In 2019, Sivaraman conjectured that every Pk-free graph has cop number at most k−3. In the same year, Liu proved this conjecture for (Pk,claw)-free graphs. Recently Chudnovsky, Norin, Seymour, and Turcotte proved this conjecture for P5-free graphs. For k≥6 the conjecture remains widely open. Let the E graph be the claw with two subdivided edges. We show that all (Pk,E)-free graphs have cop number at most ⌈k−12⌉+3, which improves and generalizes Liu's result for (Pk,claw)-free graphs. We also prove that if G is a graph whose longest path is order p, then G has cop number at most ⌈2p3⌉+3. This improves a bound of Joret, Kamiński, and Theis. Our proof relies on demonstrating that all (Pk,claw,butterfly,C4,C5)-free graphs have cop number at most ⌈k−13⌉+3.
This qualitative study examines communication in family mathematics settings, focusing on how embodied communication is constituted during parent-child interaction with a multitouch technology application, TouchCounts. Moving beyond information-transmission perspectives, the study adopts an embodied and relational approach centered on affectivity-conceptualized as the circulation of attunement, resonance, and intensity of movement and feeling across human and non-human components of the parent-child-TouchCounts assemblage. Drawing on close qualitative analysis of two selected video-recorded excerpts, the study traces how mathematical events unfold moment by moment through bodily action and material engagement, and how affective flow shapes how participants respond to one another and to the digital interface. The findings show that affectivity operates as a constitutive dimension of mathematical communication, shaping how mathematical concepts (e.g., addition toward bigness, making two by V-gesture) are enacted, oriented, sustained, and transformed as lived events. By foregrounding affect and embodiment, this study offers a novel perspective on communication in family mathematics and contributes to broader discussions on mathematical meaning-making in technology-mediated contexts.
Let G be a graph of order n and size m. If G is a connected graph of order at least 5, we show that χ(G)≤⌈mn⌉. Moreover, we prove that if the girth of G is greater than 2k+1, then χ(G)≤⌈(k+2)mk+2⌉+1. It was proved that for every graph G with no isolated vertices χ(G)≤2H(G), where H(G)=∑uv∈E(G)2d(u)+d(v). We improve this result with a shorter proof by showing that col(G)≤2H(G), where col(G)=max{δ(H)+1:H⊆G}. In the present paper, it is shown that if G is a triangle-free graph of order at least 3 with no isolated vertex, then χ(G)≤⌈H(G)⌉.
Maintaining optimal membrane hydration is critical for the performance and durability of proton exchange membrane fuel cell (PEMFC) stacks. While membrane hydration cannot be directly measured, the high-frequency resistance (HFR) obtained from electrochemical impedance spectroscopy (EIS) has been shown to be strongly correlated with membrane hydration. Recent advances in onboard embedded diagnostics have made online estimation of PEMFC membrane hydration based on HFR increasingly feasible, enabling real-time implementation in deployed systems. However, this requires that HFR be systematically mapped to well-defined reference states corresponding to known and spatially uniform membrane hydration levels. To this end an experimental investigation was conducted on a short-stack PEMFC operated under non-reactive H-2/N-2 conditions to establish this relationship. HFR measurements were obtained over a wide range of relative humidity (RH) levels (30-120%) and stack temperatures (50-80 degrees C), including both increasing and decreasing RH profiles. The results show that HFR decreases significantly with increasing RH, with up to an 87% reduction between dry and fully humidified conditions. A pronounced hysteresis of up to similar to 10% was observed between increasing and decreasing RH cycles, particularly in the 40-80% RH range, indicating a strong dependence on hydration history. The hysteresis behavior observed in the HFR-RH relationship is primarily driven by variations in RH, while the influence of temperature hysteresis loop area is comparatively less significant within the investigated operating range. Finally, a physics-informed mathematical model, inspired by the membrane conductivity relation [1], was developed to correlate HFR with membrane water activity and temperature based on empirical formulations of water uptake and ionic conductivity. To account for hysteresis effects, a sigmoid-based function was incorporated into the primary correlation to capture the path-dependent hydration behavior. The resulting model predicts HFR as a function of RH, stack temperature, and the direction of the most recent RH sequence (increasing (+) or decreasing (-)). The model demonstrated high predictive capability when validated against independent experimental datasets, achieving a root mean square error (RMSE) of 0.59 m Omega and a coefficient of determination (R-2) of 0.98.
