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GFUN: A Maple Package for the Manipulation of Generating and Holonomic Functions in One Variable
, 1992
"... We describe the gfun package which contains functions for manipulating sequences, linear recurrences or di erential equations and generating functions of various types. This document isintended both as an elementary introduction to the subject and as a reference manual for the package. ..."
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Cited by 150 (19 self)
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We describe the gfun package which contains functions for manipulating sequences, linear recurrences or di erential equations and generating functions of various types. This document isintended both as an elementary introduction to the subject and as a reference manual for the package.
Galoisian Obstructions to integrability of Hamiltonian Systems II
, 1997
"... An inconvenience of all the known galoisian formulations of Ziglin’s nonintegrability theory is the Fuchsian condition at the singular points of the variational equations. We avoid this restriction. Moreover we prove that a necessary condition for meromorphic complete integrability (in Liouville se ..."
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Cited by 44 (11 self)
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An inconvenience of all the known galoisian formulations of Ziglin’s nonintegrability theory is the Fuchsian condition at the singular points of the variational equations. We avoid this restriction. Moreover we prove that a necessary condition for meromorphic complete integrability (in Liouville sense) is that the identity component of the Galois group of the variational equation (in the complex domain) must be abelian. We test the efficacy of these new approaches on some examples. We will give some non academic applications in two following papers. 1
Analytic Variations on QuadTrees
, 1991
"... Quadtrees constitute a hierarchical data structure which permits fast access to multidimensional data. This paper presents the analysis of the expected cost of various types of searches in quadtreesfully specified and partial match queries. The data model assumes random points with independently ..."
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Cited by 31 (4 self)
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Quadtrees constitute a hierarchical data structure which permits fast access to multidimensional data. This paper presents the analysis of the expected cost of various types of searches in quadtreesfully specified and partial match queries. The data model assumes random points with independently drawn coordinate values. The analysis leads to a class of "fullhistory" divideandconquer recurrences. These recurrences are solved using generating functions, either exactly for dimension d = 2, or asymptotically for higher dimensions. The exact solutions involve hypergeometric functions. The general asymptotic solutions relie on the classification of singularities of linear differential equations with analytic coefficients, and on singularity analysis techniques. These methods are applicable to the asymptotic solution of a wide range of linear recurrences, as may occur in particular in the analysis of multidimensional searching problems.
The X Window System
 Department of Computer Engineering at the University of Istanbul. His
, 1990
"... realization and model order reduction for nonlinear ..."
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Cited by 22 (0 self)
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realization and model order reduction for nonlinear
Search costs in quadtrees and singularity perturbation asymptotics
 Discrete Comput. Geom
, 1994
"... Abstract. Quadtrees constitute a classical data structure for storing and accessing collections of points in multidimensional space. It is proved that, in any dimension, the cost of a random search in a randomly grown quadtree has logarithmic mean and variance and is asymptotically distributed as a ..."
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Cited by 21 (5 self)
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Abstract. Quadtrees constitute a classical data structure for storing and accessing collections of points in multidimensional space. It is proved that, in any dimension, the cost of a random search in a randomly grown quadtree has logarithmic mean and variance and is asymptotically distributed as a normal variable. The limit distribution property extends to quadtrees of all dimensions a result only known so far to hold for binary search trees. The analysis is based on a technique of singularity perturbation that appears to be of some generality. For quadtrees, this technique is applied to linear differential equations satisfied by intervening bivariate generating functions 1.
Symplectic manifolds and isomonodromic deformations
 Adv. Math
"... Abstract. We study moduli spaces of meromorphic connections (with arbitrary order poles) over Riemann surfaces together with the corresponding spaces of monodromy data (involving Stokes matrices). Natural symplectic structures are found and described both explicitly and from an infinite dimensional ..."
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Cited by 19 (1 self)
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Abstract. We study moduli spaces of meromorphic connections (with arbitrary order poles) over Riemann surfaces together with the corresponding spaces of monodromy data (involving Stokes matrices). Natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint (generalising the AtiyahBott approach). This enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of Jimbo, Miwa and Ueno, thereby putting the existing results for the six Painleve ́ equations and Schlesinger’s equations into a uniform framework.
Exponential Asymptotics in a Singular Limit for nLevel Scattering Systems
, 1997
"... The singular limit " ! 0 of the Smatrix associated with the equation i"d/(t)=dt = H(t)/(t) is considered, where the analytic generator H(t) 2 M n (C) is such that its spectrum is real and nondegenerate for all t 2 R. Sufficient conditions allowing to compute asymptotic formulas for the ..."
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Cited by 18 (11 self)
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The singular limit " ! 0 of the Smatrix associated with the equation i"d/(t)=dt = H(t)/(t) is considered, where the analytic generator H(t) 2 M n (C) is such that its spectrum is real and nondegenerate for all t 2 R. Sufficient conditions allowing to compute asymptotic formulas for the exponentially small offdiagonal elements of the Smatrix as " ! 0 are explicited and a wide class of generators for which these conditions are verified is defined. These generators are obtained by means of generators whose spectrum exhibits eigenvalue crossings which are perturbed in such a way that these crossings turn to avoided crossings. The exponentially small asymptotic formulas which are derived are shown to be valid up to exponentially small relative error, by means of a joint application of the complex WKB method together with superasymptotic renormalization. The application of these results to the study of quantum adiabatic transitions in the time dependent Schrodinger equation and of the s...