206,642 research outputs found
Lusternik - Schnirelman theory and dynamics
In this paper we study a new topological invariant \Cat(X,\xi), where
is a finite polyhedron and is a real cohomology class.
\Cat(X,\xi) is defined using open covers of with certain geometric
properties; it is a generalization of the classical Lusternik -- Schnirelman
category. We show that \Cat(X,\xi) depends only on the homotopy type of
. We prove that \Cat(X,\xi) allows to establish a relation between
the number of equilibrium states of dynamical systems and their global
dynamical properties (such as existence of homoclinic cycles and the structure
of the set of chain recurrent points). In the paper we give a cohomological
lower bound for \Cat(X,\xi), which uses cup-products of cohomology classes of
flat line bundles with monodromy described by complex numbers, which are not
Dirichlet units.Comment: 20 page
Double logarithms resummation in exclusive processes : the surprising behavior of DVCS
Double logarithms resummation has been much studied in inclusive as well as
exclusive processes. The Sudakov mechanism has often be the crucial tool to
exponentiate potentially large contributions to amplitudes or cross-sections
near phase-space boundaries. We report on a recent work where a very different
pattern emerges : the DVCS quark coefficient function C(x,\xi) develops, near
the particular point x=\xi, a non-alternate series in \alpha_s^n
log^{2n}(x-\xi) which may be resummed in a cosh[K sqrt \alpha_s log(x-\xi)]
factor. This result is at odds with the known result for the corresponding
coefficient function for the pion transition form factor near the end point
C(z) although they are much related through a z -> x/\xi correspondence.Comment: 9 pages, 1 figure, Presented at the Low x workshop, May 30 - June 4
2013, Rehovot and Eilat, Israe
Fractional differentiability for solutions of nonlinear elliptic equations
We study nonlinear elliptic equations in divergence form
When
has linear growth in , and assuming that enjoys smoothness, local
well-posedness is found in for certain values of
and . In the particular case
, and ,
, we obtain for each
. Our main tool in the proof is a more general result, that
holds also if has growth in , , and
asserts local well-posedness in for each , provided that
satisfies a locally uniform condition
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