Interdisciplinary Approach to mathematical network theory

2016 Fiscal Year Final Research Report

• Project/Area Number: 26310203
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 特設分野
• Research Field: Mathematical Sciences in Search of New Cooperation
• Research Institution: Tohoku University
• Principal Investigator: obata nobuaki 東北大学, 情報科学研究科, 教授 (10169360)
• Co-Investigator(Kenkyū-buntansha): 長谷川 雄央 茨城大学, 理学部, 准教授 (10528425)
• Co-Investigator(Renkei-kenkyūsha): KONNO Norio 横浜国立大学, 大学院工学研究院, 教授 (80205575) ; SEGAWA Etsuo 東北大学, 大学院情報科学研究科, 准教授 (30634547)
• Research Collaborator: NAKAZAWA Takashi 東北大学, 大学院情報科学研究科, 助教 (20726765)
• Project Period (FY): 2014-07-18 – 2017-03-31
• Keywords: ネットワーク / 関数解析 / 確率論 / 量子確率解析 / スペクトル解析 / 量子ウォーク / ランダムウォーク / 直交多項式

Outline of Final Research Achievements

Based on the non-commutative analysis, we aimed at developing an interdisciplinary approach to mathematical theory of networks. The long-time behaviour of energy levels in the lazor experiment of isotope separation was obtained by means of spectral analysis of quantum walk and we examined a good fitting to the physical experiment. We derived complex spectra of some Manhattan products of directed paths. The Gauss-Poisson distributions were derived as the asymptotic spectral distributions of strong regular graphs by means of two-variable orthogonal polynomials.　Applying the Kronecker product to two-dimensional restricted lattices, we obtained the spectral density in terms of elliptic integrals. We derived the distribution of connected components at the critical phase of percolation on the Cayley tree. We analyzed the mechanism of the outbreak for an infection model with multiple sources of infection.

Free Research Field

量子確率論・スペクトル解析