| Project/Area Number |
12554004
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 展開研究 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | KYOTO UNIVERSITY (2001-2003)
Osaka University (2000) |
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Principal Investigator |
NAKAMURA Yoshimasa Kyoto University, Graduate School of Informatics, Professor, 情報学研究科, 教授 (50172458)
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| Co-Investigator(Kenkyū-buntansha) |
IMAI Jun NTT Communication Science Laboratories, Researcher, コミュニケーション科学基礎研究所, 主任研究員
NAKAYAMA Isao Nagoya University of Commerce & Business, Faculty of Management Information Science, Professor, 経営情報学部, 教授 (80164359)
SHIROTA Norihisa Sony Corporation, Laboratories for Information & Network, Manager, インフォメーション&ネットワーク研究所, 統括部長
KONDO Koichi Doshisha University, Faculty of Engineering, Lecturer, 工学部, 専任講師 (30314397)
OKAZAKI Ryotaro Doshisha University, Faculty of Engineering, Lecturer, 工学部, 専任講師 (20268113)
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| Project Period (FY) |
2000 – 2003
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| Keywords | discrete integrable systems / continued fractions / Pade approximation / Painleve equations / Laplace transform / applied integrable systems |
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| Research Abstract |
There has not been known a continued fraction expansion of order O(N^2) for the Perron continued fraction, which emerges in the Carathe\'odory interpolation problem, such as the qd algorithm for the Chebyshev continued fraction. First Nakamura and coworkers, being based on the orthogonal polynomials on the unit circle, derived a new integrable system named the Schur flow which has a Lax representation given by the three terms recurrence relation. Secondly in terms of the discrete Schur flow they designed a new continued fraction expansion algorithm of order O(N^2) for the Perron continued fraction and its application to algorithm for computing zeros of certain algebraic equations. Consequently, the new correspondence
1)classical orthogonal polynomials -Chebyshev continued fraction -Toda equation
2)orthogonal polynomials on the unit circle -Perron continued fraction -Schur flow
is revealed.
They also considered the Thron continued fraction through the relativistic Toda equation having a Lax representation given by the three terms recurrence relation for the bi-orthogonal polynomials. An integrable discretization of the equation enable them to design a new continued fraction algorithm of order O(N^3) for the Thron fraction. This algorithm has an advantage that it computes the continued fraction for the case where the FG algorithm does not work.
Nakamura showed that a Pad\'e approximation, namely, a continued fraction expansion of the Laplace transform of the Airy function can be computed in a pure algebraic manner.
Each coefficients of the continued fraction is connected by the By\"acklund transformation of the second Painlev\'e equation PII, where one of the Lax pair is just the recurrence relation of orthogonal polynomials.
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