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Fumiharu Kato, Hiroyuki Ochiai, *Arithmetic structure of CMSZ fake projective planes*, JOURNAL OF ALGEBRA, 10.1016/j.jalgebra.2006.07.019, Vol.305, No.2, pp.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(root-3,root 5). (c) 2006 Published by Elsevier Inc.. |

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Fumiharu Kato, Hiroyuki Ochiai, *Arithmetic structure of CMSZ fake projective planes*, JOURNAL OF ALGEBRA, 10.1016/j.jalgebra.2006.07.019, Vol.305, No.2, pp.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(root-3,root 5). (c) 2006 Published by Elsevier Inc.. |

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Tetsuya Hattori, Hiroyuki Ochiai, *Scaling Limit of Successive Approximations for w(1) = -w(2)*, FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA, 10.1619/fesi.49.291, Vol.49, No.2, pp.291-319, 2006.08, We prove existence of scaling limits of sequences of functions defined by the recursion relation w(n+1)(1) (x) = -w(n)(x)(2). which is a successive approximation to w(1) (x) = -w(x)(2), a simplest non-linear ordinary differential equation whose solutions have moving singularities. Namely, the sequence approaches the exact solution as n -> infinity in an asymptotically conformal way, w(n)(x) asymptotic to q(n)w(-)(q(n)-x), for a sequence of numbers {q(n)} and a function w(-). We also discuss implication of the results in terms of random sequential bisections of a rod.. |

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Hiroyuki Ochiai, *Invariant distributions on a non-isotropic pseudo-Riemannian symmetric space of rank one*, Indagationes Mathematicae, 10.1016/S0019-3577(05)80043-6, Vol.16, No.3-4, pp.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. © 2005 Royal Netherlands Academy of Arts and Sciences.. |

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H Ochiai, *Invariant distributions on a non-isotropic pseudo-Riemannian symmetric space of rank one*, INDAGATIONES MATHEMATICAE-NEW SERIES, 10.1016/S0019-3577(05)80043-6, Vol.16, No.3-4, pp.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.. |

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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, Vol.70, No.2, pp.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.. |

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H Ochiai, *Non-commutative harmonic oscillators and the connection problem for the heun differential equation*, LETTERS IN MATHEMATICAL PHYSICS, 10.1007/s11005-004-4292-5, Vol.70, No.2, pp.133-139, 2004.11, 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 I 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.. |

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H Ochiai, M Yoshida, *Polynomials associated with the hypergeometric functions with finite monodromy groups*, INTERNATIONAL JOURNAL OF MATHEMATICS, 10.1142/S0129167X0400248X, Vol.15, No.7, pp.629-649, 2004.09, The hypergeometric equations with polyhedral monodromy groups derive 3-integral-parameter families of polynomials.. |

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H Ochiai, M Yoshida, *Polynomials associated with the hypergeometric functions with finite monodromy groups*, INTERNATIONAL JOURNAL OF MATHEMATICS, 10.1142/S0129167X0400248X, Vol.15, No.7, pp.629-649, 2004.09, The hypergeometric equations with polyhedral monodromy groups derive 3-integral-parameter families of polynomials.. |