九州大学 研究者情報
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基本情報 研究活動 教育活動 社会活動
落合 啓之(おちあい ひろゆき) データ更新日:2022.06.17



主な研究テーマ
代数解析学的手法による表現論、特殊関数の研究
キーワード:代数解析学
2009.10~2020.03.
従事しているプロジェクト研究
デジタル映像数学の構築と表現技術の革新
2010.10~2016.03, 代表者:安生健一, OLMデジタル, JST(科学技術振興機構)
コンピュータグラフィックスに代表されるデジタル映像の応用は拡大の一途をたどっています.
本研究は,作りやすさや効率を重視しつつ,従来よりさらに豊かな表現力を持つ映像の制作を可能にするために,デジタル映像表現を対象とする新たな数学分野形成の礎を築くことを目指します.
特に,人間の動作や表情と流体の表現に焦点をあて,これらの映像表現の数学的特徴づけと,作り手の意図をより的確に反映できる数学モデルの構築を推進します..
研究業績
主要著書
1. Hiroyuki Ochiai, Ken Anjyo, Mathematical Progress in Expressive Image Synthesis II, Springer-Verlag, Selected papers from the proceeding of MEIS2014.
2. Ken Anjyo, Hiroyuki Ochiai, Mathematical Progress in Expressive Image Synthesis I, Springer-Verlag, 2014.08, Selected papers from the proceeding of MEIS2013.
3. Ken Anjyo, Hiroyuki Ochiai, Mathematical Basics of Motion and Deformation in Computer Graphics, Morgan & Claypool Publishers, doi:10.2200/S00599ED1V01Y201409CGR017, 2014.10, The unique introduction of the mathematical background to the computer graphics.
主要原著論文
1. Hiroyuki Ochiai, Symmetry of Dressed Photon, Symmetry, https://doi.org/10.3390/sym13071283, 13, 7, 1283, 2021.07, 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.
2. 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.
3. 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, 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..
4. KEISUKE HAKUTA, HIROYUKI OCHIAI, AND 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, 2022.03, Batch verification for digital signature scheme is a method to verify multiple signatures
simultaneously. The complex exponent test (CE test for short) proposed by Cheon and Lee is one of
the most efficient batch verification tests for several digital signature schemes on certain types of elliptic
curves (including Koblitz curves). The security of the CE test relies essentially on the cardinality of
a subset of a residue ring of the endomorphism ring of an elliptic curve over an ideal. They have
evaluated the cardinality of the above subset, and have illustrated the effectiveness of the CE test by
using the evaluation. The aim of this paper is to point out that their evaluation contains a flaw. The
flaw is generally related to two roots of a quadratic equation which is used in their argument. We
mend the flaw of their evaluation. Our correct evaluation shows that the CE test can achieve the same
security as the underlying signature scheme on Koblitz curves. As a result, the CE test is a secure
batch verification when the underlying signature scheme uses Koblitz curves..
5. 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..
6. 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, [URL], 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..
7. Piotr Graczyk, Hideyuki Ishi, Salha Mamane, Hiroyuki Ochiai, On the Letac-Massam Conjecture on cones QAn, Proceedings of the Japan Academy Series A: Mathematical Sciences, 10.3792/pjaa.93.16, 93, 3, 16-21, 2017.03, [URL], 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..
8. Shizuo Kaji, Hiroyuki Ochiai, A concise parametrization of affine transformation, SIAM Journal on Imaging Sciences, 10.1137/16M1056936, 9, 3, 1355-1373, 2016.09, [URL], 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..
9. 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, [URL], 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..
10. 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.
11. 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.
12. 落合 啓之, Zunderiya Uuganbayar, A generalized hypergeometric system, J. Math. Sci. Univ. Tokyo, 20, 2, 285-315, 2013.06.
13. 落合 啓之, Non-commutative harmonic oscillators, Symmetries, Integrable Systems and Representations, 2013.05.
14. 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.
15. 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.
16. Kentaro Ihara and Hiroyuki Ochiai,, Symmetry on linear relations for multiple zeta values,, Nagoya Mathematical Journal, 189, 49--62, 2008.05.
17. 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.
主要総説, 論評, 解説, 書評, 報告書等
主要学会発表等
1. Hiroyuki Ochiai, An absolute version of Hasse zeta functions, Riemann-Roch in characteristic one and related topics, 2019.10.
2. 落合啓之, 超幾何関数のリー環対称性について, 研究集会「微分方程式と表現論」, 2018.12.
3. 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.
4. Hiroyuki Ochiai, Zeros of Eulerian polynomials, Various Aspects of Multiple Zeta Functions, 2017.08.
5. Hiroyuki Ochiai, Ken Anjyo and Ayumi Kimura, An Elementary Introduction to Matrix Exponential for CG, SIGGRAPH, 2016.07.
6. 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。.
7. Hiroyuki Ochiai, Ken Anjyo, An Introduction to Matrix Exponential for CG, 2016.02.
8. 落合啓之, 概説講演“CG 映像制作におけるリー理論”, 表現論シンポジウム, 2015.11.
9. 落合 啓之, 行列の数理と運動の記述, Computer Entertainment Developer Conference 2013, 2013.08, ゲーム開発者を中心とした実業界で活躍するエキスパートに向けて、数学的手法や思想、特に行列やリー群を活用した、運動の記述や補間について、分かりやすく解説した。.
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. 落合 啓之, 共変微分作用素の超幾何多項式表示, 不変性と双対性, 2015.09.
13. 落合 啓之, CG映像制作の数理, 日本数学会, 2015.09.
14. 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.
15. 落合 啓之, ユニタリ表現の分類はそろそろできるだろうか?, 2013 年度日本数学会年会, 2013.03.
16. OCHIAI HIROYUKI, Double flag variety for a symmetric pair and finiteness of orbits, Representation Theory of Chevalley Groups and Related Topics, 2012.03.
17. OCHIAI HIROYUKI, Positivity of alpha determinant, Geometrci Analysis on Euclidean Homogeneous Spaces, 2012.01, [URL].
18. Hiroyuki Ochiai, On Sekiguchi correspondence, Topics in the Theory of Weyl Groups and Root Systems, 2011.09, Kostant-Sekiguchi correspondence, so called, is the bijective correspondence between the set of nilpotent (co)adjoint orbits on a semisimple Lie algebra and the set of nilpotent KC-orbits on the complexified tangent space of the corresponding Riemannian symmetric space.
From the beginning of the theory, Sekiguchi has the correspondence on the context of non-Riemannian semisimple symmetric spaces. I will review this Sekiguchi's work, as well as an application/ interpretation to the geometric invariants of representation theory..
19. Hiroyuki Ochiai, On non-commutative harmonic oscillators, Infinite Analysis 11, 2011.07.
20. 落合啓之, Invariant hyperfunctions on some semisimple symmetric spaces,, Workshop on Harmonic Analysis and Invariant Distributions, , 2009.12.
21. 落合啓之、黒川信重, Zeta functions and Casimir energies on infinite symmetric groups,, Casimir Force, Casimir Operators and the Riemann Hypothesis, , 2009.11.
22. 落合啓之, An algebraic transformation of Gauss hypergeometric function, Differential equations and symmetric spaces, 2009.01.
23. OCHIAI HIROYUKI, Positivity of an alpha determinant, Analysis, Geometry and Group Representations for Homogeneous Spaces, 2010.11.
24. OCHIAI HIROYUKI, Invariant hyperfunctions on some semisimple symmetric space, International conference on representation theory and harmonic analysis, 2010.06.
学会活動
所属学会名
日本数学会
MSJ
日本数学会
日本アクチュアリ会
学会大会・会議・シンポジウム等における役割
2014.09.01~2014.09.05, Prehomogeneous Vector Spaces and Related Topics, co-organizer.
2015.09.25~2015.09.27, MEIS2015, co-chair.
2014.11.12~2014.11.14, MEIS2014, co-chair.
2010.03~2011.02.28, 日本数学会, 評議員.
2007.04~2010.03.31, 日本数学会, 函数解析分科会分科会委員.
学会誌・雑誌・著書の編集への参加状況
2009.07~2015.06, 「メモワールズ」日本数学会, 国内, 編集委員.
2005.07~2009.06, 雑誌「数学」日本数学会, 国内, 編集委員.
学術論文等の審査
年度 外国語雑誌査読論文数 日本語雑誌査読論文数 国際会議録査読論文数 国内会議録査読論文数 合計
2014年度 17  23 
2013年度
2012年度 12 
2011年度 12 
2010年度 12 
2009年度 12 
その他の研究活動
海外渡航状況, 海外での教育研究歴
Anaheim Convention Center, UnitedStatesofAmerica, 2016.07~2016.07.
Victoria University, NewZealand, 2014.02~2014.02.
Victoria University, NewZealand, 2015.02~2015.03.
外国人研究者等の受入れ状況
2016.03~2015.03, 2週間以上1ヶ月未満, テクニオン, Israel, 文部科学省.
2015.05~2015.06, 1ヶ月以上, OLMデジタル, France, 学内資金.
受賞
科学技術分野の文部科学大臣表彰:科学技術賞(研究部門), 文部科学省, 2014.04.
研究資金
科学研究費補助金の採択状況(文部科学省、日本学術振興会)
1990年度~1990年度, 基盤研究(C), 半単純対称空間の表現論に現われる特殊関数の研究.
1992年度~1992年度, 基盤研究(C), 実半単純代数群の指標の幾何的側面.
1993年度~1993年度, 基盤研究(C), 無限次元リ一環の指標と微分方程式.
1996年度~1996年度, 基盤研究(C), オイラーポアソン方程式の対称性.
1997年度~1998年度, 基盤研究(C), 拡張された帯球函数の微分方程式系.
1999年度~2001年度, 基盤研究(C), 簡約群の幾何学的表現論.
2003年度~2005年度, 基盤研究(C), 非可換調和振動子と特殊関数.
2003年度~2003年度, 基盤研究(C), 等質空間のプランシェレル公式.
2003年度~2006年度, 基盤研究(C), 実簡約群の表現の幾何学的不変量と積分変換.
2007年度~2010年度, 基盤研究(C), 表現論における積分と特殊関数.
2007年度~2009年度, 基盤研究(C), 多変数マーラー測度の研究.
2011年度~2013年度, 基盤研究(C), 動く映像の特徴抽出空間の構成.
2014年度~2016年度, 基盤研究(C), 映像の特徴抽出空間の構成.
2015年度~2019年度, 基盤研究(C), 表現論と特殊関数論の統合的展開.
2017年度~2018年度, 基盤研究(C), 特殊関数論の圏化.
2017年度~2018年度, 挑戦的研究(萌芽), 代表, 特殊関数論の圏化.
2015年度~2019年度, 基盤研究(B), 代表, 表現論と特殊関数論の統合的展開.
2014年度~2016年度, 挑戦的萌芽研究, 代表, 映像の特徴抽出空間の構成.
2011年度~2013年度, 挑戦的萌芽研究, 代表, 動く映像の特徴抽出空間の構成.
2007年度~2010年度, 基盤研究(A), 代表, 表現論における積分と特殊関数.
科学研究費補助金の採択状況(文部科学省、日本学術振興会以外)
2010年度~2015年度, クレスト(科学技術振興機構), 分担, デジタル映像数学の構築と表現技術の革新.

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