On cell problems for Hamilton-Jacobi equations with non-coercive Hamiltonians and its application to homogenization problems

We study a cell problem arising in homogenization for a Hamilton-Jacobi equation whose Hamiltonian is not coercive. We introduce a generalized notion of effective Hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective Hamiltonian. Under some sufficient conditions, the result is applied to the associated homogenization problem. We also show that homogenization for non-coercive equations fails in general

Eikonal equations in metric spaces

A new notion of a viscosity solution for Eikonal equations in a general metric space is introduced. A comparison principle is established. The existence of a unique solution is shown by constructing a value function of the corresponding optimal control theory. The theory applies to in- nite dimensional setting as well as topological networks, surfaces with singularities

非ユークリッド距離構造の下での粘性解の安定性について

学位の種別: 課程博士審査委員会委員 : （主査）東京大学教授 儀我 美一, 東京大学教授 片岡 清臣, 東京大学教授 山本 昌宏, 東京大学教授 中村 周, 東京大学准教授 下村 明洋, 早稲田大学教授 石井 仁司University of Tokyo(東京大学

Diurnal metabolic regulation of isoflavones and soyasaponins in soybean roots

Isoflavones and soyasaponins are major specialized metabolites accumulated in soybean roots and secreted into the rhizosphere. Unlike the biosynthetic pathway, the transporters involved in metabolite secretion remain unknown. The developmental regulation of isoflavone and soyasaponin secretions has been recently reported, but the diurnal regulation of their biosynthesis and secretion still needs to be further studied. To address these challenges, we conducted transcriptome and metabolite analysis using hydroponically grown soybean plants at 6-hr intervals for 48 hr in a 12-hr-light/12-hr-dark condition. Isoflavone and soyasaponin biosynthetic genes showed opposite patterns in the root tissues; that is, the former genes are highly expressed in the daytime, while the latter ones are strongly induced at nighttime. GmMYB176 encoding a transcription factor of isoflavone biosynthesis was upregulated from ZT0 (6:00 a.m.) to ZT6 (12:00 a.m.), followed by the induction of isoflavone biosynthetic genes at ZT6. The isoflavone aglycone content in the roots accordingly increased from ZT6 to ZT18 (0:00 a.m.). The isoflavone aglycone content in root exudates was kept consistent throughout the day, whereas that of glucosides increased at ZT6, which reflected the decreased expression of the gene encoding beta-glucosidase involved in the hydrolysis of apoplast-localized isoflavone conjugates. Co-expression analysis revealed that those isoflavone and soyasaponin biosynthetic genes formed separate clusters, which exhibited a correlation to ABC and MATE transporter genes. In summary, the results in this study indicated the diurnal regulation of isoflavone biosynthesis in soybean roots and the putative transporter genes responsible for isoflavone and soyasaponin transport