B\"acklund transformations for nonlinear evolution equations: Hilbert space approach

A new method of determining B\"acklund transformations for nonlinear partial differential equations of the evolution type is introduced. Using the Hilbert space approach the problem of finding B\"acklund transformations is brought down to the solution of an abstract equation in Hilbert space.Comment: 15 page

The Effect of Student Learning Styles on the Learning Gains Achieved When Interactive Simulations Are Coupled with Real-Time Formative Assessment via Pen-Enabled Mobile Technology

This paper describes results from a project in an undergraduate engineering physics course that coupled classroom use of interactive computer simulations with the collection of real-time formative assessment using pen-enabled mobile technology. Interactive simulations (free or textbook-based) are widely used across the undergraduate science and engineering curriculia to help actively engaged students increase their understanding of abstract concepts or phenomena which are not directly or easily observable. However, there are indications in the literature that we do not yet know the pedagogical best practices associated with their use to maximize learning. This project couples student use of interactive simulations with the gathering of real-time formative assessment via pen-enabled mobile technology (in this case, Tablet PCs). The research question addressed in this paper is: are learning gains achieved with this coupled model greater for certain types of learners in undergraduate STEM classrooms? To answer this, we correlate learning gains with various learning styles, as identified using the Index of Learning Styles (ILS) developed by Felder and Soloman. These insights will be useful for others who use interactive computer simulations in their instruction and other adopters of this pedagogical model; the insights may have broader implications about modification of instruction to address various learning styles.Comment: 6 pages 2 tables and 1 figur

Amplification arguments for large sieve inequalities

We present a new proof of the "arithmetic" large sieve inequality, starting from the corresponding "harmonic" inequality, which is based on an amplification idea. We show that this also adapts to give some new sieve inequality for modular forms, where Hecke eigenvalues are thought as the analogues of the reductions of integers modulo primes.Comment: 13 pages, 1 figure; v2, version accepted for publication in Archiv der Math

Weil numbers generated by other Weil numbers and torsion fields of abelian varieties

Using properties of the Frobenius eigenvalues, we show that, in a precise sense, ``most'' isomorphism classes of (principally polarized) simple abelian varieties over a finite field are characterized up to isogeny by the sequence of their division fields, and a similar result for ``most'' isogeny classes. Some global cases are also treated.Comment: 13 page