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Abstract We establish a modified notion of Nash equilibrium learning—convergence of the population state to the Nash equilibria set—in a generalization of the standard population games and evolutionary dynamics framework using system-theoretic passivity methods. In this setting, we allow each strategy to involve a sequence of sub-tasks that must be completed before strategy revision so long as the durations of the sub-tasks can be modeled with Erlang or exponential distributions. Furthermore, several canonical classes of natural learning rules are established and useful properties are derived. 

Math anxiety is negatively correlated with student performance and can result in avoidance of further math/STEM (science, technology, engineering, and mathematics) classes and careers. Cooperative learning (i.e., group work) is a proven strategy that can reduce math anxiety and has additional social and pedagogical benefits. However, depending on the group individuals, some peer interactions can mitigate anxiety, while others exacerbate it. We propose a mathematical modeling approach to help untangle and explore this complex dynamic. We introduce a modification of the Hegselmann–Krause bounded confidence model, including both attractive and repulsive interactions to simulate how math anxiety levels are affected by pairwise student interactions. The model is simple but provides interesting qualitative predictions. In particular, Monte Carlo simulations show that there is an optimal group size to minimize average math anxiety, and that switching group members randomly at certain frequencies can dramatically reduce math anxiety levels. The model is easily adaptable to incorporate additional personal and societal factors, making it ripe for future research. 

Abstract Mass-loss influences stellar evolution, especially for massive stars with strong winds. Stellar wind bow shock nebulae driven by Galactic OB stars can be used to measure mass-loss rates ( $\dot{M}$). The standoff distance (R₀) between the star and the bow shock is set by momentum flux balance between the stellar wind and the surrounding interstellar medium (ISM). We created the Milky Way Project: mass-loss rates for OB Stars driving infrared bow shocks (MOBStIRS) using the online Zooniverse citizen science platform. We enlisted several hundred students to measureR₀and two other projected shape parameters for 764 cataloged infrared bow shocks. MOBStIRS incorporated 1528 JPEG cutout images produced from Spitzer GLIMPSE and MIPSGAL survey data. Measurements were aggregated to compute shape parameters for each bow shock image deemed high quality by participants. The average statistical uncertainty onR₀is 12.5% but varies from <5% to ∼40% among individual bow shocks, contributing significantly to the total error budget of $\dot{M}$. The derived nebular morphologies agree well with (magneto) hydrodynamic simulations of bow shocks driven by the winds of OB stars moving atV_a = 10–40 km s⁻¹with respect to the ambient ISM. A systematic correction toR₀to account for viewing angle appears unnecessary for computing $\dot{M}$. Slightly more than half of MOBStIRS bow shocks are asymmetric, which could indicate anisotropic stellar winds, ISM clumping on sub-pc scales, time-dependent instabilities, and/or misalignments between the local ISM magnetic field and the star-bow-shock axis. 

ABSTRACT Geographic variation in ecosystem function is often attributed to differences in climate and soil properties, with biophysical constraints assumed to dictate spatial patterns in nutrient cycling, carbon storage, and plant productivity. However, biotic interactions, particularly herbivory, also vary geographically and can generate feedbacks that influence ecosystem processes. Using a replicated three‐year field experiment, we tested how population‐level functional differences in a widespread arthropod herbivore mediate geographic variation in ecosystem function. Structural equation modeling revealed that herbivores exerted strong direct effects on plant biomass, soil carbon, and nitrogen mineralization, often surpassing the influence of historical conditions and geographic variation in climate. Moreover, functionally distinct herbivore populations had divergent effects on nutrient cycling and plant diversity, demonstrating that population‐level differences introduce novel pathways of influence on ecosystem function. These findings challenge ecosystem models that prioritize abiotic constraints and highlight the need to incorporate consumer‐driven feedbacks into ecological frameworks. 

The application of statistical modeling in organic chemistry is emerging as a standard practice for probing structure-activity relationships and as a predictive tool for many optimization objectives. This review is aimed as a tutorial for those entering the area of statistical modeling in chemistry. We provide case studies to highlight the considerations and approaches that can be used to successfully analyze datasets in low data regimes, a common situation encountered given the experimental demands of organic chemistry. Statistical modeling hinges on the data (what is being modeled), descriptors (how data are represented), and algorithms (how data are modeled). Herein, we focus on how various reaction outputs (e.g., yield, rate, selectivity, solubility, stability, and turnover number) and data structures (e.g., binned, heavily skewed, and distributed) influence the choice of algorithm used for constructing predictive and chemically insightful statistical models. 

Heteroaromatics are the basis for many pharmaceuticals. The ability to modify these structures through selective core-atom transformations, or “skeletal edits”, can dramatically expand the landscape for drug discovery and development. However, despite the importance of core-atom modifications, the quantitative impact of such transformations on accessible chemical space remains undefined. Here, we report a cheminformatic platform to analyze which skeletal edits would most increase access to novel chemical space. This study underscores the significance of emerging single and multiple core-atom transformations of heteroaromatics in enhancing chemical diversity, for example, at a late-stage of a drug discovery campaign. Our findings provide a quantitative framework for prioritizing core-atom modifications in heteroaromatic structural motifs, calling for the development of new methods to achieve these types of transformations.