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-1 2 1 +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 [ 23p 1 + 3. This improves a bound of Joret, Kaminski, and Theis. Our proof relies on demonstrating that all (Pk, claw, butterfly, C4, C5)-free graphs have cop number at most [k-1 3 1 +3. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
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.
Organizational diversity practices are widely implemented to support the workplace integration of international migrants (IMs), yet their outcomes remain inconsistent. Little is known about why IMs respond differently to the same initiatives offered by their organizations or why these practices sometimes fail or backfire. Drawing on signaling and attribution theory, we argue that IMs interpret diversity practices through the lens of two key disadvantages: unfamiliarity with the destination country and exposure to discrimination within this context. We develop a dual disadvantage typology of IMs and theorize how resource and nondiscrimination practices differentially affect organizational commitment depending on migrants’ disadvantage configurations. This typology refines existing diversity frameworks by introducing a configurational view of migrant diversity and explaining why identical practices may produce divergent outcomes. Using survey data from 394 IMs working in Germany, we find that while nondiscrimination practices are consistently positively associated with commitment, the effects of resource practices vary: some IMs view them as empowering, while others interpret them as stigmatizing or misaligned with their needs. In particular, training initiatives can backfire for IMs who are already familiar with the labor market and face discrimination. Our findings demonstrate the importance of tailoring initiatives to different migrant groups. We contribute a theory-driven framework that explains how diversity signals are differentially interpreted across migrant disadvantage configurations and when diversity practices produce intended versus unintended effects.
Background and objective This study presents a computationally and physiologically grounded framework for electromyography (EMG) simulation to examine how assumed cortical and subcortical inputs shape bilateral EMG and intermuscular coherence during bimanual coordination. Methods Simulated first dorsal interosseous (FDI) EMG was generated for a 1:2 force coordination task (approximately 40% maximum voluntary contraction) using motor unit models incorporating (i) variable cortical and subcortical drive weighting, (ii) narrowband versus broadband cortical drive (13–30 vs. 13–60 Hz), and (iii) asymmetric bilateral coupling. Intermuscular EMG–EMG wavelet coherence was computed across α, β, low-γ, and high-γ bands. Model outputs were validated against experimental human EMG at the signal level, comparing simulated and recorded bipolar first dorsal interosseous traces across eight amplitude and frequency-domain signal features. Results Coherence magnitude increased with greater modeled neural crosstalk and cortical weighting. The frequency band most sensitive to drive weighting depended on cortical bandwidth: β-band coherence (13–30 Hz) best differentiated cortical and subcortical influences under narrowband drive, whereas low-γ (30–60 Hz) was most informative under broadband conditions. α-band coherence showed minimal sensitivity to subcortical modulation in this context. Conclusions Within this simulation framework, intermuscular coherence was sensitive to modeled bilateral coupling and cortical-subcortical weighting, with diagnostic sensitivity depending on the spectral bandwidth of cortical drive. Thus, the model should be interpreted as a hypothesis generating, signal level framework rather than a direct neural measurement. This open-source framework bridges computational modeling and biomarker signal analysis, supporting reproducible, hypothesis driven biomarker development for neuromotor disorders such as Parkinson’s disease, spinal cord injury, and stroke.
Let G be a graph of order nand size m. If G is a connected graph of order at least 5, we show that chi(G) <= inverted right perpendicularm/root ninverted left perpendicular. Moreover, we prove that if the girth of G is greater than 2k + 1, then chi (G) <= inverted right perpendicular(k+2)root(k + 2)minverted left perpendicular + 1. It was proved that for every graph G with no isolated vertices chi(G) <= 2H(G), where H(G) = Sigma(uv is an element of E(G)) 2/d(u)+d(v) . We improve this result with a shorter proof by showing that col(G) <= 2H(G), where col(G) = max{delta(H) + 1: H subset of 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 chi(G) <= inverted right perpendicularH(G)inverted left perpendicular. (c) 2026 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
