Kenji Kajiwara | Last modified date：2020.06.19 |

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Academic Degree

Ph.D(engineering)

Country of degree conferring institution (Overseas)

No

Field of Specialization

integrable systems, discrete differential geometry

ORCID(Open Researcher and Contributor ID)

0000-0002-0543-9384

Total Priod of education and research career in the foreign country

00years00months

Outline Activities

I have been engaged in research and related educational activities in

the area of the theory of nonlinear integrable systemｓ and discrete differential geometry. As for social activities, I gave a lecture at the public

lectures organized by graduate school of mathematics in 2002. I also

wrote articles in mathematical magazines such as "Suugaku Seminar" and

in some computer magazines.

the area of the theory of nonlinear integrable systemｓ and discrete differential geometry. As for social activities, I gave a lecture at the public

lectures organized by graduate school of mathematics in 2002. I also

wrote articles in mathematical magazines such as "Suugaku Seminar" and

in some computer magazines.

Research

**Research Interests**

- Development of discrete differential geometry by means of discrete integrable systems

keyword : integrable systems, discrete systems, soliton equations, discrete differential geometry

2008.04. - Study of the Painlev\'e systems

keyword : integrable systems, discrete systems, Painlev\'e systems, special functions

1993.04Studies of the Painleve systems. - Ultradiscretization of solvable chaotic systems and tropical geometry

keyword : solvable chaotic systems, ultradiscretization, tropical geometry

2007.12～2011.03.

**Current and Past Project**

- We originate a novel geometry of discrete surfaces having the surfaces with good properties, such as developable surfaces, as the shape elements, incorporate the theory of aesthetic shapes and construct the technique of structural analysis/optimization on it. Under this framework, we develop an innovative software platform that enables structural design with high efficiency and low cost, endowed with beauty and artisticity, and guaranteeing security and safety.Total budget: 246,700,000JPY
- Work on integrable discrete dynamical systems; discrete integrable systems to boundary value problems of practical interest in heat and mass flow; the ability to construct new solutions of the differential equations for scale-invariant curvature-driven evolving curves and scrolls
- This project aims to solve long-standing problems in discrete dynamical systems that are of particular interest to physics, by using reflection groups to reveal unexpected geometric insights. Mathematics has the power to abstract crucial patterns from complex observations. Symmetries familiar in the real world, like the hexagonal patterns of honeycombs, arise inside convoluted structures in high-dimensional systems. By revealing the structure of space-filling polytopes in integrable systems, the project seeks to find sought-after reductions of high-dimensional discrete models to two dimensions. Expected outputs include new reductions to discrete Painlevé equations, new circle patterns useful for computer graphics and discrete holomorphic functions.

**Academic Activities**

**Papers**

1. | Sampei Hirose, Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, Discrete local induction equation, Journal of Integrable Systems, 10.1093/integr/xyz003, 4, 1, xyz003, 2019.06, [URL]. |

2. | Jun ichi Inoguchi, Kenji Kajiwara, Kenjiro T. Miura, Masayuki Sato, Wolfgang K. Schief, Yasuhiro Shimizu, Log-aesthetic curves as similarity geometric analogue of Euler's elasticae, Computer Aided Geometric Design, 10.1016/j.cagd.2018.02.002, 61, 1-5, 2018.03, In this paper we consider the log-aesthetic curves and their generalization which are used in CAGD. We consider those curves under similarity geometry and characterize them as stationary integrable flow on plane curves which is governed by the Burgers equation. We propose a variational formulation of those curves whose Euler–Lagrange equation yields the stationary Burgers equation. Our result suggests that the log-aesthetic curves and their generalization can be regarded as the similarity geometric analogue of Euler's elasticae.. |

3. | Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada, Geometric aspects of Painlevé equations, Journal of Physics A: Mathematical and Theoretical, 10.1088/1751-8121/50/7/073001, 50, 7, 2017.01, In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlev equations, with a particular emphasis on the discrete Painlevï¿½ equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on P^{1} P^{1} and classified according to the degeneration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.. |

4. | Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, Motion and Bäcklund transformations of discrete plane curves, Kyushu Journal of Mathematics, doi:10.2206/kyushujm.66.303, 66, 2, 303-324, 2012.10, We construct explicit solutions to the discrete motion of discrete plane curves that has been introduced by one of the authors recently. Explicit formulas in terms of the τ function are presented. Transformation theory of the motions of both smooth and discrete curves is developed simultaneously.. |

5. | Mike Hay, Kenji Kajiwara and Tetsu Masuda, Bilinearization and special solutions to the discrete Schwarzian KdV equation, Journal of Math-for-Industry, 3, 2011A, 53-62, 2011.04. |

6. | Kenji Kajiwara, Nobutaka Nakazono and Teruhisa Tsuda, Projective reduction of the discrete Painlevé system of type (A2+A1)(1), International Mathematical Research Notices, 10.1093/imrn/rnq089, Vol. 2010, article ID: rnq089, 2010.05. |

7. | Kenji Kajiwara, Masanobu Kaneko, Atsushi Nobe and Teruhisa Tsuda, Ultradiscretization of a solvable two-dimensional chaotic map assciated with the Hesse cubic curve , Kyushu Journal of Mathematics, 63巻2号315-338ページ, 2009.09. |

8. | Kenji Kajiwara, Atsushi Nobe and Teruhisa Tsuda, Ultradiscretization of solvable one-dimensional chaotic maps, Journal of Physics A: Mathematical and Theoretical, 41巻，395202, 2008.09. |

9. | Kenji Kajiwara and Yasuhiro Ohta, Bilinearization and Casorati determinant solution to the non-autonomous discrete KdV equation, Journal of the Physical Society of Japan, 77巻, 054004, 2008.05. |

10. | Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and Yasuhiko Yamada, Point configurations, Cremona transformations and the elliptic difference Painleve equation, Séminaires et Congrès , 14巻，175-204., 2007.08. |

11. | Taro Hamamoto, Kenji Kajiwara and Nicholas S. Witte, Hypergeometric solutions to the q-Painleve equation of type $(A_1+A_1')^{(1)}$, Interenational Mathematics Research Notices, 2006, Article ID 84619, 2006.10. |

12. | Nalini Joshi, Kenji Kajiwara and Marta Mazzocco, Generating Function Associated with the Determinant Formula for the Solutions of the Painlev\'e II Equation, Ast\'erisque, 297, 67-78, Vol.274(2004) 67-78, 2005.06. |

13. | Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and Yasuhiko Yamada, Cubic Pencils and Painlev\'e Hamiltonians, Funkcialaj Ekvacioj, Vol.48(1) (2005) 147-160, 2005.04. |

14. | Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and Yasuhiko Yamada, Construction of Hypergeometric Solutions to the q-Painlev\'e Equations, International Mathematical Research Notices, 24, 1439-1463, Vol.2005(24) (2005) 1439-1463, 2005.01. |

15. | Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and Yasuhiko Yamada, Hypergeometric solutions to the q-Painleve equations, International Mathematical Research Notices, 47, 2497-2521, 2004:47 (2004) 2497-2521, 2004.08. |

16. | Kenji Kajiwara, On a q-Painleve III equation.II: rational solutions, Journal of Nonlinear Mathematical Physics, Vol.22 282-303, 2003.08. |

17. | K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, 10E9 solution to elliptic Painleve equation, Journal of Physics A: Mathematica and General, Vol.36 L263-L272, 2003.05. |

18. | Kenji Kajiwara and Kinji Kimura, On a $q$-Painlev\'e III Equation. I: Derivations, Symmetry and Riccati Type Soutions, Journal of Nonlinear Mathematical Physics, Vol.10, 86-102, 2003.02. |

19. | Kenji Kajiwara, Masatoshi Noumi and Yasuhiko Yamada, q-Painlev\'e Systems Arising from q-KP Hierarchy, Letters in Mathematical Physics, 10.1023/A:1022216308475, 62, 3, 259-268, Vol.62, 259-268., 2002.12. |

20. | Katsunori Iwasaki, Kenji Kajiwara and Toshiya Nakamura, Generating Function Associated with the Rational Solutions of the Painlev\'e II Equation, Journal of Physics A: Mathematical and General, 10.1088/0305-4470/35/16/101, 35, 16, L207-L211, Vol.35, L207-L211, 2002.04. |

21. | Kenji Kajiwara, Masatoshi Noumi and Yasuhiko Yamada, A Study on the Fourth q-Painlev\'e Equation, Journal of Physics A: Mathematical and General, Vol.34,8563-8581, 2001.10. |

22. | Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and
Yasuhiko Yamada, Determinant Formulas for the Toda and Discrete Toda Equations, Funkcialaj Ekvacioj, Vol. 44,291-307, 2001.08. |

23. | Kenji Kajiwara and Yasuhiro Ohta, Determinant Structure of the Rational Solutions for the Painlev\'e IV Equation, Journal of Physics A: Mathematical and General, Vol. 31, 2431--2446, 1998.03. |

**Presentations**

**Membership in Academic Society**

- Australian Mathematical Society

Educational

**Educational Activities**

I have been in charge of mathematical courses of graduate school of mathematics, department of mathematics, faculty of engineering. I also supervise students at seminars of department of mathematics and graduate school of mathematics. Moreover, I organize "Kyushu Integrable Systems Seminar" and "La Trobe-Kyushu Joint Seminar on Mathematics for Industry" to provide a chance for graduate students to get in touch with the most recent researches.

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