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Abstract: Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G,r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we consider percolation numbers of strong products of graphs. If $G$ is the strong product $G_1\boxtimes \cdots \boxtimes G_k$ of $k$ connected graphs, we prove that $m(G,r)=r$ as soon as $r\le 2^{k-1}$ and $|V(G)|\ge r$. As a dichotomy, we present a family of strong products of $k$ connected graphs with the $(2^{k-1}+1)$-percolation number arbitrarily large. We refine these results for strong products of graphs in which at least two factors have at least three vertices. In addition, when all factors $G_i$ have at least three vertices we prove that $m(G_1 \boxtimes \dots \boxtimes G_k,r)\leq 3^{k-1} -k$ for all $r\leq 2^k-1$, and we again get a dichotomy, since there exist families of strong products of $k$ graphs such that their $2^{k}$-percolation numbers are arbitrarily large. While $m(G\boxtimes H,3)=3$ if both $G$ and $H$ have at least three vertices, we also characterize the strong prisms $G\boxtimes K_2$ for which this equality holds. Some of the results naturally extend to infinite graphs, and we briefly consider percolation numbers of strong products of two-way infinite paths.
Keywords: bootstrap percolation, strong product of graphs, infinite path
Published in DiRROS: 20.11.2024; Views: 103; Downloads: 50
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Keywords: Sweet potato, Flesh colour, Nutritional profile, Cropping system, Genetic makeup, Linear discriminant analysis
Published in DiRROS: 19.11.2024; Views: 161; Downloads: 2594
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Published in DiRROS: 19.11.2024; Views: 180; Downloads: 2219
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Abstract: This study evaluated the effects of a neuromuscular training (NMT) warm-up program on injury incidence,neuromuscular function, and program adherence, maintenance and acceptance in adolescent basketballplayers. A total of 275 players from 20 Slovenian teams (15 ± 1.7 years of age), were randomized into anintervention group (IG, n=129) and a control group (CG, n=146). Over three months, the IG incorporatedNMT into their warm-ups, while the CG followed their usual practice. Measurements of body anthro-pometry, muscle contractile properties, and balance were taken before and after the intervention. Also,the injury incidence, training adherence and maintenance were reported. Both groups showed improvedbalance, with no significant difference between them. However, IG demonstrated reduced delay times inspecific muscles, indicating improved neuromuscular function. Injury prevalence proportion (%) duringthe whole study period was higher in the control group compared to intervention (IG: 10.9% vs. CG:23.3%), and incidence rate. Moreover, the incidence rate ratio for sustaining an injury was 2.6 on average(ranging from 0.88 to 7.07 for tendon and muscle injuries, respectively), indicating significantly lowerinjury risk in IG than CG. These findings highlight the effectiveness of NMT warm-ups in reducing injuryrisk and enhancing neuromuscular function, emphasizing the value of structured injury preventionstrategies in youth sports.
Keywords: muscle contractile properties, balance, injury prevalence, training adherence
Published in DiRROS: 18.11.2024; Views: 178; Downloads: 94
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Abstract: We consider a multiphase spectral problem on a stratified Lie group.We prove the existence of an eigenfunction of $(2, q)$-eigenvalue problem on a bounded domain. Furthermore, we also establish a Pohozaev-like identity corresponding to the problem on the Heisenberg group.
Keywords: multiphase spectral problem, stratified Lie group, Heisenberg group, left invariant vector field, (2, q)-eigenvalue problem
Published in DiRROS: 15.11.2024; Views: 187; Downloads: 84
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