientific networks and on occasional collaborations within the SMF

Dynamic simulation of the THAI heavy oil recovery process

Toe-to-Heel Air Injection (THAI) is a variant of conventional In-Situ Combustion (ISC) that uses a horizontal production well to recover mobilised partially upgraded heavy oil. It has a number of advantages over other heavy oil recovery techniques such as high recovery potential. However, existing models are unable to predict the effect of the most important operational parameters, such as fuel availability and produced oxygen concentration, which will give rise to unsafe designs. Therefore, we have developed a new model that accurately predicts dynamic conditions in the reservoir and also is easily scalable to investigate different field scenarios. The model used a three component direct conversion cracking kinetics scheme, which does not depend on the stoichiometry of the products and, thus, reduces the extent of uncertainty in the simulation results as the number of unknowns is reduced. The oil production rate and cumulative oil produced were well predicted, with the latter deviating from the experimental value by only 4%. The improved ability of the model to emulate real process dynamics meant it also accurately predicted when the oxygen was first produced, thereby enabling a more accurate assessment to be made of when it would be safe to shut-in the process, prior to oxygen breakthrough occurring. The increasing trend in produced oxygen concentration following a step change in the injected oxygen rate by 33 % was closely replicated by the model. The new simulations have now elucidated the mechanism of oxygen production during the later stages of the experiment. The model has allowed limits to be placed on the air injection rates that ensure stability of operation. Unlike previous models, the new simulations have provided better quantitative prediction of fuel laydown, which is a key phenomenon that determines whether, or not, successful operation of the THAI process can be achieved. The new model has also shown that, for completely stable operation, the combustion zone must be restricted to the upper portion of the sand pack, which can be achieved by using higher producer back pressure

LIPH Expression in Skin and Hair Follicles of Normal Coat and Rex Rabbits

Natural mutations in the LIPH gene were shown to be responsible for hair growth defects in humans and for the rex short hair phenotype in rabbits. In this species, we identified a single nucleotide deletion in LIPH (1362delA) introducing a stop codon in the C-terminal region of the protein. We investigated the expression of LIPH between normal coat and rex rabbits during critical fetal stages of hair follicle genesis, in adults and during hair follicle cycles. Transcripts were three times less expressed in both fetal and adult stages of the rex rabbits than in normal rabbits. In addition, the hair growth cycle phases affected the regulation of the transcription level in the normal and mutant phenotypes differently. LIPH mRNA and protein levels were higher in the outer root sheath (ORS) than in the inner root sheath (IRS), with a very weak signal in the IRS of rex rabbits. In vitro transfection shows that the mutant protein has a reduced lipase activity compared to the wild type form. Our results contribute to the characterization of the LIPH mode of action and confirm the crucial role of LIPH in hair production

<i>LIPH</i> expression at three different fetal stages (days 19, 25 and 29) corresponding respectively to the formation of the central primary, lateral primary and secondary hair follicles.

<p>(Bars = 20 µm). (A) Q-PCR expression of <i>LIPH</i> mRNA in the <i>orylag®</i> rabbit skin (light blue bars) and in the normal rabbit skin (dark blue bars). The Y axis represents the relative expression level of <i>LIPH</i>. (B) Histology (ROAN staining) of <i>orylag®</i> (B1) and normal rabbit skins (B2). (C) Immunohistochemistry of <i>orylag®</i> (C1) and normal rabbit skins (C2).</p

<i>LIPH</i> expression in adult <i>orylag®+, orylag®−,</i> rex, heterozygous and normal rabbit skins.

<p>(Bars = 20 µm). (A) Q-PCR expression of <i>LIPH</i> mRNA. The Y axis represents the relative expression level of <i>LIPH</i>. (B) <i>In situ</i> hybridization. (C) Immunohistochemistry.</p

Primer pairs used for PCR amplification, qPCR and ISH.

<p>Primer pairs used for PCR amplification, qPCR and ISH.</p

<i>LIPH</i> expression in <i>orylag®</i> and heterozygous skin of adult synchronized rabbits at the anagen and catagen/telogen follicle cycle stages.

<p>(Bars = 20 µm). (A) Q-PCR expression of <i>LIPH</i> mRNA. The Y axis represents the relative expression level of <i>LIPH</i>. (B) <i>In situ</i> hybridization. (C) Immunohistochemistry.</p