Geometry of arithmetic qusicrystals
| Project/Area Number |
19K03504
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Meiji University |
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Principal Investigator |
Sunada Toshikazu 明治大学, 研究・知財戦略機構(中野), 研究推進員 (20022741)
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| Project Period (FY) |
2019-04-01 – 2022-03-31
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| Project Status |
Completed (Fiscal Year 2021)
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| Budget Amount *help |
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2021: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | 離散集合 / 非周期的結晶構造 / アイゼンシュタイン数 / 準結晶 / 一般化されたリーマン和 / ポアソンの和公式 |
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| Outline of Research at the Start |
本研究では、を一般化した和公式が成立するような離散集合を準結晶と呼ぶことにして、1次の合同式を満たすという意味で算術的に定義される離散集合を考察し、それが準結晶になるかどうかを問う。さらに、テスト関数に制限を設けたときに成り立つ一般化されたポアソンの和公式を考えることにより、より広いクラスの離散集合に対して「弱」準結晶の概念を定式化できる。研究代表者によるこれまでの研究で、原始的格子点の集合は弱準結晶であることが示されている。この事実を、さらに一般的な離散集合に対しても拡張したい。
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| Outline of Final Research Achievements |
For more than two years, I was significantly ill due to an intractable disease (a type of vasculitis) and a kidney disease, and he was repeatedly hospitalized and discharged. Therefore, my research did not proceed as expected, and he could not go out due to the spread of the new corona. This is why I have not used a significant portion of his Grants-in-Aid for Scientific Research.
For this reason, unfortunately, I have not yet provided an overview of our achievements. However, studies of discrete sets in Euclidean space have yielded some results. The most interesting subject is the discrete set defined "arithmetically". The discrete set related to the Eisenstein number as an typical example has been the subject of research so far. I have obtained knowledge about the distribution from the viewpoint of generalized Riemann sum. It is also interesting as an example of aperiodic crystal composition. If I feel better, I will continue my research in this direction.
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| Academic Significance and Societal Importance of the Research Achievements |
純粋数学のみならず、結晶構造の理論を通して物質科学の分野にも大きな意義を有している。病気のため、完成には至らなかったが、拙著の増補版を通して、広く数理科学の発展に貢献している。
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Report
(4 results)
Research Products
(4 results)
