Mathematics and science teacher educators' use of representations of practice: A mixed methods study

This study sought to explore math and science teacher educators' use of various media to represent practice within methods courses. There is little understanding of why certain media is used over other representations and the rationale for these choices. Specifically, the study focused on the prevalence and familiarity of teacher educators with comics and animations, standard videos, and 360 videos. This mixed methods study utilized a survey and interviews to ascertain math and science teacher educators' level of familiarity and perceived usefulness of representations of practice. Results indicate that standard video is by far the most used representation of practice in methods classes with three themes explaining this finding: access to representations of practice, dimensions of representation, and pertinacity of using representations. Familiarity with representations of practice relates to teacher educators' perceptions of access thereby indicating a need for teacher educators to have better access to representations. Implications of this study include supporting current literature about the relationship between the level of familiarity and perceived usefulness in media along with the potential need for a central platform that houses these representations of practice resources for teacher educators.     

Representation in Computational Environments: Epistemological and Social Distance

Computational environments have the potential to provide new representational resources and new ways of supporting teaching and learning of mathematics. In this paper, we seek to characterize relationships between the representations offered by particular technologies and other representations commonly available in the classroom context, using the notion of ‘distance’. Distance between representations in different media may be epistemological, affecting the nature of the mathematical concepts available to students, or may be social, affecting pedagogic relationships in the classroom and the ease with which the technology may be adopted in particular classroom or national contexts. We illustrate these notions through examples taken from cross-experimentation of computational environments in national contexts different from those in which they were developed. Implications for the design and dissemination of computational environments for use in learning mathematics are discussed.     

Unitary Representations and Coadjoint Orbits for a Group of Germs of Real Analytic Diffeomorphisms

The method of coadjoint orbits is developed for the group of real analytic germs of diffeomorphisms φ with φ(0) = 0 and φ′(0) = 1. The form of all infinite dimensional coadjoint orbits is described. Classes U of unitary representations are constructed. In the case n = 2 these representations are related to coadjoint orbits.     

Integral equations in a diffraction problem on a locally inhomogeneous medium interface

We study the diffraction of an E-polarized field on a locally inhomogeneous interface of transparent media. We prove the unique solvability of the boundary value diffraction problem and obtain integral representations of the solution. We derive a system of integral equations equivalent to the original boundary value problem and prove a solvability theorem for this system.     

SCALE-INVARIANT SOLUTION FOR FRACTIONAL ANOMALOUS DIFFUSION EOUATION

Fractional diffusion equation for transport phenomena in fractal media is an integro-partial differential equation. For solving the problem of this kind of equation the invariants of the group of scaling transformations are given and then used for deriving the integro-ordinary differential equation for the scale-invariant solution. With the help of Mellin transform the scale-invariant solution is obtained in terms of Fox functions. And the series and asymptotic representations for the solution are considered.     

Time harmonic acoustic scattering in anisotropic media

The scattering problem of a plane or a point source generated wave is considered for the case where both the medium of propagation and the interior of the scatterer exhibit their own anisotropies. A particular redirected gradient operator is introduced, which carries all directional characteristics of the anisotropic medium. Once the fundamental solution is obtained, integral representations for the scattered as well as for the interior and the total fields are generated. For such media even the handling of the singularities, in generating integral representations, depends on the characteristics of the particular medium. A modified, also medium dependent, radiation condition is introduced. Detailed asymptotic analysis leads to an integral representation for the scattering amplitude. The associated energy functionals are presented and the relative cross sections are also defined. Copyright © 2005 John Wiley & Sons, Ltd.     

A method for computation of scattering amplitudes and Green functions of whole axis problems

A method for the computation of scattering data and of the Green function for the one‐dimensional Schrödinger operator with a decaying potential is presented. It is based on representations for the Jost solutions in the case of a compactly supported potential obtained in terms of Neumann series of Bessel functions (NSBF). The representations are used for calculating a complete orthonormal system of generalized eigenfunctions of the operator H, which in turn allow one to compute the scattering amplitudes and the Green function of the operator H?λ with .     

The local period integrals and essential vectors

By applying the formula for essential Whittaker functions established by Matringe and Miyauchi, we study five integral representations for irreducible admissible generic representations of GL_n over p-adic fields. In each case, we show that the integrals achieve local formal L-functions defined by Langlands parameters, when the test vector is associated to the new form. We give the relation between local periods involving essential Whittaker functions and special values of formal L-factors at $s=1$ for certain distinguished or unitary representations. The period integrals are also served as standard nonzero distinguished forms.     

Two approaches to the solution of coefficient inverse problems for wave equations

Two approaches to solving coefficient inverse problems for wave equations are compared. One approach is based on integral representations obtained with the help of the Green’s function for the wave equation. In the other approach, the gradient of the error functional is directly computed in terms of the solution of the adjoint problem for a partial differential equation. The methods developed are intended for finding inhomogeneities in homogeneous media and can be applied in medicine diagnostics, acoustic and seismic near surface exploration, engineering seismics, etc.     

Convergent Power Series for Fields in Positive or Negative High-Contrast Periodic Media

We obtain convergent power series representations for Bloch waves in periodic high-contrast media. The material coefficient in the inclusions can be positive or negative. The small expansion parameter is the ratio of period cell width to wavelength, and the coefficient functions are solutions of the cell problems arising from formal asymptotic expansion. In the case of positive coefficient, the dispersion relation has an infinite sequence of branches, each represented by a convergent even power series whose leading term is a branch of the dispersion relation for the homogenized medium. In the negative case, there is a single branch.     

A new approach to Cagniard's problem

Cagniard problem refers to the class of linear reflection and transmission problem for pulsed line and point sources, which have solution methods leading to exact algebraic representations of the wave fields. All previous methods have relied heavily on integral or differential transforms. We present in this paper a new and direct approach to the problem which involves only the wave equation and its associated characteristic equation. We illustrate the new method by applying it to the problem of the reflection and transmission of acoustic waves radiating from a line source in the vicinity of a plane boundary separating two uniform acoustic media.     

A Boundary Value Problem for Conjugate Conductivity Equations

We give explicit integral formulas for the solutions of planar conjugate conductivity equations in a circular domain of the right half‐plane with conductivity , . The representations are obtained via the so‐called unified transform method or Fokas method, involving a Riemann–Hilbert problem on the complex plane when p is even and on a two‐sheeted Riemann surface when p is odd. They are given in terms of the Dirichlet and Neumann data on the boundary of the domain. For even exponent p, we also show how to make the conversion from one type of conditions to the other by using the global relation that follows from the closedness of some differential form. The method used to derive our integral representations could be applied in any bounded simply connected domain of the right half‐plane with a smooth boundary.     

Effective transverse elastic moduli of composites at non-dilute concentration of a random field of aligned fibers

We consider a transversal loading of a linearly elastic isotropic media containing the identical isotropic aligned circular fibers at non-dilute concentration c. By the use of solution obtained by the Kolosov–Muskhelishvili complex potential method for two interacting circles subjected to three different applied stresses at infinity, and exact integral representations for both the stress and strain distributions in a microinhomogeneous medium, one estimates the effective moduli of the composite accurately to order c². Received: March 4, 2003; revised: August 8, 2003     

Potential scenarios for Internet use in the mathematics classroom

Research on the influence of multiple representations in mathematics education gained new momentum when personal computers and software started to become available in the mid-1980s. It became much easier for students who were not fond of algebraic representations to work with concepts such as function using graphs or tables. Research on how students use such software showed that they shaped the tools to their own needs, resulting in an intershaping relationship in which tools shape the way students know at the same time the students shape the tools and influence the design of the next generation of tools. This kind of research led to the theoretical perspective presented in this paper: knowledge is constructed by collectives of humans-with-media. In this paper, I will discuss how media have shaped the notions of problem and knowledge, and a parallel will be developed between the way that software has brought new possibilities to mathematics education and the changes that the Internet may bring to mathematics education. This paper is, therefore, a discussion about the future of mathematics education. Potential scenarios for the future of mathematics education, if the Internet becomes accepted in the classroom, will be discussed.     

On the solution of boundary value problems on an inhomogeneous plane with a crack and a barrier connected in series

We consider boundary value problems for an equation in divergence form on a plane divided into two inhomogeneous half-planes by a film inclusion in the form of a strongly permeable crack and a weakly permeable barrier connected in series; this models a contact of heterogeneous media under inhomogeneous external conditions. The desired potentials have prescribed singular points (sources, drains, etc.). The coefficients of the equation are nonconstant and may increase or decrease when moving away from the film inclusion along a family of parabolas. We obtain representations of solutions of the considered problems via harmonic functions with the corresponding singular points on the plane.     

Unitarizable Representations of Quivers

We investigate representations of *-algebras associated with posets. Unitarizable representations of the corresponding (bound) quivers (which are polystable representations for some appropriately chosen slope function) give rise to representations of these algebras. Considering posets which correspond to unbound quivers this leads to an ADE-classification which describes the unitarization behaviour of their representations. Considering posets which correspond to bound quivers, it is possible to construct unitarizable representations starting with polystable representations of related unbound quivers which can be glued together with a suitable direct sum of simple representations. Finally, we estimate the number of complex parameters parametrizing irreducible unitary non-equivalent representations of the corresponding algebras.