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

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

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

Conceptually driven and visually rich tasks in texts and teaching practice: the case of infinite series

The study we report here examines parts of what Chevallard calls the institutional dimension of the students’ learning experience of a relatively under-researched, yet crucial, concept in Analysis, the concept of infinite series. In particular, we examine how the concept is introduced to students in texts and in teaching practice. To this purpose, we employ Duval's Theory of Registers of Semiotic Representation towards the analysis of 22 texts used in Canada and UK post-compulsory courses. We also draw on interviews with in-service teachers and university lecturers in order to discuss briefly teaching practice and some of their teaching suggestions. Our analysis of the texts highlights that the presentation of the concept is largely a-historical, with few graphical representations, few opportunities to work across different registers (algebraic, graphical, verbal), few applications or intra-mathematical references to the concept's significance and few conceptually driven tasks that go beyond practising with the application of convergence tests and prepare students for the complex topics in which the concept of series is implicated. Our preliminary analysis of the teacher interviews suggests that pedagogical practice often reflects the tendencies in the texts. Furthermore, the interviews with the university lecturers point at the pedagogical potential of: illustrative examples and evocative visual representations in teaching; and, student engagement with systematic guesswork and writing explanatory accounts of their choices and applications of convergence tests

‘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

Simulating JWST deep extragalactic imaging surveys and physical parameter recovery

International audienceWe present a new prospective analysis of deep multi-band imaging with the James Webb Space Telescope (JWST). In this work, we investigate the recovery of high-redshift 5  6 and redshifts of 0   5 galaxy samples can be reduced to < 0.01 arcmin−2 with a limited impact on galaxy completeness. We investigate multiple high-redshift galaxy selection techniques and find that the best compromise between completeness and purity at 5 <  z <  10 using the full redshift posterior probability distributions. In the EGS field, the galaxy completeness remains higher than 50% at magnitudes mUV <  27.5 and at all redshifts, and the purity is maintained above 80 and 60% at z ≤ 7 and 10, respectively. The faint-end slope of the galaxy UV luminosity function is recovered with a precision of 0.1–0.25, and the cosmic star formation rate density within 0.1 dex. We argue in favor of additional observing programs covering larger areas to better constrain the bright end

european didactic traditions in mathematics aspects and examples from four selected cases

In this paper, we report on the presentations and activities from the strand on "European Didactic Traditions" during the Thematic Afternoon at ICME-13. The focal point of the first hour of this afternoon were four key features that were identified as common in all European traditions and the second and third hours were devoted to the presentation of concrete examples from four specific traditions, organised in four parallel sessions

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