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Birational symmetries, Hirota bilinear forms and special solutions of the Garnier systems in 2variables
 J. Math. Sci. Univ. Tokyo
, 2003
"... Abstract. Hirota bilinear forms of the Garnier system in 2variables, G(1, 1, 1, 1, 1), are given. By using Hirota bilinear forms we construct new birational symmetries of G(1, 1, 1, 1, 1). We obtain special solutions of the Garnier system in nvariables, which are described in terms of solutions ..."
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Abstract. Hirota bilinear forms of the Garnier system in 2variables, G(1, 1, 1, 1, 1), are given. By using Hirota bilinear forms we construct new birational symmetries of G(1, 1, 1, 1, 1). We obtain special solutions of the Garnier system in nvariables, which are described in terms of solutions of the Garnier system in (n − 1)variables. We investigate also algebraic solutions for n = 2.
ON THE REDUCTIONS AND CLASSICAL SOLUTIONS OF THE SCHLESINGER EQUATIONS.
, 2006
"... To the memory of our friend Andrei Bolibruch Abstract. The Schlesinger equations S (n,m) describe monodromy preserving deformations of order m Fuchsian systems with n+1 poles. They can be considered as a family of commuting timedependent Hamiltonian systems on the direct product of n copies of m × ..."
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To the memory of our friend Andrei Bolibruch Abstract. The Schlesinger equations S (n,m) describe monodromy preserving deformations of order m Fuchsian systems with n+1 poles. They can be considered as a family of commuting timedependent Hamiltonian systems on the direct product of n copies of m × m matrix algebras equipped with the standard linear Poisson bracket. In this paper we address the problem of reduction of particular solutions of “more complicated” Schlesinger equations S (n,m) to “simpler ” S (n ′,m ′ ) having n ′ < n or m ′ < m. Contents
Algebraic solutions of the sixth Painlevé Equation
, 2008
"... We describe all finite orbits of an action of the extended modular group ¯ Λ on conjugacy classes of SL2(C)triples. The result is used to classify all algebraic solutions ..."
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We describe all finite orbits of an action of the extended modular group ¯ Λ on conjugacy classes of SL2(C)triples. The result is used to classify all algebraic solutions
RANDOM MATRIX THEORY AND THE SIXTH PAINLEV É EQUATION
, 2006
"... Abstract. A feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices. Study of such ensembles realised by matrices with Gaussian entries leads to statistical quantities related to the eigenspectrum, such as the distribution ..."
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Abstract. A feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices. Study of such ensembles realised by matrices with Gaussian entries leads to statistical quantities related to the eigenspectrum, such as the distribution of the largest eigenvalue, which can be expressed as multidimensional integrals or equivalently as determinants. These distributions are well known to be τfunctions for Painlevé systems, allowing for the former to be characterised as the solution of certain nonlinear equations. We consider the random matrix ensembles for which the nonlinear equation is the σ form of PVI. Known results are reviewed, as is their implication by way of series expansions for the distributions. New results are given for the boundary conditions in the neighbourhood of the fixed singularities at t = 0, 1, ∞ of σPVI displayed by a generalisation of the generating function for the distributions. The structure of these expansions is related to Jimbo’s general expansions for the τfunction of σPVI in the neighbourhood of its fixed singularities, and this theory is itself put in its context of the linear isomonodromy problem relating to PVI. 1.
Apparent singularities of Fuchsian equations, and the
, 905
"... We study movable singularities of Garnier systems (and Painlevé VI equations) using the connection of the latter with isomonodromic deformations of Fuchsian systems. Questions on the existence of solutions for some inverse monodromy problems are also considered. In the middle of the XIXth century B. ..."
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We study movable singularities of Garnier systems (and Painlevé VI equations) using the connection of the latter with isomonodromic deformations of Fuchsian systems. Questions on the existence of solutions for some inverse monodromy problems are also considered. In the middle of the XIXth century B. Riemann [28] considered the problem of the construction of a linear differential equation dpu dzp + b1(z) dp−1u
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"... Abstract:Generic Painlevé VI tau function τ(t) can be interpreted as fourpoint correlator of primary fields of arbitrary dimensions in 2D CFT with c = 1. Using AGT combinatorial representation of conformal blocks and determining the corresponding structure constants, we obtain full and completely e ..."
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Abstract:Generic Painlevé VI tau function τ(t) can be interpreted as fourpoint correlator of primary fields of arbitrary dimensions in 2D CFT with c = 1. Using AGT combinatorial representation of conformal blocks and determining the corresponding structure constants, we obtain full and completely explicit expansion of τ(t) near the singular points. After a check of this expansion, we discuss examples of conformal blocks arising from Riccati, Picard, Chazy and algebraic solutions of Painlevé VI. ArXiv ePrint: arxiv:1207.0787 ar X iv