The Jacobi operator on (−1,1)(-1,1) and its various mm-functions

We offer a detailed treatment of spectral and Weyl-Titchmarsh-Kodaira theory for all self-adjoint Jacobi operator realizations of the differential expression \begin{align*} \tau_{\alpha,\beta} = - (1-x)^{-\alpha} (1+x)^{-\beta}(d/dx) \big((1-x)^{\alpha+1}(1+x)^{\beta+1}\big) (d/dx),& \\ \alpha, \beta \in \mathbb{R}, \; x \in (-1,1),& \end{align*} in $L^2\big((-1,1); (1-x)^{\alpha} (1+x)^{\beta} dx\big)$, $\alpha, \beta \in \mathbb{R}$. In addition to discussing the separated boundary conditions that lead to Jacobi orthogonal polynomials as eigenfunctions in detail, we exhaustively treat the case of coupled boundary conditions and illustrate the latter with the help of the general $\eta$-periodic and Krein--von Neumann extensions. In particular, we treat all underlying Weyl-Titchmarsh-Kodaira and Green's function induced $m$-functions and revisit their Nevanlinna-Herglotz property. We also consider connections to other differential operators associated with orthogonal polynomials such as Laguerre, Gegenbauer, and Chebyshev.Comment: 59 pages. arXiv admin note: text overlap with arXiv:2102.00685, arXiv:2110.15913, arXiv:1910.1311

Embracing the barbarian: John Chrysostom\u27s pastoral care of the Goths

This dissertation examines John Chrysostom\u27s efforts to promote the Christianization of the Goths as the bishop of Constantinople (c. 397–404 CE). Although Chrysostom\u27s engagement with the Goths has long been recognized as a defining feature of his episcopate, it remains astonishingly understudied. In fact, most modern treatments provide little more than a summary of the brief account in Theodoret\u27s Ecclesiastical History. This study seeks to provide hitherto unappreciated depth and nuance to our understanding of the bishop\u27s overall strategy for the Goths, especially with respect to what animated his interest in mission, who were the Goths with whom he interacted, what can be ascertained about the scope of the mission, and how Chrysostom justified this controversial undertaking to his fellow Roman Christians. The first chapter treats the renewed missionary aspirations of the See of Antioch during Chrysostom\u27s formative years, his adoption of this distinctive approach to mission, and his promotion of the apostle Paul as Christianity\u27s missionary exemplar to support contemporary mission in Antioch. The reappraisal of the Christianization of the Goths in chapter two demonstrates that, contrary to the belief that the Goths with whom John interacted were Arians, the bishop would have encountered a variety of Gothic ecclesiastical factions, including a strong contingent of Nicene Goths. The third chapter examines the Chrysostom\u27s establishment of a Gothic parish in Constantinople, and also refutes the prevailing view that his Gothic mission ceased after the Gaïnas crisis. The fourth chapter demonstrates that Chrysostom\u27s Gothic mission further entailed enlisting the support of the Gothic monastery in Constantinople, fostering the Goths\u27 sense of belonging in the Nicene community through their participation in his citywide liturgical processions, and overseeing separate missions to the Goths living along the Danube and in the Crimea. The fifth chapter argues that Chrysostom\u27s homily to the Goths also functioned as a defense of his barbarian mission intended for his fellow Roman Christians, which entailed his rationale for providing pastoral care to the Goths as well as his justification for embracing the Goths as full members in the Nicene church

On functions related to the spectral theory of Sturm--Liouville operators.

Functions related to the spectral theory of differential operators have been extensively studied due to their many applications in mathematics and physics. In this dissertation, we will consider spectral ζ-functions, ζ-regularized functional determinants, and Donoghue m-functions associated with Sturm--Liouville operators. We apply our results to an array of examples, including regular Schrödinger operators as well as Jacobi and generalized Bessel operators in the singular context. We begin by employing a recently developed unified approach to the computation of traces of resolvents and ζ-functions to efficiently compute values of spectral ζ-functions at positive integers associated with regular (three-coefficient) self-adjoint Sturm--Liouville differential expressions τ. Furthermore, we give the full analytic continuation of the ζ-function through a Liouville transformation and provide an explicit expression for the ζ-regularized functional determinant in terms of a particular set of a fundamental system of solutions of τy = zy. Next we turn to Donoghue m-functions. Assuming the standard local integrability hypotheses on the coefficients of the singular Sturm--Liouville differential equation τ, we study all corresponding self-adjoint realizations in L^2((a,b); rdx) and systematically construct the associated Donoghue m-functions in all cases where τ is in the limit circle case at least at one interval endpoint a or b. Finally, we construct Donoghue m-functions for the Jacobi differential operator in L^2 ((-1,1); (1-x)^alpha(1+x)^beta dx) associated with the differential expression τ alpha,beta = - (1-x)^-alpha(1+x)^-beta(d/dx)((1-x)^alpha+1((1+x)^beta+1)(d/dx), x E (-1,1), alpha,beta E R, whenever at least one endpoint, x = pm1, is in the limit circle case. In doing so, we provide a full treatment of the Jacobi operator's m-functions corresponding to coupled boundary conditions whenever both endpoints are in the limit circle case

WHEN, WHERE, AND HOW IS DIGITAL SOUND?

This panel’s first author, in discussing podcast archiving, notes that internet archives like the Wayback Machine have had much more focus on preserving visual and text content than sound. Internet Research has similarly traditionally had less engagement with sound than with other forms of digital content. This panel seeks to contribute to ongoing work to bring Sound Studies and Internet Studies into better conversation with each other, taking digital sound as a common object and examining it in different cases and through different methods to provide a richer understanding of the role sound plays in shaping our online experiences. The papers coalesce around their common object of inquiry, digital sound, providing depth of understanding about the subject matter by approaching from different directions. Moreover, the papers help to illuminate each other by taking different approaches to common themes. The first and second papers raise key questions about who tends to be included and excluded in circuits of production as well as whose digital sound tends to be seen as valuable. Papers 1, 2, and 3 all ask about how, despite rhetorics of democratization and variety, forms of digital sound may be becoming standardized through technological and social means. The first and third papers call attention to the ways the specific affordances of given digital production technologies shape (though do not determine) the kinds of production that become prevalent in a given moment. There are also methodological convergences: papers 3 and 4 take as their object of inquiry technology makers, and papers 2 and 4 both use press coverage as the site of investigation. Finally, papers 2 and 4 ask questions about what people believe is socially proper or correct in the case of digital sound. In these ways, this panel represents both an important contribution to our understanding of contemporary issues in digital sound as well as relating to broader questions central to internet research

The Krein-von Neumann extension revisited

We revisit the Krein-von Neumann extension in the case where the underlying symmetric operator is strictly positive and apply this to derive the explicit form of the Krein-von Neumann extension for singular, general (i.e., three-coefficient) Sturm-Liouville operators on arbitrary intervals. In particular, the boundary conditions for the Krein-von Neumann extension of the strictly positive minimal Sturm-Liouville operator are explicitly expressed in terms of generalized boundary values adapted to the (possible) singularity structure of the coefficients near an interval endpoint.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1910.1311