We describe a method to obtain Lax pairs for periodic reductions of a rather general class of integrable non-autonomous lattice equations. The method is applied to obtain reductions of the non-autonomous discrete Korteweg-de Vries equation and non-autonomous discrete Schwarzian Korteweg-de Vries equation, which yield a discrete analogue of the fourth Painlevé equation, a q-analogue of the sixth Painlevé equation and the q-Painlevé equation with a symmetry group of affine Weyl type E (1) 6 .1 In practice, the product follows the path of a standard staircase [44].

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We present the mathematical theory underlying systems of linear difference equations over the max-plus semi-ring. The result provides an analog of isomonodromy theory for ultradiscrete Painlevé equations, which are extended cellular automata, and provide evidence for their integrability. Our theory is analogous to that developed by Birkhoff and his school for linear q-difference equations, but stands independently of the latter. As an example, we derive linear problems in this algebra for ultradiscrete versions of the symmetric P IV equation and show how it is a necessary condition for isomonodromic deformation of a linear system.

Background Deep learning methods, where models do not use explicit features and instead rely on implicit features estimated during model training, suffer from an explainability problem. In text classification, saliency maps that reflect the importance of words in prediction are one approach toward explainability. However, little is known about whether the salient words agree with those identified by humans as important. Objectives The current study examines in‐line annotations from human annotators and saliency map annotations from a deep learning model (ELECTRA transformer) to understand how well both humans and machines provide evidence for their assigned label. Methods Data were responses to test items across a mix of United States subjects, states, and grades. Humans were trained to annotate responses to justify a crisis alert label and two model interpretability methods (LIME, Integrated Gradients) were used to obtain engine annotations. Human inter‐annotator agreement and engine agreement with the human annotators were computed and compared. Results and Conclusions Human annotators agreed with one another at similar rates to those observed in the literature on similar tasks. The annotations derived using the integrated gradients (IG) agreed with human annotators at higher rates than LIME on most metrics; however, both methods underperformed relative to the human annotators. Implications Saliency map‐based engine annotations show promise as a form of explanation, but do not reach human annotation agreement levels. Future work should examine the appropriate unit for annotation (e.g., word, sentence), other gradient based methods, and approaches for mapping the continuous saliency values to Boolean annotations.

We present theory outlining the linear problems associated with ultradiscrete equation. The appropriate domain for such problems being the max-plus semiring. Our main result being that despite the restrictive nature of this semiring, it is still possible to define a theory of monodromy analogous to that of Birkhoff and his school for systems of linear difference equations over the max-plus semiring. We use such theory to provide evidence for the integrability of some ultradiscrete difference equations.