Spectra of graph neighborhoods and scattering

Let $(G_\epsilon)_{\epsilon>0}$ be a family of '$\epsilon$-thin' Riemannian manifolds modeled on a finite metric graph $G$, for example, the $\epsilon$-neighborhood of an embedding of $G$ in some Euclidean space with straight edges. We study the asymptotic behavior of the spectrum of the Laplace-Beltrami operator on $G_\epsilon$ as $\epsilon\to 0$, for various boundary conditions. We obtain complete asymptotic expansions for the $k$th eigenvalue and the eigenfunctions, uniformly for $k\leq C\epsilon^{-1}$, in terms of scattering data on a non-compact limit space. We then use this to determine the quantum graph which is to be regarded as the limit object, in a spectral sense, of the family $(G_\epsilon)$. Our method is a direct construction of approximate eigenfunctions from the scattering and graph data, and use of a priori estimates to show that all eigenfunctions are obtained in this way.Comment: 37 pages, 3 figures, added references, added comment at end of Section 1.2, changed comment after Theorem 30; in v4: made appendix into a separate paper (arXiv:0711.2869), added reference, minor correction

Pedagogical content knowledge and preparation of string teachers

In the past few decades, there has been an increase in the percentage of non-string specialists teaching string classes. In this article, we review literature about subject-specific pedagogical content knowledge (PCK) in general and music education settings, to better understand the challenges that teachers with limited knowledge of string-specific content may face when teaching strings students. Included in this review are discussions concerning trends in the string teacher workforce, PCK in education and music, acquisition of PCK in general settings and music teacher preparation programs, and relationships between teacher content knowledge and instructional effectiveness, both in general and string education settings. Based on this review, we recommend that preservice and professional development curricula for music teachers include comprehensive preparation in both content-specific and pedagogical-specific knowledge for teaching strings

Pedagogical content knowledge for SHIFTING: More than a toolbox of tricks

The heterogeneous string classroom can often present challenges to string teachers in knowing how to help a variety of students develop complex string technique such as shifting and vibrato. Just like teaching any skill in any subject, teaching string-specific technique requires specific types of knowledge, and long-term success depends largely on ensuring that technical fundamentals are well taught. In this two-part series, we will address the issues of pedagogical content knowledge (PCK)—the integration of content knowledge and pedagogical knowledge—in regard to shifting and vibrato in the heterogeneous string classroom, to demonstrate how knowledge of technique works hand-in-hand with knowledge of teaching. “We explore common shifting challenges and realistic teaching strategies that take into consideration the large heterogeneous string classroom.” In this first article, we focus on shifting technique, and in the second, we will discuss vibrato. Previous research suggests that teachers who have PCK to teach a concept or skill can help students deepen the understanding of complex skills and concepts. In this article, we discuss various teaching strategies from the pre-shifting exercises to early shifting exercises. We explore common shifting challenges and realistic teaching strategies that take into consideration the large heterogeneous string classroom

A Parametrix Construction for the Laplacian on Q-rank 1 Locally Symmetric Space

This paper presents the construction of parametrices for the Gauss-Bonnet and Hodge Laplace operators on noncompact manifolds modelled on Q-rank 1 locally symmetric spaces. These operators are, up to a scalar factor, $\phi$-differential operators, that is, they live in the generalised $\phi$-calculus studied by the authors in a previous paper, which extends work of Melrose and Mazzeo. However, because they are not totally elliptic elements in this calculus, it is not possible to construct parametrices for these operators within the $\phi$-calculus. We construct parametrices for them in this paper using a combination of the $b$-pseudodifferential operator calculus of R. Melrose and the $\phi$-pseudodifferential operator calculus. The construction simplifies and generalizes the construction done by Vaillant in his thesis for the Dirac operator. In addition, we study the mapping properties of these operators and determine the appropriate Hlibert spaces between which the Gauss-Bonnet and Hodge Laplace operators are Fredholm. Finally, we establish regularity results for elements of the kernels of these operators.Comment: 29 pages; in revision: improved exposition, unified use of rescaled bundles and half densities, corrected misprint