Purdue University Fort Wayne

This study extends prior efforts to assess mathematical knowledge for teaching (MKT) by advancing the MKT-Fractions instrument, focusing specifically on teachers’ knowledge for teaching fractions. Building an earlier measure centered on Knowledge of Content and Students (KCS), this phase integrates items targeting Specialized Content Knowledge (SCK) and Knowledge of Content and Teaching (KCT). Drawing on responses from 103 preservice teachers across two teacher preparation programs, we employed Rasch modeling to examine evidence for internal structure and item functioning. Results indicated strong item reliability, unidimensionality, and a coherent item hierarchy aligned with a construct map that represents the developmental trajectory of knowledge needed to teach fractions. The instrument effectively differentiated item difficulty across domains without significant variation in performance across programs. These findings support the use of the MKT-Fractions instrument as a tool for assessing preservice teachers’ readiness and for informing teacher education practices focused on fraction instruction.

Search and rescue operations save thousands of lives yearly, and robotics can further improve survival rates in hazardous, partially observable environments. This article introduces a trustworthy human-autonomy teaming (HAT) navigation framework that fuses human situational awareness with algorithmic planning across both global and local layers. At the global level, a hybrid graph integrates generalized Voronoi diagrams (GVDs), for high-clearance, topology-preserving roadmaps interpretable to operators, with a spatiotemporal graph that dynamically updates edges and node salience as obstacles and mission priorities evolve. Human input is incorporated through a node optimization protocol that prioritizes access nodes, adjusts waypoint sequences, and invokes RRT*-informative path planning when jump-search shortcuts become unreliable, thereby aligning exploration with operator-defined areas of interest and mission constraints. At the local level, a bioinspired neural network (BNN) with an adaptive window strategy provides reactive obstacle avoidance and target tracking, ensuring smooth and safe motion in cluttered, dynamic scenes. Simulation results across indoor and urban-like environments demonstrate that the proposed framework enhances path efficiency, robustness to environmental change, and responsiveness while reducing unnecessary re-planning. By unifying GVD-based global structure, human-guided decision-making, and BNN-based local reactivity, the HAT framework advances autonomous navigation toward greater safety, adaptability, and effectiveness in real-world search and rescue operations.

In this paper, we provide necessary and sufficient conditions for the existence and multiplicity of periodic smooth solutions to a hybrid system of differential equations that describes animal running at constant angular velocity during stance. We show that solutions to this system exist if and only if the angular velocity during stance is lower than the angular velocity of the corresponding linear harmonic oscillator and satisfies a non-resonance condition. The proof of our main theorem relies on a combination of tools from oscillation theory for ODEs, intersection theory for real plane conics and the analysis of Poincaré maps. Furthermore, we demonstrate the validity of our result with numerical simulations for combinations of parameters corresponding to periodic, resonant, and unbounded solutions. Finally, we show that our model fits accurately publicly available data on human running on different conditions, including comfortable and fast running on a treadmill and overground.

Abstract Digital twins (DT) have been widely applied to model complex manufacturing processes and products for various purposes such as implementing controls, monitoring operations, and identifying bottlenecks of manufacturing processes. This paper aims to develop a DT model for a precise wire drawing process with dozens of passes. Metal microwires are required to possess high quality and precision since they are mostly used in micro-electro-mechanical systems (MEMS) in health care. Since microwires are drawn through dozens of passes to reduce their sizes gradually, it is unavoidable that the whole wire-drawing process is affected by numerous factors, and thus might cause defects in production. The developed DT model will help to analyze the effects of main design factors on products, predict the performance of a process, and identify critical ones when a defect occurs. Distinguishing from existing works, this paper emphasizes the effect of existing residual stresses and strains generated in previous passes on these in sequential passes, and it is our finding that only residual strains will be accumulated pass by pass; while the ranges of residual stresses stay at a similar level. This further implies that over-accumulated residual strains, rather than residual stresses, should be emphasized when a defect occurs to a sequential pass.