OCHIAI HIROYUKI | Last modified date：2023.06.09 |

Graduate School

Undergraduate School

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website for research, education of Hiroyuki Ochiai .

Academic Degree

Ph.D(Math. Sci.)

Country of degree conferring institution (Overseas)

No

Field of Specialization

Algebraic Analysis

Total Priod of education and research career in the foreign country

01years00months

Outline Activities

Research interest is Algebraic Analysis, Representation Theory and Special Functions. I give courses on these topics as well as undergraduate linear algebra, complex variables, and differential equations for undergraduate and those for graduate students.

Research

**Research Interests**

- Representation theory of real reductive Lie groups

keyword : Algebraic Analysis

2009.10～2033.03.

**Academic Activities**

**Books**

1. | Ken Anjyo, Hiroyuki Ochiai, Mathematical Basics of Motion and Deformation in Computer Graphics, Second Edition, Morgan and Claypool, https://doi.org/10.1007/978-3-031-02592-1, 2017.04, [URL]. |

2. | Hiroyuki Ochiai, Ken Anjyo, Mathematical Progress in Expressive Image Synthesis II, Springer-Verlag, 2015.06, Selected papers from the proceeding of MEIS2014. |

3. | Ken Anjyo, Hiroyuki Ochiai, Mathematical Progress in Expressive Image Synthesis I, Springer-Verlag, 2014.08, Selected papers from the proceeding of MEIS2013. |

4. | Ken Anjyo, Hiroyuki Ochiai, Mathematical Basics of Motion and Deformation in Computer Graphics, Morgan & Claypool Publishers, doi:10.2200/S00599ED1V01Y201409CGR017, 2014.10, [URL], The unique introduction of the mathematical background to the computer graphics. |

**Papers**

1. | Keisuke Hakuta, Hiroyuki Ochiai, Tsuyoshi Takagi, Comments on efficient batch verification test for digital signatures based on elliptic curves, MATHEMATICA SLOVACA, 10.1515/ms-2022-0038, 72, 3, 575-590, 2022.06. |

2. | Hiroyuki Ochiai, Yoshiyuki Sekiguchi, Hayato Waki
, Exact convergence rates of alternating projections for nontransversal intersections, JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS, 10.1007/s13160-023-00584-9, 2023.04. |

3. | Akihito Ebisu, Yoshishige Haraoka, Masanobu Kaneko, Hiroyuki Ochiai, Takeshi Sasaki and Masaaki Yoshida, A study of a Fuchsian system of rank 8 in 3 variables and the ordinary differential equations as its restrictions, , Osaka Math. J., accepted for publication;math.arXiv:2005.04465, 2022.06. |

4. | Shin-Ichiro Ei, Hiroyuki Ochiai, and Yoshitaro Tanaka
, Method of the fundamental solution for the Neumann problems of the modified Helmholtz equation in disk domain, Journal of Computational and Applied Mathematics, 10.1016/j.cam.2021.113795, 402, 2022.03, [URL], The method of the fundamental solutions (MFS) is used to construct an approximate solution for a partial differential equation in a bounded domain. It is demonstrated by combining the fundamental solutions shifted to the points outside the domain and determining the coefficients of the linear sum to satisfy the boundary condition on the finite points of the boundary. In this paper, the existence of the approximate solution by the MFS for the Neumann problems of the modified Helmholtz equation in disk domains is rigorously demonstrated. We reveal the sufficient condition of the existence of the approximate solution. Applying the Green formula to the Neumann problem of the modified Helmholtz equation, we bound the error between the approximate solution and exact solution into the difference of the function of the boundary condition and the normal derivative of the approximate solution by boundary integrations. Using this estimate of the error, we show the convergence of the approximate solution by the MFS to the exact solution with exponential order, that is, N2aN order, where a is a positive constant less than one and N is the number of collocation points. Furthermore, it is demonstrated that the error tends to 0 in exponential order in the numerical simulations with increasing number of collocation points N.. |

5. | Hiroyuki Ochiai, Symmetry of Dressed Photon, Symmetry, https://doi.org/10.3390/sym13071283, 13, 7, 1283, 2021.07, [URL], Motivated by describing the symmetry of a theoretical model of dressed photons, we introduce several spaces with Lie group actions and the morphisms between them depending on three integer parameters n≥r≥s on dimensions. We discuss the symmetry on these spaces using classical invariant theory, orbit decomposition of prehomogeneous vector spaces, and compact reductive homogeneous space such as Grassmann manifold and flag variety. Finally, we go back to the original dressed photon with n=4,r=2,s=1. |

6. | Yasuaki Hiraoka, Hiroyuki Ochiai, and Tomoyuki Shirai, Zeta functions of periodic cubical lattices and cyclotomic-like polynomials, Advanced Studies in Pure Mathematics, 84, 93-121, 2020.04, Zeta functions of periodic cubical lattices are explicitly derived by computing all the eigenvalues of the adjacency operators and their characteristic polynomials. We introduce cyclotomic-like polynomials to give factorization of the zeta function in terms of them and count the number of orbits of the Galois action associated with each cyclotomic-like polynomial to obtain its further factorization. We also give a necessary and sufficient condition for such a polynomial to be irreducible and discuss its irreducibility from this point of view.. |

7. | Khongorzul Dorjgotov, Hiroyuki Ochiai, Uuganbayar Zunderiya, Exact solutions to a class of time fractional evolution systems with variable coefficients, Journal of Mathematical Physics, 10.1063/1.5035392, 59, 8, 2018.08, We explicitly give new group invariant solutions to a class of Riemann-Liouville time fractional evolution systems with variable coefficients. These solutions are derived from every element in an optimal system of Lie algebras generated by infinitesimal symmetries of evolution systems in the class. We express the solutions in terms of Mittag-Leffler functions, generalized Wright functions, and Fox H-functions and show that these solutions solve diffusion-wave equations with variable coefficients. These solutions contain previously known solutions as particular cases. Some plots of solutions subject to the order of the fractional derivative are illustrated.. |

8. | Piotr Graczyk, Hideyuki Ishi, Salha Mamane, Hiroyuki Ochiai, On the Letac-Massam Conjecture on cones Q_{An}, Proceedings of the Japan Academy Series A: Mathematical Sciences, 10.3792/pjaa.93.16, 93, 3, 16-21, 2017.03, We prove, for graphical models for nearest neighbour interactions, a conjecture stated by Letac and Massam in 2007. Our result is important in the analysis of Wishart distributions on cones related to graphical models and in its statistical applications.. |

9. | Shizuo Kaji, Hiroyuki Ochiai, A concise parametrization of affine transformation, SIAM Journal on Imaging Sciences, 10.1137/16M1056936, 9, 3, 1355-1373, 2016.09, Good parametrizations of affine transformations are essential to interpolation, deformation, and analysis of shape, motion, and animation. It has been one of the central research topics in computer graphics. However, there is no single perfect method and each one has both advantages and disadvantages. In this paper, we propose a novel parametrization of affine transformations, which is a generalization to or an improvement of existing methods. Our method adds yet another choice to the existing toolbox and shows better performance in some applications. A C++ implementation is available to make our framework ready to use in various applications.. |

10. | Ken Anjyo, Hiroyuki Ochiai, Mathematical basics of motion and deformation in computer graphics, Synthesis Lectures on Computer Graphics and Animation, 10.2200/S00599ED1V01Y201409CGR017, 6, 3, 1-85, 2015.01, This synthesis lecture presents an intuitive introduction to the mathematics of motion and deformation in computer graphics. Starting with familiar concepts in graphics, such as Euler angles, quaternions, and affine transformations, we illustrate that a mathematical theory behind these concepts enables us to develop the techniques for efficient/effective creation of computer animation. This book, therefore, serves as a good guidepost to mathematics (differential geometry and Lie theory) for students of geometric modeling and animation in computer graphics. Experienced developers and researchers will also benefit from this book, since it gives a comprehensive overview of mathematical approaches that are particularly useful in character modeling, deformation, and animation.. |

11. | 落合 啓之, Dominic Lanphier, Howard Skogman, Values of twisted tensor L-functions of automorphic forms over imaginary quadratic fields, Canadian J. Math., http://dx.doi.org/10.4153/CJM-2013-047-5, 66, 5, 1078-1109, 2014.04. |

12. | 落合 啓之, He Xuhua, Nishiyama Kyo, Oshima Yoshiki, On orbits in double flag varieties for symmetric pairs, Transformation Groups, 18, 4, 1091-1136, 2013.06. |

13. | 落合 啓之, Zunderiya Uuganbayar, A generalized hypergeometric system, J. Math. Sci. Univ. Tokyo, 20, 2, 285-315, 2013.06. |

14. | 落合 啓之, Non-commutative harmonic oscillators, Symmetries, Integrable Systems and Representations, 2013.05. |

15. | Kyo Nishiyama and Hiroyuki Ochiai, Double flag varieties for a symmetric pair and finiteness of orbits, Journal of Lie Theory, 21, 79--99, 2011.01. |

16. | Nobushige Kurokawa and Hiroyuki Ochiai, Zeta functions and Casimir energies on infinite symmetric groups II, Casimir Force, Casimir operators and Riemann hypothesis, 57--63, de Gruyter, 2010.12. |

17. | Kentaro Ihara and Hiroyuki Ochiai,, Symmetry on linear relations for multiple zeta values,, Nagoya Mathematical Journal, 189, 49--62, 2008.05. |

18. | 落合啓之, A special value of the spectral zeta function of the non-commutative harmonic oscillators, The Ramanujan Journal, {\bf 15} (2008) 31--36, 2008.01. |

**Presentations**

1. | Hiroyuki Ochiai, A non-holonomic D-module with infinite-dimensional solutions, workshop "D-modules and hyperplane arrangements",, 2023.02. |

2. | Hiroyuki Ochiai, Math meets CG, IPMU Colloquium, 2023.02. |

3. | Hiroyuki Ochiai, Ken Anjyo, Opening the black box of mathematics for CG, SIGGRAPH ASIA 2021, 2021.12. |

4. | Hiroyuki Ochiai, An absolute version of Hasse zeta functions, Riemann-Roch in characteristic one and related topics, 2019.10. |

5. | Hiroyuki Ochiai, Making a bridge between Ibukiyama and Kobayashi, 第20回整数論オータムワークショップ, 2017.09. |

6. | Hiroyuki Ochiai, Zeros of Eulerian polynomials, Various Aspects of Multiple Zeta Functions, 2017.08. |

7. | Hiroyuki Ochiai, Ken Anjyo and Ayumi Kimura, An Elementary Introduction to Matrix Exponential for CG, SIGGRAPH, 2016.07. |

8. | Hiroyuki Ochiai, Covariant differential operators and Heckman-Opdam hypergeometric systems, International Conference for Korean Mathematical Society 70th Anniversary,, 2016.10, 保形形式に作用する共変な微分作用素を多変数の場合に超幾何関数を用いて記述した。. |

9. | Hiroyuki Ochiai, Ken Anjyo, An Introduction to Matrix Exponential for CG, 2016.02. |

10. | 落合 啓之, Covariant differential operators and Heckman–Opdam hypergeometric systems, Analytic Representation Theory of Lie Groups, 2015.07. |

11. | 落合 啓之, Computer graphics and mathematics, Computational and Geometric Approaches for Nonlinear Phenomena, 2015.08. |

12. | Ken Anjyo, Hiroyuki Ochiai, Mathematical basics of motion and deformation in computer graphics, ACM SIGGRAPH, 2014.08, This is a course lecture on mathematical basics to graphics community. |

13. | OCHIAI HIROYUKI, Double flag variety for a symmetric pair and finiteness of orbits, Representation Theory of Chevalley Groups and Related Topics, 2012.03. |

14. | OCHIAI HIROYUKI, Positivity of alpha determinant, Geometrci Analysis on Euclidean Homogeneous Spaces, 2012.01. |

15. | , [URL]. |

16. | , [URL]. |

17. | 12/14 と 12/16 の２コマの連続講演. |

18. | An algebraic transformation of Gauss hypergeometric function. |

19. | OCHIAI HIROYUKI, Positivity of an alpha determinant, Analysis, Geometry and Group Representations for Homogeneous Spaces, 2010.11. |

20. | OCHIAI HIROYUKI, Invariant hyperfunctions on some semisimple symmetric space, International conference on representation theory and harmonic analysis, 2010.06. |

**Membership in Academic Society**

- MSJ
- Japan Mathematical Society

**Awards**

- On the study of mathematical models of computer graphics

Educational

Social

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