53,673 research outputs found
Resurgence of the Euler-MacLaurin summation formula
The Euler-MacLaurin summation formula relates a sum of a function to a
corresponding integral, with a remainder term. The remainder term has an
asymptotic expansion, and for a typical analytic function, it is a divergent
(Gevrey-1) series. Under some decay assumptions of the function in a half-plane
(resp. in the vertical strip containing the summation interval), Hardy (resp.
Abel-Plana) prove that the asymptotic expansion is a Borel summable series, and
give an exact Euler-MacLaurin summation formula.
Using a mild resurgence hypothesis for the function to be summed, we give a
Borel summable transseries expression for the remainder term, as well as a
Laplace integral formula, with an explicit integrand which is a resurgent
function itself. In particular, our summation formula allows for resurgent
functions with singularities in the vertical strip containing the summation
interval.
Finally, we give two applications of our results. One concerns the
construction of solutions of linear difference equations with a small
parameter. And another concerns the problem of proving resurgence of formal
power series associated to knotted objects.Comment: AMS-LaTeX, 15 pages with 2 figure
New approach to Borel summation of divergent series and critical exponent estimates for an N-vector cubic model in three dimensions from five-loop \epsilon expansions
A new approach to summation of divergent field-theoretical series is
suggested. It is based on the Borel transformation combined with a conformal
mapping and does not imply the exact asymptotic parameters to be known. The
method is tested on functions expanded in their asymptotic power series. It is
applied to estimating the critical exponent values for an N-vector field model,
describing magnetic and structural phase transitions in cubic and tetragonal
crystals, from five-loop \epsilon expansions.Comment: 9 pages, LaTeX, 3 PostScript figure
On parametric Gevrey asymptotics for some Cauchy problems in quasiperiodic function spaces
We investigate Gevrey asymptotics for solutions to nonlinear parameter
depending Cauchy problems with -periodic coefficients, for initial data
living in a space of quasiperiodic functions. By means of the Borel-Laplace
summation procedure, we construct sectorial holomorphic solutions which are
shown to share the same formal power series as asymptotic expansion in the
perturbation parameter. We observe a small divisor phenomenon which emerges
from the quasiperiodic nature of the solutions space and which is the origin of
the Gevrey type divergence of this formal series. Our result rests on the
classical Ramis-Sibuya theorem which asks to prove that the difference of any
two neighboring constructed solutions satisfies some exponential decay. This is
done by an asymptotic study of a Dirichlet-like series whose exponents are
positive real numbers which accumulate to the origin
Clustering in mixing flows
We calculate the Lyapunov exponents for particles suspended in a random
three-dimensional flow, concentrating on the limit where the viscous damping
rate is small compared to the inverse correlation time. In this limit Lyapunov
exponents are obtained as a power series in epsilon, a dimensionless measure of
the particle inertia. Although the perturbation generates an asymptotic series,
we obtain accurate results from a Pade-Borel summation. Our results prove that
particles suspended in an incompressible random mixing flow can show pronounced
clustering when the Stokes number is large and we characterise two distinct
clustering effects which occur in that limit.Comment: 5 pages, 1 figur
Hyperbolic low-dimensional invariant tori and summations of divergent series
We consider a class of a priori stable quasi-integrable analytic Hamiltonian
systems and study the regularity of low-dimensional hyperbolic invariant tori
as functions of the perturbation parameter. We show that, under natural
nonresonance conditions, such tori exist and can be identified through the
maxima or minima of a suitable potential. They are analytic inside a disc
centered at the origin and deprived of a region around the positive or negative
real axis with a quadratic cusp at the origin. The invariant tori admit an
asymptotic series at the origin with Taylor coefficients that grow at most as a
power of a factorial and a remainder that to any order N is bounded by the
(N+1)-st power of the argument times a power of . We show the existence of
a summation criterion of the (generically divergent) series, in powers of the
perturbation size, that represent the parametric equations of the tori by
following the renormalization group methods for the resummations of
perturbative series in quantum field theoryComment: 32 pages, 5 figure
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