Quantum mechanics in finite dimensional Hilbert space

The quantum mechanical formalism for position and momentum of a particle in a one dimensional cyclic lattice is constructively developed. Some mathematical features characteristic of the finite dimensional Hilbert space are compared with the infinite dimensional case. The construction of an unbiased basis for state determination is discussed.Comment: 14 pages, no figure

Condensation and Slow Dynamics of Polar Nanoregions in Lead Relaxors

It is now well established that the unique properties of relaxor ferroelectrics are due to the presence of polar nanoregions (PNR's). We present recent results from Neutron and Raman scattering of single crystals of PZN, PZN-xPT, and PMN. Both sets of measurements provide information on the condensation of the PNR's and on their slow dynamics, directly through the central peak and, indirectly, through their coupling to transverse phonons. A comparative analysis of these results allows identification of three stages in the evolution of the PNR's with decreasing temperature: a purely dynamic stage, a quasi-static stage with reorientational motion and a frozen stage. A model is proposed, based on a prior study of KTN, which explains the special behavior of the transverse phonons (TO and TA) in terms of their mutual coupling through the rotations of the PNR's.Comment: AIP 6x9 style files, 10 pages, 4 figures, Conference-Fundamental Physics of Ferroelectrics 200

Building reflective practices in a pre-service math and science teacher education course that focuses on qualitative video analysis

The use of video for in-service and pre-service teacher development has been gaining acceptance, and yet video remains a challenging and understudied tool. Many projects have used video to help pre-service and in-service teachers reflect on their own teaching processes, examine teacher–student interactions, and develop their professional vision. But rarely has video been used in ways more akin to qualitative education research that is focused on student learning. Even more rarely has this focus occurred at the earliest stages of pre-service teaching when students have not yet decided to pursue teaching careers. Yet here we argue that there are benefits to our approach. We examine a course for prospective pre-service math and science teachers at the University of California, Berkeley, that engages participants in qualitative video analysis to foster their reflective practice. This course is unique in that the prospective pre-service teachers engage in qualitative video analysis at a level characteristic of professional educational research, in that their analysis focuses on student learning of math and science content. We describe classroom activities that provide opportunities for the preservice teacher participants to better observe, notice, and interpret their students’ sociocognitive activity. The course culmination project involves participants developing and teaching lessons in a high school classroom. The participants then videotape the lessons and conduct qualitative video analysis. Results include detailed examples of two selected prospective pre-service teachers demonstrating coherent and effective approaches to conceptualizing the learning and teaching of mathematical and science content along with some potential design principles for building reflective practices through qualitative video projects. © 2018 Association for Science Teacher Education

Sums of arithmetic functions over values of binary forms

Given a suitable arithmetic function h, we investigate the average order of h as it ranges over the values taken by an integral binary form F. A general upper bound is obtained for this quantity, in which the dependence upon the coefficients of F is made completely explicit.Comment: 12 page

Binary forms as sums of two squares and Ch\^atelet surfaces

The representation of integral binary forms as sums of two squares is discussed and applied to establish the Manin conjecture for certain Ch\^atelet surfaces defined over the rationals.Comment: 33 page