3,907 research outputs found
Singularity analysis via the iterated kernel method
We provide exact and asymptotic counting formulas for five singular lattice path models in the quarter plane. Furthermore, we prove that these models have a non D-finite generating function
Partially directed paths in a wedge
The enumeration of lattice paths in wedges poses unique mathematical
challenges. These models are not translationally invariant, and the absence of
this symmetry complicates both the derivation of a functional recurrence for
the generating function, and solving for it. In this paper we consider a model
of partially directed walks from the origin in the square lattice confined to
both a symmetric wedge defined by , and an asymmetric wedge defined
by the lines and Y=0, where is an integer. We prove that the
growth constant for all these models is equal to , independent of
the angle of the wedge. We derive functional recursions for both models, and
obtain explicit expressions for the generating functions when . From these
we find asymptotic formulas for the number of partially directed paths of
length in a wedge when .
The functional recurrences are solved by a variation of the kernel method,
which we call the ``iterated kernel method''. This method appears to be similar
to the obstinate kernel method used by Bousquet-Melou. This method requires us
to consider iterated compositions of the roots of the kernel. These
compositions turn out to be surprisingly tractable, and we are able to find
simple explicit expressions for them. However, in spite of this, the generating
functions turn out to be similar in form to Jacobi -functions, and have
natural boundaries on the unit circle.Comment: 26 pages, 5 figures. Submitted to JCT
An improvement of the product integration method for a weakly singular Hammerstein equation
We present a new method to solve nonlinear Hammerstein equations with weakly
singular kernels. The process to approximate the solution, followed usually,
consists in adapting the discretization scheme from the linear case in order to
obtain a nonlinear system in a finite dimensional space and solve it by any
linearization method. In this paper, we propose to first linearize, via Newton
method, the nonlinear operator equation and only then to discretize the
obtained linear equations by the product integration method. We prove that the
iterates, issued from our method, tends to the exact solution of the nonlinear
Hammerstein equation when the number of Newton iterations tends to infinity,
whatever the discretization parameter can be. This is not the case when the
discretization is done first: in this case, the accuracy of the approximation
is limited by the mesh size discretization. A Numerical example is given to
confirm the theorical result
Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral
We introduce a class of iterated integrals that generalize multiple
polylogarithms to elliptic curves. These elliptic multiple polylogarithms are
closely related to similar functions defined in pure math- ematics and string
theory. We then focus on the equal-mass and non-equal-mass sunrise integrals,
and we develop a formalism that enables us to compute these Feynman integrals
in terms of our iterated integrals on elliptic curves. The key idea is to use
integration-by-parts identities to identify a set of integral kernels, whose
precise form is determined by the branch points of the integral in question.
These kernels allow us to express all iterated integrals on an elliptic curve
in terms of them. The flexibility of our approach leads us to expect that it
will be applicable to a large variety of integrals in high-energy physics.Comment: 22 page
Regularity theory for the spatially homogeneous Boltzmann equation with cut-off
We develop the regularity theory of the spatially homogeneous Boltzmann
equation with cut-off and hard potentials (for instance, hard spheres), by (i)
revisiting the Lp-theory to obtain constructive bounds, (ii) establishing
propagation of smoothness and singularities, (iii) obtaining estimates about
the decay of the sin- gularities of the initial datum. Our proofs are based on
a detailed study of the "regularity of the gain operator". An application to
the long-time behavior is presented.Comment: 47 page
Multi-Regge kinematics and the moduli space of Riemann spheres with marked points
We show that scattering amplitudes in planar N = 4 Super Yang-Mills in
multi-Regge kinematics can naturally be expressed in terms of single-valued
iterated integrals on the moduli space of Riemann spheres with marked points.
As a consequence, scattering amplitudes in this limit can be expressed as
convolutions that can easily be computed using Stokes' theorem. We apply this
framework to MHV amplitudes to leading-logarithmic accuracy (LLA), and we prove
that at L loops all MHV amplitudes are determined by amplitudes with up to L +
4 external legs. We also investigate non-MHV amplitudes, and we show that they
can be obtained by convoluting the MHV results with a certain helicity flip
kernel. We classify all leading singularities that appear at LLA in the Regge
limit for arbitrary helicity configurations and any number of external legs.
Finally, we use our new framework to obtain explicit analytic results at LLA
for all MHV amplitudes up to five loops and all non-MHV amplitudes with up to
eight external legs and four loops.Comment: 104 pages, six awesome figures and ancillary files containing the
results in Mathematica forma
On the low-x NLO evolution of 4 point colorless operators
The NLO evolution equations for quadrupole and double dipole operators have
been obtained within the high energy operator expansion method. The
corresponding quasi-conformal evolution equations for the composite operators
were constructed
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