OCHIAI HIROYUKI | Last modified date：2024.05.02 |

Professor /
Division of Fundamental mathematics /
Institute of Mathematics for Industry

**Papers**

1. | Hiroyuki Ochiai, Yoshiyuki Sekiguchi, Hayato Waki,
, Exact Convergence Rates of Alternating Projections for Nontransversal Intersections, Japan Journal of Industrial and Applied Mathematics, 41, 57-83, 2024.04. |

2. | 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. |

3. | 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. |

4. | 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., 60, 151-204, math.arXiv:2005.04465, 2023.04. |

5. | 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.. |

6. | 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. |

7. | 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.. |

8. | Khongorzul Dorjgotov, Hiroyuki Ochiai, Uuganbayar Zunderiya, On solutions of linear fractional differential equations and systems thereof, Fractional Calculus and Applied Analysis, 10.1515/fca-2019-0028, 22, 2, 479-494, 2019.04, We derive exact solutions to classes of linear fractional differential equations and systems thereof expressed in terms of generalized Wright functions and Fox H-functions. These solutions are invariant solutions of diffusion-wave equations obtained through certain transformations, which are briefly discussed. We show that the solutions given in this work contain previously known results as particular cases.. |

9. | 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.. |

10. | Khongorzul Dorjgotov, Hiroyuki Ochiai, Uuganbayar Zunderiya, Lie symmetry analysis of a class of time fractional nonlinear evolution systems, Applied Mathematics and Computation, 10.1016/j.amc.2018.01.056, 329, 105-117, 2018.07, We study a class of nonlinear evolution systems of time fractional partial differential equations using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of group invariant solutions of this class of systems. We find that the class of systems of differential equations studied is naturally divided into two cases on the basis of the type of a function that they contain. In each case, the dimension of the Lie algebra generated by the infinitesimal symmetries is greater than 2, and for this reason we present the structures and one-dimensional optimal systems of these Lie algebras. The reduced systems corresponding to the optimal systems are also obtained. Explicit group invariant solutions are found for particular cases.. |

11. | 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.. |

12. | 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.. |

13. | Hiroyuki Ochiai, Ken Anjyo, Ayumi Kimura, An elementary introduction to matrix exponential for CG, ACM International Conference on Computer Graphics and Interactive Techniques, SIGGRAPH 2016
ACM SIGGRAPH 2016 Courses, SIGGRAPH 2016, 10.1145/2897826.2927338, 2016.07. |

14. | 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.. |

15. | Nobushige Kurokawa, Hiroyuki Ochiai, Zeta Functions of Representations, Commentarii mathematici Universitatis Sancti Pauli = Rikkyo Daigaku sugaku zasshi, 10.14992/00010884, 63, 1, 215-222, 2014.12. |

16. | Dominic Lanphier, Howard Skogman, Hiroyuki Ochiai, Values of Twisted Tensor L-functions of Automorphic Forms Over Imaginary Quadratic Fields, CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 10.4153/CJM-2013-047-5, 66, 5, 1078-1109, 2014.10, Let K be a complex quadratic extension of Q and let A(K) denote the adeles of K. We find special values at all of the critical points of twisted tensor L-functions attached to cohomological cuspforms on GL(2)(AK) and establish Galois equivariance of the values. To investigate the values, we determine the archimedean factors of a class of integral representations of these L-functions, thus proving a conjecture due to Ghate. We also investigate analytic properties of these L-functions, such as their functional equations.. |

17. | Keisuke Kondo, Kyo Nishiyama, Hiroyuki Ochiai, Kenji Taniguchi, Closed orbits on partial flag varieties and double flag variety of finite type, Kyushu J. Math., 68, 1, 113-119, 2014.04. |

18. | 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. |

19. | Kensuke Kondo, Kyo Nishiyama, Hiroyuki Ochiai, Kenji Taniguchi, Closed orbits on partial flag varieties and double flag variety of finite type, Kyushu Journal of Mathematics, 10.2206/kyushujm.68.113, 68, 1, 113-119, 2014.01, Let G be a connected reductive algebraic group over C. We denote by K = (G^{θ})_{0}the identity component of the fixed points of an involutive automorphism θ of G. The pair (G, K) is called a symmetric pair. Let Q be a parabolic subgroup of K. We want to find a pair of parabolic subgroups P _{1}, P _{2}of G such that (i) P _{1}∩ P _{2}= Q and (ii) P _{1} P _{2}is dense in G. The main result of this article states that, for a simple group G, we can find such a pair if and only if (G, K) is a Hermitian symmetric pair. The conditions (i) and (ii) imply that the K -orbit through the origin (eP _{1}, eP _{2}) of G/P _{1}× G/P _{2}is closed and it generates an open dense G -orbit on the product of partial flag variety. From this point of view, we also give a complete classification of closed K -orbits on G/P _{1}× G/P _{2}.. |

20. | Hiroyuki Ochiai, Ken Anjyo, Mathematical basics of motion and deformation in computer graphics, ACM Special Interest Group on Computer Graphics and Interactive Techniques Conference, SIGGRAPH 2014
ACM SIGGRAPH 2014 Courses, SIGGRAPH 2014, 10.1145/2614028.2615386, 2014.01. |

21. | Ken Anjyo, Hiroyuki Ochiai, Mathematical basics of motion and deformation in computer graphics, Mathematical Basics of Motion and Deformation in Computer Graphics, 1-82, 2014.01, 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.. |

22. | Syuhei Sato, Yoshinori Dobashi, Kei Iwasaki, Hiroyuki Ochiai, Tsuyoshi Yamamoto, Tomoyuki Nishita, Generating various flow fields using principal component analysis, ACM Special Interest Group on Computer Graphics and Interactive Techniques Conference, SIGGRAPH 2014
ACM SIGGRAPH 2014 Posters, SIGGRAPH 2014, 10.1145/2614217.2630575, 2014. |

23. | Hiroyuki Ochiai, Ken Anjyo, Mathematical description of motion and deformation - From basics to graphics applications, SIGGRAPH Asia 2013 Courses, SA 2013
SIGGRAPH Asia 2013 Courses, SA 2013, 10.1145/2542266.2542268, 2013.12, While many technical terms, such as Euler angle, quaternion, and affine transformation, now become quite popular in computer graphics, their graphical meanings are sometimes a bit far from the original mathematical entities, which might cause misunderstanding or misuse of the mathematical techniques. This course presents an intuitive introduction to several mathematical concepts that are quite useful for describing motion and deformation of geometric objects. The concepts are inherited mostly from differential geometry and Lie theory, and now commonly used in various aspects of computer graphics, including curve/surface editing, deformation and animation of geometric objects. The objective of this course is to fill the gap between the original mathematical concepts and the practical meanings in computer graphics without assuming any prior knowledge of pure mathematics. We then illustrate practical usefulness of deep understanding of the mathematical concepts. Moreover this course demonstrates our current/ongoing research work, which is benefited from our mathematical formulation.. |

24. | Xuhua He, Hiroyuki Ochiai, Kyo Nishiyama, Yoshiki Oshima, On Orbits in Double Flag Varieties for Symmetric Pairs, Transformation Groups, 10.1007/s00031-013-9243-8, 18, 4, 1091-1136, 2013.12, Let G be a connected, simply connected semisimple algebraic group over the complex number field, and let K be the fixed point subgroup of an involutive automorphism of G so that (G, K) is a symmetric pair. We take parabolic subgroups P of G and Q of K, respectively, and consider the product of partial flag varieties G/P and K/Q with diagonal K-action, which we call a double flag variety for a symmetric pair. It is said to be of finite type if there are only finitely many K-orbits on it. In this paper, we give a parametrization of K-orbits on G/P × K/Q in terms of quotient spaces of unipotent groups without assuming the finiteness of orbits. If one of P ⊂ G or Q ⊂ K is a Borel subgroup, the finiteness of orbits is closely related to spherical actions. In such cases, we give a complete classification of double flag varieties of finite type, namely, we obtain classifications of K-spherical flag varieties G/P and G-spherical homogeneous spaces G/Q.. |

25. | Nobushige Kurokawa, Hiroyuki Ochiai, ZEROS OF WITTEN ZETA FUNCTIONS AND ABSOLUTE LIMIT, KODAI MATHEMATICAL JOURNAL, 36, 3, 440-454, 2013.10. |

26. | Nobushige Kurokawa, Hiroyuki Ochiai, Dualities for absolute zeta functions and multiple gamma functions, Proceedings of the Japan Academy Series A: Mathematical Sciences, 10.3792/pjaa.89.75, 89, 7, 75-79, 2013.07, We study absolute zeta functions from the view point of a canonical normalization. We introduce the absolute Hurwitz zeta function for the normalization. In particular, we show that the theory of multiple gamma and sine functions gives good normalizations in cases related to the Kurokawa tensor product. In these cases, the functional equation of the absolute zeta function turns out to be equivalent to the simplicity of the associated non-classical multiple sine function of negative degree.. |

27. | 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. |

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

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

30. | Hiroyuki Ochiai, Uuganbayar Zunderiya, A generalized hypergeometric system, Journal of Mathematical Sciences (Japan), 20, 2, 285-315, 2013, We give a combinatorial formula of the dimension of global solutions to a generalization of Gauss-Aomoto-Gelfand hypergeometric system, where the quadratic differential operators are replaced by higher order operators. We also derive a polynomial estimate of the dimension of global solutions for the case in 3 × 3 variables.. |

31. | Syuhei Sato, Yoshinori Dobashi, Kei Iwasaki, Hiroyuki Ochiai, Tsuyoshi Yamamoto, Generating flow fields variations by modulating amplitude and resizing simulation space, SIGGRAPH Asia 2013 Technical Briefs, SA 2013
SIGGRAPH Asia 2013 Technical Briefs, SA 2013, 10.1145/2542355.2542371, 2013, The visual simulation of fluids has become an important element in many applications, such as movies and computer games. In these applications, large-scale fluid scenes, such as fire in a village, are often simulated by repeatedly rendering multiple small-scale fluid flows. In these cases, animators are requested to generate many variations of a small-scale fluid flow. This paper presents a method to help animators meet such requirements. Our method enables the user to generate flow field variations from a single simulated dataset obtained by fluid simulation. The variations are generated in both the frequency and spatial domains. Fluid velocity fields are represented using Laplacian eigenfunctions which ensure that the flow field is always incompressible. In generating the variations in the frequency domain, we modulate the coefficients (amplitudes) of the basis functions. To generate variations in the spatial domain, our system expands or contracts the simulation space, then the flow is calculated by solving a minimization problem subject to the resized velocity field. Using our method, the user can easily create various animations from a single dataset calculated by fluid simulation. 2013 Copyright held by the Owner/Author.. |

32. | Hiroyuki Ochiai, Non-commutative harmonic oscillators, Symmetries, Integrable Systems and Representations, 10.1007/978-1-4471-4863-0_19, 40, 483-490, 2013, This is a survey on the non-commutative harmonic oscillator, which is a generalization of usual (scalar) harmonic oscillators to the system introduced by Parmeggiani and Wakayama.With the definitions and the basic properties, we summarize the positivity of several related operators with s^{l}_{2} interpretations. We also mention some unsolved questions, in order to clarify the current status of the problems and expected further development.. |

33. | Nobushige Kurokawa, Hiroyuki Ochiai, Zeros of Witten zeta functions and absolute limit, Kodai Mathematical Journal, 10.2996/kmj/1383660691, 36, 3, 440-454, 2013. |

34. | Tomoyoshi Ibukiyama, Takako Kuzumaki, Hiroyuki Ochiai, Holonomic systems of Gegenbauer type polynomials of matrix arguments related with Siegel modular forms, Journal of the Mathematical Society of Japan, 10.2969/jmsj/06410273, 64, 1, 273-316, 2012.10, Differential operators on Siegel modular forms which behave well under the restriction of the domain are essentially intertwining operators of the tensor product of holomorphic discrete series to its irreducible components. These are characterized by polynomials in the tensor of pluriharmonic polynomials with some invariance properties. We give a concrete study of such polynomials in the case of the restriction from Siegel upper half space of degree 2n to the product of degree n. These generalize the Gegenbauer polynomials which appear for n = 1. We also describe their radial parts parametrization and differential equations which they satisfy, and show that these differential equations give holonomic systems of rank 2 ^{n}.. |

35. | Tomoyoshi Ibukiyama, Takako Kuzumaki, Hiroyuki Ochiai, Holonomic system of Gegenbauer type polynomials , Journal of Mathematical Society of Japan, 64, 1, , 2012.01. |

36. | Tomoyoshi Ibukiyama, Takako Kuzumaki, Hiroyuki Ochiai, Holonomic systems of Gegenbauer type polynomials of matrix arguments related with Siegel modular forms, JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, 10.2969/jmsj/06410273, 64, 1, 273-316, 2012.01, Differential operators on Siegel modular forms which behave well under the restriction of the domain are essentially intertwining operators of the tensor product of holomorphic discrete series to its irreducible components. These are characterized by polynomials in the tensor of pluriharmonic polynomials with some invariance properties. We give a concrete study of such polynomials in the case of the restriction from Siegel upper half space of degree 2n to the product of degree n. These generalize the Gegenbauer polynomials which appear for n = 1. We also describe their radial parts parametrization and differential equations which they satisfy, and show that these differential equations give holonomic systems of rank 2(n).. |

37. | 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. |

38. | 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. |

39. | Nobushige Kurokawa, Hiroyuki Ochiai, A multivariable Euler product of Igusa type and its applications, Journal of Number Theory, 10.1016/j.jnt.2008.10.008, 129, 8, 1919-1930, 2009.08, Knowing the number of solutions for a Diophantine equation is an important step to study various arithmetic problems. Igusa originated the study of Igusa zeta functions associated to local Diophantine problems. Multiplying all these local Igusa zeta functions we obtain the global version in the natural way. Unfortunately, investigations on global Igusa zeta functions are rare up to now. Reformulating the global Igusa zeta function via the number of morphisms between algebraic systems we discover a new aspect: the multivariable Euler product of Igusa type and its applications. A purpose of this paper is to encourage experts for further studies on global Igusa zeta functions by treating a simple interesting example.. |

40. | Nobushige Kurokawa and Hiroyuki Ochiai,, Spectra of alternating Hilbert operators,, Spectral Analysis in Geometry and Number Theory,
Contemporary Mathematics, 484, 89--101, 2009.05. |

41. | Kohji Matsumoto, Takashi Nakamura, Hiroyuki Ochiai, Hirofumi Tsumura, On value-relations, functional relations and singularities of Mordell-Tornheim and related triple zeta-functions, Acta Arithmetica, 10.4064/aa132-2-1, 132, 2, 99-125, 2008.08. |

42. | Nobushige Kurokawa, Matilde Lalin and Hiroyuki Ochiai,, Higher Mahler measures and zeta functions,, Acta Arithmetica, 135, 269--297, 2008.05. |

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

44. | 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. |

45. | Hiroyuki Ochiai, A special value of the spectral zeta function of the non-commutative harmonic oscillators, Ramanujan Journal, 10.1007/s11139-007-9065-1, 15, 1, 31-36, 2008.01, The non-commutative harmonic oscillator is a 2×2-system of harmonic oscillators with a non-trivial correlation. We write down explicitly the special value at s=2 of the spectral zeta function of the non-commutative harmonic oscillator in terms of the complete elliptic integral of the first kind, which is a special case of a hypergeometric function.. |

46. | N. Kurokawa, M. Lalín, Hiroyuki Ochiai, Higher Mahler measures and zeta functions, Acta Arithmetica, 10.4064/aa135-3-5, 135, 3, 269-297, 2008. |

47. | Kentaro Ihara, Hiroyuki Ochiai, Symmetry on linear relations for multiple zeta values, Nagoya Mathematical Journal, 189, 49-62, 2008, We find a symmetry for the reflection groups in the double shuffle space of depth three. The space was introduced by Ihara, Kaneko and Zagier and consists of polynomials in three variables satisfying certain identities which are connected with the double shuffle relations for multiple zeta values. Goncharov has defined a space essentially equivalent to the double shuffle space and has calculated the dimension. In this paper we relate the structure among multiple zeta values of depth three with the invariant theory for the reflection groups and discuss the dimension of the double shuffle space in this view point.. |

48. | Michihiko Fujii, Hiroyuki Ochiai, An expression of harmonic vector fields on hyperbolic 3-cone-manifolds in terms of hypergeometric functions, Publications of the Research Institute for Mathematical Sciences, 10.2977/prims/1201012040, 43, 3, 727-761, 2007.09, Let V be a neighborhood of a singular locus of a hyperbolic 3-cone-manifold, which is a quotient space of the 3-dimensional hyperbolic space. In this paper we give an explicit expression of a harmonic vector field v on the hyperbolic manifold V in terms of hypergeometric functions. The expression is obtained by solving a system of ordinary differential equations which is induced by separation of the variables in the vector-valued partial differential equation (Δ + 4)τ = 0, where Δ is the Laplacian of V and τ is the dual 1-form of v. We transform this system of ordinary differential equations to single-component differential equations by elimination of unknown functions and solve these equations. The most important step in solving them consists of two parts, decomposing their differential operators into differential operators of the type appearing in Riemann's P-equation in the ring of differential operators and then describing the projections to the components of this decomposition in terms of differential operators that are also of the type appearing in Riemann's P-equation.. |

49. | Michihiko Fujii, Hiroyuki Ochiai, An expression of harmonic vector fields on hyperbolic 3-cone-manifolds in terms of hypergeometric functions, PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES, 10.2977/prims/1201012040, 43, 3, 727-761, 2007.09, Lot V be a neighborhood of a singular locus of a hyperbolic 3-cone-manifold, which is a quotient space of the 3-dimensional hyperbolic space. In this paper we give an explicit expression of a harmonic vector field v on the hyperbolic manifold V in terms of hypergeometric functions. The expression is obtained by solving a system of ordinary differential equations which is induced by separation of the variables in the vector-valued partial differential equation (A + 4)tau = 0, where Delta is the Laplacian of V and tau is the dual 1-form of v. We transform this system of ordinary differential equations to single-component differential equations by elimination of unknown functions and solve these equations. The most important step in solving them consists of two parts, decomposing their differential operators into differential operators of the type appearing in Riemann's P-equation in the ring of differential operators and then describing the projections to the components of this decomposition in terms of differential operators that are also of the type appearing in Riemann's P-equation.. |

50. | Fumiharu Kato, Hiroyuki Ochiai, Arithmetic structure of CMSZ fake projective planes, Journal of Algebra, 10.1016/j.jalgebra.2006.07.019, 305, 2, 1166-1185, 2006.11, We show that the fake projective planes that are constructed from dyadic discrete subgroups discovered by Cartwright, Mantero, Steger, and Zappa are realized as connected components of certain unitary Shimura surfaces. As a corollary we show that these fake projective planes have models defined over the number field Q (sqrt(-3), sqrt(5)).. |

51. | Kyo Nishiyama, Hiroyuki Ochiai, Chen Bo Zhu, Theta lifting of nilpotent orbits for symmetric pairs, Transactions of the American Mathematical Society, 10.1090/S0002-9947-05-03826-2, 358, 6, 2713-2734, 2006.06, We consider a reductive dual pair (G, G′) in the stable range with G′ the smaller member and of Hermitian symmetric type. We study the theta lifting of nilpotent K′_{ℂ}-orbits, where K′ is a maximal compact subgroup of G′ and we describe the precise K _{ℂ}-module structure of the regular function ring of the closure of the lifted nilpotent orbit of the symmetric pair (G, K). As an application, we prove sphericality and normality of the closure of certain nilpotent K _{ℂ}-orbits obtained in this way. We also give integral formulas for their degrees.. |

52. | Nobushige Kurokawa, Hiroyuki Ochiai, Generalized Kinkelin's formulas, KODAI MATHEMATICAL JOURNAL, 10.2996/kmj/1183475511, 30, 2, 195-212, 2007.06. |

53. | Tetsuya Hattori, Hiroyuki Ochiai, Scaling Limit of Successive Approximations for w′=-w^{2}, Funkcialaj Ekvacioj, 10.1619/fesi.49.291, 49, 2, 291-319, 2006.01, We prove existence of scaling limits of sequences of functions defined by the recursion relation w′_{n+1}(ϰ)= -w_{n}(ϰ)^{2}. which is a successive approximation to w′(ϰ)= -w_{n}(ϰ)^{2} a simplest non-linear ordinary differential equation whose solutions have moving singularities. Namely, the sequence approaches the exact solution as n→ √ in an asymptotically conformal way, [formula omitted], for a sequence of numbers {qn} and a function [formula omitted]. We also discuss implication of the results in terms of random sequential bisections of a rod.. |

54. | Hiroyuki Ochiai, Invariant distributions on a non-isotropic pseudo-Riemannian symmetric space of rank one, Indagationes Mathematicae, 10.1016/S0019-3577(05)80043-6, 16, 3-4, 631-638, 2005.12, We investigate the structure of invariant distributions on a non-isotropic non-Riemannian symmetric space of rank one. Especially, the J-criterion related to the generalized Gelfand pair is shown for this space without imposing the condition on the eigenfuction of the Laplace-Bertrami operator.. |

55. | Hiroyuki Ochiai, Non-commutative harmonic oscillators and the connection problem for the heun differential equation, Letters in Mathematical Physics, 10.1007/s11005-004-4292-5, 70, 2, 133-139, 2004.12, We consider the connection problem for the Heun differential equation, which is a Fuchsian differential equation that has four regular singular points. We consider the case in which the parameters in this equation satisfy a certain set of conditions coming from the eigenvalue problem of the non-commutative harmonic oscillators. As an application, we describe eigenvalues with multiplicities greater than 1 and the corresponding odd eigenfunctions of the non-commutative harmonic oscillators. The existence of a rational or a certain algebraic solution of the Heun equation implies that the corresponding eigenvalues has multiplicities greater than 1.. |

56. | Hiroyuki Ochiai, Masaaki Yoshida, Polynomials associated with the hypergeometric functions with finite monodromy groups, International Journal of Mathematics, 10.1142/S0129167X0400248X, 15, 7, 629-649, 2004.09, The hypergeometric equations with polyhedral monodromy groups derive 3-integral-parameter families of polynomials.. |

57. | Hiroyuki Ochiai, Toshio Oshima, Commuting Differential Operators of Type B_{2}, Funkcialaj Ekvacioj, 10.1619/fesi.46.297, 46, 2, 297-336, 2003.01. |

58. | Masanobu Kaneko, Hiroyuki Ochiai, On coefficients of Yablonskii-Vorob'ev polynomials, Journal of the Mathematical Society of Japan, 10.2969/jmsj/1191418760, 55, 4, 985-993, 2003.01, We give a formula for the coefficients of the Yablonskii-Vorob'ev polynomial. Also the reduction modulo a prime number of the polynomial is studied.. |

59. | Katsuhisa Mimachi, Hiroyuki Ochiai, Masaaki Yoshida, Intersection theory for loaded cycles IV - Resonant cases, Mathematische Nachrichten, 10.1002/mana.200310105, 260, 67-77, 2003, We study the structure of the twisted homology groups attached to weighted lines in the plane with resonant singular points, and find possible intersection pairings. As a typical example, we treat homology groups attached to 2-dimensional Selberg-type integrals.. |

60. | Hiroyuki Ochiai, Non-commutative harmonic oscillators and Fuchsian ordinary differential operators, Communications in Mathematical Physics, 10.1007/s002200100362, 217, 2, 357-373, 2001.12, The spectral problem for non-commutative harmonic oscillators is shown to be equivalent to solve Fuchsian ordinary differential equations with four regular singular points in a complex domain.. |

61. | Kyo Nishiyama, Hiroyuki Ochiai, Kenji Taniguchi, Bernstein degree and associated cycles of Harish-Chandra modules - Hermitian symmetric case, Asterisque, 273, 13-80, 2001, Let G̃ be the metaplectic double cover of Sp(2n,ℝ), U(p,q) or O*(2p). we study the Bernstein degrees and the associated cycles of the irreducible unitary highest weight representations of G̃, by using the theta correspondence of dual pairs. The first part of this article is a summary of fundamental properties and known results of the Bernstein degrees and the associated cycles. Our first result is a comparison theorem between the K-module structures of the following two spaces; one is the theta lift of the trivial representation and the other is the ring of regular functions on its associated variety. Secondarily, we obtain the explicit values of the degrees of some small nilpotent K_{C}-orbits by means of representation theory. The main result of this article is the determination of the associated cycles of singular unitary highest weight representations, which are the theta lifts of irreducible representations of certain compact groups. In the proofs of these results, the multiplicity free property of spherical subgroups and the stability of the branching coefficients play important roles.. |

62. | Sholiei Kato, Hiroyuki Ochiai, The degrees of orbits of the multiplicity-free actions, Asterisque, 273, 139-158, 2001, We give a formula for the degrees of orbits of the irreducible representations with multiplicity-free action. In particular, we obtain the Bernstein degree and the associated cycle of the irreducible unitary highest weight modules of the scalar type for arbitrary hermitian Lie algebras.. |

63. | Kyo Nishiyama, Hiroyuki Ochiai, Bernstein degree of singular unitary highest weight representations of the metaplectic group, Proceedings of the Japan Academy Series A: Mathematical Sciences, 10.3792/pjaa.75.9, 75, 2, 9-11, 1999. |

64. | Hiroyuki Ochiai, Quotients of some prehomogeneous vector spaces, Journal of Algebra, 10.1006/jabr.1996.6979, 192, 1, 61-73, 1997.06, We give quotientsV//G=Spec(C[V]^{G}) for some prehomogeneous vector spaces (G×H,V).. |

65. | Hiroyuki Ochiai, Commuting differential operators of rank two, Indagationes Mathematicae, 10.1016/0019-3577(96)85093-2, 7, 2, 243-255, 1996.06, Commuting differential operators with symbols ∂^{2}_{1} + ∂^{2}_{2}, ∂^{2}_{1}∂^{2}_{2} are related to the functional equation V_{1}(x_{1})U_{+}(x_{1} + x_{2}) + V_{1}(x_{1})U_{-}(x_{1} - x_{2}) + V_{2}(x_{2})U_{+}(x_{1} + x_{2}) - V_{2}(x_{2})U_{-}(x_{1} - x_{2}) = F_{-}(x_{1} + x_{2}) + F_{-}(x_{1} - x_{2}) + G_{1}(x_{1}) + G_{2}(x_{2}). We discuss several properties, such as a meromorphic extension of solutions and periodicity, of this functional equation and commuting differential operators.. |

66. | Hiroyuki Ochiai, Characters and character cycles, Journal of the Mathematical Society of Japan, 10.2969/jmsj/04540583, 45, 4, 583-598, 1993. |

67. | Hiroyuki Ochiai, A p-adic property of the Taylor series of exp(x + x^{p}/p), Hokkaido Mathematical Journal, 10.14492/hokmj/1351001078, 28, 1, 71-85, 1999.01, The p-adic norms of the Taylor coefficients of the function exp(x + x^{p}/p) are expressed in terms of a p-adic analytic function for p ≤ 23.. |

68. | K Nishiyama, H Ochiai, Bernstein degree of singular unitary highest weight representations of the metaplectic group, PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES, 75, 2, 9-11, 1999.02. |

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