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We present a linear system of difference equations whose entries are expressed in terms of theta functions. This linear system is singular at $${4m+12}$$ points for $${m \geq 1}$$ , which appear in pairs due to a symmetry condition. We parameterize this linear system in terms of a set of kernels at the singular points. We regard the system of discrete isomonodromic deformations as an elliptic analogue of the Garnier system. We identify the special case in which m = 1 with the elliptic Painlevé equation; hence, this work provides an explicit form and Lax pair for the elliptic Painlevé equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the matrix generalization of ultradiscrete systems. Specifically, we establish a matrix generalization of the ultradiscrete fourth Painlevé equation (ud-PIV). Well-defined multicomponent systems that permit ultradiscretization are obtained using an approach that relies on a group defined by constraints imposed by the requirement of a consistent evolution of the systems. The ultradiscrete limit of these systems yields coupled multicomponent ultradiscrete systems that generalize ud-PIV. The dynamics, irreducibility, and integrability of the matrix-valued ultradiscrete systems are studied. [ABSTRACT FROM AUTHOR]

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 PIV equation and show how it is a necessary condition for isomonodromic deformation of a linear system. [ABSTRACT FROM AUTHOR]

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. Lay Description: What is known about subject matter: Explainability of deep learning models is complicated by their large size, design, and complexity.Explainability methods in deep learning exist but have not been rigorously tested.Saliency maps, as a form of explainability, can be used to generate engine annotations.Human annotations can serve as basis for comparison. What paper adds to knowledge: How well human annotations work in one domain (crisis paper identification).How well two engine annotation methods match human annotations. Implications of study findings: Human annotations have reasonable reliability.Engine annotations have lower reliability than humans.Model‐based engine annotations perform better than model‐agnostic engine annotations. [ABSTRACT FROM AUTHOR]