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Understanding howAI recommendationswork can help the younger generation become more informed and critical consumers of the vast amount of information they encounter daily. However, young learners with limited math and computing knowledge often find AI concepts too abstract. To address this, we developed Briteller, a light-based recommendation system that makes learning tangible. By exploring and manipulating light beams, Briteller enables children to understand an AI recommender system’s core algorithmic building block, the dot product, through hands-on interactions. Initial evaluations with ten middle school students demonstrated the effectiveness of this approach, using embodied metaphors, such as "merging light" to represent addition. To overcome the limitations of the physical optical setup, we further explored how AR could embody multiplication, expand data vectors with more attributes, and enhance contextual understanding. Our findings provide valuable insights for designing embodied and tangible learning experiences that make AI concepts more accessible to young learners. 

Abstract Recently we constructed Mahler discrete residues for rational functions and showed they comprise a complete obstruction to the Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $$g(x^{p})-g(x)$$ for some rational function $g(x)$ and an integer $p> 1$. Here we develop a notion of $$\lambda $$-twisted Mahler discrete residues for $$\lambda \in \mathbb{Z}$$, and show that they similarly comprise a complete obstruction to the twisted Mahler summability problem of deciding whether a given rational function $f(x)$ is of the form $$p^{\lambda } g(x^{p})-g(x)$$ for some rational function $g(x)$ and an integer $p>1$. We provide some initial applications of twisted Mahler discrete residues to differential creative telescoping problems for Mahler functions and to the differential Galois theory of linear Mahler equations. 

Coagulation/flocculation is a widely used water and wastewater treatment process due to its low cost, simplicity, and effectiveness. However, the process is not effective in the treatment of per- and polyfluoroalkyl substances (PFAS), the presence and treatment of which is an ongoing challenge for water providers. Here, we explore cationic surfactant-enhanced coagulation as a process modification to target the removal of PFAS in existing coagulation/flocculation systems. Batch experiments, in jar testing apparatus, were performed to assess the removal of two short-chain and two long-chain PFAS at an initial concentration of 10 µg/L with the addition of cetyltrimethylammoniumg chloride (CTAC) as the coagulant-aid. Our findings suggest that elevated coagulant dose (60 mg/L of alum or 100 mg/L of FeCl3) coupled with the addition of a cationic surfactant (1 mg/L of CTAC) significantly enhanced the removal of both short-chain (perfluorobutane sulfonate: PFBS removal to >40%) and long-chain PFAS (perfluorooctanoic acid: PFOA and perfluorooctane sulfonate: PFOS removal to >80%), with FeCl3 showing better performance than alum. Sulfonates (PFBS, PFOS) were shown to be removed more efficiently compared to carboxylates (PFBA, PFOA), presumably due to their higher hydrophobicity leading to better interactions with the flocs. Furthermore, CTAC in combination with traditionally used additives such as Powdered Activated Carbon (PAC), served as a better aid for PFAS treatment and improved the removal of PFBS, PFOA, and PFOS to >98%. This study highlights that introducing a cost-effective pre-treatment with a cationic surfactant to existing conventional treatment systems can improve the performance efficiency in treating PFAS-contaminated waters.