Towards new mathematical tools for understanding chemical reaction networks
| Project/Area Number |
18K18723
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| Research Category |
Grant-in-Aid for Challenging Research (Exploratory)
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| Allocation Type | Multi-year Fund |
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| Review Section |
Medium-sized Section 12:Analysis, applied mathematics, and related fields
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| Research Institution | Chubu University |
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Principal Investigator |
Arai Zin 中部大学, 創発学術院, 教授 (80362432)
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| Project Period (FY) |
2018-06-29 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥6,370,000 (Direct Cost: ¥4,900,000、Indirect Cost: ¥1,470,000)
Fiscal Year 2020: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2019: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000)
Fiscal Year 2018: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
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| Keywords | 反応経路網 / 同変モース理論 / コンレイ指数 / 計算トポロジー / グラフ理論 |
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| Outline of Final Research Achievements |
We have developed a mathematical framework for visualizing and realizing quantum chemical reaction networks. Following our recent work to reduce a dimension of a set of reference structures along the intrinsic reaction coordinate by a classical multidimensional scaling approach, we proposed the method to project on-the-fly trajectories into a reduced-dimension subspace determined by the network. By extending the equivariant Morse theory, we have also developed an equivariant version of the Conley index theory for dynamical systems with a group action. It is then applied to the study of complex dynamical systems and used to prove the topological properties of the Julia set of the quadratic map and the Henon map.
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| Academic Significance and Societal Importance of the Research Achievements |
これまで，研究者の経験と直観によっていた，量子化学計算の可視化に対して，数理的に明確な方法によるアプローチが確立された．これにより，詳細な反応軌道のデータを，化学的にも数学的にも自然にグラフ内に埋め込むことができるようになり，反応自体のより深い理解に繋がるものと期待できる．同変コンレイ指数理論を展開して複素力学系に応用した成果についていうと，この結果は，複素力学系に限定したものではなく，群作用がある力学系に対して一般的に適用できると期待され，対称性のある力学系の分岐理論など，将来のさらなる応用に繋がる可能性を持っている．
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Report
(6 results)
Research Products
(12 results)
