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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 174 (20 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.
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 44 (6 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.
The Social System
"... into cally to a process of national selection. This article describes the process used, presents feedback from the assessors and candidates involved, and discusses possible improvements for future rounds. ..."
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Cited by 39 (0 self)
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into cally to a process of national selection. This article describes the process used, presents feedback from the assessors and candidates involved, and discusses possible improvements for future rounds.
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 34 (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.
Quasilinear stokes phenomenon for the Painlevé first equation
 J. Phys. A: Math. Gen
"... Abstract. Using the RiemannHilbert approach, the Ψfunction corresponding to the solution of the first Painlevé equation yxx = 6y 2 + x with the asymptotic behavior y ∼ ± √ −x/6 as x  → ∞ is constructed. The exponentially small jump in the dominant solution and the coefficient asymptotics in ..."
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Cited by 27 (1 self)
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Abstract. Using the RiemannHilbert approach, the Ψfunction corresponding to the solution of the first Painlevé equation yxx = 6y 2 + x with the asymptotic behavior y ∼ ± √ −x/6 as x  → ∞ is constructed. The exponentially small jump in the dominant solution and the coefficient asymptotics in the powerlike expansion to the latter are found. The Painlevé first equation [1] 1.
On the nonholonomic character of logarithms, powers, and the nth prime function
, 2005
"... We establish that the sequences formed by logarithms and by “fractional” powers of integers, as well as the sequence of prime numbers, are nonholonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction ..."
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Cited by 27 (7 self)
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We establish that the sequences formed by logarithms and by “fractional” powers of integers, as well as the sequence of prime numbers, are nonholonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction of the Structure Theorem for singularities of solutions to linear differential equations and of an Abelian theorem. A brief discussion is offered regarding the scope of singularitybased methods and several naturally occurring sequences are proved to be nonholonomic.
On The Borel Summability Of Divergent Solutions Of The Heat Equation
 Nagoya Math. J
, 1997
"... this paper is to improve the asymptotic results by discussing the summation of formal solutions ..."
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Cited by 25 (1 self)
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this paper is to improve the asymptotic results by discussing the summation of formal solutions
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 22 (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.