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We introduce and study a relative cancellation property for associative algebras. We also prove a characterization result for polynomial rings which partially answers a question of Kraft. 

We study Poisson structures on a weighted polynomial algebra A of three variables defined by a homogeneous potential. We start with classifying potentials w and list all with isolated singularity. Based on the classification, we study the rigidity of A in terms of graded twistings and classify Poisson fraction fields of A/(w) for irreducible potentials w. Using Poisson valuations, we characterize the Poisson automorphism group of A when it has an isolated singularity extending a nice result of Makar-Limanov-Turusbekova-Umirbaev. Finally, Poisson cohomology groups are computed for new classes of Poisson polynomial algebras. 

Dimension reduction (DR) is inherently non- unique: multiple embeddings can preserve the structure of high-dimensional data equally well while differing in layout or geometry. In this paper, we formally define the Rashomon set for DR—the collection of ‘good’ embed- dings—and show how embracing this multi- plicity leads to more powerful and trustworthy representations. Specifically, we pursue three goals. First, we introduce PCA-informed alignment to steer embeddings toward prin- cipal components, making axes interpretable without distorting local neighborhoods. Sec- ond, we design concept-alignment regulariza- tion that aligns an embedding dimension with external knowledge, such as class labels or user-defined concepts. Third, we propose a method to extract common knowledge across the Rashomon set by identifying trustworthy and persistent nearest-neighbor relationships, which we use to construct refined embeddings withimprovedlocalstructurewhilepreserving global relationships. By moving beyond a sin- gle embedding and leveraging the Rashomon set, we provide a flexible framework for build- ing interpretable, robust, and goal-aligned visualizations. 

The formation of engineering identity was found to play a pivotal role in engineering students’ learning trajectories[1], motivation and persistence[2], sense of belonging[3], and long-term commitment to an engineering career[4]. Focusing on engineering courses with design projects, research on engineering identity produced convincing evidence that links design experiences to engineering identity formation[5]. However, students are not heavily involved in design projects in a typical engineering program until junior and senior years. Freshman and sophomore courses focusing on foundational knowledge rarely provide any design experience and thus have limited contribution to engineering identity development. Introducing Design Thinking (DT) to these foundational courses could address this shortcoming. Evolving from the practice of design, DT has been generalized as a creative, iterative, human-centric process for solving complex problems[6]. Researchers are exploring DT as a pedagogical framework to teach students how to think like a designer[7]. It was found that “intentional implementation, including organization and framing of design thinking pedagogy, was an essential foundation for fostering student interest.”[8] A peer learning and teaching approach is suggested for DT pedagogy[9]. Peer-Led-Team-Learning (PLTL) is an active learning pedagogy that has been shown to improve student performance, retention, and commitment to engineering[10]. PLTL is typically employed as a supplement to regular classroom teaching, in which students voluntarily sign up and participate in PLTL sessions out of class time. As such, only a portion of students in a class offering PLTL can participate[11]. This paper presents a pedagogical framework that integrates in-class PLTL (iPLTL) and DT to foster engineering identity in a two freshman courses for Mechanical and Aerospace Engineering (MAE) students. Preliminary data show a significant improvement in students’ learning outcomes and engagement. 

A variety of imaging systems are in use in oceanographic surveys, and the opto-mechanical configurations have become highly sophisticated. However, much less consideration has been given to the accurate reconstruction of imaging data. To improve reconstruction of particles captured by Focused Shadowgraph Imaging (FoSI)—a system that excels at visualizing low-optical-density objects, we developed a novel object detection algorithm to process images with a resolution of ~ 12 μm per pixel. Suggested improvements to conventional edge-detection methods are relatively simple and time-efficient, and more accurately render the sizes and shapes of small particles ranging from 24 to 500 μm. In addition, we introduce a gradient of neutral density filters as a part of the protocol serving to calibrate recorded gray levels and thus determine the absolute values of detection thresholds. Set to intermediate detection threshold levels, particle numbers were highly correlated with beam attenuation (c_p) measured independently. The utility of our method was underscored by its ability to remove imperfections (dirt, scratches and uneven illumination), and by capturing the transparent particle features such as found in gelatinous plankton, marine snow and a portion of the oceanic gel phase. 

Following International Ocean Discovery Program (IODP) Expedition 397, split archive halves from Site 397-U1385 were analyzed at 1 cm resolution using an Avaatech X-ray fluorescence (XRF) core scanner at the University of Cambridge (United Kingdom). The resulting Ca/Ti profiles were used to revise the composite depth scales of Integrated Ocean Drilling Program Sites 339-U1385 and 397-U1385. Expedition 339 recovered a nearly continuous sediment sequence to 151.5 meters below seafloor (mbsf) at Site U1385, except for a gap spanning Marine Isotope Stage (MIS) 11/Termination V at 53.06 mbsf. This interval was successfully retrieved during Expedition 397, completing the 1.45 My section. The existing depth scale for Site 339-U1385 was revised to integrate this missing sequence. Using dynamic programming, the Ca/Ti signal from Site 397-U1385 was stretched and compressed to align with the revised depth scale of Site 339-U1385. This alignment enables direct transfer of age models and isotopic records between the two. Below 1.45 Ma (148.83 m core depth below seafloor, Method A [CSF-A], or 176.64 m core composite depth below seafloor [CCSF]), the chronology at Site 397-U1385 is established through benthic oxygen isotope stratigraphy and orbital tuning. The isotope record aligns with Site 339-U1385 for MISs 41–47 and extends to MIS 61, where a hiatus (~1.72–1.87 Ma) removed MISs 62–70. Below the hiatus, pronounced precession-driven variations in Ca/Ti serve as the basis for astronomical tuning, extending the chronology to 5.3 Ma. The Site 397-U1385 record provides a near-continuous record of changes in sediment composition at millennial resolution to the base of the Pliocene.