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1. A quadratic growth learning trajectory

   https://doi.org/10.1016/j.jmathb.2020.100795  

   Fonger, N. (October 2020, The Journal of mathematical behavior)

   This paper introduces a quadratic growth learning trajectory, a series of transitions in students’ ways of thinking (WoT) and ways of understanding (WoU) quadratic growth in response to instructional supports emphasizing change in linked quantities. We studied middle grade (ages 12–13) students’ conceptions during a small-scale teaching experiment aimed at fostering an understanding of quadratic growth as phenomenon of constantly-changing rate of change. We elaborate the duality, necessity, repeated reasoning framework, and methods of creating learning trajectories. We report five WoT: Variation, Early Coordinated Change, Explicitly Quantified Coordinated Change, Dependency Relations of Change, and Correspondence. We also articulate instructional supports that engendered transitions across these WoT: teacher moves, norms, and task design features. Our integration of instructional supports and transitions in students’ WoT extend current research on quadratic function. A visual metaphor is leveraged to discuss the role of learning trajectories research in unifying research on teaching and learning. 

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2. GTI: A Scalable Graph-based Trajectory Imputation

   https://doi.org/10.1145/3589132.3625620  

   Isufaj, Keivin; Elshrif, Mohamed Mokhtar; Abbar, Sofiane; Mokbel, Mohamed (November 2023, ACM SIGSPATIAL 2023)
3. Investigating Trajectory Based Combustion Control Using a Controlled Trajectory Rapid Compression and Expansion Machine (CT-RCEM)

   Tripathi, A.; Sun, Z. (January 2020, Proceedings of the ASME Dynamic Systems and Control Conference)
4. A steeply-inclined trajectory for the Chicxulub impact

   https://doi.org/10.1038/s41467-020-15269-x  

   Collins, G. S.; Patel, N.; Davison, T. M.; Rae, A. S.; Morgan, J. V.; Gulick, S. P. (December 2020, Nature Communications)
5. Testing Symmetric Markov Chains from a Single Trajectory

   Constantinos Daskalakis, Nishanth Dikkala (July 2018, Conference on Learning Theory (COLT))

   The paper’s abstract in valid LaTeX, without non-standard macros or \cite commands. Classical distribution testing assumes access to i.i.d. samples from the distribution that is being tested. We initiate the study of Markov chain testing, assuming access to a {\em single trajectory of a Markov Chain.} In particular, we observe a single trajectory X0,…,Xt,… of an unknown, symmetric, and finite state Markov Chain M. We do not control the starting state X0, and we cannot restart the chain. Given our single trajectory, the goal is to test whether M is identical to a model Markov Chain M′, or far from it under an appropriate notion of difference. We propose a measure of difference between two Markov chains, motivated by the early work of Kazakos [78], which captures the scaling behavior of the total variation distance between trajectories sampled from the Markov chains as the length of these trajectories grows. We provide efficient testers and information-theoretic lower bounds for testing identity of symmetric Markov chains under our proposed measure of difference, which are tight up to logarithmic factors if the hitting times of the model chain M′ is O~(n) in the size of the state space n. 

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