2018 Fiscal Year Final Research Report
The relationship between the behaviour of random walks and the spectral structure via some combinatorial geometries of graphs
| Project/Area Number |
25400208
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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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| Research Field |
Foundations of mathematics/Applied mathematics
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| Research Institution | Showa University |
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Principal Investigator |
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| Research Collaborator |
SEGAWA Etsuo
NOMURA Yuji
OGURISU Osamu
Sato Iwao
KONNO Norio
ANDO Kazunori
MORIOKA Hisashi
SUZUKI Akito
TANAKA Mikihiro
Portugal Rento
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| Project Period (FY) |
2013-04-01 – 2019-03-31
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| Keywords | グラフ理論 / スペクトル幾何 / ラプラシアン / 酔歩 / 量子ウォーク / 状態密度函数 / 被覆構造 / 脳内辞書ネットワーク |
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| Outline of Final Research Achievements |
Let us show three topics in our research supported by this grant. One is about quantum walks: we showed the spectral relationship between quantum walk and its underlying random walk. This kind of relation is now called “Spectral mapping theorem in quantum walk” and widely applied in this area. Next one is about some resonance of discrete Laplacians on a graph. We can characterize the intensities of potentials to keep the existence of edge-eigenvalues, which implies some eigenvalues whose value coincides with the lowest of continuous spectrum. Moreover we succeeded in construction of algebraic variety composed by such intensites. Finally, we tried to apply our basic results to one of other fields than mathematics: mental lexicon, which is one of topics in psycholinguistics.
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| Free Research Field |
離散スペクトル幾何
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| Academic Significance and Societal Importance of the Research Achievements |
酔歩がより速く伝播するグラフの特徴付けは，未だ未解決な部分が多い双曲的なグラフの解明，さらには現実社会におけるネットワークの構築に貢献できるものである．一方，酔歩よりも速い拡散性を持つ量子ウォークへの研究成果は，今後より現実的に扱われるだろう量子コンピューティングなどの“量子化”された媒体での実装に貢献できると考えている．つまり当該研究は現状と未来を見据えた意義をもたらすと期待しても良いだろう．
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