106,178 research outputs found
Melnikov's approximation dominance. Some examples
We continue a previous paper to show that Mel'nikov's first order formula for
part of the separatrix splitting of a pendulum under fast quasi periodic
forcing holds, in special examples, as an asymptotic formula in the forcing
rapidity.Comment: 46 Kb; 9 pages, plain Te
Singularity analysis, Hadamard products, and tree recurrences
We present a toolbox for extracting asymptotic information on the
coefficients of combinatorial generating functions. This toolbox notably
includes a treatment of the effect of Hadamard products on singularities in the
context of the complex Tauberian technique known as singularity analysis. As a
consequence, it becomes possible to unify the analysis of a number of
divide-and-conquer algorithms, or equivalently random tree models, including
several classical methods for sorting, searching, and dynamically managing
equivalence relationsComment: 47 pages. Submitted for publicatio
A variation norm Carleson theorem
We strengthen the Carleson-Hunt theorem by proving estimates for the
-variation of the partial sum operators for Fourier series and integrals,
for . Four appendices are concerned with transference, a
variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear
Fourier transforms and ergodic theory.Comment: 41 page
Galois symmetries of fundamental groupoids and noncommutative geometry
We define motivic iterated integrals on the affine line, and give a simple
proof of the formula for the coproduct in the Hopf algebra of they make. We
show that it encodes the group law in the automorphism group of certain
non-commutative variety. We relate the coproduct with the coproduct in the Hopf
algebra of decorated rooted planar trivalent trees - a planar decorated version
of the Hopf algebra defined by Connes and Kreimer. As an application we derive
explicit formulas for the coproduct in the motivic multiple polylogarithm Hopf
algebra. We give a criteria for a motivic iterated integral to be unramified at
a prime ideal, and use it to estimate from above the space spanned by the
values of iterated integrals. In chapter 7 we discuss some general principles
relating Feynman integrals and mixed motives.Comment: 51 pages, The final version to appear in Duke Math.
Algorithms for Combinatorial Systems: Well-Founded Systems and Newton Iterations
We consider systems of recursively defined combinatorial structures. We give
algorithms checking that these systems are well founded, computing generating
series and providing numerical values. Our framework is an articulation of the
constructible classes of Flajolet and Sedgewick with Joyal's species theory. We
extend the implicit species theorem to structures of size zero. A quadratic
iterative Newton method is shown to solve well-founded systems combinatorially.
From there, truncations of the corresponding generating series are obtained in
quasi-optimal complexity. This iteration transfers to a numerical scheme that
converges unconditionally to the values of the generating series inside their
disk of convergence. These results provide important subroutines in random
generation. Finally, the approach is extended to combinatorial differential
systems.Comment: 61 page
Uniqueness for the signature of a path of bounded variation and the reduced path group
We introduce the notions of tree-like path and tree-like equivalence between
paths and prove that the latter is an equivalence relation for paths of finite
length. We show that the equivalence classes form a group with some similarity
to a free group, and that in each class there is one special tree reduced path.
The set of these paths is the Reduced Path Group. It is a continuous analogue
to the group of reduced words. The signature of the path is a power series
whose coefficients are definite iterated integrals of the path. We identify the
paths with trivial signature as the tree-like paths, and prove that two paths
are in tree-like equivalence if and only if they have the same signature. In
this way, we extend Chen's theorems on the uniqueness of the sequence of
iterated integrals associated with a piecewise regular path to finite length
paths and identify the appropriate extended meaning for reparameterisation in
the general setting. It is suggestive to think of this result as a
non-commutative analogue of the result that integrable functions on the circle
are determined, up to Lebesgue null sets, by their Fourier coefficients. As a
second theme we give quantitative versions of Chen's theorem in the case of
lattice paths and paths with continuous derivative, and as a corollary derive
results on the triviality of exponential products in the tensor algebra.Comment: 52 pages - considerably extended and revised version of the previous
version of the pape
On the classification of rank two representations of quasiprojective fundamental groups
Suppose is a smooth quasiprojective variety over \cc and \rho : \pi
_1(X,x) \to SL(2,\cc) is a Zariski-dense representation with quasiunipotent
monodromy at infinity. Then factors through a map with
either a DM-curve or a Shimura modular stack.Comment: minor changes in exposition, citation
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