Zig-zag instability of an Ising wall in liquid crystals

We present a theoretical explanation for the interfacial zigzag instability that appears in anisotropic systems. Such an instability has been experimentally highlighted for an Ising wall formed in a nematic liquid crystal cell under homeotropic anchoring conditions. From an envelope equation, relevant close to the Freedericksz transition, we have derived an asymptotic equation describing the interface dynamics in the vicinity of its bifurcation. The asymptotic limit used accounts for a strong difference between two of the elastic constants. The model is characterized by a conservative order parameter which satisfies a Cahn-Hilliard equation. It provides a good qualitative understanding of the experiments.Comment: 4 pagess, 4 figures, lette

Single polymer dynamics: coil-stretch transition in a random flow

By quantitative studies of statistics of polymer stretching in a random flow and of a flow field we demonstrate that the stretching of polymer molecules in a 3D random flow occurs rather sharply via the coil-stretch transition at the value of the criterion close to theoretically predicted.Comment: 4 pages, 5 figure

Ring-Pattern Dynamics in Smectic-C* and Smectic-C_A* Freely Suspended Liquid Crystal Films

Ring patterns of concentric 2pi-solitons in molecular orientation, form in freely suspended chiral smectic-C films in response to an in-plane rotating electric field. We present measurements of the zero-field relaxation of ring patterns and of the driven dynamics of ring formation under conditions of synchronous winding, and a simple model which enables their quantitative description in low polarization DOBAMBC. In smectic C_A* TFMHPOBC we observe an odd-even layer number effect, with odd number layer films exhibiting order of magnitude slower relaxation rates than even layer films. We show that this rate difference is due to much larger spontaneous polarization in odd number layer films.Comment: 4 RevTeX pgs, 4 eps figures, submitted to Phys. Rev. Let

Coalescence in the 1D Cahn-Hilliard model

We present an approximate analytical solution of the Cahn-Hilliard equation describing the coalescence during a first order phase transition. We have identified all the intermediate profiles, stationary solutions of the noiseless Cahn-Hilliard equation. Using properties of the soliton lattices, periodic solutions of the Ginzburg-Landau equation, we have construct a family of ansatz describing continuously the processus of destabilization and period doubling predicted in Langer's self similar scenario

‘Warrant’ revisited: Integrating mathematics teachers’ pedagogical and epistemological considerations into Toulmin’s model for argumentation

In this paper, we propose an approach to analysing teacher arguments that takes into account field dependence—namely, in Toulmin’s sense, the dependence of warrants deployed in an argument on the field of activity to which the argument relates. Freeman, to circumvent issues that emerge when we attempt to determine the field(s) that an argument relates to, proposed a classification of warrants (a priori, empirical, institutional and evaluative). Our approach to analysing teacher arguments proposes an adaptation of Freeman’s classification that distinguishes between: epistemological and pedagogical a priori warrants, professional and personal empirical warrants, epistemological and curricular institutional warrants, and evaluative warrants. Our proposition emerged from analyses conducted in the course of a written response and interview study that engages secondary mathematics teachers with classroom scenarios from the mathematical areas of analysis and algebra. The scenarios are hypothetical, grounded on seminal learning and teaching issues, and likely to occur in actual practice. To illustrate our proposed approach to analysing teacher arguments here, we draw on the data we collected through the use of one such scenario, the Tangent Task. We demonstrate how teacher arguments, not analysed for their mathematical accuracy only, can be reconsidered, arguably more productively, in the light of other teacher considerations and priorities: pedagogical, curricular, professional and personal

Elastic turbulence in curvilinear flows of polymer solutions

Following our first report (A. Groisman and V. Steinberg, \sl Nature $\bf 405$, 53 (2000)) we present an extended account of experimental observations of elasticity induced turbulence in three different systems: a swirling flow between two plates, a Couette-Taylor (CT) flow between two cylinders, and a flow in a curvilinear channel (Dean flow). All three set-ups had high ratio of width of the region available for flow to radius of curvature of the streamlines. The experiments were carried out with dilute solutions of high molecular weight polyacrylamide in concentrated sugar syrups. High polymer relaxation time and solution viscosity ensured prevalence of non-linear elastic effects over inertial non-linearity, and development of purely elastic instabilities at low Reynolds number (Re) in all three flows. Above the elastic instability threshold, flows in all three systems exhibit features of developed turbulence. Those include: (i)randomly fluctuating fluid motion excited in a broad range of spatial and temporal scales; (ii) significant increase in the rates of momentum and mass transfer (compared to those expected for a steady flow with a smooth velocity profile). Phenomenology, driving mechanisms, and parameter dependence of the elastic turbulence are compared with those of the conventional high Re hydrodynamic turbulence in Newtonian fluids.Comment: 23 pages, 26 figure

From resource to document: Scaffolding content and organising student learning in teachers’ documentation work on the teaching of series

We examine teachers’ use of resources as they prepare to teach the topic of numerical series of real numbers in order to identify how their personal relationship with mathematical content—and its teaching—interacts with their use of a commonly used textbook. We describe this interplay between textbook and personal relationship, a term coined in the Anthropological Theory of the Didactic (ATD, Chevallard, 2003), in the terms of documentation work (resources, aims, rules of action, operational invariants), a key construct from the documentational approach (DA, Gueudet & Trouche, 2009). We do so in the case of five post-secondary teachers who use the same textbook as a main resource to teach the topic. Documentational analysis of interviews with the teachers led to the identification of their aims and rules of action (the what and how of their resource use as they organise their teaching of the topic) as well as the operational invariants (the why for this organisation of their teaching). We describe the teachers’ documentation work in two sets of aims/rules of action: scaffolding mathematical content (series as a stepping stone to learning about Taylor polynomials and Maclaurin series) and organising student learning about series through drill exercises, visualisation, examples, and applications. Our bridging (networking) of theoretical constructs originating in one theoretical framework (personal relationship, ATD) with the constructs of a different, yet compatible, framework (documentation work, DA) aims to enrich the latter (teachers’ documentation work) with the individual agency (teachers’ personal relationship with the topic) provided by the former

The French Didactic Tradition in Mathematics

This chapter presents the French didactic tradition. It first describes theemergence and development of this tradition according to four key features (role ofmathematics and mathematicians, role of theories, role of design of teaching andlearning environments, and role of empirical research), and illustrates it through two case studies respectively devoted to research carried out within this traditionon algebra and on line symmetry-reflection. It then questions the influence of thistradition through the contributions of four researchers from Germany, Italy, Mexicoand Tunisia, before ending with a short epilogue

Limpet Shells from the Aterian Level 8 of El Harhoura 2 Cave (Témara, Morocco): Preservation State of Crossed-Foliated Layers

International audienceThe exploitation of mollusks by the first anatomically modern humans is a central question for archaeologists. This paper focuses on level 8 (dated around * 100 ka BP) of El Har-houra 2 Cave, located along the coastline in the Rabat-Témara region (Morocco). The large quantity of Patella sp. shells found in this level highlights questions regarding their origin and preservation. This study presents an estimation of the preservation status of these shells. We focus here on the diagenetic evolution of both the microstructural patterns and organic components of crossed-foliated shell layers, in order to assess the viability of further investigations based on shell layer minor elements, isotopic or biochemical compositions. The results show that the shells seem to be well conserved, with microstructural patterns preserved down to sub-micrometric scales, and that some organic components are still present in situ. But faint taphonomic degradations affecting both mineral and organic components are nonetheless evidenced, such as the disappearance of organic envelopes surrounding crossed-foliated lamellae, combined with a partial recrystallization of the lamellae. Our results provide a solid case-study of the early stages of the diagenetic evolution of crossed-foliated shell layers. Moreover, they highlight the fact that extreme caution must be taken before using fossil shells for palaeoenvironmental or geochronological reconstructions. Without thorough investigation, the alteration patterns illustrated here would easily have gone unnoticed. However, these degradations are liable to bias any proxy based on the elemental, isotopic or biochemical composition of the shells. This study also provides significant data concerning human subsistence behavior: the presence of notches and the good preservation state of limpet shells (no dissolution/recrystallization, no bioerosion and no abrasion/fragmentation aspects) would attest that limpets were gathered alive with tools by Middle Palaeolithic (Aterian) populations in North Africa for consumption

Roteiros de aprendizagem a partir da transposição didática reflexiva

Este artigo busca socializar o produto da dissertação de mestrado profissional em Ensino de Matemática. Apresentamos roteiros de aprendizagem como uma possibilidade educacional para que professores de matemática do Ensino Médio possam aplicá-los de modo a desenvolver a matemática a partir de dados reais. Os roteiros foram adaptados de trabalhos oriundos do Projeto de Iniciação Científica do IFC - Campus Rio do Sul. A pesquisa foi embasada na teoria da Transposição Didática e na Educação Matemática Crítica. Essa abordagem foi denominada de Transposição Didática Reflexiva, com o intuito de unir as duas teorias num processo em que o saber a ser ensinado seja adaptado de forma a provocar reflexões, instituindo um cenário para investigação que propicie a participação do aluno no processo, provoque discussões e tomada de decisão, de modo a instigar o sujeito a ser transformador de sua própria realidade. De modo particular, o artigo apresenta, em linhas gerais, um dos roteiros detalhados na dissertação, aplicado em uma turma do IFC - Campus Rio do Sul