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Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω.Let$G_0$G_{0}be the stabilizer of a point 𝛼 in Ω.The rank of 𝐺 is defined as the number of orbits of$G_0$G_{0}in Ω, including the trivial orbit$\{\alpha \}$\{\alpha\}.In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.

This study introduces a novel method to enhance numerical simulation accuracy for high-speed flows by refining the weighted essentially non-oscillatory (WENO) flux with higher-order corrections like the modified weighted compact scheme (MWCS). Numerical experiments demonstrate improved sharpness in capturing shock waves and stability in complex conditions like two interacting blast waves. Key highlights include simultaneous capture of small-scale smooth fluctuations and shock waves with precision surpassing the original WENO and MWCS methods. Despite the significantly improved accuracy, the extra computational cost brought by the new method is only marginally increased compared to the original WENO, and it outperforms MWCS in both accuracy and efficiency. Overall, this method enhances simulation fidelity and effectively balances accuracy and computational efficiency across various problems.

Chromoplasts are plant organelles with a unique ability to sequester and store massive carotenoids. Chromoplasts have been hypothesized to enable high levels of carotenoid accumulation due to enhanced sequestration ability or sequestration substructure formation. However, the regulators that control the substructure component accumulation and substructure formation in chromoplasts remain unknown. In melon (Cucumis melo) fruit, β-carotene accumulation in chromoplasts is governed by ORANGE (OR), a key regulator for carotenoid accumulation in chromoplasts. By using comparative proteomic analysis of a high β-carotene melon variety and its isogenic line low-β mutant that is defective in CmOr with impaired chromoplast formation, we identified carotenoid sequestration protein FIBRILLIN1 (CmFBN1) as differentially expressed. CmFBN1 expresses highly in melon fruit tissue. Overexpression of CmFBN1 in transgenic Arabidopsis (Arabidopsis thaliana) containing ORHis that genetically mimics CmOr significantly enhances carotenoid accumulation, demonstrating its involvement in CmOR-induced carotenoid accumulation. Both in vitro and in vivo evidence showed that CmOR physically interacts with CmFBN1. Such an interaction occurs in plastoglobules and results in promoting CmFBN1 accumulation. CmOR greatly stabilizes CmFBN1, which stimulates plastoglobule proliferation and subsequently carotenoid accumulation in chromoplasts. Our findings show that CmOR directly regulates CmFBN1 protein levels and suggest a fundamental role of CmFBN1 in facilitating plastoglobule proliferation for carotenoid sequestration. This study also reveals an important genetic tool to further enhance OR-induced carotenoid accumulation in chromoplasts in crops.