We consider star-shaped, strictly mean convex and closed hypersurfaces expanding by a class of inverse mean curvature flows in R n + 1 \mathbb {R}^{n+1} , and we prove that this evolution exists for all time and the evolving hypersurfaces converge smoothly to a round sphere after rescaling.

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Plant polyploidization is frequently associated with changes in nutrient contents. However, the possible contribution of metabolites to this change has not been investigated by characterizing the metabolite contents of diploid and tetraploid forms of rice (Oryza sativa L.). We compared the metabolites of a group of diploid–tetraploid japonica brown rice and a group of diploid–tetraploid indica brown rice based on liquid chromatography–tandem mass spectrometry. In total, 401 metabolites were identified; of these, between the two diploid–tetraploid groups, 180 showed opposite expression trends, but 221 showed the same trends (147 higher abundance vs. 74 lower abundance). Hierarchical cluster analysis of differential metabolites between diploid and tetraploid species showed a clear grouping pattern, in which the expression abundance of lipids, amino acids and derivatives, and phenolic acids increased in tetraploids. Further analysis revealed that the lipids in tetraploid rice increased significantly, especially unsaturated fatty acids and phospholipids. This study provides further basis for understanding the changes in rice nutritional quality following polyploidization and may serve as a new theoretical reference for breeding eutrophic or functional rice varieties via polyploidization.

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