3,857 research outputs found
A periodicity criterion and the section problem on the Mapping Class Group
Some years ago, V. Markovic proved that there is no section of the Mapping
Class Group for a closed surface of genus g larger than 5 (in the case of
homeomorphims) and more recently generalized this result with D. Saric to the
case where g is larger than 1. We will state a periodicity criterion and will
use it to simplify some of the arguments given by Markovic and Saric in the
proof of their theorem. The periodicity criterion tells us that a homeomorphism
of a connected surface must be periodic if the set of connected periodic open
sets generates the topology of the surface.Comment: 40 page
A kinetic eikonal equation
We analyse the linear kinetic transport equation with a BGK relaxation
operator. We study the large scale hyperbolic limit (t,x)\to (t/\eps,x/\eps).
We derive a new type of limiting Hamilton-Jacobi equation, which is analogous
to the classical eikonal equation derived from the heat equation with small
diffusivity. We prove well-posedness of the phase problem and convergence
towards the viscosity solution of the Hamilton-Jacobi equation. This is a
preliminary work before analysing the propagation of reaction fronts in kinetic
equations
Fast algorithmic Nielsen-Thurston classification of four-strand braids
We give an algorithm which decides the Nielsen-Thurston type of a given
four-strand braid. The complexity of our algorithm is quadratic with respect to
word length. The proof of its validity is based on a result which states that
for a reducible 4-braid which is as short as possible within its conjugacy
class (short in the sense of Garside), reducing curves surrounding three
punctures must be round or almost round.Comment: One minor error corrected (Example 4.2 was wrong
Optimal growth for linear processes with affine control
We analyse an optimal control with the following features: the dynamical
system is linear, and the dependence upon the control parameter is affine. More
precisely we consider , where
and are matrices with some prescribed structure. In the
case of constant control , we show the existence of an
optimal Perron eigenvalue with respect to varying under some
assumptions. Next we investigate the Floquet eigenvalue problem associated to
time-periodic controls . Finally we prove the existence of an
eigenvalue (in the generalized sense) for the optimal control problem. The
proof is based on the results by [Arisawa 1998, Ann. Institut Henri Poincar\'e]
concerning the ergodic problem for Hamilton-Jacobi equations. We discuss the
relations between the three eigenvalues. Surprisingly enough, the three
eigenvalues appear to be numerically the same
A one-dimensional Keller-Segel equation with a drift issued from the boundary
We investigate in this note the dynamics of a one-dimensional Keller-Segel
type model on the half-line. On the contrary to the classical configuration,
the chemical production term is located on the boundary. We prove, under
suitable assumptions, the following dichotomy which is reminiscent of the
two-dimensional Keller-Segel system. Solutions are global if the mass is below
the critical mass, they blow-up in finite time above the critical mass, and
they converge to some equilibrium at the critical mass. Entropy techniques are
presented which aim at providing quantitative convergence results for the
subcritical case. This note is completed with a brief introduction to a more
realistic model (still one-dimensional).Comment: short version, 8 page
Blow-up dynamics of self-attracting diffusive particles driven by competing convexities
In this paper, we analyze the dynamics of an particles system evolving
according the gradient flow of an energy functional. The particle system is a
consistent approximation of the Lagrangian formulation of a one parameter
family of non-local drift-diffusion equations in one spatial dimension. We
shall prove the global in time existence of the trajectories of the particles
(under a sufficient condition on the initial distribution) and give two blow-up
criteria. All these results are consequences of the competition between the
discrete entropy and the discrete interaction energy. They are also consistent
with the continuous setting, that in turn is a one dimension reformulation of
the parabolic-elliptic Keller-Segel in high dimensions
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