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1. Classifying photonic topology using the spectral localizer and numerical K -theory

   https://doi.org/10.1063/5.0239018  

   Cerjan, Alexander; Loring, Terry_A (November 2024, APL Photonics)

   Recently, the spectral localizer framework has emerged as an efficient approach for classifying topology in photonic systems featuring local nonlinearities and radiative environments. In nonlinear systems, this framework provides rigorous definitions for concepts such as topological solitons and topological dynamics, where a system’s occupation induces a local change in its topology due to nonlinearity. For systems embedded in radiative environments that do not possess a shared bulk spectral gap, this framework enables the identification of local topology and shows that local topological protection is preserved despite the lack of a common gap. However, as the spectral localizer framework is rooted in the mathematics of C*-algebras, and not vector bundles, understanding and using this framework requires developing intuition for a somewhat different set of underlying concepts than those that appear in traditional approaches for classifying material topology. In this tutorial, we introduce the spectral localizer framework from a ground-up perspective and provide physically motivated arguments for understanding its local topological markers and associated local measure of topological protection. In doing so, we provide numerous examples of the framework’s application to a variety of topological classes, including crystalline and higher-order topology. We then show how Maxwell’s equations can be reformulated to be compatible with the spectral localizer framework, including the possibility of radiative boundary conditions. To aid in this introduction, we also provide a physics-oriented introduction to multi-operator pseudospectral methods and numerical K-theory, two mathematical concepts that form the foundation for the spectral localizer framework. Finally, we provide some mathematically oriented comments on the C*-algebraic origins of this framework, including a discussion of real C*-algebras and graded C*-algebras that are necessary for incorporating physical symmetries. Looking forward, we hope that this tutorial will serve as an approachable starting point for learning the foundations of the spectral localizer framework. 

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2. Towards a Framework for Smart Classrooms that Teach Instructors to Teach

   Gerritsen, David; Zimmerman, John; Ogan, Amy (July 2018, International Conference of the Learning Sciences)

   Teaching Assistants (TAs) play a major role in higher education; however, they receive little if any training on how to teach. Quality training requires access to grounded feedback and relevant suggestions for improvement. We developed a framework for using features of a smart classroom. This work reframes the instructor as the learner. It provides training on discursive practices with feedback based on the instructor’s in-class behaviors. We built and deployed a system based on this framework to five STEM TAs as part of a larger study. This paper: discusses the action-reflection-planning framework we used, provides evidence for how the framework addresses TA learning goals, and discusses how other researchers might make use of the framework. 

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3. Theoretical report: A framework for examining prospective teachers’ use of mathematical knowledge for teaching in mathematics courses.

   Lai, Y.; Strayer, J.; Lischka, A.; Anhalt., C.; Quinn, C.; Reed, S. (January 2019, Proceedings of the 22nd Annual Conference on Research in Undergraduate Mathematics Education)

   This theoretical report addresses the challenge and promise of improving prospective secondary mathematics teachers’ experiences in undergraduate mathematics courses through tasks embedded in pedagogical contexts. The objective of this approach, used by multiple nationally- funded projects, is to enhance the development of teachers’ MKT. We report on the construction of a framework for observing and analyzing the development of teachers’ MKT. This framework is the result of integrating several existing frameworks and analyzing a sample of prospective secondary teachers’ responses to tasks embedded in pedagogical contexts. We discuss the methods used to build this framework, the strengths and weaknesses of the framework, and the potential of the framework for informing future work in curriculum design and implementation. 

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4. Theoretical report: A framework for examining prospective teachers’ use of mathematical knowledge for teaching in mathematics courses

   Lai, Y.; Lischka, A.; Strayer, J.; Anhalt, C.; Quinn, C.; Reed, S. (January 2019, In A. Weinberg, D. Moore-Russo, H. Soto, & M. Wawro (Eds.), Proceedings of the 22nd Annual Conference on Research in Undergraduate Mathematics Education)

   This theoretical report addresses the challenge and promise of improving prospective secondary mathematics teachers’ experiences in undergraduate mathematics courses through tasks embedded in pedagogical contexts. The objective of this approach, used by multiple nationally funded projects, is to enhance the development of teachers’ MKT. We report on the construction of a framework for observing and analyzing the development of teachers’ MKT. This framework is the result of integrating several existing frameworks and analyzing a sample of prospective secondary teachers’ responses to tasks embedded in pedagogical contexts. We discuss the methods used to build this framework, the strengths and weaknesses of the framework, and the potential of the framework for informing future work in curriculum design and implementation. 

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5. AoI, Timely-Throughput, and Beyond: A Theory of Second-Order Wireless Network Optimization

   https://doi.org/10.1109/TNET.2024.3432655  

   Guo, Daojing; Nakhleh, Khaled; Hou, I-Hong; Kompella, Sastry; Kam, Clement (December 2024, IEEE/ACM Transactions on Networking)

   This paper introduces a new theoretical framework for optimizing second-order behaviors of wireless networks. Unlike existing techniques for network utility maximization, which only consider first-order statistics, this framework models every random process by its mean and temporal variance. The inclusion of temporal variance makes this framework well-suited for modeling Markovian fading wireless channels and emerging network performance metrics such as age-of-information (AoI) and timely-throughput. Using this framework, we sharply characterize the second-order capacity region of wireless access networks. We also propose a simple scheduling policy and prove that it can achieve every interior point in the second-order capacity region. To demonstrate the utility of this framework, we apply it to an unsolved network optimization problem where some clients wish to minimize AoI while others wish to maximize timely-throughput. We show that this framework accurately characterizes AoI and timely-throughput. Moreover, it leads to a tractable scheduling policy that outperforms other existing work. 

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