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On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing.
 Proc. Roy. Soc. London A 452
"... We prove that for divergent series solutions of nonlinear (or linear) differential systems near a generic irregular singularity, the common prescription of summation to the least term is, if properly interpreted, meaningful and correct, and we extend this method to transseries solutions. In every di ..."
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Cited by 11 (5 self)
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We prove that for divergent series solutions of nonlinear (or linear) differential systems near a generic irregular singularity, the common prescription of summation to the least term is, if properly interpreted, meaningful and correct, and we extend this method to transseries solutions. In every direction in the complex plane at the singularity (Stokes directions not excepted) there exists a nonempty set of solutions whose difference from the "optimally" (i.e., near the least term) truncated asymptotic series is of the same (exponentially small) order of magnitude as the least term of the series. There is a family of generalized Borel summation formulas B which commute with the usual algebraic and analytic operations (addition, multiplication, differentiation, etc). We show that there is exactly one of them, B0 , such that for any formal series solution ~ f, B0 ( ~ f) differs from the optimal truncation of ~ f by at most the order of the least term of ~ f . We show in addition that the Berry (1989) smoothing phenomenon is universal within this class of differential systems. Whenever the terms "beyond all orders" change in crossing a Stokes line, these terms vary smoothly on the Berry scale arg(x) jxj \Gamma1=2 and the transition is always given by the error function; under the same conditions we show that Dingle's rule of signs for Stokes transitions holds. 1
Symbolic Asymptotics: Multiseries of Inverse Functions
, 1997
"... We give an algorithm to compute an asymptotic expansion of multiseries type for the inverse of any given explog function. An example of the use of this algorithm to compute asymptotic expansions in combinatorics via the saddlepoint method is then treated in detail. ..."
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Cited by 8 (0 self)
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We give an algorithm to compute an asymptotic expansion of multiseries type for the inverse of any given explog function. An example of the use of this algorithm to compute asymptotic expansions in combinatorics via the saddlepoint method is then treated in detail.
UNIVERSALITY AND ASYMPTOTICS OF GRAPH COUNTING PROBLEMS IN UNORIENTED SURFACES
, 2009
"... BenderCanfield showed that a plethora of graph counting problems in oriented/unoriented surfaces involve two constants tg and pg for the oriented and the unoriented case respectively. T.T.Q. Le and the authors recently discovered a hidden relation between the sequence tg and a formal power series ..."
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Cited by 4 (0 self)
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BenderCanfield showed that a plethora of graph counting problems in oriented/unoriented surfaces involve two constants tg and pg for the oriented and the unoriented case respectively. T.T.Q. Le and the authors recently discovered a hidden relation between the sequence tg and a formal power series solution u(z) of the Painlevé I equation which, among other things, allows to give exact asymptotic expansion of tg to all orders in 1/g for large g. The paper introduces a formal power series solution v(z) of a Riccati equation, gives a nonlinear recursion for its coefficients and an exact asymptotic expansion to all orders in g for large g, using the theory of Borel transforms. In addition, we conjecture a precise relation between the sequence pg and v(z). Our conjecture is motivated by the enumerative aspects of a quartic matrix model for real symmetric matrices, and the analytic properties of its double scaling limit. In particular, the matrix model provides a computation of the number of rooted quadrangulations in the 2dimensional projective plane. Our conjecture implies analyticity of the O(N) and Sp(N)types of free energy of an arbitrary closed 3manifold in a neighborhood of zero. Finally, we give a matrix model calculation of the Stokes constants, pose several problems that can be answered by the RiemannHilbert approach, and provide ample numerical evidence for our results.
Analyticity of the free energy of a closed 3manifold
 AND ASYMPTOTICS OF GRAPH COUNTING PROBLEMS IN UNORIENTED SURFACES 23
"... Abstract. The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative ChernSimons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold ..."
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Cited by 3 (1 self)
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Abstract. The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative ChernSimons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold is uniformly Gevrey1. As a corollary, it follows that the genus g part of the free energy is convergent in a neighborhood of zero, independent of the genus. Our results follow from an estimate of the LMO invariant, in a particular gauge, and from recent results of BenderGaoRichmond on the asymptotics of the number of rooted maps for arbitrary genus. We illustrate our results with an explicit formula for the free energy of a Lens space. In addition, using the Painlevé differential equation, we obtain an asymptotic expansion for the number of cubic graphs to all orders, stengthening the results of BenderGaoRichmond.
Movable singularities of solutions of difference equations in relation to solvability, and study of a superstable fixed point Theoretical and
 Mathematical Physics
, 2002
"... Abstract. We overview applications exponential asymptotics and analyzable function theory to difference equations, in defining an analog of the Painlevé property for them and we sketch the conclusions with respect to the solvability properties of first order autonomous ones. It turns out that if the ..."
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Cited by 2 (1 self)
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Abstract. We overview applications exponential asymptotics and analyzable function theory to difference equations, in defining an analog of the Painlevé property for them and we sketch the conclusions with respect to the solvability properties of first order autonomous ones. It turns out that if the Painlevé property is present the equations are explicitly solvable and in the contrary case, under further assumptions, the integrals of motion develop singularity barriers. We apply the method to the logistic map xn+1 = axn(1 − xn) where it turns out that the only cases with the Painlevé property are a = −2, 0, 2 and 4 for which explicit solutions indeed exist; in the opposite case an associated conjugation map develops singularity barriers. 1.
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 080, 20 pages Analyticity of the Free Energy of a Closed 3Manifold ⋆
, 809
"... Abstract. The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative Chern–Simons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold ..."
Abstract
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Abstract. The free energy of a closed 3manifold is a 2parameter formal power series which encodes the perturbative Chern–Simons invariant (also known as the LMO invariant) of a closed 3manifold with gauge group U(N) for arbitrary N. We prove that the free energy of an arbitrary closed 3manifold is uniformly Gevrey1. As a corollary, it follows that the genus g part of the free energy is convergent in a neighborhood of zero, independent of the genus. Our results follow from an estimate of the LMO invariant, in a particular gauge, and from recent results of Bender–Gao–Richmond on the asymptotics of the number of rooted maps for arbitrary genus. We illustrate our results with an explicit formula for the free energy of a Lens space. In addition, using the Painlevé differential equation, we obtain an asymptotic expansion for the number of cubic graphs to all orders, stengthening the results of Bender–Gao–Richmond.