Constructing Knowledge Via a Peer Interaction in a CAS Environment with Tasks Designed from a Task–Technique–Theory Perspective

Our research project aimed at understanding the complexity of the construction of knowledge in a CAS environment. Basing our work on the French instrumental approach, in particular the Task–Technique–Theory (T–T–T) theoretical frame as adapted from Chevallard’s Anthropological Theory of Didactics, we were mindful that a careful task design process was needed in order to promote in students rich and meaningful learning. In this paper, we explore further Lagrange’s (2000) conjecture that the learning of techniques can foster conceptual understanding by investigating at close range the task-based activity of a pair of 10th grade students—activity that illustrates the ways in which the use of symbolic calculators along with appropriate tasks can stimulate the emergence of epistemic actions within technique-oriented algebraic activity.     

Evolution of the Probability Measure for the Majda Model: New Invariant Measures and Breathing PDFs

In 1993, Majda proposed a simple, random shear model from which scalar intermittency was rigorously predicted for the invariant probability measure of passive tracers. In this work, we present an integral formulation for the tracer measure, which leads to a new, comprehensive study on its temporal evolution based on Monte Carlo simulation and direct numerical integration. An interesting, non-monotonic “breathing” phenomenon is discovered from these results and carefully defined, with a solid example for special initial data to predict such phenomenon. The signature of this phenomenon may persist at long time, characterized by the approach of the PDF core to its infinite time, invariant value. We find that this approach may be strongly dependent on the non-dimensional Péclet number, of which the invariant measure itself is independent. Further, the “breathing” PDF is recovered as a new invariant measure in a distinguished time scale in the diffusionless limit. Rigorous asymptotic analysis is also performed to identify the Gaussian core of the invariant measures, and the critical rate at which the heavy, stretched exponential regime propagates towards the tail as a function of time is calculated.     

Semiclassical Limit of Focusing NLS for a Family of Square Barrier Initial Data

The small dispersion limit of the focusing nonlinear Schrödinger equation (NLS) exhibits a rich structure of sharply separated regions exhibiting disparate rapid oscillations at microscopic scales. The non‐self‐adjoint scattering problem and ill‐posed limiting Whitham equations associated to focusing NLS make rigorous asymptotic results difficult. Previous studies have focused on special classes of analytic initial data for which the limiting elliptic Whitham equations are wellposed. In this paper we consider another exactly solvable family of initial data,the family of square barriers,ψ 0(x) = qχ[?L,L] for real amplitudes q. Using Riemann‐Hilbert techniques, we obtain rigorous pointwise asymptotics for the semiclassical limit of focusing NLS globally in space and up to an O(1) maximal time. In particular, we show that the discontinuities in our initial data regularize by the immediate generation of genus‐one oscillations emitted into the support of the initial data. To the best of our knowledge, this is the first case in which the genus structure of the semiclassical asymptotics for focusing NLS have been calculated for nonanalytic initial data. © 2013 Wiley Periodicals, Inc.     

Attractors and transients for a perturbed periodic KdV equation: A nonlinear spectral analysis

Summary In this paper we rigorously show the existence and smoothness inε of traveling wave solutions to a periodic Korteweg-deVries equation with a Kuramoto-Sivashinsky-type perturbation for sufficiently small values of the perturbation parameterε. The shape and the spectral transforms of these traveling waves are calculated perturbatively to first order. A linear stability theory using squared eigenfunction bases related to the spectral theory of the KdV equation is proposed and carried out numerically. Finally, the inverse spectral transform is used to study the transient and asymptotic stages of the dynamics of the solutions.     

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights w_n(x)dx = e^−nV(x) dx on the line as n → ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel‐Rotach‐type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann‐Hilbert problem following [19, 20]. The Riemann‐Hilbert problem is analyzed in turn using the steepest‐descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμ_V for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.     

Strong asymptotics of orthogonal polynomials with respect to exponential weights

We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx = e^−Q(x) dx on the real line, where Q(x) = Σ q_k x^k, q_2m > 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann‐Hilbert problem following [22, 23]. We employ the steepest‐descent‐type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel‐Rotach‐type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials. © 1999 John Wiley & Sons, Inc.     

Homoclinic orbits and chaos in discretized perturbed NLS systems: Part I. Homoclinic orbits

Summary The existence of homocliic orbits, for a finite-difference discretized form of a damped and driven perturbation of the focusing nonlinear Schroedinger equation under even periodic boundary conditions, is established. More specifically, for external parameters on a codimension 1 submanifold, the existence of homoclinic orbits is established through an argument which combines Melnikov analysis with a geometric singular perturbation theory and a purely geometric argument (called the “second measurement” in the paper). The geometric singular perturbation theory deals with persistence of invariant manifolds and fibration of the persistent invariant manifolds. The approximate location of the codimension 1 submanifold of parameters is calculated. (This is the material in Part I.) Then, in a neighborhood of these homoclinic orbits, the existence of “Smale horseshoes” and the corresponding symbolic dynamics are established in Part II [21].